### (0) Obligation:

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

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(p(s(x))))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
fac(s(x)) →+ times(s(x), fac(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
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:

p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(s(x), fac(p(s(x))))

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
times/0

### (6) Obligation:

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

p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(fac(p(s(x))))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(fac(p(s(x))))

Types:
p :: s:0':times → s:0':times
s :: s:0':times → s:0':times
fac :: s:0':times → s:0':times
0' :: s:0':times
times :: s:0':times → s:0':times
hole_s:0':times1_0 :: s:0':times
gen_s:0':times2_0 :: Nat → s:0':times

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

Heuristically decided to analyse the following defined symbols:
fac

### (10) Obligation:

TRS:
Rules:
p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(fac(p(s(x))))

Types:
p :: s:0':times → s:0':times
s :: s:0':times → s:0':times
fac :: s:0':times → s:0':times
0' :: s:0':times
times :: s:0':times → s:0':times
hole_s:0':times1_0 :: s:0':times
gen_s:0':times2_0 :: Nat → s:0':times

Generator Equations:
gen_s:0':times2_0(0) ⇔ 0'
gen_s:0':times2_0(+(x, 1)) ⇔ s(gen_s:0':times2_0(x))

The following defined symbols remain to be analysed:
fac

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

Proved the following rewrite lemma:
fac(gen_s:0':times2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Induction Base:
fac(gen_s:0':times2_0(0))

Induction Step:
fac(gen_s:0':times2_0(+(n4_0, 1))) →RΩ(1)
times(fac(p(s(gen_s:0':times2_0(n4_0))))) →RΩ(1)
times(fac(gen_s:0':times2_0(n4_0))) →IH
times(*3_0)

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

### (13) Obligation:

TRS:
Rules:
p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(fac(p(s(x))))

Types:
p :: s:0':times → s:0':times
s :: s:0':times → s:0':times
fac :: s:0':times → s:0':times
0' :: s:0':times
times :: s:0':times → s:0':times
hole_s:0':times1_0 :: s:0':times
gen_s:0':times2_0 :: Nat → s:0':times

Lemmas:
fac(gen_s:0':times2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_s:0':times2_0(0) ⇔ 0'
gen_s:0':times2_0(+(x, 1)) ⇔ s(gen_s:0':times2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
fac(gen_s:0':times2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

### (16) Obligation:

TRS:
Rules:
p(s(x)) → x
fac(0') → s(0')
fac(s(x)) → times(fac(p(s(x))))

Types:
p :: s:0':times → s:0':times
s :: s:0':times → s:0':times
fac :: s:0':times → s:0':times
0' :: s:0':times
times :: s:0':times → s:0':times
hole_s:0':times1_0 :: s:0':times
gen_s:0':times2_0 :: Nat → s:0':times

Lemmas:
fac(gen_s:0':times2_0(n4_0)) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_s:0':times2_0(0) ⇔ 0'
gen_s:0':times2_0(+(x, 1)) ⇔ s(gen_s:0':times2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
fac(gen_s:0':times2_0(n4_0)) → *3_0, rt ∈ Ω(n40)