(0) Obligation:

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

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

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
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
ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c2
ACTIVE(isNat(0)) → c3
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c26
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
PROPER(0) → c32
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(U11(tt, z0)) → c
ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(and(tt, z0)) → c2
ACTIVE(isNat(0)) → c3
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(tt) → c26
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
PROPER(0) → c32
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35

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

Removed 5 trailing nodes:

ACTIVE(U11(tt, z0)) → c
PROPER(0) → c32
PROPER(tt) → c26
ACTIVE(and(tt, z0)) → c2
ACTIVE(isNat(0)) → c3

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c7(U21'(and(isNat(z1), isNat(z0)), z1, z0), AND(isNat(z1), isNat(z0)), ISNAT(z1), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35

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

Removed 2 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c7(ISNAT(z1), ISNAT(z0))
S tuples:

ACTIVE(U21(tt, z0, z1)) → c1(S(plus(z1, z0)), PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c4(AND(isNat(z0), isNat(z1)), ISNAT(z0), ISNAT(z1))
ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, 0)) → c6(U11'(isNat(z0), z0), ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, s(z1))) → c7(ISNAT(z1), ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c1, c4, c5, c6, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c7

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, U11, U21, s, plus, and, proper, isNat, top

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c

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

Use narrowing to replace ACTIVE(U11(z0, z1)) → c8(U11'(active(z0), z1), ACTIVE(z0)) by

ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8

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

Removed 2 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c8(U11'(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c8(U11'(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1

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

Use narrowing to replace ACTIVE(U21(z0, z1, z2)) → c9(U21'(active(z0), z1, z2), ACTIVE(z0)) by

ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(U11(tt, z0)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2), ACTIVE(isNat(0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9

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

Removed 2 trailing tuple parts

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c8(U11'(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c9(U21'(mark(s(plus(z1, z0))), x1, x2), ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c9(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c9(U21'(mark(isNat(z0)), x1, x2), ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9

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

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2

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

Use narrowing to replace ACTIVE(s(z0)) → c10(S(active(z0)), ACTIVE(z0)) by

ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)), ACTIVE(U11(tt, z0)))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)), ACTIVE(isNat(0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10

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

Removed 2 trailing tuple parts

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(U21(tt, z0, z1))) → c10(S(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(isNat(plus(z0, z1)))) → c10(S(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c10(S(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10

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

Split RHS of tuples not part of any SCC

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3

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

Use narrowing to replace ACTIVE(and(z0, z1)) → c13(AND(active(z0), z1), ACTIVE(z0)) by

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

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2), ACTIVE(and(tt, z0)))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)), ACTIVE(and(tt, z0)))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(U11(tt, z0)))
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1), ACTIVE(and(tt, z0)))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1), ACTIVE(isNat(0)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13

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

Removed 6 trailing tuple parts

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(U21(tt, z0, z1), x1)) → c13(AND(mark(s(plus(z1, z0))), x1), ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c13(AND(mark(and(isNat(z0), isNat(z1))), x1), ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c13(AND(mark(isNat(z0)), x1), ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13

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

Split RHS of tuples not part of any SCC

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4

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

Use narrowing to replace PROPER(U11(z0, z1)) → c25(U11'(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25

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

Removed 4 trailing tuple parts

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c27, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25

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

Use narrowing to replace PROPER(U21(z0, z1, z2)) → c27(U21'(proper(z0), proper(z1), proper(z2)), PROPER(z0), PROPER(z1), PROPER(z2)) by

PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1), PROPER(tt))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1), PROPER(0))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(tt), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(0), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(tt), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(0), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27

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

Removed 6 trailing tuple parts

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c28, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27

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

Use narrowing to replace PROPER(s(z0)) → c28(S(proper(z0)), PROPER(z0)) by

PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(tt)) → c28(S(ok(tt)), PROPER(tt))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(0)) → c28(S(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28

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

Removed 2 trailing tuple parts

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c29, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28

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

Use narrowing to replace PROPER(plus(z0, z1)) → c29(PLUS(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29

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

Removed 4 trailing tuple parts

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c30, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29

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

Use narrowing to replace PROPER(and(z0, z1)) → c30(AND(proper(z0), proper(z1)), PROPER(z0), PROPER(z1)) by

PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0), PROPER(tt))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0), PROPER(0))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(tt), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(0), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30

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

Removed 4 trailing tuple parts

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, PROPER, ISNAT, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c31, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30

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

Use narrowing to replace PROPER(isNat(z0)) → c31(ISNAT(proper(z0)), PROPER(z0)) by

PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)), PROPER(tt))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31

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

Removed 2 trailing tuple parts

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c34, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31

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

Use narrowing to replace TOP(mark(z0)) → c34(TOP(proper(z0)), PROPER(z0)) by

TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(tt)) → c34(TOP(ok(tt)), PROPER(tt))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(0)) → c34(TOP(ok(0)), PROPER(0))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34

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

Removed 2 trailing tuple parts

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34

(63) 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.

TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
We considered the (Usable) Rules:

active(plus(z0, z1)) → plus(active(z0), z1)
active(U11(z0, z1)) → U11(active(z0), z1)
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
s(mark(z0)) → mark(s(z0))
active(isNat(0)) → mark(tt)
active(and(z0, z1)) → and(active(z0), z1)
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
active(s(z0)) → s(active(z0))
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
U11(mark(z0), z1) → mark(U11(z0, z1))
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
and(mark(z0), z1) → mark(and(z0, z1))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, z1)) → plus(z0, active(z1))
plus(mark(z0), z1) → mark(plus(z0, z1))
active(U11(tt, z0)) → mark(z0)
s(ok(z0)) → ok(s(z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(and(tt, z0)) → mark(z0)
and(ok(z0), ok(z1)) → ok(and(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
isNat(ok(z0)) → ok(isNat(z0))
active(isNat(s(z0))) → mark(isNat(z0))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
And the Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVE(x1)) = 0   
POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(PROPER(x1)) = 0   
POL(S(x1)) = 0   
POL(TOP(x1)) = [4]x1   
POL(U11(x1, x2)) = [4]   
POL(U11'(x1, x2)) = 0   
POL(U21(x1, x2, x3)) = [4]   
POL(U21'(x1, x2, x3)) = 0   
POL(active(x1)) = x1   
POL(and(x1, x2)) = [4]   
POL(c(x1)) = x1   
POL(c1(x1)) = x1   
POL(c10(x1)) = x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c11(x1, x2)) = x1 + x2   
POL(c12(x1, x2)) = x1 + x2   
POL(c13(x1)) = x1   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c2(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c25(x1, x2)) = x1 + x2   
POL(c25(x1, x2, x3)) = x1 + x2 + x3   
POL(c27(x1, x2, x3)) = x1 + x2 + x3   
POL(c27(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c28(x1)) = x1   
POL(c28(x1, x2)) = x1 + x2   
POL(c29(x1, x2)) = x1 + x2   
POL(c29(x1, x2, x3)) = x1 + x2 + x3   
POL(c3(x1)) = x1   
POL(c30(x1, x2)) = x1 + x2   
POL(c30(x1, x2, x3)) = x1 + x2 + x3   
POL(c31(x1)) = x1   
POL(c31(x1, x2)) = x1 + x2   
POL(c33(x1)) = x1   
POL(c34(x1)) = x1   
POL(c34(x1, x2)) = x1 + x2   
POL(c35(x1, x2)) = x1 + x2   
POL(c4(x1)) = x1   
POL(c5(x1)) = x1   
POL(c8(x1)) = x1   
POL(c8(x1, x2)) = x1 + x2   
POL(c9(x1)) = x1   
POL(c9(x1, x2)) = x1 + x2   
POL(isNat(x1)) = [4]   
POL(mark(x1)) = [4]   
POL(ok(x1)) = x1   
POL(plus(x1, x2)) = [4]   
POL(proper(x1)) = 0   
POL(s(x1)) = [4]   
POL(tt) = 0   

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
K tuples:

TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, TOP, PROPER

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c35, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34

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

Use narrowing to replace TOP(ok(z0)) → c35(TOP(active(z0)), ACTIVE(z0)) by

TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:

TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, PROPER, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c34, c35

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

Removed 3 trailing nodes:

TOP(ok(isNat(0))) → c35(TOP(mark(tt)), ACTIVE(isNat(0)))
TOP(mark(tt)) → c34(TOP(ok(tt)))
TOP(mark(0)) → c34(TOP(ok(0)))

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
S tuples:

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

ACTIVE, U11', U21', S, PLUS, AND, ISNAT, PROPER, TOP

Compound Symbols:

c5, c11, c12, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33, c, c8, c8, c1, c9, c9, c2, c10, c10, c3, c13, c13, c4, c25, c25, c27, c27, c28, c28, c29, c29, c30, c30, c31, c31, c34, c35

(69) CdtUnreachableProof (EQUIVALENT transformation)

