Runtime Complexity (innermost) proof of /tmp/tmpGvorb_/PEANO_nokinds_noand

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 94 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 (⇔, 156 ms)

↳10 CdtProblem

↳11 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 40 ms)

↳12 CdtProblem

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

↳14 CdtProblem

↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 13 ms)

↳16 CdtProblem

↳17 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 0 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), 10 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), 0 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), 24 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), 8 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), 0 ms)

↳48 CdtProblem

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

↳50 CdtProblem

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

↳52 CdtProblem

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

↳54 CdtProblem

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

↳56 CdtProblem

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

↳58 CdtProblem

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

↳60 CdtProblem

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

↳62 CdtProblem

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

↳64 CdtProblem

↳65 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 14 ms)

↳66 CdtProblem

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

↳68 CdtProblem

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

↳70 CdtProblem

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

↳72 CdtProblem

↳73 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 58 ms)

↳74 CdtProblem

↳75 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 18 ms)

↳76 CdtProblem

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

↳78 CdtProblem

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

↳80 CdtProblem

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

↳82 CdtProblem

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

↳84 CdtProblem

↳85 CdtNarrowingProof (BOTH BOUNDS(ID, ID), 99 ms)

↳86 CdtProblem

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

↳88 CdtProblem

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

↳90 CdtProblem

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

↳92 CdtProblem

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

↳94 CdtProblem

↳95 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 23 ms)

↳96 CdtProblem

↳97 CdtUnreachableProof (⇔, 0 ms)

↳98 CdtProblem

↳99 CdtUsableRulesProof (⇔, 0 ms)

↳100 CdtProblem

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

↳102 CdtProblem

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

↳104 CdtProblem

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

↳106 CdtProblem

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

↳108 CdtProblem

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

↳110 CdtProblem

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

↳112 CdtProblem

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

↳114 CdtProblem

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

↳116 CdtProblem

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

↳118 CdtProblem

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

↳120 CdtProblem

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

↳122 CdtProblem

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

↳124 BOUNDS(O(1), O(1))

(0) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U12(tt)) → c1
ACTIVE(U21(tt)) → c2
ACTIVE(U31(tt, z0)) → c3
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(0)) → c6
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(U21'(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c38
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c47
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U12(tt)) → c1
ACTIVE(U21(tt)) → c2
ACTIVE(U31(tt, z0)) → c3
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(0)) → c6
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(U21'(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c38
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(0) → c47
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U12, U21, U31, U41, U42, s, plus, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, 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

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

Removed 6 trailing nodes:

PROPER(tt) → c38
ACTIVE(U12(tt)) → c1
ACTIVE(U21(tt)) → c2
ACTIVE(U31(tt, z0)) → c3
PROPER(0) → c47
ACTIVE(isNat(0)) → c6

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(U21'(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))

S tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(U21'(isNat(z0)), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U12, U21, U31, U41, U42, s, plus, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

c, c4, c5, 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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))

S tuples:

ACTIVE(U11(tt, z0)) → c(U12'(isNat(z0)), ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c5(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ACTIVE(plus(z0, 0)) → c9(U31'(isNat(z0), z0), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))

K tuples:none
Defined Rule Symbols:

active, U11, U12, U21, U31, U41, U42, s, plus, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

c, c4, c5, c7, c9, 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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)
isNat(ok(z0)) → ok(isNat(z0))
top(mark(z0)) → top(proper(z0))
top(ok(z0)) → top(active(z0))

Tuples:

ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))

S tuples:

ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, U11, U12, U21, U31, U41, U42, s, plus, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))

S tuples:

ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1

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

Use narrowing to replace ACTIVE(U11(z0, z1)) → c11(U11'(active(z0), z1), ACTIVE(z0)) by

ACTIVE(U11(U11(tt, z0), x1)) → c11(U11'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c11(U11'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c11(U11'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c11(U11'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c11(U11'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c11(U11'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c11(U11'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c11(U11'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c11(U11'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c11(U11'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c11(U11'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c11(U11'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))

S tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c11(U11'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c11(U11'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c11(U11'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0))
ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c

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

Use narrowing to replace ACTIVE(U12(z0)) → c12(U12'(active(z0)), ACTIVE(z0)) by

ACTIVE(U12(U11(tt, z0))) → c12(U12'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c12(U12'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c12(U12'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U11(tt, z0))) → c12(U12'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c12(U12'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c12(U12'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U11(tt, z0))) → c12(U12'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c12(U12'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c12(U12'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c, c12

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

Removed 3 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U11(tt, z0))) → c12(U12'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c12(U12'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c12(U12'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))

S tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U11(tt, z0))) → c12(U12'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c12(U12'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c12(U12'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c, c12, c12

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0))
ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c, c12, c12, c2

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

Use narrowing to replace ACTIVE(U21(z0)) → c13(U21'(active(z0)), ACTIVE(z0)) by

ACTIVE(U21(U11(tt, z0))) → c13(U21'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c13(U21'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c13(U21'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0))) → c13(U21'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c13(U21'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c13(U21'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0))) → c13(U21'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c13(U21'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c13(U21'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c, c12, c12, c2, c13

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

Removed 5 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0))) → c13(U21'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c13(U21'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c13(U21'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))

S tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0))) → c13(U21'(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c13(U21'(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c13(U21'(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c39, c40, c41, c42, c43, c44, c45, c46, c48, c49, c50, c8, c10, c1, c11, c11, c, c12, c12, c2, c13, c13

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0))
ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(U31(z0, z1)) → c14(U31'(active(z0), z1), ACTIVE(z0)) by

