### (0) Obligation:

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

g(0, f(x, x)) → x
g(x, s(y)) → g(f(x, y), 0)
g(s(x), y) → g(f(x, y), 0)
g(f(x, y), 0) → f(g(x, 0), g(y, 0))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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:

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

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

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

Heuristically decided to analyse the following defined symbols:
g

### (8) Obligation:

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

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

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

The following defined symbols remain to be analysed:
g

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

Proved the following rewrite lemma:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) → *3_0, rt ∈ Ω(n40)

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

Induction Step:
g(gen_0':f:s2_0(+(1, +(n4_0, 1))), gen_0':f:s2_0(0)) →RΩ(1)
f(g(0', 0'), g(gen_0':f:s2_0(+(1, n4_0)), 0')) →IH
f(g(0', 0'), *3_0)

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

### (11) Obligation:

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

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

Lemmas:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) → *3_0, rt ∈ Ω(n40)

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

No more defined symbols left to analyse.

### (12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) → *3_0, rt ∈ Ω(n40)

### (14) Obligation:

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

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

Lemmas:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) → *3_0, rt ∈ Ω(n40)

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

No more defined symbols left to analyse.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':f:s2_0(+(1, n4_0)), gen_0':f:s2_0(0)) → *3_0, rt ∈ Ω(n40)