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

(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) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative 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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
isNat, activate, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(6) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, isNat, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

Induction Base:
activate(gen_n__0:n__plus:n__s3_3(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s3_3(0)

Induction Step:
activate(gen_n__0:n__plus:n__s3_3(+(n5_3, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s3_3(n5_3)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s3_3(c6_3), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s3_3(n5_3), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s3_3(n5_3), n__0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, isNat, U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol plus.

(11) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
U31, isNat

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U31.

(13) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
isNat

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Induction Base:
isNat(gen_n__0:n__plus:n__s3_3(0)) →_R^Ω(1)
tt

Induction Step:
isNat(gen_n__0:n__plus:n__s3_3(+(n2331_3, 1))) →_R^Ω(1)
U11(isNat(activate(gen_n__0:n__plus:n__s3_3(n2331_3))), activate(n__0)) →_L^{Ω(1 + n2331₃)}
U11(isNat(gen_n__0:n__plus:n__s3_3(n2331_3)), activate(n__0)) →_IH
U11(tt, activate(n__0)) →_L^Ω(1)
U11(tt, gen_n__0:n__plus:n__s3_3(0)) →_R^Ω(1)
U12(isNat(activate(gen_n__0:n__plus:n__s3_3(0)))) →_L^Ω(1)
U12(isNat(gen_n__0:n__plus:n__s3_3(0))) →_R^Ω(1)
U12(tt) →_R^Ω(1)
tt

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(15) Complex Obligation (BEST)

(16) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, U31, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s3_3(n3960_3)) → gen_n__0:n__plus:n__s3_3(n3960_3), rt ∈ Ω(1 + n3960₃)

Induction Base:
activate(gen_n__0:n__plus:n__s3_3(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s3_3(0)

Induction Step:
activate(gen_n__0:n__plus:n__s3_3(+(n3960_3, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s3_3(n3960_3)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s3_3(c3961_3), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s3_3(n3960_3), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s3_3(n3960_3), n__0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(18) Complex Obligation (BEST)

(19) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n3960_3)) → gen_n__0:n__plus:n__s3_3(n3960_3), rt ∈ Ω(1 + n3960₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol plus.

(21) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n3960_3)) → gen_n__0:n__plus:n__s3_3(n3960_3), rt ∈ Ω(1 + n3960₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

The following defined symbols remain to be analysed:
U31

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U31
isNat = plus
activate = U31
activate = plus
U31 = plus

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U31.

(23) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n3960_3)) → gen_n__0:n__plus:n__s3_3(n3960_3), rt ∈ Ω(1 + n3960₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n3960_3)) → gen_n__0:n__plus:n__s3_3(n3960_3), rt ∈ Ω(1 + n3960₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

(28) BOUNDS(n^2, INF)

(29) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s3_3(n2331_3)) → tt, rt ∈ Ω(1 + n2331₃ + n2331₃²)

(31) BOUNDS(n^2, INF)

(32) Obligation:

Innermost TRS:
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

Types:
U11 :: tt → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U41 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U42 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
s :: n__0:n__plus:n__s → n__0:n__plus:n__s
plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__0 :: n__0:n__plus:n__s
n__plus :: n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
n__s :: n__0:n__plus:n__s → n__0:n__plus:n__s
0' :: n__0:n__plus:n__s
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s2_3 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_3 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

Generator Equations:
gen_n__0:n__plus:n__s3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_3(x), n__0)

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__0:n__plus:n__s3_3(n5_3)) → gen_n__0:n__plus:n__s3_3(n5_3), rt ∈ Ω(1 + n5₃)

(34) BOUNDS(n^1, INF)