Runtime Complexity (innermost) proof of /tmp/tmps_pnFa/MYNAT_nokinds-noand

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

0 CpxTRS

↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CdtProblem

↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)

↳4 CdtProblem

↳5 CdtKnowledgeProof (⇔, 0 ms)

↳6 CdtProblem

↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1035 ms)

↳8 CdtProblem

↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1126 ms)

↳10 CdtProblem

↳11 CdtKnowledgeProof (⇔, 0 ms)

↳12 CdtProblem

↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 1098 ms)

↳14 CdtProblem

↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 431 ms)

↳16 CdtProblem

↳17 SIsEmptyProof (⇔, 0 ms)

↳18 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(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, 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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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(z0, z1))
ACTIVATE(n__s(z0)) → c25(S(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(z0, 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(z0, z1))
ACTIVATE(n__s(z0)) → c25(S(z0))
ACTIVATE(n__x(z0, z1)) → c26(X(z0, 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) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 4 leading nodes:

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))

Removed 8 trailing nodes:

ACTIVATE(n__s(z0)) → c25(S(z0))
U61'(tt) → c8(0')
ACTIVATE(n__0) → c23(0')
ACTIVATE(n__plus(z0, z1)) → c24(PLUS(z0, z1))
ACTIVATE(n__x(z0, z1)) → c26(X(z0, z1))
U41'(tt, z0) → c5(ACTIVATE(z0))
U52'(tt, z0, z1) → c7(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
U72'(tt, z0, z1) → c10(PLUS(x(activate(z1), activate(z0)), activate(z1)), X(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(5) CdtKnowledgeProof (EQUIVALENT transformation)

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

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))

(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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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))

K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), 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:

U11', U31', U51', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(7) 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(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, z1)
activate(z0) → z0
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
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0

And the 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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

POL(0) = [3]
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂)) = [3] + [5]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = [4] + x₂
POL(U12(x₁)) = [3] + [2]x₁
POL(U12'(x₁)) = 0
POL(U21(x₁)) = [2] + [2]x₁
POL(U21'(x₁)) = 0
POL(U31(x₁, x₂)) = [2] + [2]x₁ + [3]x₂
POL(U31'(x₁, x₂)) = x₂
POL(U32(x₁)) = [4] + [5]x₁
POL(U32'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [4]x₁ + [2]x₂ + x₃
POL(U52'(x₁, x₂, x₃)) = [5] + x₂
POL(U71'(x₁, x₂, x₃)) = [2] + [3]x₁ + [5]x₃
POL(U72'(x₁, x₂, x₃)) = [3] + x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c14(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(c9(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(isNat(x₁)) = 0
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) = [2]
POL(x(x₁, x₂)) = x₁ + x₂

(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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

S tuples:

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))

K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(U12'(isNat(activate(z0))), 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:

U11', U31', U51', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(9) 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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, z1)
activate(z0) → z0
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
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0

And the 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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

POL(0) = 0
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = [2]x₁
POL(U11(x₁, x₂)) = [2] + [2]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = [2]x₂
POL(U12(x₁)) = [5] + [2]x₁
POL(U12'(x₁)) = 0
POL(U21(x₁)) = [2] + [2]x₁
POL(U21'(x₁)) = 0
POL(U31(x₁, x₂)) = [3]x₁ + [5]x₂
POL(U31'(x₁, x₂)) = [2]x₂
POL(U32(x₁)) = [1]
POL(U32'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [4]x₁ + [5]x₂ + [2]x₃
POL(U52'(x₁, x₂, x₃)) = 0
POL(U71'(x₁, x₂, x₃)) = [3]x₁ + [2]x₂ + [4]x₃
POL(U72'(x₁, x₂, x₃)) = 0
POL(activate(x₁)) = x₁
POL(c(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c14(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(c9(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(isNat(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__x(x₁, x₂)) = [2] + x₁ + x₂
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0
POL(x(x₁, x₂)) = [2] + x₁ + x₂

(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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

S tuples:

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))

K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(U12'(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))

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', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

U31'(tt, z0) → c3(U32'(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))

(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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

S tuples:

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))

K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(U12'(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))
U31'(tt, z0) → c3(U32'(isNat(activate(z0))), 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:

U11', U31', U51', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(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.

