The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔)
↳4 CdtProblem
↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳6 CdtProblem
↳7 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳10 CdtProblem
↳11 CdtKnowledgeProof (⇔)
↳12 CdtProblem
↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳14 CdtProblem
↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳16 CdtProblem
↳17 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳18 CdtProblem
↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳20 CdtProblem
↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))))
↳22 CdtProblem
↳23 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳24 CdtProblem
↳25 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳26 CdtProblem
↳27 SIsEmptyProof (⇔)
↳28 BOUNDS(O(1), O(1))

(0) Obligation:

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

U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
activate(n__x(X1, X2)) → x(activate(X1), activate(X2))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, PLUS, X, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c15, c16, c18, c19, c23, c24, c25, c26

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PLUS(z0, 0) → c15(U41'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c16(U51'(isNat(z1), z1, z0), ISNAT(z1))
X(z0, 0) → c18(U61'(isNat(z0)), ISNAT(z0))
X(z0, s(z1)) → c19(U71'(isNat(z1), z1, z0), ISNAT(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0) → c5(ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U61'(tt) → c8(0')
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U51', U52', U61', U71', U72', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c5, c6, c7, c8, c9, c10, c12, c13, c14, c23, c24, c25, c26

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 15 dangling nodes:

ACTIVATE(n__0) → c23(0')
U61'(tt) → c8(0')
U41'(tt, z0) → c5(ACTIVATE(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', U51', U52', U71', U72', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c6, c7, c9, c10, c12, c13, c14, c24, c25, c26

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

Split RHS of tuples not part of any SCC

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(S(plus(activate(z1), activate(z0))))
U52'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(PLUS(x(activate(z1), activate(z0)), activate(z1)))
U72'(tt, z0, z1) → c1(X(activate(z1), activate(z0)))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(S(plus(activate(z1), activate(z0))))
U52'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(PLUS(x(activate(z1), activate(z0)), activate(z1)))
U72'(tt, z0, z1) → c1(X(activate(z1), activate(z0)))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

U11', U31', ISNAT, ACTIVATE, U51', U52', U71', U72'

Compound Symbols:

c, c3, c12, c13, c14, c24, c25, c26, c1

(9) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)