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

ACTIVE(isNat(s(z0))) → c5(ISNAT(z0))
ACTIVE(plus(z0, z1)) → c11(PLUS(active(z0), z1), ACTIVE(z0))
ACTIVE(plus(z0, z1)) → c12(PLUS(z0, active(z1)), ACTIVE(z1))
ACTIVE(U21(tt, z0, z1)) → c(S(plus(z1, z0)))
ACTIVE(U21(tt, z0, z1)) → c(PLUS(z1, z0))
ACTIVE(isNat(plus(z0, z1))) → c(AND(isNat(z0), isNat(z1)))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z0))
ACTIVE(isNat(plus(z0, z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, 0)) → c(U11'(isNat(z0), z0))
ACTIVE(plus(z0, 0)) → c(ISNAT(z0))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z1))
ACTIVE(plus(z0, s(z1))) → c(ISNAT(z0))
ACTIVE(U11(plus(z0, 0), x1)) → c8(U11'(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(U11(plus(z0, s(z1)), x1)) → c8(U11'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(U11(U11(z0, z1), x1)) → c8(U11'(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(U11(U21(z0, z1, z2), x1)) → c8(U11'(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U11(s(z0), x1)) → c8(U11'(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(plus(z0, z1), x1)) → c8(U11'(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(U11(and(z0, z1), x1)) → c8(U11'(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(U11(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U11(isNat(0), x1)) → c8(U11'(mark(tt), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(U11'(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(U11(isNat(plus(z0, z1)), x1)) → c1(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(U11'(mark(isNat(z0)), x1))
ACTIVE(U11(isNat(s(z0)), x1)) → c1(ACTIVE(isNat(s(z0))))
ACTIVE(U21(plus(z0, 0), x1, x2)) → c9(U21'(mark(U11(isNat(z0), z0)), x1, x2), ACTIVE(plus(z0, 0)))
ACTIVE(U21(plus(z0, s(z1)), x1, x2)) → c9(U21'(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1, x2), ACTIVE(plus(z0, s(z1))))
ACTIVE(U21(U11(z0, z1), x1, x2)) → c9(U21'(U11(active(z0), z1), x1, x2), ACTIVE(U11(z0, z1)))
ACTIVE(U21(U21(z0, z1, z2), x1, x2)) → c9(U21'(U21(active(z0), z1, z2), x1, x2), ACTIVE(U21(z0, z1, z2)))
ACTIVE(U21(s(z0), x1, x2)) → c9(U21'(s(active(z0)), x1, x2), ACTIVE(s(z0)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(active(z0), z1), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(plus(z0, z1), x1, x2)) → c9(U21'(plus(z0, active(z1)), x1, x2), ACTIVE(plus(z0, z1)))
ACTIVE(U21(and(z0, z1), x1, x2)) → c9(U21'(and(active(z0), z1), x1, x2), ACTIVE(and(z0, z1)))
ACTIVE(U21(U11(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(U21(isNat(0), x1, x2)) → c9(U21'(mark(tt), x1, x2))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(U11'(mark(s(plus(z1, z0))), x1))
ACTIVE(U11(U21(tt, z0, z1), x1)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(U21'(mark(s(plus(z1, z0))), x1, x2))
ACTIVE(U21(U21(tt, z0, z1), x1, x2)) → c2(ACTIVE(U21(tt, z0, z1)))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(U21'(mark(and(isNat(z0), isNat(z1))), x1, x2))
ACTIVE(U21(isNat(plus(z0, z1)), x1, x2)) → c2(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(U21'(mark(isNat(z0)), x1, x2))
ACTIVE(U21(isNat(s(z0)), x1, x2)) → c2(ACTIVE(isNat(s(z0))))
ACTIVE(s(plus(z0, 0))) → c10(S(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
ACTIVE(s(plus(z0, s(z1)))) → c10(S(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
ACTIVE(s(U11(z0, z1))) → c10(S(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
ACTIVE(s(U21(z0, z1, z2))) → c10(S(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
ACTIVE(s(s(z0))) → c10(S(s(active(z0))), ACTIVE(s(z0)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
ACTIVE(s(plus(z0, z1))) → c10(S(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
ACTIVE(s(and(z0, z1))) → c10(S(and(active(z0), z1)), ACTIVE(and(z0, z1)))
ACTIVE(s(U11(tt, z0))) → c10(S(mark(z0)))
ACTIVE(s(isNat(0))) → c10(S(mark(tt)))
ACTIVE(s(U21(tt, z0, z1))) → c3(S(mark(s(plus(z1, z0)))))
