### (0) Obligation:

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

f(0) → s(0)
f(s(0)) → s(s(0))
f(s(0)) → *(s(s(0)), f(0))
f(+(x, s(0))) → +(s(s(0)), f(x))
f(+(x, y)) → *(f(x), f(y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
f(+(x, s(0))) →+ +(s(s(0)), f(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / +(x, s(0))].
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(0') → s(0')
f(s(0')) → s(s(0'))
f(s(0')) → *'(s(s(0')), f(0'))
f(+'(x, s(0'))) → +'(s(s(0')), f(x))
f(+'(x, y)) → *'(f(x), f(y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

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

Types:
f :: 0':s:*':+' → 0':s:*':+'
0' :: 0':s:*':+'
s :: 0':s:*':+' → 0':s:*':+'
*' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
+' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
hole_0':s:*':+'1_0 :: 0':s:*':+'
gen_0':s:*':+'2_0 :: Nat → 0':s:*':+'

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

Heuristically decided to analyse the following defined symbols:
f

### (8) Obligation:

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

Types:
f :: 0':s:*':+' → 0':s:*':+'
0' :: 0':s:*':+'
s :: 0':s:*':+' → 0':s:*':+'
*' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
+' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
hole_0':s:*':+'1_0 :: 0':s:*':+'
gen_0':s:*':+'2_0 :: Nat → 0':s:*':+'

Generator Equations:
gen_0':s:*':+'2_0(0) ⇔ 0'
gen_0':s:*':+'2_0(+(x, 1)) ⇔ s(gen_0':s:*':+'2_0(x))

The following defined symbols remain to be analysed:
f

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

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

### (10) Obligation:

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

Types:
f :: 0':s:*':+' → 0':s:*':+'
0' :: 0':s:*':+'
s :: 0':s:*':+' → 0':s:*':+'
*' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
+' :: 0':s:*':+' → 0':s:*':+' → 0':s:*':+'
hole_0':s:*':+'1_0 :: 0':s:*':+'
gen_0':s:*':+'2_0 :: Nat → 0':s:*':+'

Generator Equations:
gen_0':s:*':+'2_0(0) ⇔ 0'
gen_0':s:*':+'2_0(+(x, 1)) ⇔ s(gen_0':s:*':+'2_0(x))

No more defined symbols left to analyse.