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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, z1)
activate(z0) → z0
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
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0

And the 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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

POL(0) = [2]
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = [2]x₁
POL(U11(x₁, x₂)) = [1] + [3]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = [2]x₂
POL(U12(x₁)) = [5]
POL(U12'(x₁)) = 0
POL(U21(x₁)) = [2] + [2]x₁
POL(U21'(x₁)) = 0
POL(U31(x₁, x₂)) = [2] + [2]x₂
POL(U31'(x₁, x₂)) = [2]x₂
POL(U32(x₁)) = [3] + [4]x₁
POL(U32'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [2] + [4]x₁ + [3]x₃
POL(U52'(x₁, x₂, x₃)) = [2]
POL(U71'(x₁, x₂, x₃)) = [4] + [4]x₁ + [4]x₂ + [2]x₃
POL(U72'(x₁, x₂, x₃)) = [1]
POL(activate(x₁)) = x₁
POL(c(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c14(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(c9(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(isNat(x₁)) = 0
POL(n__0) = [2]
POL(n__plus(x₁, x₂)) = [2] + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__x(x₁, x₂)) = [5] + x₁ + x₂
POL(plus(x₁, x₂)) = [2] + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0
POL(x(x₁, x₂)) = [5] + 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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

S tuples:

ISNAT(n__s(z0)) → c13(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))

K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(U12'(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))
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))

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', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(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(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))

We considered the (Usable) Rules:

activate(n__0) → 0
activate(n__plus(z0, z1)) → plus(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, z1)
activate(z0) → z0
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
x(z0, z1) → n__x(z0, z1)
s(z0) → n__s(z0)
plus(z0, z1) → n__plus(z0, z1)
0 → n__0

And the 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

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

POL(0) = [4]
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = [4]x₁
POL(U11(x₁, x₂)) = [4]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = [1] + [4]x₂
POL(U12(x₁)) = [2] + [4]x₁
POL(U12'(x₁)) = [1]
POL(U21(x₁)) = [3]
POL(U21'(x₁)) = 0
POL(U31(x₁, x₂)) = [4] + [2]x₁ + [3]x₂
POL(U31'(x₁, x₂)) = [4]x₂
POL(U32(x₁)) = [3] + [2]x₁
POL(U32'(x₁)) = 0
POL(U51'(x₁, x₂, x₃)) = [2]x₂ + [5]x₃
POL(U52'(x₁, x₂, x₃)) = x₂ + x₃
POL(U71'(x₁, x₂, x₃)) = [2] + [2]x₁ + [2]x₂ + [4]x₃
POL(U72'(x₁, x₂, x₃)) = 0
POL(activate(x₁)) = x₁
POL(c(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c12(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c13(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c14(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c3(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(c6(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(c9(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + x₄ + x₅
POL(isNat(x₁)) = 0
POL(n__0) = [4]
POL(n__plus(x₁, x₂)) = [4] + x₁ + x₂
POL(n__s(x₁)) = [4] + x₁
POL(n__x(x₁, x₂)) = x₁ + x₂
POL(plus(x₁, x₂)) = [4] + x₁ + x₂
POL(s(x₁)) = [4] + x₁
POL(tt) = 0
POL(x(x₁, x₂)) = 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(z0, z1)
activate(n__s(z0)) → s(z0)
activate(n__x(z0, z1)) → x(z0, 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))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), 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))

S tuples:none
K tuples:

U51'(tt, z0, z1) → c6(U52'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U71'(tt, z0, z1) → c9(U72'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(U12'(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))
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))

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', U71', ISNAT

Compound Symbols:

c, c3, c6, c9, c12, c13, c14

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

(4) Obligation:

(5) CdtKnowledgeProof (EQUIVALENT transformation)

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtKnowledgeProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

(17) SIsEmptyProof (EQUIVALENT transformation)

(18) BOUNDS(O(1), O(1))