ACTIVE(s(U21(tt, z0, z1))) → c3(ACTIVE(U21(tt, z0, z1)))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(S(mark(and(isNat(z0), isNat(z1)))))
ACTIVE(s(isNat(plus(z0, z1)))) → c3(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(s(isNat(s(z0)))) → c3(S(mark(isNat(z0))))
ACTIVE(s(isNat(s(z0)))) → c3(ACTIVE(isNat(s(z0))))
ACTIVE(and(plus(z0, 0), x1)) → c13(AND(mark(U11(isNat(z0), z0)), x1), ACTIVE(plus(z0, 0)))
ACTIVE(and(plus(z0, s(z1)), x1)) → c13(AND(mark(U21(and(isNat(z1), isNat(z0)), z1, z0)), x1), ACTIVE(plus(z0, s(z1))))
ACTIVE(and(U11(z0, z1), x1)) → c13(AND(U11(active(z0), z1), x1), ACTIVE(U11(z0, z1)))
ACTIVE(and(U21(z0, z1, z2), x1)) → c13(AND(U21(active(z0), z1, z2), x1), ACTIVE(U21(z0, z1, z2)))
ACTIVE(and(s(z0), x1)) → c13(AND(s(active(z0)), x1), ACTIVE(s(z0)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(active(z0), z1), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(plus(z0, z1), x1)) → c13(AND(plus(z0, active(z1)), x1), ACTIVE(plus(z0, z1)))
ACTIVE(and(and(z0, z1), x1)) → c13(AND(and(active(z0), z1), x1), ACTIVE(and(z0, z1)))
ACTIVE(U11(and(tt, z0), x1)) → c8(U11'(mark(z0), x1))
ACTIVE(U21(and(tt, z0), x1, x2)) → c9(U21'(mark(z0), x1, x2))
ACTIVE(s(and(tt, z0))) → c10(S(mark(z0)))
ACTIVE(and(U11(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(and(tt, z0), x1)) → c13(AND(mark(z0), x1))
ACTIVE(and(isNat(0), x1)) → c13(AND(mark(tt), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(AND(mark(s(plus(z1, z0))), x1))
ACTIVE(and(U21(tt, z0, z1), x1)) → c4(ACTIVE(U21(tt, z0, z1)))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(AND(mark(and(isNat(z0), isNat(z1))), x1))
ACTIVE(and(isNat(plus(z0, z1)), x1)) → c4(ACTIVE(isNat(plus(z0, z1))))
ACTIVE(and(isNat(s(z0)), x1)) → c4(AND(mark(isNat(z0)), x1))
ACTIVE(and(isNat(s(z0)), x1)) → c4(ACTIVE(isNat(s(z0))))
PROPER(U11(x0, U11(z0, z1))) → c25(U11'(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(U11(x0, U21(z0, z1, z2))) → c25(U11'(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(U11(x0, s(z0))) → c25(U11'(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(U11(x0, plus(z0, z1))) → c25(U11'(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(U11(x0, and(z0, z1))) → c25(U11'(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(U11(x0, isNat(z0))) → c25(U11'(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(U11(U11(z0, z1), x1)) → c25(U11'(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(U11(U21(z0, z1, z2), x1)) → c25(U11'(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(U11(s(z0), x1)) → c25(U11'(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(U11(plus(z0, z1), x1)) → c25(U11'(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(U11(and(z0, z1), x1)) → c25(U11'(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(U11(isNat(z0), x1)) → c25(U11'(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(U11(x0, tt)) → c25(U11'(proper(x0), ok(tt)), PROPER(x0))
PROPER(U11(x0, 0)) → c25(U11'(proper(x0), ok(0)), PROPER(x0))
PROPER(U11(tt, x1)) → c25(U11'(ok(tt), proper(x1)), PROPER(x1))
PROPER(U11(0, x1)) → c25(U11'(ok(0), proper(x1)), PROPER(x1))
PROPER(U21(x0, x1, U11(z0, z1))) → c27(U21'(proper(x0), proper(x1), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(U11(z0, z1)))
PROPER(U21(x0, x1, U21(z0, z1, z2))) → c27(U21'(proper(x0), proper(x1), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(x1), PROPER(U21(z0, z1, z2)))
PROPER(U21(x0, x1, s(z0))) → c27(U21'(proper(x0), proper(x1), s(proper(z0))), PROPER(x0), PROPER(x1), PROPER(s(z0)))
PROPER(U21(x0, x1, plus(z0, z1))) → c27(U21'(proper(x0), proper(x1), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(plus(z0, z1)))
PROPER(U21(x0, x1, and(z0, z1))) → c27(U21'(proper(x0), proper(x1), and(proper(z0), proper(z1))), PROPER(x0), PROPER(x1), PROPER(and(z0, z1)))
PROPER(U21(x0, x1, isNat(z0))) → c27(U21'(proper(x0), proper(x1), isNat(proper(z0))), PROPER(x0), PROPER(x1), PROPER(isNat(z0)))
