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

(0) Obligation:

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → U42(isNat(N), M, N)
U42(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)

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

Tuples:

U11'(tt, z0) → c(U12'(isNat(z0)), ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ISNAT(s(z0)) → c8(U21'(isNat(z0)), ISNAT(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))

S tuples:

U11'(tt, z0) → c(U12'(isNat(z0)), ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ISNAT(s(z0)) → c8(U21'(isNat(z0)), ISNAT(z0))
PLUS(z0, 0) → c9(U31'(isNat(z0), z0), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U11', U41', U42', ISNAT, PLUS

Compound Symbols:

c, c4, c5, c7, c8, c9, c10

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U11'(tt, z0) → c(U12'(isNat(z0)), ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ISNAT(s(z0)) → c8(U21'(isNat(z0)), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(U31'(isNat(z0), z0))
PLUS(z0, 0) → c1(ISNAT(z0))

S tuples:

U11'(tt, z0) → c(U12'(isNat(z0)), ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
ISNAT(s(z0)) → c8(U21'(isNat(z0)), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(U31'(isNat(z0), z0))
PLUS(z0, 0) → c1(ISNAT(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U11', U41', U42', ISNAT, PLUS

Compound Symbols:

c, c4, c5, c7, c8, c10, c1

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

K tuples:none
Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(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.

PLUS(z0, 0) → c1

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = 0   
POL(ISNAT(x₁)) = 0   
POL(PLUS(x₁, x₂)) = [4] + x₁   
POL(U11(x₁, x₂)) = [3]x₁ + [4]x₂   
POL(U11'(x₁, x₂)) = 0   
POL(U12(x₁)) = 0   
POL(U21(x₁)) = [1]   
POL(U41'(x₁, x₂, x₃)) = [4] + x₃   
POL(U42'(x₁, x₂, x₃)) = [4] + x₃   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = [2] + [2]x₁   
POL(plus(x₁, x₂)) = [2] + [3]x₁ + [4]x₂   
POL(s(x₁)) = [3]   
POL(tt) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(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.

PLUS(z0, 0) → c1(ISNAT(z0))

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = [3]   
POL(ISNAT(x₁)) = 0   
POL(PLUS(x₁, x₂)) = [1] + x₁   
POL(U11(x₁, x₂)) = [4] + [3]x₁ + [2]x₂   
POL(U11'(x₁, x₂)) = 0   
POL(U12(x₁)) = [2] + [2]x₁   
POL(U21(x₁)) = [5] + x₁   
POL(U41'(x₁, x₂, x₃)) = [1] + x₃   
POL(U42'(x₁, x₂, x₃)) = [1] + x₃   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = [4]   
POL(plus(x₁, x₂)) = [2] + x₁ + [3]x₂   
POL(s(x₁)) = [2]   
POL(tt) = [2]   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(11) 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.

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = [2]   
POL(ISNAT(x₁)) = 0   
POL(PLUS(x₁, x₂)) = [2] + x₂   
POL(U11(x₁, x₂)) = 0   
POL(U11'(x₁, x₂)) = [2]x₁   
POL(U12(x₁)) = [2]x₁   
POL(U21(x₁)) = 0   
POL(U41'(x₁, x₂, x₃)) = [3] + [3]x₁ + x₂   
POL(U42'(x₁, x₂, x₃)) = [2] + [4]x₁ + x₂   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = 0   
POL(plus(x₁, x₂)) = [2] + [3]x₁ + [2]x₂   
POL(s(x₁)) = [1] + x₁   
POL(tt) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

U42'(tt, z0, z1) → c5(PLUS(z1, z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
PLUS(z0, 0) → c1
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(15) 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.

ISNAT(s(z0)) → c8(ISNAT(z0))

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = [2]   
POL(ISNAT(x₁)) = x₁   
POL(PLUS(x₁, x₂)) = [1] + x₂² + [2]x₁·x₂   
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] + x₃ + [2]x₂·x₃ + x₂²   
POL(U42'(x₁, x₂, x₃)) = [1] + [2]x₂·x₃ + x₂²   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = 0   
POL(plus(x₁, x₂)) = x₁ + x₂ + [2]x₂² + [3]x₁·x₂   
POL(s(x₁)) = [1] + x₁   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ISNAT(s(z0)) → c8(ISNAT(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(17) 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.

U11'(tt, z0) → c(ISNAT(z0))

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = [1]   
POL(ISNAT(x₁)) = x₁   
POL(PLUS(x₁, x₂)) = [3]x₂ + x₂² + x₁·x₂   
POL(U11(x₁, x₂)) = x₂² + [2]x₁·x₂   
POL(U11'(x₁, x₂)) = [2]x₁ + x₂   
POL(U12(x₁)) = x₁   
POL(U21(x₁)) = [2]   
POL(U41'(x₁, x₂, x₃)) = [3]x₂ + [2]x₃ + x₂·x₃ + x₂²   
POL(U42'(x₁, x₂, x₃)) = [3]x₂ + x₂·x₃ + x₂²   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = x₁²   
POL(plus(x₁, x₂)) = x₁ + [2]x₂ + x₂² + [2]x₁²   
POL(s(x₁)) = [2] + x₁   
POL(tt) = [1]   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:

ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))

K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ISNAT(s(z0)) → c8(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, 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.

ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))

We considered the (Usable) Rules:

isNat(0) → tt
isNat(plus(z0, z1)) → U11(isNat(z0), z1)
isNat(s(z0)) → U21(isNat(z0))
U21(tt) → tt
U11(tt, z0) → U12(isNat(z0))
U12(tt) → tt

And the Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

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

POL(0) = [2]   
POL(ISNAT(x₁)) = x₁   
POL(PLUS(x₁, x₂)) = x₂² + x₁·x₂   
POL(U11(x₁, x₂)) = 0   
POL(U11'(x₁, x₂)) = x₂   
POL(U12(x₁)) = 0   
POL(U21(x₁)) = 0   
POL(U41'(x₁, x₂, x₃)) = [2]x₂ + x₃ + x₂·x₃ + x₂²   
POL(U42'(x₁, x₂, x₃)) = [2]x₂ + x₂·x₃ + x₂²   
POL(c(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c4(x₁, x₂)) = x₁ + x₂   
POL(c5(x₁)) = x₁   
POL(c7(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(isNat(x₁)) = 0   
POL(plus(x₁, x₂)) = [2] + x₁ + x₂   
POL(s(x₁)) = [2] + x₁   
POL(tt) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
PLUS(z0, 0) → c1(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(s(z0)) → c8(ISNAT(z0))
PLUS(z0, 0) → c1

S tuples:none
K tuples:

PLUS(z0, 0) → c1
PLUS(z0, 0) → c1(ISNAT(z0))
U41'(tt, z0, z1) → c4(U42'(isNat(z1), z0, z1), ISNAT(z1))
U42'(tt, z0, z1) → c5(PLUS(z1, z0))
PLUS(z0, s(z1)) → c10(U41'(isNat(z1), z1, z0), ISNAT(z1))
ISNAT(s(z0)) → c8(ISNAT(z0))
U11'(tt, z0) → c(ISNAT(z0))
ISNAT(plus(z0, z1)) → c7(U11'(isNat(z0), z1), ISNAT(z0))

Defined Rule Symbols:

U11, U12, U21, U31, U41, U42, isNat, plus

Defined Pair Symbols:

U41', U42', ISNAT, PLUS, U11'

Compound Symbols:

c4, c5, c7, c10, c1, c, c8, c1

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))