The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(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: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) 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: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
times/0

(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(fac(p(s(x))))

S is empty.
Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Innermost 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

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

Heuristically decided to analyse the following defined symbols:
fac

(8) Obligation:

Innermost 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

(9) 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))) →_R^Ω(1)
times(fac(p(s(gen_s:0':times2_0(n4_0))))) →_R^Ω(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 irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost 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.

(12) 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)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost 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.

(15) 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) BOUNDS(n^1, INF)