### (0) Obligation:

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

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

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

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

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

They will be analysed ascendingly in the following order:
f < g

### (8) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

They will be analysed ascendingly in the following order:
f < g

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

Proved the following rewrite lemma:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)

Induction Base:
f(gen_0':s3_0(0), gen_0':s3_0(0)) →RΩ(1)
s(0')

Induction Step:
f(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →RΩ(1)
s(f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0))) →IH
s(gen_0':s3_0(+(1, c6_0)))

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

### (11) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
g

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

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

### (13) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)

### (16) Obligation:

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

Types:
f :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
g :: 0':s → 0':s → g
hole_0':s1_0 :: 0':s
hole_g2_0 :: g
gen_0':s3_0 :: Nat → 0':s

Lemmas:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
f(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(+(1, n5_0)), rt ∈ Ω(1 + n50)