Removed 10 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(11) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(13) CdtPolyRedPairProof (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'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]   
POL(ACTIVATE(x1)) = 0   
POL(ISNAT(x1)) = [2]x1   
POL(U11(x1, x2)) = [4] + [3]x1 + [3]x2   
POL(U11'(x1, x2)) = [4] + [2]x2   
POL(U12(x1)) = [4]   
POL(U21(x1)) = [5] + [2]x1   
POL(U31(x1, x2)) = [3] + [3]x1   
POL(U31'(x1, x2)) = [2]x2   
POL(U32(x1)) = [2] + [3]x1   
POL(U51'(x1, x2, x3)) = [3] + [2]x3   
POL(U52'(x1, x2, x3)) = x3   
POL(U71'(x1, x2, x3)) = [5] + [4]x1 + [3]x2 + [4]x3   
POL(U72'(x1, x2, x3)) = [3]   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = [2]   
POL(n__0) = [3]   
POL(n__plus(x1, x2)) = [4] + x1 + x2   
POL(n__s(x1)) = x1   
POL(n__x(x1, x2)) = x1 + x2   
POL(plus(x1, x2)) = [4] + x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(x(x1, x2)) = x1 + x2   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]   
POL(ACTIVATE(x1)) = 0   
POL(ISNAT(x1)) = x1   
POL(U11(x1, x2)) = [3] + [2]x1 + x2   
POL(U11'(x1, x2)) = x2   
POL(U12(x1)) = [2] + [2]x1   
POL(U21(x1)) = [5] + [2]x1   
POL(U31(x1, x2)) = [3] + [2]x1   
POL(U31'(x1, x2)) = x2   
POL(U32(x1)) = [2] + [3]x1   
POL(U51'(x1, x2, x3)) = [3] + [3]x1 + [4]x2 + [4]x3   
POL(U52'(x1, x2, x3)) = [3]x2   
POL(U71'(x1, x2, x3)) = [3] + [5]x1 + [4]x2 + [5]x3   
POL(U72'(x1, x2, x3)) = 0   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(n__0) = [2]   
POL(n__plus(x1, x2)) = [4] + x1 + x2   
POL(n__s(x1)) = [1] + x1   
POL(n__x(x1, x2)) = [4] + x1 + x2   
POL(plus(x1, x2)) = [4] + x1 + x2   
POL(s(x1)) = [1] + x1   
POL(tt) = 0   
POL(x(x1, x2)) = [4] + x1 + x2   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(17) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVATE(x1)) = 0   
POL(ISNAT(x1)) = [4]x1   
POL(U11(x1, x2)) = x1   
POL(U11'(x1, x2)) = [3] + [2]x1 + [4]x2   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = x1   
POL(U31'(x1, x2)) = [4]x2   
POL(U32(x1)) = 0   
POL(U51'(x1, x2, x3)) = [3] + [3]x1 + [2]x2 + [4]x3   
POL(U52'(x1, x2, x3)) = [2]x2   
POL(U71'(x1, x2, x3)) = [2] + [3]x1 + [5]x2 + [4]x3   
POL(U72'(x1, x2, x3)) = [2] + [3]x2 + [4]x3   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = [2]   
POL(n__0) = 0   
POL(n__plus(x1, x2)) = [2] + x1 + x2   
POL(n__s(x1)) = [2] + x1   
POL(n__x(x1, x2)) = x1 + x2   
POL(plus(x1, x2)) = [2] + x1 + x2   
POL(s(x1)) = [2] + x1   
POL(tt) = 0   
POL(x(x1, x2)) = x1 + x2   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(19) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVATE(x1)) = 0   
POL(ISNAT(x1)) = [4]x1   
POL(U11(x1, x2)) = [3]x1 + [3]x2   
POL(U11'(x1, x2)) = [2] + [4]x2   
POL(U12(x1)) = [4] + [2]x1   
POL(U21(x1)) = [2] + [2]x1   
POL(U31(x1, x2)) = [5] + [2]x1 + [2]x2   
POL(U31'(x1, x2)) = [1] + [4]x2   
POL(U32(x1)) = [5] + [5]x1   
POL(U51'(x1, x2, x3)) = [3]x1 + [5]x2 + [5]x3   
POL(U52'(x1, x2, x3)) = [5]x2 + x3   
POL(U71'(x1, x2, x3)) = [1] + [3]x1 + [5]x2 + [4]x3   
POL(U72'(x1, x2, x3)) = [1] + [2]x3   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(n__0) = 0   
POL(n__plus(x1, x2)) = [2] + x1 + x2   
POL(n__s(x1)) = x1   
POL(n__x(x1, x2)) = [4] + x1 + x2   
POL(plus(x1, x2)) = [2] + x1 + x2   
POL(s(x1)) = x1   
POL(tt) = 0   
POL(x(x1, x2)) = [4] + x1 + x2   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(21) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVATE(x1)) = 0   
POL(ISNAT(x1)) = x1   
POL(U11(x1, x2)) = [1] + [3]x1 + [2]x2   
POL(U11'(x1, x2)) = [2] + x2   
POL(U12(x1)) = [3]   
POL(U21(x1)) = [2] + [2]x1   
POL(U31(x1, x2)) = [3] + [5]x1   
POL(U31'(x1, x2)) = x2   
POL(U32(x1)) = [2] + [4]x1   
POL(U51'(x1, x2, x3)) = [5] + [4]x1 + [5]x2 + x3   
POL(U52'(x1, x2, x3)) = [3] + x2 + x3   
POL(U71'(x1, x2, x3)) = [4]x1 + [3]x2 + [2]x3   
POL(U72'(x1, x2, x3)) = x3   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(n__0) = 0   
POL(n__plus(x1, x2)) = [2] + x1 + x2   
POL(n__s(x1)) = x1   
POL(n__x(x1, x2)) = [2] + x1 + x2   
POL(plus(x1, x2)) = [2] + x1 + x2   
POL(s(x1)) = x1   
POL(tt) = [2]   
POL(x(x1, x2)) = [2] + x1 + x2   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVATE(x1)) = x1   
POL(ISNAT(x1)) = [2]x12   
POL(U11(x1, x2)) = 0   
POL(U11'(x1, x2)) = [2]x2 + [2]x22   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U31'(x1, x2)) = [2]x2 + [2]x22   
POL(U32(x1)) = 0   
POL(U51'(x1, x2, x3)) = [3]x2 + x3 + [3]x32 + [3]x2·x3 + x22   
POL(U52'(x1, x2, x3)) = [2]x2 + x3   
POL(U71'(x1, x2, x3)) = x2 + [2]x3 + [2]x32 + [3]x2·x3 + [2]x22   
POL(U72'(x1, x2, x3)) = x2 + x3 + [2]x22   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(n__0) = 0   
POL(n__plus(x1, x2)) = [2] + x1 + x2   
POL(n__s(x1)) = [2] + x1   
POL(n__x(x1, x2)) = [2] + x1 + x2   
POL(plus(x1, x2)) = [2] + x1 + x2   
POL(s(x1)) = [2] + x1   
POL(tt) = 0   
POL(x(x1, x2)) = [2] + x1 + x2   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(25) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
And the Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(ACTIVATE(x1)) = [1] + x1   
POL(ISNAT(x1)) = x12   
POL(U11(x1, x2)) = 0   
POL(U11'(x1, x2)) = [2] + [3]x2 + x22   
POL(U12(x1)) = 0   
POL(U21(x1)) = 0   
POL(U31(x1, x2)) = 0   
POL(U31'(x1, x2)) = [2] + [2]x2 + x22   
POL(U32(x1)) = 0   
POL(U51'(x1, x2, x3)) = [1] + x2 + x3 + [2]x32 + [2]x22   
POL(U52'(x1, x2, x3)) = [1] + x2 + x3   
POL(U71'(x1, x2, x3)) = [1] + [2]x2 + [2]x3 + x32 + [3]x2·x3 + x22   
POL(U72'(x1, x2, x3)) = [1] + [2]x2 + x3 + x22   
POL(activate(x1)) = x1   
POL(c(x1, x2)) = x1 + x2   
POL(c1) = 0   
POL(c1(x1)) = x1   
POL(c12(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c13(x1, x2)) = x1 + x2   
POL(c14(x1, x2, x3, x4)) = x1 + x2 + x3 + x4   
POL(c24(x1, x2)) = x1 + x2   
POL(c25(x1)) = x1   
POL(c26(x1, x2)) = x1 + x2   
POL(c3(x1, x2)) = x1 + x2   
POL(isNat(x1)) = 0   
POL(n__0) = 0   
POL(n__plus(x1, x2)) = [2] + x1 + x2   
POL(n__s(x1)) = [1] + x1   
POL(n__x(x1, x2)) = [2] + x1 + x2   
POL(plus(x1, x2)) = [2] + x1 + x2   
POL(s(x1)) = [1] + x1   
POL(tt) = 0   
POL(x(x1, x2)) = [2] + x1 + x2   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → U32(isNat(activate(z0)))
U32(tt) → tt
U41(tt, z0) → activate(z0)
U51(tt, z0, z1) → U52(isNat(activate(z1)), activate(z0), activate(z1))
U52(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
U61(tt) → 0
U71(tt, z0, z1) → U72(isNat(activate(z1)), activate(z0), activate(z1))
U72(tt, z0, z1) → plus(x(activate(z1), activate(z0)), activate(z1))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
isNat(n__x(z0, z1)) → U31(isNat(activate(z0)), activate(z1))
plus(z0, 0) → U41(isNat(z0), z0)
plus(z0, s(z1)) → U51(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(z0, z1)
x(z0, 0) → U61(isNat(z0))
x(z0, s(z1)) → U71(isNat(z1), z1, z0)
x(z0, z1) → n__x(z0, z1)
0 → n__0
s(z0) → n__s(z0)
activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(activate(z0), activate(z1))
activate(n__s(z0)) → s(activate(z0))
activate(n__x(z0, z1)) → x(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
S tuples:none
K tuples:

