The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, 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, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0) → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0) → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0 → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
isNat(n__plus(V1, V2)) →⁺ U11(isNat(V1), activate(V2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [V1 / n__plus(V1, V2)].
The result substitution is [ ].

(2) BOUNDS(n^1, 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, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

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

Heuristically decided to analyse the following defined symbols:
isNat, activate, U41, plus, x

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(8) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, isNat, U41, plus, x

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, isNat, U41, x

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
U41, isNat, x

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U41.

(14) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

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

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Induction Base:
isNat(gen_n__0:n__plus:n__s:n__x3_3(0)) →_R^Ω(1)
tt

Induction Step:
isNat(gen_n__0:n__plus:n__s:n__x3_3(+(n225_3, 1))) →_R^Ω(1)
U11(isNat(activate(gen_n__0:n__plus:n__s:n__x3_3(n225_3))), activate(n__0)) →_R^Ω(1)
U11(isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)), activate(n__0)) →_IH
U11(tt, activate(n__0)) →_R^Ω(1)
U11(tt, n__0) →_R^Ω(1)
U12(isNat(activate(n__0))) →_R^Ω(1)
U12(isNat(n__0)) →_R^Ω(1)
U12(tt) →_R^Ω(1)
tt

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

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
x, activate, U41, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
activate, U41, plus

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

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

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

(21) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
plus, U41

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

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

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

(23) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

The following defined symbols remain to be analysed:
U41

They will be analysed ascendingly in the following order:
isNat = activate
isNat = U41
isNat = plus
isNat = x
activate = U41
activate = plus
activate = x
U41 = plus
U41 = x
plus = x

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol U41.

(25) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
U11(tt, V2) → U12(isNat(activate(V2)))
U12(tt) → tt
U21(tt) → tt
U31(tt, V2) → U32(isNat(activate(V2)))
U32(tt) → tt
U41(tt, N) → activate(N)
U51(tt, M, N) → U52(isNat(activate(N)), activate(M), activate(N))
U52(tt, M, N) → s(plus(activate(N), activate(M)))
U61(tt) → 0'
U71(tt, M, N) → U72(isNat(activate(N)), activate(M), activate(N))
U72(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
isNat(n__0) → tt
isNat(n__plus(V1, V2)) → U11(isNat(activate(V1)), activate(V2))
isNat(n__s(V1)) → U21(isNat(activate(V1)))
isNat(n__x(V1, V2)) → U31(isNat(activate(V1)), activate(V2))
plus(N, 0') → U41(isNat(N), N)
plus(N, s(M)) → U51(isNat(M), M, N)
x(N, 0') → U61(isNat(N))
x(N, s(M)) → U71(isNat(M), M, N)
0' → n__0
plus(X1, X2) → n__plus(X1, X2)
s(X) → n__s(X)
x(X1, X2) → n__x(X1, X2)
activate(n__0) → 0'
activate(n__plus(X1, X2)) → plus(X1, X2)
activate(n__s(X)) → s(X)
activate(n__x(X1, X2)) → x(X1, X2)
activate(X) → X

Types:
U11 :: tt → n__0:n__plus:n__s:n__x → tt
tt :: tt
U12 :: tt → tt
isNat :: n__0:n__plus:n__s:n__x → tt
activate :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U21 :: tt → tt
U31 :: tt → n__0:n__plus:n__s:n__x → tt
U32 :: tt → tt
U41 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U51 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U52 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U61 :: tt → n__0:n__plus:n__s:n__x
0' :: n__0:n__plus:n__s:n__x
U71 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
U72 :: tt → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__0 :: n__0:n__plus:n__s:n__x
n__plus :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__s :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
n__x :: n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x → n__0:n__plus:n__s:n__x
hole_tt1_3 :: tt
hole_n__0:n__plus:n__s:n__x2_3 :: n__0:n__plus:n__s:n__x
gen_n__0:n__plus:n__s:n__x3_3 :: Nat → n__0:n__plus:n__s:n__x

Lemmas:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

Generator Equations:
gen_n__0:n__plus:n__s:n__x3_3(0) ⇔ n__0
gen_n__0:n__plus:n__s:n__x3_3(+(x, 1)) ⇔ n__plus(gen_n__0:n__plus:n__s:n__x3_3(x), n__0)

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
isNat(gen_n__0:n__plus:n__s:n__x3_3(n225_3)) → tt, rt ∈ Ω(1 + n225₃)

(30) BOUNDS(n^1, INF)