PROPER(U21(x0, U11(z0, z1), x2)) → c27(U21'(proper(x0), U11(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(U11(z0, z1)), PROPER(x2))
PROPER(U21(x0, U21(z0, z1, z2), x2)) → c27(U21'(proper(x0), U21(proper(z0), proper(z1), proper(z2)), proper(x2)), PROPER(x0), PROPER(U21(z0, z1, z2)), PROPER(x2))
PROPER(U21(x0, s(z0), x2)) → c27(U21'(proper(x0), s(proper(z0)), proper(x2)), PROPER(x0), PROPER(s(z0)), PROPER(x2))
PROPER(U21(x0, plus(z0, z1), x2)) → c27(U21'(proper(x0), plus(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(plus(z0, z1)), PROPER(x2))
PROPER(U21(x0, and(z0, z1), x2)) → c27(U21'(proper(x0), and(proper(z0), proper(z1)), proper(x2)), PROPER(x0), PROPER(and(z0, z1)), PROPER(x2))
PROPER(U21(x0, isNat(z0), x2)) → c27(U21'(proper(x0), isNat(proper(z0)), proper(x2)), PROPER(x0), PROPER(isNat(z0)), PROPER(x2))
PROPER(U21(U11(z0, z1), x1, x2)) → c27(U21'(U11(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(U11(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(U21(z0, z1, z2), x1, x2)) → c27(U21'(U21(proper(z0), proper(z1), proper(z2)), proper(x1), proper(x2)), PROPER(U21(z0, z1, z2)), PROPER(x1), PROPER(x2))
PROPER(U21(s(z0), x1, x2)) → c27(U21'(s(proper(z0)), proper(x1), proper(x2)), PROPER(s(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(plus(z0, z1), x1, x2)) → c27(U21'(plus(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(plus(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(and(z0, z1), x1, x2)) → c27(U21'(and(proper(z0), proper(z1)), proper(x1), proper(x2)), PROPER(and(z0, z1)), PROPER(x1), PROPER(x2))
PROPER(U21(isNat(z0), x1, x2)) → c27(U21'(isNat(proper(z0)), proper(x1), proper(x2)), PROPER(isNat(z0)), PROPER(x1), PROPER(x2))
PROPER(U21(x0, x1, tt)) → c27(U21'(proper(x0), proper(x1), ok(tt)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, x1, 0)) → c27(U21'(proper(x0), proper(x1), ok(0)), PROPER(x0), PROPER(x1))
PROPER(U21(x0, tt, x2)) → c27(U21'(proper(x0), ok(tt), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(x0, 0, x2)) → c27(U21'(proper(x0), ok(0), proper(x2)), PROPER(x0), PROPER(x2))
PROPER(U21(tt, x1, x2)) → c27(U21'(ok(tt), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(U21(0, x1, x2)) → c27(U21'(ok(0), proper(x1), proper(x2)), PROPER(x1), PROPER(x2))
PROPER(s(U11(z0, z1))) → c28(S(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(s(U21(z0, z1, z2))) → c28(S(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(s(s(z0))) → c28(S(s(proper(z0))), PROPER(s(z0)))
PROPER(s(plus(z0, z1))) → c28(S(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(s(and(z0, z1))) → c28(S(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(s(isNat(z0))) → c28(S(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(s(tt)) → c28(S(ok(tt)))
PROPER(s(0)) → c28(S(ok(0)))
PROPER(plus(x0, U11(z0, z1))) → c29(PLUS(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(plus(x0, U21(z0, z1, z2))) → c29(PLUS(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(plus(x0, s(z0))) → c29(PLUS(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(plus(x0, plus(z0, z1))) → c29(PLUS(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(plus(x0, and(z0, z1))) → c29(PLUS(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(plus(x0, isNat(z0))) → c29(PLUS(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(plus(U11(z0, z1), x1)) → c29(PLUS(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(plus(U21(z0, z1, z2), x1)) → c29(PLUS(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(plus(s(z0), x1)) → c29(PLUS(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(plus(plus(z0, z1), x1)) → c29(PLUS(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(plus(and(z0, z1), x1)) → c29(PLUS(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(plus(isNat(z0), x1)) → c29(PLUS(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(plus(x0, tt)) → c29(PLUS(proper(x0), ok(tt)), PROPER(x0))