ACTIVE(U31(U11(tt, z0), x1)) → c14(U31'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c14(U31'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c14(U31'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U11(tt, z0), x1)) → c14(U31'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c14(U31'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c14(U31'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U11(tt, z0), x1)) → c14(U31'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U12(tt)))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1), ACTIVE(U21(tt)))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1), ACTIVE(U31(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c14(U31'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c14(U31'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 7 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U11(tt, z0), x1)) → c14(U31'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c14(U31'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c14(U31'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))

S tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U11(tt, z0), x1)) → c14(U31'(mark(U12(isNat(z0))), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c14(U31'(mark(U11(isNat(z0), z1)), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c14(U31'(mark(U21(isNat(z0))), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(U41(z0, z1, z2)) → c15(U41'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U41(U11(tt, z0), x1, x2)) → c15(U41'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c15(U41'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c15(U41'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c15(U41'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c15(U41'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c15(U41'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c15(U41'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c15(U41'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c15(U41'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c15(U41'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c15(U41'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c15(U41'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c15(U41'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c15(U41'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c15(U41'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c15(U41'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))

S tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c11(U11'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U41(tt, z0, z1))) → c12(U12'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U41(tt, z0, z1))) → c13(U21'(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c14(U31'(mark(U42(isNat(z1), z0, z1)), x1), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c15(U41'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c15(U41'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(isNat(plus(z0, z1)), x1, x2)) → c15(U41'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c15(U41'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(U42(z0, z1, z2)) → c16(U42'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U42(U11(tt, z0), x1, x2)) → c16(U42'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c16(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c16(U42'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c16(U42'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c16(U42'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c16(U42'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c16(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c16(U42'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c16(U42'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c16(U42'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c16(U42'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U12(tt)))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(U21(tt)))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2), ACTIVE(U31(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c16(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c16(U42'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c16(U42'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c16(U42'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c16(U42'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c16(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c16(U42'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c16(U42'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c16(U42'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))

S tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c11(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(U42(tt, z0, z1))) → c12(U12'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U42(tt, z0, z1))) → c13(U21'(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c14(U31'(mark(s(plus(z1, z0))), x1), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c15(U41'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c16(U42'(mark(U12(isNat(z0))), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c16(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2), ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c16(U42'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c16(U42'(mark(U11(isNat(z0), z1)), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c16(U42'(mark(U21(isNat(z0))), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

(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(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace ACTIVE(s(z0)) → c17(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(U11(tt, z0))) → c17(S(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c17(S(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c17(S(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c17(S(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c17(S(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c17(S(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c17(S(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c17(S(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c17(S(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c17(S(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c17(S(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)), ACTIVE(U12(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)), ACTIVE(U21(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)), ACTIVE(U31(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c17(S(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c17(S(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c17(S(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c17(S(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c17(S(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c17(S(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c17(S(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c17(S(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c17(S(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c17(S(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c17(S(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c17(S(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c17(S(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c17(S(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Split RHS of tuples not part of any SCC

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U11(z0, z1)) → c37(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U12(z0)) → c39(U12'(proper(z0)), PROPER(z0)) by

PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)), PROPER(tt))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(0)) → c39(U12'(ok(0)), PROPER(0))

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)), PROPER(tt))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(0)) → c39(U12'(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)), PROPER(tt))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(0)) → c39(U12'(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(isNat(z0)) → c40(ISNAT(proper(z0)), PROPER(z0)) by

PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)), PROPER(0))

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U21(z0)) → c41(U21'(proper(z0)), PROPER(z0)) by

PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)), PROPER(tt))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(0)) → c41(U21'(ok(0)), PROPER(0))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)), PROPER(tt))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(0)) → c41(U21'(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)), PROPER(tt))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(0)) → c41(U21'(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U31(z0, z1)) → c42(U31'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U41(z0, z1, z2)) → c43(U41'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 6 trailing tuple parts

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(U42(z0, z1, z2)) → c44(U42'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 6 trailing tuple parts

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(s(z0)) → c45(S(proper(z0)), PROPER(z0)) by

PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c45(S(ok(0)), PROPER(0))

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c45(S(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)), PROPER(tt))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(0)) → c45(S(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, PROPER, ISNAT, TOP

Compound Symbols:

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

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

Use narrowing to replace PROPER(plus(z0, z1)) → c46(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, TOP, PROPER

Compound Symbols:

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

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

Removed 4 trailing tuple parts

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(mark(z0)) → c49(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c49(TOP(ok(0)), PROPER(0))

(90) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c49(TOP(ok(0)), PROPER(0))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)), PROPER(tt))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(0)) → c49(TOP(ok(0)), PROPER(0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, TOP, PROPER

Compound Symbols:

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

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

Removed 2 trailing tuple parts

(92) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)))
TOP(mark(0)) → c49(TOP(ok(0)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)))
TOP(mark(0)) → c49(TOP(ok(0)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, TOP, PROPER

Compound Symbols:

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

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

Use narrowing to replace TOP(ok(z0)) → c50(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U12(tt))) → c50(TOP(mark(tt)), ACTIVE(U12(tt)))
TOP(ok(U21(tt))) → c50(TOP(mark(tt)), ACTIVE(U21(tt)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(0))) → c50(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

(94) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)))
TOP(mark(0)) → c49(TOP(ok(0)))
TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U12(tt))) → c50(TOP(mark(tt)), ACTIVE(U12(tt)))
TOP(ok(U21(tt))) → c50(TOP(mark(tt)), ACTIVE(U21(tt)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(0))) → c50(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(tt)) → c49(TOP(ok(tt)))
TOP(mark(0)) → c49(TOP(ok(0)))
TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U12(tt))) → c50(TOP(mark(tt)), ACTIVE(U12(tt)))
TOP(ok(U21(tt))) → c50(TOP(mark(tt)), ACTIVE(U21(tt)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(0))) → c50(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, PROPER, TOP

