Runtime Complexity (innermost) proof of /tmp/tmpwn5stO/MYNAT_nokinds

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 86 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CdtProblem

↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳6 CdtProblem

↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CdtProblem

↳9 CdtUsableRulesProof (⇔, 149 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 71 ms)

↳12 CdtProblem

↳13 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CdtProblem

↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 48 ms)

↳18 CdtProblem

↳19 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳20 CdtProblem

↳21 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳22 CdtProblem

↳23 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 5 ms)

↳24 CdtProblem

↳25 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳26 CdtProblem

↳27 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳28 CdtProblem

↳29 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 25 ms)

↳30 CdtProblem

↳31 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳32 CdtProblem

↳33 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳34 CdtProblem

↳35 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 56 ms)

↳36 CdtProblem

↳37 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳38 CdtProblem

↳39 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳40 CdtProblem

↳41 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 23 ms)

↳42 CdtProblem

↳43 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳44 CdtProblem

↳45 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳46 CdtProblem

↳47 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 32 ms)

↳48 CdtProblem

↳49 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳50 CdtProblem

↳51 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 93 ms)

↳52 CdtProblem

↳53 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳54 CdtProblem

↳55 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳56 CdtProblem

↳57 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳58 CdtProblem

↳59 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 97 ms)

↳60 CdtProblem

↳61 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳62 CdtProblem

↳63 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳64 CdtProblem

↳65 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳66 CdtProblem

↳67 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 177 ms)

↳68 CdtProblem

↳69 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 37 ms)

↳70 CdtProblem

↳71 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 89 ms)

↳72 CdtProblem

↳73 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳74 CdtProblem

↳75 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 106 ms)

↳76 CdtProblem

↳77 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 29 ms)

↳78 CdtProblem

↳79 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 100 ms)

↳80 CdtProblem

↳81 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)

↳82 CdtProblem

↳83 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳84 CdtProblem

↳85 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 9 ms)

↳86 CdtProblem

↳87 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 86 ms)

↳88 CdtProblem

↳89 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)

↳90 CdtProblem

↳91 CdtUnreachableProof (⇔, 0 ms)

↳92 CdtProblem

↳93 CdtUsableRulesProof (⇔, 0 ms)

↳94 CdtProblem

↳95 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 15 ms)

↳96 CdtProblem

↳97 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 1 ms)

↳98 CdtProblem

↳99 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 18 ms)

↳100 CdtProblem

↳101 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 16 ms)

↳102 CdtProblem

↳103 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 16 ms)

↳104 CdtProblem

↳105 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 13 ms)

↳106 CdtProblem

↳107 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳108 CdtProblem

↳109 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳110 CdtProblem

↳111 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳112 CdtProblem

↳113 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 9 ms)

↳114 CdtProblem

↳115 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳116 CdtProblem

↳117 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 0 ms)

↳118 CdtProblem

↳119 CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)), 13 ms)

↳120 CdtProblem

↳121 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)

↳122 BOUNDS(1, 1)

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(U31(tt)) → mark(0)
active(U41(tt, M, N)) → mark(plus(x(N, M), N))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(isNat(x(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
active(x(N, 0)) → mark(U31(isNat(N)))
active(x(N, s(M))) → mark(U41(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U31(X)) → U31(active(X))
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U31(mark(X)) → mark(U31(X))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U31(X)) → U31(proper(X))
proper(0) → ok(0)
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U31(ok(X)) → ok(U31(X))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(tt, z0)) → c
ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U31(tt)) → c2
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(and(tt, z0)) → c4
ACTIVE(isNat(0)) → c5
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(U41'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c42
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(0) → c47
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U11(tt, z0)) → c
ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U31(tt)) → c2
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(and(tt, z0)) → c4
ACTIVE(isNat(0)) → c5
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(U41'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c42
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(0) → c47
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, U31, U41, x, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing nodes:

