### (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)))
plus(N, 0) → N
plus(N, s(M)) → U11(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(M279_3), N) →+ s(U11(tt, M279_3, activate(activate(N))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [M279_3 / s(M279_3)].
The result substitution is [N / activate(activate(N))].

### (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)))
plus(N, 0') → N
plus(N, s(M)) → U11(tt, M, N)
activate(X) → X

S is empty.
Rewrite Strategy: FULL

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)))
plus(N, 0') → N
plus(N, s(M)) → U11(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'
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

### (8) Obligation:

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

### (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)) →RΩ(1)
gen_s:0'3_0(a)

Induction Step:
plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n5_0, 1))) →RΩ(1)
U11(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →RΩ(1)
U12(tt, activate(gen_s:0'3_0(n5_0)), activate(gen_s:0'3_0(a))) →RΩ(1)
U12(tt, gen_s:0'3_0(n5_0), activate(gen_s:0'3_0(a))) →RΩ(1)
U12(tt, gen_s:0'3_0(n5_0), gen_s:0'3_0(a)) →RΩ(1)
s(plus(activate(gen_s:0'3_0(a)), activate(gen_s:0'3_0(n5_0)))) →RΩ(1)
s(plus(gen_s:0'3_0(a), activate(gen_s:0'3_0(n5_0)))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0') → N
plus(N, s(M)) → U11(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'
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.

### (12) 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)

### (14) Obligation:

TRS:
Rules:
U11(tt, M, N) → U12(tt, activate(M), activate(N))
U12(tt, M, N) → s(plus(activate(N), activate(M)))
plus(N, 0') → N
plus(N, s(M)) → U11(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'
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.

### (15) 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)