Compound Symbols:

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

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

Removed 5 trailing nodes:

TOP(ok(isNat(0))) → c50(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(mark(tt)) → c49(TOP(ok(tt)))
TOP(mark(0)) → c49(TOP(ok(0)))
TOP(ok(U12(tt))) → c50(TOP(mark(tt)), ACTIVE(U12(tt)))
TOP(ok(U21(tt))) → c50(TOP(mark(tt)), ACTIVE(U21(tt)))

(96) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

S tuples:

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

ACTIVE, U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT, PROPER, TOP

Compound Symbols:

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

(97) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(plus(z0, z1)) → c18(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c19(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(isNat(s(z0))) → c8(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c10(ISNAT(z1))
ACTIVE(U11(tt, z0)) → c1(U12'(isNat(z0)))
ACTIVE(U11(tt, z0)) → c1(ISNAT(z0))
ACTIVE(U41(tt, z0, z1)) → c1(U42'(isNat(z1), z0, z1))
ACTIVE(U41(tt, z0, z1)) → c1(ISNAT(z1))
ACTIVE(U42(tt, z0, z1)) → c1(S(plus(z1, z0)))
ACTIVE(U42(tt, z0, z1)) → c1(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c1(U11'(isNat(z0), z1))
ACTIVE(isNat(plus(z0, z1))) → c1(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c1(U31'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c1(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c11(U11'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c11(U11'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c11(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U12(z0), x1)) → c11(U11'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U11(U21(z0), x1)) → c11(U11'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U11(U31(z0, z1), x1)) → c11(U11'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U11(U41(z0, z1, z2), x1)) → c11(U11'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U11(U42(z0, z1, z2), x1)) → c11(U11'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c11(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c11(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(isNat(0), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(U11'(mark(U12(isNat(z0))), x1))
ACTIVE(U11(U11(tt, z0), x1)) → c(ACTIVE(U11(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(U11'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c(U11'(mark(U21(isNat(z0))), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c(ACTIVE(isNat(s(z0))))
ACTIVE(U12(plus(z0, 0))) → c12(U12'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U12(plus(z0, s(z1)))) → c12(U12'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U12(U11(z0, z1))) → c12(U12'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U12(U12(z0))) → c12(U12'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U12(U21(z0))) → c12(U12'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U12(U31(z0, z1))) → c12(U12'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U12(U41(z0, z1, z2))) → c12(U12'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U12(U42(z0, z1, z2))) → c12(U12'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U12(s(z0))) → c12(U12'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U12(plus(z0, z1))) → c12(U12'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U12(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U12(tt))) → c12(U12'(mark(tt)))
ACTIVE(U12(isNat(0))) → c12(U12'(mark(tt)))
ACTIVE(U12(U11(tt, z0))) → c2(U12'(mark(U12(isNat(z0)))))
ACTIVE(U12(U11(tt, z0))) → c2(ACTIVE(U11(tt, z0)))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(U12'(mark(U11(isNat(z0), z1))))
ACTIVE(U12(isNat(plus(z0, z1)))) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U12(isNat(s(z0)))) → c2(U12'(mark(U21(isNat(z0)))))
ACTIVE(U12(isNat(s(z0)))) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0))) → c13(U21'(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)))) → c13(U21'(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1))) → c13(U21'(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U12(z0))) → c13(U21'(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(U21(U21(z0))) → c13(U21'(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(U21(U31(z0, z1))) → c13(U21'(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(U21(U41(z0, z1, z2))) → c13(U21'(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U21(U42(z0, z1, z2))) → c13(U21'(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U21(s(z0))) → c13(U21'(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1))) → c13(U21'(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U21(tt), x1)) → c11(U11'(mark(tt), x1))
ACTIVE(U12(U21(tt))) → c12(U12'(mark(tt)))
ACTIVE(U21(U12(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(U21(tt))) → c13(U21'(mark(tt)))
ACTIVE(U21(isNat(0))) → c13(U21'(mark(tt)))
ACTIVE(U21(U11(tt, z0))) → c3(U21'(mark(U12(isNat(z0)))))
ACTIVE(U21(U11(tt, z0))) → c3(ACTIVE(U11(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(U21'(mark(U11(isNat(z0), z1))))
ACTIVE(U21(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)))) → c3(U21'(mark(U21(isNat(z0)))))
ACTIVE(U21(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(U31(plus(z0, 0), x1)) → c14(U31'(mark(U31(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U31(plus(z0, s(z1)), x1)) → c14(U31'(mark(U41(isNat(z1), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U31(U11(z0, z1), x1)) → c14(U31'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U31(U12(z0), x1)) → c14(U31'(U12(active(z0)), x1), ACTIVE(U12(z0)))
ACTIVE(U31(U21(z0), x1)) → c14(U31'(U21(active(z0)), x1), ACTIVE(U21(z0)))
ACTIVE(U31(U31(z0, z1), x1)) → c14(U31'(U31(active(z0), z1), x1), ACTIVE(U31(z0, z1)))
ACTIVE(U31(U41(z0, z1, z2), x1)) → c14(U31'(U41(active(z0), z1, z2), x1), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U31(U42(z0, z1, z2), x1)) → c14(U31'(U42(active(z0), z1, z2), x1), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U31(s(z0), x1)) → c14(U31'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U31(plus(z0, z1), x1)) → c14(U31'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(U31(tt, z0), x1)) → c11(U11'(mark(z0), x1))
ACTIVE(U12(U31(tt, z0))) → c12(U12'(mark(z0)))
ACTIVE(U21(U31(tt, z0))) → c13(U21'(mark(z0)))
ACTIVE(U31(U12(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U21(tt), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U31(tt, z0), x1)) → c14(U31'(mark(z0), x1))
ACTIVE(U31(isNat(0), x1)) → c14(U31'(mark(tt), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(U31'(mark(U12(isNat(z0))), x1))
ACTIVE(U31(U11(tt, z0), x1)) → c4(ACTIVE(U11(tt, z0)))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(U31'(mark(U11(isNat(z0), z1)), x1))
ACTIVE(U31(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(U31'(mark(U21(isNat(z0))), x1))
ACTIVE(U31(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
