The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

(0) Obligation:

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

p(s(x)) → x
fact(0) → s(0)
fact(s(x)) → *(s(x), fact(p(s(x))))
*(0, y) → 0
*(s(x), y) → +(*(x, y), y)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

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
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
fact, *', +'

They will be analysed ascendingly in the following order:
*' < fact
+' < *'

(6) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s: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:
+', fact, *'

They will be analysed ascendingly in the following order:
*' < fact
+' < *'

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
+'(gen_s:0'2_0(a), gen_s:0'2_0(0)) →_R^Ω(1)
gen_s:0'2_0(a)

Induction Step:
+'(gen_s:0'2_0(a), gen_s:0'2_0(+(n4_0, 1))) →_R^Ω(1)
s(+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0))) →_IH
s(gen_s:0'2_0(+(a, c5_0)))

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

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

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:
*', fact

They will be analysed ascendingly in the following order:
*' < fact

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

Induction Base:
*'(gen_s:0'2_0(0), gen_s:0'2_0(b)) →_R^Ω(1)
0'

Induction Step:
*'(gen_s:0'2_0(+(n527_0, 1)), gen_s:0'2_0(b)) →_R^Ω(1)
+'(*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)), gen_s:0'2_0(b)) →_IH
+'(gen_s:0'2_0(*(c528_0, b)), gen_s:0'2_0(b)) →_L^{Ω(1 + b)}
gen_s:0'2_0(+(b, *(n527_0, b)))

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

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:
fact

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

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.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

(16) BOUNDS(n^2, INF)

(17) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

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.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_s:0'2_0(n527_0), gen_s:0'2_0(b)) → gen_s:0'2_0(*(n527_0, b)), rt ∈ Ω(1 + b·n527₀ + n527₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
p(s(x)) → x
fact(0') → s(0')
fact(s(x)) → *'(s(x), fact(p(s(x))))
*'(0', y) → 0'
*'(s(x), y) → +'(*'(x, y), y)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))

Types:
p :: s:0' → s:0'
s :: s:0' → s:0'
fact :: s:0' → s:0'
0' :: s:0'
*' :: s:0' → s:0' → s:0'
+' :: s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
gen_s:0'2_0 :: Nat → s:0'

Lemmas:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

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.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
+'(gen_s:0'2_0(a), gen_s:0'2_0(n4_0)) → gen_s:0'2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(22) BOUNDS(n^1, INF)