### (0) Obligation:

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

plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0), y) → s(y)
plus(0, y) → y
ack(0, y) → s(y)
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
ack(s(s(x52929_1)), s(s(s(x52701_1)))) →+ ack(x52929_1, plus(plus(x52701_1, s(ack(s(s(x52929_1)), s(s(x52701_1))))), ack(s(x52929_1), plus(x52701_1, s(ack(s(s(x52929_1)), s(s(x52701_1))))))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,1,0].
The pumping substitution is [x52701_1 / s(x52701_1)].
The result substitution is [ ].

The rewrite sequence
ack(s(s(x52929_1)), s(s(s(x52701_1)))) →+ ack(x52929_1, plus(plus(x52701_1, s(ack(s(s(x52929_1)), s(s(x52701_1))))), ack(s(x52929_1), plus(x52701_1, s(ack(s(s(x52929_1)), s(s(x52701_1))))))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,1,1,0].
The pumping substitution is [x52701_1 / s(x52701_1)].
The result substitution is [ ].

### (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:

plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

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

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

### (8) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

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

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

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

Proved the following rewrite lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)

Induction Base:
plus(gen_s:0'2_0(+(1, *(2, 0))), gen_s:0'2_0(b)) →RΩ(1)
s(gen_s:0'2_0(b))

Induction Step:
plus(gen_s:0'2_0(+(1, *(2, +(n4_0, 1)))), gen_s:0'2_0(b)) →RΩ(1)
s(plus(gen_s:0'2_0(+(1, *(2, n4_0))), s(gen_s:0'2_0(b)))) →IH
s(gen_s:0'2_0(+(+(1, +(b, 1)), *(2, c5_0))))

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

### (11) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

The following defined symbols remain to be analysed:
ack

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

Proved the following rewrite lemma:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n804_0))) → *3_0, rt ∈ Ω(n8040)

Induction Base:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, 0)))

Induction Step:
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, +(n804_0, 1)))) →RΩ(1)
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n804_0)), ack(s(gen_s:0'2_0(0)), gen_s:0'2_0(+(1, n804_0))))) →IH
ack(gen_s:0'2_0(0), plus(gen_s:0'2_0(+(1, n804_0)), *3_0))

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

### (14) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n804_0))) → *3_0, rt ∈ Ω(n8040)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_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:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)

### (17) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)
ack(gen_s:0'2_0(1), gen_s:0'2_0(+(1, n804_0))) → *3_0, rt ∈ Ω(n8040)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_0(x))

No more defined symbols left to analyse.

### (18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)

### (20) Obligation:

TRS:
Rules:
plus(s(s(x)), y) → s(plus(x, s(y)))
plus(x, s(s(y))) → s(plus(s(x), y))
plus(s(0'), y) → s(y)
plus(0', y) → y
ack(0', y) → s(y)
ack(s(x), 0') → ack(x, s(0'))
ack(s(x), s(y)) → ack(x, plus(y, ack(s(x), y)))

Types:
plus :: s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
ack :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
plus(gen_s:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_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:0'2_0(+(1, *(2, n4_0))), gen_s:0'2_0(b)) → gen_s:0'2_0(+(+(1, *(2, n4_0)), b)), rt ∈ Ω(1 + n40)