The Runtime Complexity (full) 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, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0) → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0) → 0
x(N, s(M)) → U21(tt, M, N)
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
U11(tt, s(M521_3), N) →⁺ s(U11(tt, M521_3, activate(activate(N))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [M521_3 / s(M521_3)].
The result substitution is [N / activate(activate(N))].

(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, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

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

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

They will be analysed ascendingly in the following order:
plus < x

(8) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

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

They will be analysed ascendingly in the following order:
plus < x

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

Proved the following rewrite lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Induction Base:
plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) →_R^Ω(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) →_R^Ω(1)
U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →_R^Ω(1)
U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) →_R^Ω(1)
U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) →_R^Ω(1)
U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →_R^Ω(1)
s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) →_R^Ω(1)
s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) →_R^Ω(1)
s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0))) →_IH
s(gen_s:0'3_0(+(a, c6_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, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
x

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n580₀ + n580₀)

Induction Base:
x(gen_s:0'3_0(a), gen_s:0'3_0(0)) →_R^Ω(1)
0'

Induction Step:
x(gen_s:0'3_0(a), gen_s:0'3_0(+(n580_0, 1))) →_R^Ω(1)
U21(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) →_R^Ω(1)
U22(tt, activate(gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) →_R^Ω(1)
U22(tt, gen_s:0'3_0(n580_0), activate(gen_s:0'3_0(a))) →_R^Ω(1)
U22(tt, gen_s:0'3_0(n580_0), gen_s:0'3_0(a)) →_R^Ω(1)
plus(x(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) →_R^Ω(1)
plus(x(gen_s:0'3_0(a), activate(gen_s:0'3_0(n580_0))), activate(gen_s:0'3_0(a))) →_R^Ω(1)
plus(x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)), activate(gen_s:0'3_0(a))) →_IH
plus(gen_s:0'3_0(*(c581_0, a)), activate(gen_s:0'3_0(a))) →_R^Ω(1)
plus(gen_s:0'3_0(*(n580_0, a)), gen_s:0'3_0(a)) →_L^{Ω(1 + a)}
gen_s:0'3_0(+(a, *(n580_0, a)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n580₀ + n580₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n580₀ + n580₀)

(16) BOUNDS(n^2, INF)

(17) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n580₀ + n580₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
x(gen_s:0'3_0(a), gen_s:0'3_0(n580_0)) → gen_s:0'3_0(*(n580_0, a)), rt ∈ Ω(1 + a·n580₀ + n580₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
U21(tt, M, N) → U22(tt, activate(M), activate(N))
U22(tt, M, N) → plus(x(activate(N), activate(M)), activate(N))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
x(N, 0') → 0'
x(N, s(M)) → U21(tt, M, N)
activate(X) → X

Types:
U11 :: tt → s:0' → s:0' → s:0'
tt :: tt
U12 :: tt → s:0' → s:0' → s:0'
activate :: s:0' → s:0'
s :: s:0' → s:0'
plus :: s:0' → s:0' → s:0'
U21 :: tt → s:0' → s:0' → s:0'
U22 :: tt → s:0' → s:0' → s:0'
x :: s:0' → s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
hole_tt2_0 :: tt
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_s:0'3_0(a), gen_s:0'3_0(n5_0)) → gen_s:0'3_0(+(n5_0, a)), rt ∈ Ω(1 + n5₀)

(22) BOUNDS(n^1, INF)