ACTIVE(U41(plus(z0, 0), x1, x2)) → c15(U41'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U41(plus(z0, s(z1)), x1, x2)) → c15(U41'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U41(U11(z0, z1), x1, x2)) → c15(U41'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U41(U12(z0), x1, x2)) → c15(U41'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U41(U21(z0), x1, x2)) → c15(U41'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U41(U31(z0, z1), x1, x2)) → c15(U41'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U41(U41(z0, z1, z2), x1, x2)) → c15(U41'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U41(U42(z0, z1, z2), x1, x2)) → c15(U41'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U41(s(z0), x1, x2)) → c15(U41'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(plus(z0, z1), x1, x2)) → c15(U41'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U41(U12(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U21(tt), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U41(U31(tt, z0), x1, x2)) → c15(U41'(mark(z0), x1, x2))
ACTIVE(U41(isNat(0), x1, x2)) → c15(U41'(mark(tt), x1, x2))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(U11'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U11(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U12(U41(tt, z0, z1))) → c5(U12'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U12(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U21(U41(tt, z0, z1))) → c5(U21'(mark(U42(isNat(z1), z0, z1))))
ACTIVE(U21(U41(tt, z0, z1))) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(U31'(mark(U42(isNat(z1), z0, z1)), x1))
ACTIVE(U31(U41(tt, z0, z1), x1)) → c5(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(U41'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U41(U11(tt, z0), x1, x2)) → c5(ACTIVE(U11(tt, z0)))
ACTIVE(U41(U41(tt, z0, z1), x1, x2)) → c5(U41'(mark(U42(isNat(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(U11(isNat(z0), 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(U21(isNat(z0))), x1, x2))
ACTIVE(U41(isNat(s(z0)), x1, x2)) → c5(ACTIVE(isNat(s(z0))))
ACTIVE(U42(plus(z0, 0), x1, x2)) → c16(U42'(mark(U31(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U42(plus(z0, s(z1)), x1, x2)) → c16(U42'(mark(U41(isNat(z1), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U42(U11(z0, z1), x1, x2)) → c16(U42'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U42(U12(z0), x1, x2)) → c16(U42'(U12(active(z0)), x1, x2), ACTIVE(U12(z0)))
ACTIVE(U42(U21(z0), x1, x2)) → c16(U42'(U21(active(z0)), x1, x2), ACTIVE(U21(z0)))
ACTIVE(U42(U31(z0, z1), x1, x2)) → c16(U42'(U31(active(z0), z1), x1, x2), ACTIVE(U31(z0, z1)))
ACTIVE(U42(U41(z0, z1, z2), x1, x2)) → c16(U42'(U41(active(z0), z1, z2), x1, x2), ACTIVE(U41(z0, z1, z2)))
ACTIVE(U42(U42(z0, z1, z2), x1, x2)) → c16(U42'(U42(active(z0), z1, z2), x1, x2), ACTIVE(U42(z0, z1, z2)))
ACTIVE(U42(s(z0), x1, x2)) → c16(U42'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(plus(z0, z1), x1, x2)) → c16(U42'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U42(U12(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U21(tt), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U42(U31(tt, z0), x1, x2)) → c16(U42'(mark(z0), x1, x2))
ACTIVE(U42(isNat(0), x1, x2)) → c16(U42'(mark(tt), x1, x2))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U12(U42(tt, z0, z1))) → c6(U12'(mark(s(plus(z1, z0)))))
ACTIVE(U12(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U21(U42(tt, z0, z1))) → c6(U21'(mark(s(plus(z1, z0)))))
ACTIVE(U21(U42(tt, z0, z1))) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(U31'(mark(s(plus(z1, z0))), x1))
ACTIVE(U31(U42(tt, z0, z1), x1)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(U41'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U41(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(U42'(mark(U12(isNat(z0))), x1, x2))
ACTIVE(U42(U11(tt, z0), x1, x2)) → c6(ACTIVE(U11(tt, z0)))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(U42'(mark(U42(isNat(z1), z0, z1)), x1, x2))
ACTIVE(U42(U41(tt, z0, z1), x1, x2)) → c6(ACTIVE(U41(tt, z0, z1)))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(U42'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U42(U42(tt, z0, z1), x1, x2)) → c6(ACTIVE(U42(tt, z0, z1)))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(U42'(mark(U11(isNat(z0), z1)), x1, x2))
ACTIVE(U42(isNat(plus(z0, z1)), x1, x2)) → c6(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(U42'(mark(U21(isNat(z0))), x1, x2))
ACTIVE(U42(isNat(s(z0)), x1, x2)) → c6(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c17(S(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c17(S(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c17(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U12(z0))) → c17(S(U12(active(z0))), ACTIVE(U12(z0)))
ACTIVE(s(U21(z0))) → c17(S(U21(active(z0))), ACTIVE(U21(z0)))
ACTIVE(s(U31(z0, z1))) → c17(S(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
ACTIVE(s(U41(z0, z1, z2))) → c17(S(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
ACTIVE(s(U42(z0, z1, z2))) → c17(S(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
ACTIVE(s(s(z0))) → c17(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c17(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(U12(tt))) → c17(S(mark(tt)))
ACTIVE(s(U21(tt))) → c17(S(mark(tt)))
ACTIVE(s(U31(tt, z0))) → c17(S(mark(z0)))
ACTIVE(s(isNat(0))) → c17(S(mark(tt)))
ACTIVE(s(U11(tt, z0))) → c7(S(mark(U12(isNat(z0)))))
ACTIVE(s(U11(tt, z0))) → c7(ACTIVE(U11(tt, z0)))
ACTIVE(s(U41(tt, z0, z1))) → c7(S(mark(U42(isNat(z1), z0, z1))))
ACTIVE(s(U41(tt, z0, z1))) → c7(ACTIVE(U41(tt, z0, z1)))
ACTIVE(s(U42(tt, z0, z1))) → c7(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U42(tt, z0, z1))) → c7(ACTIVE(U42(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(S(mark(U11(isNat(z0), z1))))
ACTIVE(s(isNat(plus(z0, z1)))) → c7(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c7(S(mark(U21(isNat(z0)))))
ACTIVE(s(isNat(s(z0)))) → c7(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c37(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U12(z0))) → c37(U11'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U11(x0, isNat(z0))) → c37(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, U21(z0))) → c37(U11'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U11(x0, U31(z0, z1))) → c37(U11'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U11(x0, U41(z0, z1, z2))) → c37(U11'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U11(x0, U42(z0, z1, z2))) → c37(U11'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c37(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c37(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(U11(z0, z1), x1)) → c37(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U12(z0), x1)) → c37(U11'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c37(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(U21(z0), x1)) → c37(U11'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U11(U31(z0, z1), x1)) → c37(U11'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U11(U41(z0, z1, z2), x1)) → c37(U11'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U11(U42(z0, z1, z2), x1)) → c37(U11'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c37(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c37(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(x0, tt)) → c37(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c37(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c37(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c37(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U12(U11(z0, z1))) → c39(U12'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U12(U12(z0))) → c39(U12'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U12(isNat(z0))) → c39(U12'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U12(U21(z0))) → c39(U12'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U12(U31(z0, z1))) → c39(U12'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U12(U41(z0, z1, z2))) → c39(U12'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U12(U42(z0, z1, z2))) → c39(U12'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U12(s(z0))) → c39(U12'(s(proper(z0))), PROPER(s(z0)))