ACTIVE(U11(tt, z0)) → c
PROPER(0) → c47
ACTIVE(U31(tt)) → c2
ACTIVE(isNat(0)) → c5
ACTIVE(and(tt, z0)) → c4
PROPER(tt) → c42

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(U41'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(U41'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, U31, U41, x, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c3, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(ISNAT(z1), ISNAT(z0))

S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c3(PLUS(x(z1, z0), z1), X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c6(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c8(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(plus(z0, 0)) → c9(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(x(z0, 0)) → c11(U31'(isNat(z0)), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1), ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c12(ISNAT(z1), ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, U31, U41, x, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c3, c6, c7, c8, c9, c11, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c10, c12

(7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, U31, U41, x, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c

(11) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(U11(z0, z1)) → c13(U11'(active(z0), z1), ACTIVE(z0)) by

ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c13(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c13(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c13(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13

(13) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c13(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c13(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c13(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13

(15) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1

(17) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(U21(z0, z1, z2)) → c14(U21'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c14(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c14(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c14(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c14(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c14(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c14(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14

(19) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c14(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c14(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c13(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c14(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c14(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c14(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14

(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2

(23) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(s(z0)) → c15(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c15(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c15(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c15(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c15(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c15(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c15(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15

(25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U21(tt, z0, z1))) → c15(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c15(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U21(tt, z0, z1))) → c15(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c15(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c15(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15

(27) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(U31(z0)) → c18(U31'(active(z0)), ACTIVE(z0)) by

ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U21(tt, z0, z1))) → c18(U31'(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)), ACTIVE(U31(tt)))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c18(U31'(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U21(tt, z0, z1))) → c18(U31'(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)), ACTIVE(U31(tt)))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c18(U31'(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U31(tt))) → c15(S(mark(0)), ACTIVE(U31(tt)))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U21(tt, z0, z1))) → c18(U31'(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)), ACTIVE(U31(tt)))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c18(U31'(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18

(31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U21(tt, z0, z1))) → c18(U31'(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c18(U31'(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U21(tt, z0, z1))) → c18(U31'(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c18(U31'(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c18(U31'(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18

(33) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(U41(z0, z1, z2)) → c19(U41'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c19(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c19(U41'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c19(U41'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c19(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c19(U41'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c19(U41'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c19(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2), ACTIVE(U31(tt)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c19(U41'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c19(U41'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19

(37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 3 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c19(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c19(U41'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c19(U41'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c13(U11'(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c14(U21'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c15(S(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c18(U31'(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c19(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c19(U41'(mark(plus(x(z1, z0), z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c19(U41'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c19(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19

(39) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace ACTIVE(and(z0, z1)) → c22(AND(active(z0), z1), ACTIVE(z0)) by

ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(and(U21(tt, z0, z1), x1)) → c22(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c22(AND(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c22(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(and(U21(tt, z0, z1), x1)) → c22(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c22(AND(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c22(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(and(U21(tt, z0, z1), x1)) → c22(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1), ACTIVE(U31(tt)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c22(AND(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c22(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22

(43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 9 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(U21(tt, z0, z1), x1)) → c22(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c22(AND(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c22(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(U21(tt, z0, z1), x1)) → c22(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c22(AND(mark(plus(x(z1, z0), z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c22(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c22(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22

(45) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(U11(z0, z1)) → c41(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41

(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c43, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(U21(z0, z1, z2)) → c43(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43

(53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c44, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(s(z0)) → c44(S(proper(z0)), PROPER(z0)) by

PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c44(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(0)) → c44(S(ok(0)), PROPER(0))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c44(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(0)) → c44(S(ok(0)), PROPER(0))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c44(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(0)) → c44(S(ok(0)), PROPER(0))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c45, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(plus(z0, z1)) → c45(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45

(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c46, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(U31(z0)) → c46(U31'(proper(z0)), PROPER(z0)) by

PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(tt)) → c46(U31'(ok(tt)), PROPER(tt))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(0)) → c46(U31'(ok(0)), PROPER(0))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(tt)) → c46(U31'(ok(tt)), PROPER(tt))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(0)) → c46(U31'(ok(0)), PROPER(0))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(tt)) → c46(U31'(ok(tt)), PROPER(tt))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(0)) → c46(U31'(ok(0)), PROPER(0))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46

(65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c48, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(U41(z0, z1, z2)) → c48(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48

(69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 6 trailing tuple parts

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c49, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(x(z0, z1)) → c49(X(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49

(73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c50, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(and(z0, z1)) → c50(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50

(77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing tuple parts

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, PROPER, ISNAT, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c51, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace PROPER(isNat(z0)) → c51(ISNAT(proper(z0)), PROPER(z0)) by

PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)), PROPER(0))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, TOP, PROPER

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51

(81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, TOP, PROPER

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c53, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51, c51

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(mark(z0)) → c53(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c53(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(0)) → c53(TOP(ok(0)), PROPER(0))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c53(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(0)) → c53(TOP(ok(0)), PROPER(0))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c53(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(0)) → c53(TOP(ok(0)), PROPER(0))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, TOP, PROPER

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51, c51, c53

(85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 2 trailing tuple parts

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c53(TOP(ok(tt)))
TOP(mark(0)) → c53(TOP(ok(0)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c53(TOP(ok(tt)))
TOP(mark(0)) → c53(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, TOP, PROPER

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c54, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51, c51, c53, c53

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Use narrowing to replace TOP(ok(z0)) → c54(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U31(tt))) → c54(TOP(mark(0)), ACTIVE(U31(tt)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c54(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c53(TOP(ok(tt)))
TOP(mark(0)) → c53(TOP(ok(0)))
TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U31(tt))) → c54(TOP(mark(0)), ACTIVE(U31(tt)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c54(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c53(TOP(ok(tt)))
TOP(mark(0)) → c53(TOP(ok(0)))
TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U31(tt))) → c54(TOP(mark(0)), ACTIVE(U31(tt)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c54(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, PROPER, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51, c51, c53, c53, c54

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

TOP(mark(tt)) → c53(TOP(ok(tt)))
TOP(ok(isNat(0))) → c54(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(mark(0)) → c53(TOP(ok(0)))
TOP(ok(U31(tt))) → c54(TOP(mark(0)), ACTIVE(U31(tt)))

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

S tuples:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, U31', U41', X, AND, ISNAT, PROPER, TOP

Compound Symbols:

c7, c16, c17, c20, c21, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52, c, c13, c13, c1, c14, c14, c2, c15, c15, c3, c18, c18, c4, c19, c19, c5, c22, c22, c6, c41, c41, c43, c43, c44, c44, c45, c45, c46, c46, c48, c48, c49, c49, c50, c50, c51, c51, c53, c54