PROPER(plus(x0, 0)) → c29(PLUS(proper(x0), ok(0)), PROPER(x0))
PROPER(plus(tt, x1)) → c29(PLUS(ok(tt), proper(x1)), PROPER(x1))
PROPER(plus(0, x1)) → c29(PLUS(ok(0), proper(x1)), PROPER(x1))
PROPER(and(x0, U11(z0, z1))) → c30(AND(proper(x0), U11(proper(z0), proper(z1))), PROPER(x0), PROPER(U11(z0, z1)))
PROPER(and(x0, U21(z0, z1, z2))) → c30(AND(proper(x0), U21(proper(z0), proper(z1), proper(z2))), PROPER(x0), PROPER(U21(z0, z1, z2)))
PROPER(and(x0, s(z0))) → c30(AND(proper(x0), s(proper(z0))), PROPER(x0), PROPER(s(z0)))
PROPER(and(x0, plus(z0, z1))) → c30(AND(proper(x0), plus(proper(z0), proper(z1))), PROPER(x0), PROPER(plus(z0, z1)))
PROPER(and(x0, and(z0, z1))) → c30(AND(proper(x0), and(proper(z0), proper(z1))), PROPER(x0), PROPER(and(z0, z1)))
PROPER(and(x0, isNat(z0))) → c30(AND(proper(x0), isNat(proper(z0))), PROPER(x0), PROPER(isNat(z0)))
PROPER(and(U11(z0, z1), x1)) → c30(AND(U11(proper(z0), proper(z1)), proper(x1)), PROPER(U11(z0, z1)), PROPER(x1))
PROPER(and(U21(z0, z1, z2), x1)) → c30(AND(U21(proper(z0), proper(z1), proper(z2)), proper(x1)), PROPER(U21(z0, z1, z2)), PROPER(x1))
PROPER(and(s(z0), x1)) → c30(AND(s(proper(z0)), proper(x1)), PROPER(s(z0)), PROPER(x1))
PROPER(and(plus(z0, z1), x1)) → c30(AND(plus(proper(z0), proper(z1)), proper(x1)), PROPER(plus(z0, z1)), PROPER(x1))
PROPER(and(and(z0, z1), x1)) → c30(AND(and(proper(z0), proper(z1)), proper(x1)), PROPER(and(z0, z1)), PROPER(x1))
PROPER(and(isNat(z0), x1)) → c30(AND(isNat(proper(z0)), proper(x1)), PROPER(isNat(z0)), PROPER(x1))
PROPER(and(x0, tt)) → c30(AND(proper(x0), ok(tt)), PROPER(x0))
PROPER(and(x0, 0)) → c30(AND(proper(x0), ok(0)), PROPER(x0))
PROPER(and(tt, x1)) → c30(AND(ok(tt), proper(x1)), PROPER(x1))
PROPER(and(0, x1)) → c30(AND(ok(0), proper(x1)), PROPER(x1))
PROPER(isNat(U11(z0, z1))) → c31(ISNAT(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
PROPER(isNat(U21(z0, z1, z2))) → c31(ISNAT(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
PROPER(isNat(s(z0))) → c31(ISNAT(s(proper(z0))), PROPER(s(z0)))
PROPER(isNat(plus(z0, z1))) → c31(ISNAT(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
PROPER(isNat(and(z0, z1))) → c31(ISNAT(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
PROPER(isNat(isNat(z0))) → c31(ISNAT(isNat(proper(z0))), PROPER(isNat(z0)))
PROPER(isNat(tt)) → c31(ISNAT(ok(tt)))
PROPER(isNat(0)) → c31(ISNAT(ok(0)))
TOP(mark(U11(z0, z1))) → c34(TOP(U11(proper(z0), proper(z1))), PROPER(U11(z0, z1)))
TOP(mark(U21(z0, z1, z2))) → c34(TOP(U21(proper(z0), proper(z1), proper(z2))), PROPER(U21(z0, z1, z2)))
TOP(mark(s(z0))) → c34(TOP(s(proper(z0))), PROPER(s(z0)))
TOP(mark(plus(z0, z1))) → c34(TOP(plus(proper(z0), proper(z1))), PROPER(plus(z0, z1)))
TOP(mark(and(z0, z1))) → c34(TOP(and(proper(z0), proper(z1))), PROPER(and(z0, z1)))
TOP(mark(isNat(z0))) → c34(TOP(isNat(proper(z0))), PROPER(isNat(z0)))
TOP(ok(U11(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(U11(tt, z0)))
TOP(ok(U21(tt, z0, z1))) → c35(TOP(mark(s(plus(z1, z0)))), ACTIVE(U21(tt, z0, z1)))
TOP(ok(and(tt, z0))) → c35(TOP(mark(z0)), ACTIVE(and(tt, z0)))
TOP(ok(isNat(plus(z0, z1)))) → c35(TOP(mark(and(isNat(z0), isNat(z1)))), ACTIVE(isNat(plus(z0, z1))))
TOP(ok(isNat(s(z0)))) → c35(TOP(mark(isNat(z0))), ACTIVE(isNat(s(z0))))
TOP(ok(plus(z0, 0))) → c35(TOP(mark(U11(isNat(z0), z0))), ACTIVE(plus(z0, 0)))
TOP(ok(plus(z0, s(z1)))) → c35(TOP(mark(U21(and(isNat(z1), isNat(z0)), z1, z0))), ACTIVE(plus(z0, s(z1))))
TOP(ok(U11(z0, z1))) → c35(TOP(U11(active(z0), z1)), ACTIVE(U11(z0, z1)))
TOP(ok(U21(z0, z1, z2))) → c35(TOP(U21(active(z0), z1, z2)), ACTIVE(U21(z0, z1, z2)))
TOP(ok(s(z0))) → c35(TOP(s(active(z0))), ACTIVE(s(z0)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(active(z0), z1)), ACTIVE(plus(z0, z1)))
TOP(ok(plus(z0, z1))) → c35(TOP(plus(z0, active(z1))), ACTIVE(plus(z0, z1)))
TOP(ok(and(z0, z1))) → c35(TOP(and(active(z0), z1)), ACTIVE(and(z0, z1)))

