Runtime Complexity (innermost) proof of /tmp/tmpVcwoGV/PEANO_nokinds-noand

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 CdtKnowledgeProof (⇔)

↳16 CdtProblem

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

↳18 CdtProblem

↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳20 CdtProblem

↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))

↳22 CdtProblem

↳23 SIsEmptyProof (⇔)

↳24 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, N) → activate(N)
U41(tt, M, N) → U42(isNat(activate(N)), activate(M), activate(N))
U42(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(activate(X1), activate(X2))
activate(n__s(X)) → s(activate(X))
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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U42', ISNAT, PLUS, ACTIVATE

Compound Symbols:

c, c3, c4, c5, c7, c8, c9, c10, c14, c15, c16

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U31'(tt, z0) → c3(ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__0) → c14(0')
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U31', U41', U42', ISNAT, ACTIVATE

Compound Symbols:

c, c3, c4, c5, c7, c8, c14, c15, c16

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 9 dangling nodes:

ACTIVATE(n__0) → c14(0')
U31'(tt, z0) → c3(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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(activate(z1)), activate(z0), activate(z1)), ISNAT(activate(z1)), ACTIVATE(z1), ACTIVATE(z0), ACTIVATE(z1))
U42'(tt, z0, z1) → c5(S(plus(activate(z1), activate(z0))), PLUS(activate(z1), activate(z0)), ACTIVATE(z1), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', U41', U42', ISNAT, ACTIVATE

Compound Symbols:

c, c4, c5, c7, c8, c15, c16

(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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(S(plus(activate(z1), activate(z0))))
U42'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))

S tuples:

U11'(tt, z0) → c(U12'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
ISNAT(n__s(z0)) → c8(U21'(isNat(activate(z0))), ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(PLUS(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(S(plus(activate(z1), activate(z0))))
U42'(tt, z0, z1) → c1(PLUS(activate(z1), activate(z0)))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

U11', ISNAT, ACTIVATE, U41', U42'

Compound Symbols:

c, c7, c8, c15, c16, c1

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

Removed 6 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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, c1

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
U42'(tt, z0, z1) → c1
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
U42'(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) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))

K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, 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.

ISNAT(n__plus(z0, z1)) → c7(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(z0) → z0
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)))
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt

And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

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

POL(0) = [5]
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂)) = [3] + [3]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = x₂
POL(U12(x₁)) = [2]x₁
POL(U21(x₁)) = [3]x₁
POL(U41'(x₁, x₂, x₃)) = [4]x₁ + [5]x₂ + x₃
POL(U42'(x₁, x₂, x₃)) = [4]x₂ + x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(n__0) = [5]
POL(n__plus(x₁, x₂)) = [1] + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(plus(x₁, x₂)) = [1] + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))

K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, c1

(15) CdtKnowledgeProof (EQUIVALENT transformation)

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

U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))

K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, 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__s(z0)) → c8(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(z0) → z0
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)))
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt

And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

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

POL(0) = [1]
POL(ACTIVATE(x₁)) = 0
POL(ISNAT(x₁)) = x₁
POL(U11(x₁, x₂)) = [3] + [3]x₁ + [3]x₂
POL(U11'(x₁, x₂)) = x₂
POL(U12(x₁)) = [2] + [3]x₁
POL(U21(x₁)) = [3] + [2]x₁
POL(U41'(x₁, x₂, x₃)) = [4] + [4]x₁ + [4]x₂ + [2]x₃
POL(U42'(x₁, x₂, x₃)) = [4]x₂ + [2]x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = [1] + [3]x₁
POL(n__0) = [1]
POL(n__plus(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = [4] + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = [4] + x₁
POL(tt) = [4]

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))

K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, c1

(19) 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__s(z0)) → c16(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(z0) → z0
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)))
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt

And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

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

POL(0) = 0
POL(ACTIVATE(x₁)) = [2] + x₁
POL(ISNAT(x₁)) = [2]x₁²
POL(U11(x₁, x₂)) = 0
POL(U11'(x₁, x₂)) = [3] + [3]x₂ + [2]x₂²
POL(U12(x₁)) = 0
POL(U21(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [3] + [2]x₁ + [2]x₂ + [3]x₃ + [2]x₃² + x₂·x₃ + x₁·x₃ + [3]x₁² + [2]x₁·x₂ + [2]x₂²
POL(U42'(x₁, x₂, x₃)) = [3] + [2]x₂ + x₃ + x₃² + x₂²
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(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(plus(x₁, x₂)) = [2] + x₁ + x₂
POL(s(x₁)) = [2] + x₁
POL(tt) = [1]

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:

ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))

K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, c1

(21) 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)) → c15(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(z0) → z0
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)))
U21(tt) → tt
U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt

And the Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(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)) = x₂ + [2]x₂²
POL(U12(x₁)) = 0
POL(U21(x₁)) = 0
POL(U41'(x₁, x₂, x₃)) = [2] + [2]x₂ + x₃ + [2]x₃² + [2]x₂·x₃ + [3]x₂²
POL(U42'(x₁, x₂, x₃)) = [1] + x₂ + x₃
POL(activate(x₁)) = x₁
POL(c(x₁, x₂)) = x₁ + x₂
POL(c1) = 0
POL(c1(x₁)) = x₁
POL(c15(x₁, x₂)) = x₁ + x₂
POL(c16(x₁)) = x₁
POL(c7(x₁, x₂, x₃, x₄)) = x₁ + x₂ + x₃ + x₄
POL(c8(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(n__0) = 0
POL(n__plus(x₁, x₂)) = [1] + x₁ + x₂
POL(n__s(x₁)) = [2] + x₁
POL(plus(x₁, x₂)) = [1] + x₁ + x₂
POL(s(x₁)) = [2] + x₁
POL(tt) = 0

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0) → U12(isNat(activate(z0)))
U12(tt) → tt
U21(tt) → tt
U31(tt, z0) → activate(z0)
U41(tt, z0, z1) → U42(isNat(activate(z1)), activate(z0), activate(z1))
U42(tt, z0, z1) → s(plus(activate(z1), activate(z0)))
isNat(n__0) → tt
isNat(n__plus(z0, z1)) → U11(isNat(activate(z0)), activate(z1))
isNat(n__s(z0)) → U21(isNat(activate(z0)))
plus(z0, 0) → U31(isNat(z0), z0)
plus(z0, s(z1)) → U41(isNat(z1), z1, z0)
plus(z0, z1) → n__plus(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(z0) → z0

Tuples:

ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U42'(tt, z0, z1) → c1

S tuples:none
K tuples:

U41'(tt, z0, z1) → c1(U42'(isNat(activate(z1)), activate(z0), activate(z1)))
U41'(tt, z0, z1) → c1(ISNAT(activate(z1)))
U41'(tt, z0, z1) → c1(ACTIVATE(z1))
U41'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1(ACTIVATE(z1))
U42'(tt, z0, z1) → c1(ACTIVATE(z0))
U42'(tt, z0, z1) → c1
ISNAT(n__plus(z0, z1)) → c7(U11'(isNat(activate(z0)), activate(z1)), ISNAT(activate(z0)), ACTIVATE(z0), ACTIVATE(z1))
U11'(tt, z0) → c(ISNAT(activate(z0)), ACTIVATE(z0))
ISNAT(n__s(z0)) → c8(ISNAT(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
ACTIVATE(n__plus(z0, z1)) → c15(ACTIVATE(z0), ACTIVATE(z1))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus, 0, s, activate

Defined Pair Symbols:

ISNAT, U41', U42', U11', ACTIVATE

Compound Symbols:

c7, c1, c, c8, c15, c16, c1

(23) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(0) Obligation:

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

(2) Obligation:

(3) CdtUnreachableProof (EQUIVALENT transformation)

(4) Obligation:

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

(6) Obligation:

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

(8) Obligation:

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

(10) Obligation:

(11) CdtKnowledgeProof (EQUIVALENT transformation)

(12) Obligation:

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

(14) Obligation:

(15) CdtKnowledgeProof (EQUIVALENT transformation)

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

(23) SIsEmptyProof (EQUIVALENT transformation)

(24) BOUNDS(O(1), O(1))