(91) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

ACTIVE(isNat(s(z0))) → c7(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c16(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c17(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(x(z0, z1)) → c20(X(active(z0), z1), ACTIVE(z0))
ACTIVE(x(z0, z1)) → c21(X(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(U41(tt, z0, z1)) → c(PLUS(x(z1, z0), z1))
ACTIVE(U41(tt, z0, z1)) → c(X(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(isNat(x(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(x(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(x(z0, 0)) → c(U31'(isNat(z0)))
ACTIVE(x(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(x(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c13(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c13(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(x(z0, 0), x1)) → c13(U11'(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(U11(x(z0, s(z1)), x1)) → c13(U11'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c13(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c13(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c13(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c13(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(z0), x1)) → c13(U11'(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c13(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(x(z0, z1), x1)) → c13(U11'(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c13(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c13(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(x(z0, z1)), x1)) → c1(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c14(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c14(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(x(z0, 0), x1, x2)) → c14(U21'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U21(x(z0, s(z1)), x1, x2)) → c14(U21'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c14(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c14(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c14(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c14(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(U31(z0), x1, x2)) → c14(U21'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U21(U41(z0, z1, z2), x1, x2)) → c14(U21'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(x(z0, z1), x1, x2)) → c14(U21'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c14(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c14(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(x(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(x(z0, z1))))
ACTIVE(s(plus(z0, 0))) → c15(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c15(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(x(z0, 0))) → c15(S(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(s(x(z0, s(z1)))) → c15(S(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c15(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c15(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c15(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c15(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U31(z0))) → c15(S(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(s(U41(z0, z1, z2))) → c15(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(x(z0, z1))) → c15(S(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(s(x(z0, z1))) → c15(S(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(s(and(z0, z1))) → c15(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c15(S(mark(z0)))
ACTIVE(s(isNat(0))) → c15(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(x(z0, z1)))) → c3(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U31(plus(z0, 0))) → c18(U31'(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)))) → c18(U31'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(x(z0, 0))) → c18(U31'(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
ACTIVE(U31(x(z0, s(z1)))) → c18(U31'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
ACTIVE(U31(U11(z0, z1))) → c18(U31'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U21(z0, z1, z2))) → c18(U31'(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U31(s(z0))) → c18(U31'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1))) → c18(U31'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U31(U31(z0))) → c18(U31'(U31(active(z0))), ACTIVE(U31(z0)))
ACTIVE(U31(U41(z0, z1, z2))) → c18(U31'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(active(z0), z1)), ACTIVE(x(z0, z1)))
ACTIVE(U31(x(z0, z1))) → c18(U31'(x(z0, active(z1))), ACTIVE(x(z0, z1)))
ACTIVE(U31(and(z0, z1))) → c18(U31'(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(U11(U31(tt), x1)) → c13(U11'(mark(0), x1))
ACTIVE(U21(U31(tt), x1, x2)) → c14(U21'(mark(0), x1, x2))
ACTIVE(s(U31(tt))) → c15(S(mark(0)))
ACTIVE(U31(U11(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U31(U31(tt))) → c18(U31'(mark(0)))
ACTIVE(U31(isNat(0))) → c18(U31'(mark(tt)))
ACTIVE(U31(U21(tt, z0, z1))) → c4(U31'(mark(s(plus(z1, z0)))))
ACTIVE(U31(U21(tt, z0, z1))) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(plus(z0, z1)))) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)))) → c4(U31'(mark(isNat(z0))))
ACTIVE(U31(isNat(s(z0)))) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(U31'(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(U31(isNat(x(z0, z1)))) → c4(ACTIVE(isNat(x(z0, z1))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c19(U41'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c19(U41'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(x(z0, 0), x1, x2)) → c19(U41'(mark(U31(isNat(z0))), x1, x2), ACTIVE(x(z0, 0)))
ACTIVE(U41(x(z0, s(z1)), x1, x2)) → c19(U41'(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(x(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c19(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U21(z0, z1, z2), x1, x2)) → c19(U41'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c19(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c19(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U31(z0), x1, x2)) → c19(U41'(U31(active(z0)), x1, x2), ACTIVE(U31(z0)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c19(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(active(z0), z1), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(x(z0, z1), x1, x2)) → c19(U41'(x(z0, active(z1)), x1, x2), ACTIVE(x(z0, z1)))
ACTIVE(U41(and(z0, z1), x1, x2)) → c19(U41'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(U41(U31(tt), x1, x2)) → c19(U41'(mark(0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c19(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(U21'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U21(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U41(tt, z0, z1))) → c5(S(mark(plus(x(z1, z0), z1))))
ACTIVE(s(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1))) → c5(U31'(mark(plus(x(z1, z0), z1))))
ACTIVE(U31(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U21(tt, z0, z1), x1, x2)) → c5(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(plus(x(z1, z0), z1)), x1, x2))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(U41'(mark(isNat(z0)), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(U41'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U41(isNat(x(z0, z1)), x1, x2)) → c5(ACTIVE(isNat(x(z0, z1))))
ACTIVE(and(plus(z0, 0), x1)) → c22(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c22(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(x(z0, 0), x1)) → c22(AND(mark(U31(isNat(z0))), x1), ACTIVE(x(z0, 0)))
ACTIVE(and(x(z0, s(z1)), x1)) → c22(AND(mark(U41(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(x(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c22(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c22(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c22(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c22(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(U31(z0), x1)) → c22(AND(U31(active(z0)), x1), ACTIVE(U31(z0)))