PROPER(U12(plus(z0, z1))) → c39(U12'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U12(tt)) → c39(U12'(ok(tt)))
PROPER(U12(0)) → c39(U12'(ok(0)))
PROPER(isNat(U11(z0, z1))) → c40(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U12(z0))) → c40(ISNAT(U12(proper(z0))), PROPER(U12(z0)))
PROPER(isNat(isNat(z0))) → c40(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(U21(z0))) → c40(ISNAT(U21(proper(z0))), PROPER(U21(z0)))
PROPER(isNat(U31(z0, z1))) → c40(ISNAT(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(isNat(U41(z0, z1, z2))) → c40(ISNAT(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(isNat(U42(z0, z1, z2))) → c40(ISNAT(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(isNat(s(z0))) → c40(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c40(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(tt)) → c40(ISNAT(ok(tt)))
PROPER(isNat(0)) → c40(ISNAT(ok(0)))
PROPER(U21(U11(z0, z1))) → c41(U21'(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(U21(U12(z0))) → c41(U21'(U12(proper(z0))), PROPER(U12(z0)))
PROPER(U21(isNat(z0))) → c41(U21'(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(U21(U21(z0))) → c41(U21'(U21(proper(z0))), PROPER(U21(z0)))
PROPER(U21(U31(z0, z1))) → c41(U21'(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(U21(U41(z0, z1, z2))) → c41(U21'(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(U21(U42(z0, z1, z2))) → c41(U21'(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(U21(s(z0))) → c41(U21'(s(proper(z0))), PROPER(s(z0)))
PROPER(U21(plus(z0, z1))) → c41(U21'(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(U21(tt)) → c41(U21'(ok(tt)))
PROPER(U21(0)) → c41(U21'(ok(0)))
PROPER(U31(x0, U11(z0, z1))) → c42(U31'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U31(x0, U12(z0))) → c42(U31'(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(U31(x0, isNat(z0))) → c42(U31'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U31(x0, U21(z0))) → c42(U31'(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(U31(x0, U31(z0, z1))) → c42(U31'(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(U31(x0, U41(z0, z1, z2))) → c42(U31'(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(U31(x0, U42(z0, z1, z2))) → c42(U31'(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(U31(x0, s(z0))) → c42(U31'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U31(x0, plus(z0, z1))) → c42(U31'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U31(U11(z0, z1), x1)) → c42(U31'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U31(U12(z0), x1)) → c42(U31'(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(U31(isNat(z0), x1)) → c42(U31'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U31(U21(z0), x1)) → c42(U31'(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(U31(U31(z0, z1), x1)) → c42(U31'(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(U31(U41(z0, z1, z2), x1)) → c42(U31'(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(U31(U42(z0, z1, z2), x1)) → c42(U31'(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(U31(s(z0), x1)) → c42(U31'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U31(plus(z0, z1), x1)) → c42(U31'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U31(x0, tt)) → c42(U31'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U31(x0, 0)) → c42(U31'(proper(x0), ok(0)), PROPER(x0))
PROPER(U31(tt, x1)) → c42(U31'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U31(0, x1)) → c42(U31'(ok(0), proper(x1)), PROPER(x1))
PROPER(U41(x0, x1, U11(z0, z1))) → c43(U41'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U41(x0, x1, U12(z0))) → c43(U41'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U41(x0, x1, isNat(z0))) → c43(U41'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U41(x0, x1, U21(z0))) → c43(U41'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U41(x0, x1, U31(z0, z1))) → c43(U41'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U41(x0, x1, U41(z0, z1, z2))) → c43(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, U42(z0, z1, z2))) → c43(U41'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U41(x0, x1, s(z0))) → c43(U41'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U41(x0, x1, plus(z0, z1))) → c43(U41'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U41(x0, U11(z0, z1), x2)) → c43(U41'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U41(x0, U12(z0), x2)) → c43(U41'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U41(x0, isNat(z0), x2)) → c43(U41'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U41(x0, U21(z0), x2)) → c43(U41'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U41(x0, U31(z0, z1), x2)) → c43(U41'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U41(x0, U41(z0, z1, z2), x2)) → c43(U41'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, U42(z0, z1, z2), x2)) → c43(U41'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U41(x0, s(z0), x2)) → c43(U41'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U41(x0, plus(z0, z1), x2)) → c43(U41'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U41(U11(z0, z1), x1, x2)) → c43(U41'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U12(z0), x1, x2)) → c43(U41'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(isNat(z0), x1, x2)) → c43(U41'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U21(z0), x1, x2)) → c43(U41'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(U31(z0, z1), x1, x2)) → c43(U41'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(U41(z0, z1, z2), x1, x2)) → c43(U41'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(U42(z0, z1, z2), x1, x2)) → c43(U41'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U41(s(z0), x1, x2)) → c43(U41'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U41(plus(z0, z1), x1, x2)) → c43(U41'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U41(x0, x1, tt)) → c43(U41'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, x1, 0)) → c43(U41'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U41(x0, tt, x2)) → c43(U41'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(x0, 0, x2)) → c43(U41'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U41(tt, x1, x2)) → c43(U41'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U41(0, x1, x2)) → c43(U41'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, U11(z0, z1))) → c44(U42'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U42(x0, x1, U12(z0))) → c44(U42'(proper(x0), proper(x1), U12(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U12(z0)))
PROPER(U42(x0, x1, isNat(z0))) → c44(U42'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U42(x0, x1, U21(z0))) → c44(U42'(proper(x0), proper(x1), U21(proper(z0))), PROPER(x0), PROPER(x1), PROPER(U21(z0)))
PROPER(U42(x0, x1, U31(z0, z1))) → c44(U42'(proper(x0), proper(x1), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U31(z0, z1)))
PROPER(U42(x0, x1, U41(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U41(z0, z1, z2)))
PROPER(U42(x0, x1, U42(z0, z1, z2))) → c44(U42'(proper(x0), proper(x1), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U42(z0, z1, z2)))
PROPER(U42(x0, x1, s(z0))) → c44(U42'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U42(x0, x1, plus(z0, z1))) → c44(U42'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U42(x0, U11(z0, z1), x2)) → c44(U42'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U42(x0, U12(z0), x2)) → c44(U42'(proper(x0), U12(proper(z0)), proper(x2)), PROPER(x0), PROPER(U12(z0)), PROPER(x2))
PROPER(U42(x0, isNat(z0), x2)) → c44(U42'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U42(x0, U21(z0), x2)) → c44(U42'(proper(x0), U21(proper(z0)), proper(x2)), PROPER(x0), PROPER(U21(z0)), PROPER(x2))
PROPER(U42(x0, U31(z0, z1), x2)) → c44(U42'(proper(x0), U31(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U31(z0, z1)), PROPER(x2))
PROPER(U42(x0, U41(z0, z1, z2), x2)) → c44(U42'(proper(x0), U41(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U41(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, U42(z0, z1, z2), x2)) → c44(U42'(proper(x0), U42(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U42(z0, z1, z2)), PROPER(x2))
PROPER(U42(x0, s(z0), x2)) → c44(U42'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U42(x0, plus(z0, z1), x2)) → c44(U42'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U42(U11(z0, z1), x1, x2)) → c44(U42'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U12(z0), x1, x2)) → c44(U42'(U12(proper(z0)), proper(x1), proper(x2)), PROPER(U12(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(isNat(z0), x1, x2)) → c44(U42'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U21(z0), x1, x2)) → c44(U42'(U21(proper(z0)), proper(x1), proper(x2)), PROPER(U21(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(U31(z0, z1), x1, x2)) → c44(U42'(U31(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U31(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(U41(z0, z1, z2), x1, x2)) → c44(U42'(U41(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U41(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(U42(z0, z1, z2), x1, x2)) → c44(U42'(U42(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U42(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U42(s(z0), x1, x2)) → c44(U42'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U42(plus(z0, z1), x1, x2)) → c44(U42'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U42(x0, x1, tt)) → c44(U42'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, x1, 0)) → c44(U42'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U42(x0, tt, x2)) → c44(U42'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(x0, 0, x2)) → c44(U42'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U42(tt, x1, x2)) → c44(U42'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U42(0, x1, x2)) → c44(U42'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c45(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U12(z0))) → c45(S(U12(proper(z0))), PROPER(U12(z0)))
PROPER(s(isNat(z0))) → c45(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(U21(z0))) → c45(S(U21(proper(z0))), PROPER(U21(z0)))
PROPER(s(U31(z0, z1))) → c45(S(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
PROPER(s(U41(z0, z1, z2))) → c45(S(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
PROPER(s(U42(z0, z1, z2))) → c45(S(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
PROPER(s(s(z0))) → c45(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c45(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(tt)) → c45(S(ok(tt)))
PROPER(s(0)) → c45(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c46(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U12(z0))) → c46(PLUS(proper(x0), U12(proper(z0))), PROPER(x0), PROPER(U12(z0)))
PROPER(plus(x0, isNat(z0))) → c46(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, U21(z0))) → c46(PLUS(proper(x0), U21(proper(z0))), PROPER(x0), PROPER(U21(z0)))
PROPER(plus(x0, U31(z0, z1))) → c46(PLUS(proper(x0), U31(proper(z0), proper(z1))), PROPER(x0), PROPER(U31(z0, z1)))
PROPER(plus(x0, U41(z0, z1, z2))) → c46(PLUS(proper(x0), U41(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U41(z0, z1, z2)))
PROPER(plus(x0, U42(z0, z1, z2))) → c46(PLUS(proper(x0), U42(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U42(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c46(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c46(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(U11(z0, z1), x1)) → c46(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U12(z0), x1)) → c46(PLUS(U12(proper(z0)), proper(x1)), PROPER(U12(z0)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c46(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(U21(z0), x1)) → c46(PLUS(U21(proper(z0)), proper(x1)), PROPER(U21(z0)), PROPER(x1))
PROPER(plus(U31(z0, z1), x1)) → c46(PLUS(U31(proper(z0), proper(z1)), proper(x1)), PROPER(U31(z0, z1)), PROPER(x1))
PROPER(plus(U41(z0, z1, z2), x1)) → c46(PLUS(U41(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U41(z0, z1, z2)), PROPER(x1))
PROPER(plus(U42(z0, z1, z2), x1)) → c46(PLUS(U42(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U42(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c46(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c46(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(x0, tt)) → c46(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c46(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c46(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c46(PLUS(ok(0), proper(x1)), PROPER(x1))
TOP(mark(U11(z0, z1))) → c49(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U12(z0))) → c49(TOP(U12(proper(z0))), PROPER(U12(z0)))
TOP(mark(isNat(z0))) → c49(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(U21(z0))) → c49(TOP(U21(proper(z0))), PROPER(U21(z0)))
TOP(mark(U31(z0, z1))) → c49(TOP(U31(proper(z0), proper(z1))), PROPER(U31(z0, z1)))
TOP(mark(U41(z0, z1, z2))) → c49(TOP(U41(proper(z0), proper(z1), proper(z2))), PROPER(U41(z0, z1, z2)))
TOP(mark(U42(z0, z1, z2))) → c49(TOP(U42(proper(z0), proper(z1), proper(z2))), PROPER(U42(z0, z1, z2)))
TOP(mark(s(z0))) → c49(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c49(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(ok(U11(tt, z0))) → c50(TOP(mark(U12(isNat(z0)))), ACTIVE(U11(tt, z0)))
TOP(ok(U31(tt, z0))) → c50(TOP(mark(z0)), ACTIVE(U31(tt, z0)))
TOP(ok(U41(tt, z0, z1))) → c50(TOP(mark(U42(isNat(z1), z0, z1))), ACTIVE(U41(tt, z0, z1)))
TOP(ok(U42(tt, z0, z1))) → c50(TOP(mark(s(plus(z1, z0)))), ACTIVE(U42(tt, z0, z1)))
TOP(ok(isNat(plus(z0, z1)))) → c50(TOP(mark(U11(isNat(z0), z1))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c50(TOP(mark(U21(isNat(z0)))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c50(TOP(mark(U31(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c50(TOP(mark(U41(isNat(z1), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c50(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U12(z0))) → c50(TOP(U12(active(z0))), ACTIVE(U12(z0)))
TOP(ok(U21(z0))) → c50(TOP(U21(active(z0))), ACTIVE(U21(z0)))
TOP(ok(U31(z0, z1))) → c50(TOP(U31(active(z0), z1)), ACTIVE(U31(z0, z1)))
TOP(ok(U41(z0, z1, z2))) → c50(TOP(U41(active(z0), z1, z2)), ACTIVE(U41(z0, z1, z2)))
TOP(ok(U42(z0, z1, z2))) → c50(TOP(U42(active(z0), z1, z2)), ACTIVE(U42(z0, z1, z2)))
TOP(ok(s(z0))) → c50(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c50(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))

(98) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

active, U12, isNat, U42, s, plus, U11, U31, U21, U41, proper

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

(99) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

active(U11(tt, z0)) → mark(U12(isNat(z0)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, z0)) → mark(z0)
active(U41(tt, z0, z1)) → mark(U42(isNat(z1), z0, z1))
active(U42(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(U11(isNat(z0), z1))
active(isNat(s(z0))) → mark(U21(isNat(z0)))
active(plus(z0, 0)) → mark(U31(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U41(isNat(z1), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U12(z0)) → U12(active(z0))
active(U21(z0)) → U21(active(z0))
active(U31(z0, z1)) → U31(active(z0), z1)
active(U41(z0, z1, z2)) → U41(active(z0), z1, z2)
active(U42(z0, z1, z2)) → U42(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))
U12(mark(z0)) → mark(U12(z0))
U12(ok(z0)) → ok(U12(z0))
isNat(ok(z0)) → ok(isNat(z0))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(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))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U21(mark(z0)) → mark(U21(z0))
U21(ok(z0)) → ok(U21(z0))
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(U12(z0)) → U12(proper(z0))
proper(isNat(z0)) → isNat(proper(z0))
proper(U21(z0)) → U21(proper(z0))
proper(U31(z0, z1)) → U31(proper(z0), proper(z1))
proper(U41(z0, z1, z2)) → U41(proper(z0), proper(z1), proper(z2))
proper(U42(z0, z1, z2)) → U42(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(0) → ok(0)

(100) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

(101) CdtRuleRemovalProof (UPPER BOUND (ADD(O(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)) → c21(U11'(z0, z1))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = 0
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = 0
POL(ok(x₁)) = [2] + x₁

(102) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₁ + [3]x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = 0
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(104) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [3]x₁ + [2]x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [2] + x₁

(106) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = x₁ + [3]x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₂
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = [4]x₂ + [3]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [3] + x₁
POL(ok(x₁)) = [2] + x₁

(108) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [2]x₂
POL(U12'(x₁)) = [4]x₁
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = [4]x₁
POL(U41'(x₁, x₂, x₃)) = [4]x₁
POL(U42'(x₁, x₂, x₃)) = [2]x₁ + [3]x₂
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [4] + x₁
POL(ok(x₁)) = x₁

(110) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

ISNAT(ok(z0)) → c48(ISNAT(z0))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = [4]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [2]x₂
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = [2]x₂
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = [2]x₂ + [3]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [2]
POL(ok(x₁)) = [4] + x₁

(112) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = [3]x₁
POL(PLUS(x₁, x₂)) = x₂
POL(S(x₁)) = [4]x₁
POL(U11'(x₁, x₂)) = [2]x₁ + x₂
POL(U12'(x₁)) = [4]x₁
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = x₁ + [3]x₂
POL(U41'(x₁, x₂, x₃)) = 0
POL(U42'(x₁, x₂, x₃)) = [4]x₁ + [3]x₂ + [2]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [2] + x₁
POL(ok(x₁)) = [4] + x₁

(114) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))
U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = 0
POL(U12'(x₁)) = 0
POL(U21'(x₁)) = 0
POL(U31'(x₁, x₂)) = 0
POL(U41'(x₁, x₂, x₃)) = x₂
POL(U42'(x₁, x₂, x₃)) = [3]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [5]
POL(ok(x₁)) = [2] + x₁

(116) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))
U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U21'(ok(z0)) → c25(U21'(z0))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = [2]x₁
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = [4]x₁
POL(U11'(x₁, x₂)) = x₂
POL(U12'(x₁)) = [2]x₁
POL(U21'(x₁)) = [2]x₁
POL(U31'(x₁, x₂)) = [2]x₂
POL(U41'(x₁, x₂, x₃)) = [4]x₃
POL(U42'(x₁, x₂, x₃)) = [4]x₁ + x₂ + [2]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(ok(x₁)) = [1] + x₁

(118) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

U21'(mark(z0)) → c24(U21'(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))
U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U21'(ok(z0)) → c25(U21'(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

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

U21'(mark(z0)) → c24(U21'(z0))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = 0
POL(PLUS(x₁, x₂)) = 0
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [2]x₁ + [5]x₂
POL(U12'(x₁)) = [4]x₁
POL(U21'(x₁)) = x₁
POL(U31'(x₁, x₂)) = [2]x₁
POL(U41'(x₁, x₂, x₃)) = [2]x₃
POL(U42'(x₁, x₂, x₃)) = [2]x₂ + [4]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(120) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:

PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))
U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U21'(ok(z0)) → c25(U21'(z0))
U21'(mark(z0)) → c24(U21'(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

(121) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

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

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

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

POL(ISNAT(x₁)) = [4]x₁
POL(PLUS(x₁, x₂)) = [2]x₁
POL(S(x₁)) = 0
POL(U11'(x₁, x₂)) = [4]x₂
POL(U12'(x₁)) = [2]x₁
POL(U21'(x₁)) = [2]x₁
POL(U31'(x₁, x₂)) = [4]x₂
POL(U41'(x₁, x₂, x₃)) = [3]x₁ + [4]x₃
POL(U42'(x₁, x₂, x₃)) = [2]x₁ + [3]x₂ + [2]x₃
POL(c20(x₁)) = x₁
POL(c21(x₁)) = x₁
POL(c22(x₁)) = 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(c48(x₁)) = x₁
POL(mark(x₁)) = [1] + x₁
POL(ok(x₁)) = x₁

(122) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c20(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U12'(ok(z0)) → c23(U12'(z0))
U21'(mark(z0)) → c24(U21'(z0))
U21'(ok(z0)) → c25(U21'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
ISNAT(ok(z0)) → c48(ISNAT(z0))

S tuples:none
K tuples:

U11'(ok(z0), ok(z1)) → c21(U11'(z0, z1))
U11'(mark(z0), z1) → c20(U11'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c31(U42'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c27(U31'(z0, z1))
U12'(mark(z0)) → c22(U12'(z0))
U31'(mark(z0), z1) → c26(U31'(z0, z1))
U41'(mark(z0), z1, z2) → c28(U41'(z0, z1, z2))
U42'(mark(z0), z1, z2) → c30(U42'(z0, z1, z2))
ISNAT(ok(z0)) → c48(ISNAT(z0))
U12'(ok(z0)) → c23(U12'(z0))
S(mark(z0)) → c32(S(z0))
S(ok(z0)) → c33(S(z0))
PLUS(z0, mark(z1)) → c35(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c36(PLUS(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c29(U41'(z0, z1, z2))
U21'(ok(z0)) → c25(U21'(z0))
U21'(mark(z0)) → c24(U21'(z0))
PLUS(mark(z0), z1) → c34(PLUS(z0, z1))

Defined Rule Symbols:none

Defined Pair Symbols:

U11', U12', U21', U31', U41', U42', S, PLUS, ISNAT

Compound Symbols:

c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c48

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

The set S is empty