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

(0) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U51(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X1 / n__plus(X1, n__0)].
The result substitution is [ ].

The rewrite sequence
activate(n__plus(X1, n__0)) →⁺ U51(isNat(activate(X1)), activate(X1))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [X1 / n__plus(X1, n__0)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
isNatKind, activate, isNat, U51, U52, plus

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(8) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
activate, isNatKind, isNat, U51, U52, plus

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Induction Base:
activate(gen_n__0:n__plus:n__s3_7(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s3_7(0)

Induction Step:
activate(gen_n__0:n__plus:n__s3_7(+(n5_7, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s3_7(n5_7)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s3_7(c6_7), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s3_7(n5_7), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s3_7(n5_7), n__0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
plus, isNatKind, isNat, U51, U52

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

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

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

(13) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
U51, isNatKind, isNat, U52

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U51.

(15) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
U52, isNatKind, isNat

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U52.

(17) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

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

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Induction Base:
isNatKind(gen_n__0:n__plus:n__s3_7(0)) →_R^Ω(1)
tt

Induction Step:
isNatKind(gen_n__0:n__plus:n__s3_7(+(n7788_7, 1))) →_R^Ω(1)
U31(isNatKind(activate(gen_n__0:n__plus:n__s3_7(n7788_7))), activate(n__0)) →_L^{Ω(1 + n7788₇)}
U31(isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)), activate(n__0)) →_IH
U31(tt, activate(n__0)) →_L^Ω(1)
U31(tt, gen_n__0:n__plus:n__s3_7(0)) →_R^Ω(1)
U32(isNatKind(activate(gen_n__0:n__plus:n__s3_7(0)))) →_L^Ω(1)
U32(isNatKind(gen_n__0:n__plus:n__s3_7(0))) →_R^Ω(1)
U32(tt) →_R^Ω(1)
tt

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

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
isNat, activate, U51, U52, plus

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
activate, U51, U52, plus

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)

Induction Base:
activate(gen_n__0:n__plus:n__s3_7(0)) →_R^Ω(1)
gen_n__0:n__plus:n__s3_7(0)

Induction Step:
activate(gen_n__0:n__plus:n__s3_7(+(n9702_7, 1))) →_R^Ω(1)
plus(activate(gen_n__0:n__plus:n__s3_7(n9702_7)), activate(n__0)) →_IH
plus(gen_n__0:n__plus:n__s3_7(c9703_7), activate(n__0)) →_R^Ω(1)
plus(gen_n__0:n__plus:n__s3_7(n9702_7), n__0) →_R^Ω(1)
n__plus(gen_n__0:n__plus:n__s3_7(n9702_7), n__0)

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

(24) Complex Obligation (BEST)

(25) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
plus, U51, U52

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(27) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
U51, U52

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U51.

(29) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

The following defined symbols remain to be analysed:
U52

They will be analysed ascendingly in the following order:
isNatKind = activate
isNatKind = isNat
isNatKind = U51
isNatKind = U52
isNatKind = plus
activate = isNat
activate = U51
activate = U52
activate = plus
isNat = U51
isNat = U52
isNat = plus
U51 = U52
U51 = plus
U52 = plus

(30) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U52.

(31) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

(33) BOUNDS(n^2, INF)

(34) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n9702_7)) → gen_n__0:n__plus:n__s3_7(n9702_7), rt ∈ Ω(1 + n9702₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

(36) BOUNDS(n^2, INF)

(37) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

No more defined symbols left to analyse.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
isNatKind(gen_n__0:n__plus:n__s3_7(n7788_7)) → tt, rt ∈ Ω(1 + n7788₇ + n7788₇²)

(39) BOUNDS(n^2, INF)

(40) Obligation:

TRS:
Rules:
U11(tt, V1, V2) → U12(isNatKind(activate(V1)), activate(V1), activate(V2))
U12(tt, V1, V2) → U13(isNatKind(activate(V2)), activate(V1), activate(V2))
U13(tt, V1, V2) → U14(isNatKind(activate(V2)), activate(V1), activate(V2))
U14(tt, V1, V2) → U15(isNat(activate(V1)), activate(V2))
U15(tt, V2) → U16(isNat(activate(V2)))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(activate(V1)), activate(V1))
U22(tt, V1) → U23(isNat(activate(V1)))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(activate(V2)))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(activate(N)), activate(N))
U52(tt, N) → activate(N)
U61(tt, M, N) → U62(isNatKind(activate(M)), activate(M), activate(N))
U62(tt, M, N) → U63(isNat(activate(N)), activate(M), activate(N))
U63(tt, M, N) → U64(isNatKind(activate(N)), activate(M), activate(N))
U64(tt, M, N) → s(plus(activate(N), activate(M)))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNatKind(activate(V1)), activate(V1), activate(V2))
isNat(n__s(V1)) → U21(isNatKind(activate(V1)), activate(V1))
isNatKind(n__0) → tt
isNatKind(n__plus(V1, V2)) → U31(isNatKind(activate(V1)), activate(V2))
isNatKind(n__s(V1)) → U41(isNatKind(activate(V1)))
plus(N, 0') → U51(isNat(N), N)
plus(N, s(M)) → U61(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 → n__0:n__plus:n__s → tt
tt :: tt
U12 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
isNatKind :: n__0:n__plus:n__s → tt
activate :: n__0:n__plus:n__s → n__0:n__plus:n__s
U13 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U14 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → tt
U15 :: tt → n__0:n__plus:n__s → tt
isNat :: n__0:n__plus:n__s → tt
U16 :: tt → tt
U21 :: tt → n__0:n__plus:n__s → tt
U22 :: tt → n__0:n__plus:n__s → tt
U23 :: tt → tt
U31 :: tt → n__0:n__plus:n__s → tt
U32 :: tt → tt
U41 :: tt → tt
U51 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U52 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s
U61 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U62 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U63 :: tt → n__0:n__plus:n__s → n__0:n__plus:n__s → n__0:n__plus:n__s
U64 :: 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_7 :: tt
hole_n__0:n__plus:n__s2_7 :: n__0:n__plus:n__s
gen_n__0:n__plus:n__s3_7 :: Nat → n__0:n__plus:n__s

Lemmas:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

Generator Equations:
gen_n__0:n__plus:n__s3_7(0) ⇔ n__0
gen_n__0:n__plus:n__s3_7(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s3_7(x), n__0)

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
activate(gen_n__0:n__plus:n__s3_7(n5_7)) → gen_n__0:n__plus:n__s3_7(n5_7), rt ∈ Ω(1 + n5₇)

(42) BOUNDS(n^1, INF)