ACTIVE(and(U41(z0, z1, z2), x1)) → c22(AND(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(active(z0), z1), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(x(z0, z1), x1)) → c22(AND(x(z0, active(z1)), x1), ACTIVE(x(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c22(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c13(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c14(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c15(S(mark(z0)))
ACTIVE(U31(and(tt, z0))) → c18(U31'(mark(z0)))
ACTIVE(U41(and(tt, z0), x1, x2)) → c19(U41'(mark(z0), x1, x2))
ACTIVE(and(U11(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(U31(tt), x1)) → c22(AND(mark(0), x1))
ACTIVE(and(and(tt, z0), x1)) → c22(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c22(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c6(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(AND(mark(plus(x(z1, z0), z1)), x1))
ACTIVE(and(U41(tt, z0, z1), x1)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c6(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(x(z0, z1)), x1)) → c6(ACTIVE(isNat(x(z0, z1))))
PROPER(U11(x0, U11(z0, z1))) → c41(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c41(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c41(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c41(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, U31(z0))) → c41(U11'(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(U11(x0, U41(z0, z1, z2))) → c41(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, x(z0, z1))) → c41(U11'(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c41(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c41(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c41(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c41(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c41(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c41(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(U31(z0), x1)) → c41(U11'(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c41(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(x(z0, z1), x1)) → c41(U11'(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c41(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c41(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c41(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c41(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c41(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c41(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c43(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c43(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c43(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, U31(z0))) → c43(U21'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U21(x0, x1, U41(z0, z1, z2))) → c43(U21'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U21(x0, x1, x(z0, z1))) → c43(U21'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c43(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c43(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c43(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c43(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c43(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c43(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, U31(z0), x2)) → c43(U21'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U21(x0, U41(z0, z1, z2), x2)) → c43(U21'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, x(z0, z1), x2)) → c43(U21'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c43(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c43(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c43(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c43(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c43(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c43(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U31(z0), x1, x2)) → c43(U21'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(U41(z0, z1, z2), x1, x2)) → c43(U21'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(x(z0, z1), x1, x2)) → c43(U21'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c43(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c43(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c43(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c43(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c43(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c43(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c43(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c43(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c44(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c44(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c44(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c44(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(U31(z0))) → c44(S(U31(proper(z0))), PROPER(U31(z0)))
PROPER(s(U41(z0, z1, z2))) → c44(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(x(z0, z1))) → c44(S(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(s(and(z0, z1))) → c44(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c44(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c44(S(ok(tt)))
PROPER(s(0)) → c44(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c45(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c45(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c45(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c45(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, U31(z0))) → c45(PLUS(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(plus(x0, U41(z0, z1, z2))) → c45(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, x(z0, z1))) → c45(PLUS(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c45(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c45(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c45(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c45(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c45(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c45(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(U31(z0), x1)) → c45(PLUS(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c45(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(x(z0, z1), x1)) → c45(PLUS(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c45(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c45(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c45(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c45(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c45(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c45(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(U31(U11(z0, z1))) → c46(U31'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U31(U21(z0, z1, z2))) → c46(U31'(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(U31(s(z0))) → c46(U31'(s(proper(z0))), PROPER(s(z0)))
PROPER(U31(plus(z0, z1))) → c46(U31'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U31(U31(z0))) → c46(U31'(U31(proper(z0))), PROPER(U31(z0)))
PROPER(U31(U41(z0, z1, z2))) → c46(U31'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U31(x(z0, z1))) → c46(U31'(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(U31(and(z0, z1))) → c46(U31'(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(U31(isNat(z0))) → c46(U31'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U31(tt)) → c46(U31'(ok(tt)))
PROPER(U31(0)) → c46(U31'(ok(0)))
PROPER(U41(x0, x1, U11(z0, z1))) → c48(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U21(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c48(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c48(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, U31(z0))) → c48(U41'(proper(x0), proper(x1), U31(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U31(z0)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c48(U41'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U41(x0, x1, x(z0, z1))) → c48(U41'(proper(x0), proper(x1), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(x(z0, z1)))