U51'(tt, z0, z1) → c1(U52'(isNat(activate(z1)), activate(z0), activate(z1)))
U51'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U51'(tt, z0, z1) → c1(ACTIVATE(z1))
U51'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1(ACTIVATE(z1))
U52'(tt, z0, z1) → c1(ACTIVATE(z0))
U71'(tt, z0, z1) → c1(U72'(isNat(activate(z1)), activate(z0), activate(z1)))
U71'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U71'(tt, z0, z1) → c1(ACTIVATE(z1))
U71'(tt, z0, z1) → c1(ACTIVATE(z0))
U72'(tt, z0, z1) → c1(ACTIVATE(z1))
U72'(tt, z0, z1) → c1(ACTIVATE(z0))
U52'(tt, z0, z1) → c1
U72'(tt, z0, z1) → c1
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c13(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c12(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U31'(tt, z0) → c3(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__x(z0, z1)) → c14(U31'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__x(z0, z1)) → c26(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__plus(z0, z1)) → c24(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c25(ACTIVATE(z0))
Defined Rule Symbols:

U11, U12, U21, U31, U32, U41, U51, U52, U61, U71, U72, isNat, plus, x, 0, s, activate

Defined Pair Symbols:

ISNAT, U51', U52', U71', U72', U11', U31', ACTIVATE

Compound Symbols:

c12, c14, c1, c, c3, c13, c24, c25, c26, c1

(27) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(28) BOUNDS(O(1), O(1))