### (0) Obligation:

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

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5)
f(0, 0, 0, 0, 0) → 0

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(x1), x2, x3, x4, x5) →+ f(x1, x2, x3, x4, x5)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x1 / s(x1)].
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(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)
f(0', 0', 0', 0', 0') → 0'

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)
f(0', 0', 0', 0', 0') → 0'

Types:
f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
f

### (8) Obligation:

TRS:
Rules:
f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)
f(0', 0', 0', 0', 0') → 0'

Types:
f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Generator Equations:
gen_s:0'2_0(0) ⇔ 0'
gen_s:0'2_0(+(x, 1)) ⇔ s(gen_s:0'2_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(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0', s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0', 0', s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0', 0', 0', s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0', 0', 0', 0', s(x5)) → f(x5, x5, x5, x5, x5)
f(0', 0', 0', 0', 0') → 0'

Types:
f :: s:0' → s:0' → s:0' → s:0' → s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

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

No more defined symbols left to analyse.