PROPER(U41(x0, x1, and(z0, z1))) → c48(U41'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U41(x0, x1, isNat(z0))) → c48(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, U11(z0, z1), x2)) → c48(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U21(z0, z1, z2), x2)) → c48(U41'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c48(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c48(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, U31(z0), x2)) → c48(U41'(proper(x0), U31(proper(z0)), proper(x2)), PROPER(x0), PROPER(U31(z0)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c48(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, x(z0, z1), x2)) → c48(U41'(proper(x0), x(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(x(z0, z1)), PROPER(x2))
PROPER(U41(x0, and(z0, z1), x2)) → c48(U41'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c48(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c48(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0, z1, z2), x1, x2)) → c48(U41'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c48(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c48(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0), x1, x2)) → c48(U41'(U31(proper(z0)), proper(x1), proper(x2)), PROPER(U31(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c48(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(x(z0, z1), x1, x2)) → c48(U41'(x(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(x(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(and(z0, z1), x1, x2)) → c48(U41'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c48(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c48(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c48(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c48(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c48(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c48(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c48(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(x(x0, U11(z0, z1))) → c49(X(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(x(x0, U21(z0, z1, z2))) → c49(X(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(x(x0, s(z0))) → c49(X(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(x(x0, plus(z0, z1))) → c49(X(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(x(x0, U31(z0))) → c49(X(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(x(x0, U41(z0, z1, z2))) → c49(X(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(x(x0, x(z0, z1))) → c49(X(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(x(x0, and(z0, z1))) → c49(X(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(x(x0, isNat(z0))) → c49(X(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(x(U11(z0, z1), x1)) → c49(X(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(x(U21(z0, z1, z2), x1)) → c49(X(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(x(s(z0), x1)) → c49(X(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(x(plus(z0, z1), x1)) → c49(X(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(x(U31(z0), x1)) → c49(X(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(x(U41(z0, z1, z2), x1)) → c49(X(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(x(x(z0, z1), x1)) → c49(X(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(x(and(z0, z1), x1)) → c49(X(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(x(isNat(z0), x1)) → c49(X(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(x(x0, tt)) → c49(X(proper(x0), ok(tt)), PROPER(x0))
PROPER(x(x0, 0)) → c49(X(proper(x0), ok(0)), PROPER(x0))
PROPER(x(tt, x1)) → c49(X(ok(tt), proper(x1)), PROPER(x1))
PROPER(x(0, x1)) → c49(X(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c50(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c50(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c50(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c50(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, U31(z0))) → c50(AND(proper(x0), U31(proper(z0))), PROPER(x0), PROPER(U31(z0)))
PROPER(and(x0, U41(z0, z1, z2))) → c50(AND(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(and(x0, x(z0, z1))) → c50(AND(proper(x0), x(proper(z0), proper(z1))), PROPER(x0), PROPER(x(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c50(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c50(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c50(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c50(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c50(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c50(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(U31(z0), x1)) → c50(AND(U31(proper(z0)), proper(x1)), PROPER(U31(z0)), PROPER(x1))
PROPER(and(U41(z0, z1, z2), x1)) → c50(AND(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(and(x(z0, z1), x1)) → c50(AND(x(proper(z0), proper(z1)), proper(x1)), PROPER(x(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c50(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c50(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c50(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c50(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c50(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c50(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c51(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c51(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c51(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c51(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(U31(z0))) → c51(ISNAT(U31(proper(z0))), PROPER(U31(z0)))
PROPER(isNat(U41(z0, z1, z2))) → c51(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(x(z0, z1))) → c51(ISNAT(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
PROPER(isNat(and(z0, z1))) → c51(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c51(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c51(ISNAT(ok(tt)))
PROPER(isNat(0)) → c51(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c53(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c53(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c53(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c53(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(U31(z0))) → c53(TOP(U31(proper(z0))), PROPER(U31(z0)))
TOP(mark(U41(z0, z1, z2))) → c53(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(x(z0, z1))) → c53(TOP(x(proper(z0), proper(z1))), PROPER(x(z0, z1)))
TOP(mark(and(z0, z1))) → c53(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c53(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c54(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(U41(tt, z0, z1))) → c54(TOP(mark(plus(x(z1, z0), z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c54(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c54(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(isNat(x(z0, z1)))) → c54(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(x(z0, z1))))
TOP(ok(plus(z0, 0))) → c54(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c54(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(x(z0, 0))) → c54(TOP(mark(U31(isNat(z0)))), ACTIVE(x(z0, 0)))
TOP(ok(x(z0, s(z1)))) → c54(TOP(mark(U41(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(x(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c54(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c54(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c54(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c54(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(U31(z0))) → c54(TOP(U31(active(z0))), ACTIVE(U31(z0)))
TOP(ok(U41(z0, z1, z2))) → c54(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(x(z0, z1))) → c54(TOP(x(active(z0), z1)), ACTIVE(x(z0, z1)))
TOP(ok(x(z0, z1))) → c54(TOP(x(z0, active(z1))), ACTIVE(x(z0, z1)))
TOP(ok(and(z0, z1))) → c54(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, s, plus, x, and, isNat, U11, U31, U21, U41, proper

