### (0) Obligation:

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

ack(0, y) → s(y)
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, ack(s(x), y))
f(s(x), y) → f(x, s(x))
f(x, s(y)) → f(y, x)
f(x, y) → ack(x, y)
ack(s(x), y) → f(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
ack(s(s(x39_1)), 0) →+ ack(x39_1, ack(s(x39_1), 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x39_1 / s(x39_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:

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

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

They will be analysed ascendingly in the following order:
ack = f

### (8) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

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

They will be analysed ascendingly in the following order:
ack = f

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

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

### (10) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
ack

They will be analysed ascendingly in the following order:
ack = f

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

Proved the following rewrite lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)

Induction Base:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, 0)))

Induction Step:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, +(n1557_0, 1)))) →RΩ(1)
ack(gen_0':s2_0(0), ack(s(gen_0':s2_0(0)), gen_0':s2_0(+(1, n1557_0)))) →IH
ack(gen_0':s2_0(0), *3_0)

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

### (13) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
ack = f

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

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

### (15) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)

### (18) Obligation:

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

Types:
ack :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n1557_0))) → *3_0, rt ∈ Ω(n15570)