### (0) Obligation:

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

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

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

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
f :: 0':s → g → f
s :: 0':s → 0':s
0' :: 0':s
g :: 0':s → g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_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:
g < f

### (8) Obligation:

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

Types:
f :: 0':s → g → f
s :: 0':s → 0':s
0' :: 0':s
g :: 0':s → g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Induction Base:
g(gen_0':s4_0(+(1, 0)))

Induction Step:
g(gen_0':s4_0(+(1, +(n6_0, 1)))) →RΩ(1)
g(gen_0':s4_0(+(1, n6_0))) →IH
*5_0

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

### (11) Obligation:

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

Types:
f :: 0':s → g → f
s :: 0':s → 0':s
0' :: 0':s
g :: 0':s → g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (13) Obligation:

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

Types:
f :: 0':s → g → f
s :: 0':s → 0':s
0' :: 0':s
g :: 0':s → g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

### (16) Obligation:

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

Types:
f :: 0':s → g → f
s :: 0':s → 0':s
0' :: 0':s
g :: 0':s → g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat → 0':s

Lemmas:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
g(gen_0':s4_0(+(1, n6_0))) → *5_0, rt ∈ Ω(n60)