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(93) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(U31(tt)) → mark(0)
active(U41(tt, z0, z1)) → mark(plus(x(z1, z0), z1))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(isNat(x(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(x(z0, 0)) → mark(U31(isNat(z0)))
active(x(z0, s(z1))) → mark(U41(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(U31(z0)) → U31(active(z0))
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(x(z0, z1)) → x(active(z0), z1)
active(x(z0, z1)) → x(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
x(mark(z0), z1) → mark(x(z0, z1))
x(z0, mark(z1)) → mark(x(z0, z1))
x(ok(z0), ok(z1)) → ok(x(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0)) → mark(U31(z0))
U31(ok(z0)) → ok(U31(z0))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(U31(z0)) → U31(proper(z0))
proper(0) → ok(0)
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(x(z0, z1)) → x(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = [2]x₁
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [2]x₁
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = [2]x₁
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = [4]x₁
POL(U41'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [4] + x₁

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = [4]x₁
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = x₁

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₁
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = x₃
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = x₃
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [3]
POL(ok(x₁)) = [1] + x₁

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U21'(x₁, x₂, x₃)) = [4]x₂
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [3]x₂ + [2]x₃
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2]
POL(ok(x₁)) = [4] + x₁

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

AND(ok(z0), ok(z1)) → c40(AND(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [2]x₂
POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U21'(x₁, x₂, x₃)) = [2]x₂ + [3]x₃
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [4]x₃
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [1]
POL(ok(x₁)) = [2] + x₁

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ISNAT(ok(z0)) → c52(ISNAT(z0))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = 0
POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U21'(x₁, x₂, x₃)) = [4]x₃
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₂ + [4]x₃
POL(X(x₁, x₂)) = 0
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2]
POL(ok(x₁)) = [1] + x₁

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [3]x₂
POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = [2]x₂
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(X(x₁, x₂)) = x₂
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [1] + x₁