(70) Obligation:

Complexity Dependency Tuples Problem
Rules:

active(U11(tt, z0)) → mark(z0)
active(U21(tt, z0, z1)) → mark(s(plus(z1, z0)))
active(and(tt, z0)) → mark(z0)
active(isNat(0)) → mark(tt)
active(isNat(plus(z0, z1))) → mark(and(isNat(z0), isNat(z1)))
active(isNat(s(z0))) → mark(isNat(z0))
active(plus(z0, 0)) → mark(U11(isNat(z0), z0))
active(plus(z0, s(z1))) → mark(U21(and(isNat(z1), isNat(z0)), z1, z0))
active(U11(z0, z1)) → U11(active(z0), z1)
active(U21(z0, z1, z2)) → U21(active(z0), z1, z2)
active(s(z0)) → s(active(z0))
active(plus(z0, z1)) → plus(active(z0), z1)
active(plus(z0, z1)) → plus(z0, active(z1))
active(and(z0, z1)) → and(active(z0), z1)
s(mark(z0)) → mark(s(z0))
s(ok(z0)) → ok(s(z0))
plus(mark(z0), z1) → mark(plus(z0, z1))
plus(z0, mark(z1)) → mark(plus(z0, z1))
plus(ok(z0), ok(z1)) → ok(plus(z0, z1))
and(mark(z0), z1) → mark(and(z0, z1))
and(ok(z0), ok(z1)) → ok(and(z0, z1))
isNat(ok(z0)) → ok(isNat(z0))
U11(mark(z0), z1) → mark(U11(z0, z1))
U11(ok(z0), ok(z1)) → ok(U11(z0, z1))
U21(mark(z0), z1, z2) → mark(U21(z0, z1, z2))
U21(ok(z0), ok(z1), ok(z2)) → ok(U21(z0, z1, z2))
proper(U11(z0, z1)) → U11(proper(z0), proper(z1))
proper(tt) → ok(tt)
proper(U21(z0, z1, z2)) → U21(proper(z0), proper(z1), proper(z2))
proper(s(z0)) → s(proper(z0))
proper(plus(z0, z1)) → plus(proper(z0), proper(z1))
proper(and(z0, z1)) → and(proper(z0), proper(z1))
proper(isNat(z0)) → isNat(proper(z0))
proper(0) → ok(0)
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
K tuples:none
Defined Rule Symbols:

active, s, plus, and, isNat, U11, U21, proper

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(71) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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

(72) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(73) 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.

AND(mark(z0), z1) → c23(AND(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [4]x1   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = 0   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = x1   

(74) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(75) 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)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = x2   
POL(ISNAT(x1)) = x1   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = [2]x2   
POL(U21'(x1, x2, x3)) = [4]x2   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = 0   
POL(ok(x1)) = [2] + x1   

(76) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(77) 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) → c14(U11'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [2]x1 + [4]x2   
POL(ISNAT(x1)) = [4]x1   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = x1   
POL(U21'(x1, x2, x3)) = 0   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = x1   

(78) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(79) 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) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [4]x1 + [2]x2   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = x1   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = 0   
POL(U21'(x1, x2, x3)) = [4]x2   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [2] + x1   

(80) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(81) 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), z1, z2) → c16(U21'(z0, z1, z2))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [4]x2   
POL(ISNAT(x1)) = 0   
POL(PLUS(x1, x2)) = [2]x1   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = [2]x1 + x2   
POL(U21'(x1, x2, x3)) = [2]x1 + x2 + [2]x3   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = [2] + x1   

(82) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(83) 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(z0, mark(z1)) → c21(PLUS(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [2]x2   
POL(ISNAT(x1)) = [2]x1   
POL(PLUS(x1, x2)) = [4]x2   
POL(S(x1)) = 0   
POL(U11'(x1, x2)) = [4]x2   
POL(U21'(x1, x2, x3)) = [2]x2   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [4] + x1   
POL(ok(x1)) = [3] + x1   

(84) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(85) 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.

S(ok(z0)) → c19(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = [4]x1 + [2]x2   
POL(ISNAT(x1)) = [2]x1   
POL(PLUS(x1, x2)) = 0   
POL(S(x1)) = [4]x1   
POL(U11'(x1, x2)) = [4]x1 + [4]x2   
POL(U21'(x1, x2, x3)) = [4]x1 + [3]x2 + [3]x3   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = x1   
POL(ok(x1)) = [1] + x1   

(86) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:

S(mark(z0)) → c18(S(z0))
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
S(ok(z0)) → c19(S(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

(87) 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.

S(mark(z0)) → c18(S(z0))
We considered the (Usable) Rules:none
And the Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AND(x1, x2)) = 0   
POL(ISNAT(x1)) = [3]x1   
POL(PLUS(x1, x2)) = [4]x2   
POL(S(x1)) = x1   
POL(U11'(x1, x2)) = [3]x2   
POL(U21'(x1, x2, x3)) = [2]x3   
POL(c14(x1)) = x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(c17(x1)) = x1   
POL(c18(x1)) = x1   
POL(c19(x1)) = x1   
POL(c20(x1)) = x1   
POL(c21(x1)) = x1   
POL(c22(x1)) = x1   
POL(c23(x1)) = x1   
POL(c24(x1)) = x1   
POL(c33(x1)) = x1   
POL(mark(x1)) = [1] + x1   
POL(ok(x1)) = x1   

(88) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

U11'(mark(z0), z1) → c14(U11'(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
S(mark(z0)) → c18(S(z0))
S(ok(z0)) → c19(S(z0))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
AND(mark(z0), z1) → c23(AND(z0, z1))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
S tuples:none
K tuples:

AND(mark(z0), z1) → c23(AND(z0, z1))
U11'(ok(z0), ok(z1)) → c15(U11'(z0, z1))
U21'(ok(z0), ok(z1), ok(z2)) → c17(U21'(z0, z1, z2))
AND(ok(z0), ok(z1)) → c24(AND(z0, z1))
ISNAT(ok(z0)) → c33(ISNAT(z0))
U11'(mark(z0), z1) → c14(U11'(z0, z1))
PLUS(mark(z0), z1) → c20(PLUS(z0, z1))
PLUS(ok(z0), ok(z1)) → c22(PLUS(z0, z1))
U21'(mark(z0), z1, z2) → c16(U21'(z0, z1, z2))
PLUS(z0, mark(z1)) → c21(PLUS(z0, z1))
S(ok(z0)) → c19(S(z0))
S(mark(z0)) → c18(S(z0))
Defined Rule Symbols:none

Defined Pair Symbols:

U11', U21', S, PLUS, AND, ISNAT

Compound Symbols:

c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c33

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

The set S is empty

(90) BOUNDS(O(1), O(1))