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
AND(mark(z0), z1) → c39(AND(z0, z1))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U31'(mark(z0)) → c32(U31'(z0))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [2]x₂
POL(ISNAT(x₁)) = [3]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = [2]x₁
POL(U11'(x₁, x₂)) = [2]x₁
POL(U21'(x₁, x₂, x₃)) = [2]x₃
POL(U31'(x₁)) = [2]x₁
POL(U41'(x₁, x₂, x₃)) = [2]x₁ + [2]x₂
POL(X(x₁, x₂)) = [2]x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = [2] + x₁

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [2]x₂
POL(ISNAT(x₁)) = [3]x₁
POL(PLUS(x₁, x₂)) = x₂
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2]x₁
POL(X(x₁, x₂)) = [4]x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(117) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [2]x₁
POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = x₁
POL(S(x₁)) = [3]x₁
POL(U11'(x₁, x₂)) = [2]x₁
POL(U21'(x₁, x₂, x₃)) = 0
POL(U31'(x₁)) = [3]x₁
POL(U41'(x₁, x₂, x₃)) = [4]x₁ + [3]x₂
POL(X(x₁, x₂)) = [4]x₁
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [1] + x₁

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))

K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x₁, x₂)) = [4]x₂
POL(ISNAT(x₁)) = [3]x₁
POL(PLUS(x₁, x₂)) = [5]x₁
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [3]x₂
POL(U21'(x₁, x₂, x₃)) = x₁
POL(U31'(x₁)) = [3]x₁
POL(U41'(x₁, x₂, x₃)) = [2]x₁ + [3]x₂
POL(X(x₁, x₂)) = [4]x₁ + [4]x₂
POL(c23(x₁)) = x₁
POL(c24(x₁)) = x₁
POL(c25(x₁)) = x₁
POL(c26(x₁)) = x₁
POL(c27(x₁)) = x₁
POL(c28(x₁)) = x₁
POL(c29(x₁)) = x₁
POL(c30(x₁)) = x₁
POL(c31(x₁)) = x₁
POL(c32(x₁)) = x₁
POL(c33(x₁)) = x₁
POL(c34(x₁)) = x₁
POL(c35(x₁)) = x₁
POL(c36(x₁)) = x₁
POL(c37(x₁)) = x₁
POL(c38(x₁)) = x₁
POL(c39(x₁)) = x₁
POL(c40(x₁)) = x₁
POL(c52(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
S(mark(z0)) → c27(S(z0))
S(ok(z0)) → c28(S(z0))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
U31'(ok(z0)) → c33(U31'(z0))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
X(mark(z0), z1) → c36(X(z0, z1))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))

S tuples:none
K tuples:

U11'(mark(z0), z1) → c23(U11'(z0, z1))
PLUS(mark(z0), z1) → c29(PLUS(z0, z1))
S(ok(z0)) → c28(S(z0))
U31'(ok(z0)) → c33(U31'(z0))
S(mark(z0)) → c27(S(z0))
X(mark(z0), z1) → c36(X(z0, z1))
U41'(mark(z0), z1, z2) → c34(U41'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c26(U21'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c35(U41'(z0, z1, z2))
U11'(ok(z0), ok(z1)) → c24(U11'(z0, z1))
AND(ok(z0), ok(z1)) → c40(AND(z0, z1))
ISNAT(ok(z0)) → c52(ISNAT(z0))
X(z0, mark(z1)) → c37(X(z0, z1))
X(ok(z0), ok(z1)) → c38(X(z0, z1))
U31'(mark(z0)) → c32(U31'(z0))
PLUS(z0, mark(z1)) → c30(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c31(PLUS(z0, z1))
AND(mark(z0), z1) → c39(AND(z0, z1))
U21'(mark(z0), z1, z2) → c25(U21'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, U31', U41', X, AND, ISNAT

Compound Symbols:

c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c52

(121) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty