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:

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

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

(2) BOUNDS(n^1, INF)

(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:

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

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus, times, p, fac

They will be analysed ascendingly in the following order:
plus < times
times < fac
p < fac

(8) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
plus, times, p, fac

They will be analysed ascendingly in the following order:
plus < times
times < fac
p < fac

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Induction Base:
plus(gen_0':s2_0(a), gen_0':s2_0(0)) →_R^Ω(1)
gen_0':s2_0(a)

Induction Step:
plus(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(plus(gen_0':s2_0(a), gen_0':s2_0(n4_0))) →_IH
s(gen_0':s2_0(+(a, c5_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:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
times, p, fac

They will be analysed ascendingly in the following order:
times < fac
p < fac

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

Induction Base:
times(gen_0':s2_0(0), gen_0':s2_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s2_0(+(n555_0, 1)), gen_0':s2_0(b)) →_R^Ω(1)
plus(times(gen_0':s2_0(n555_0), gen_0':s2_0(b)), gen_0':s2_0(b)) →_IH
plus(gen_0':s2_0(*(c556_0, b)), gen_0':s2_0(b)) →_L^{Ω(1 + b)}
gen_0':s2_0(+(b, *(n555_0, b)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
p, fac

They will be analysed ascendingly in the following order:
p < fac

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
p(gen_0':s2_0(+(1, n1276_0))) → gen_0':s2_0(n1276_0), rt ∈ Ω(1 + n1276₀)

Induction Base:
p(gen_0':s2_0(+(1, 0))) →_R^Ω(1)
0'

Induction Step:
p(gen_0':s2_0(+(1, +(n1276_0, 1)))) →_R^Ω(1)
s(p(s(gen_0':s2_0(n1276_0)))) →_IH
s(gen_0':s2_0(c1277_0))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)
p(gen_0':s2_0(+(1, n1276_0))) → gen_0':s2_0(n1276_0), rt ∈ Ω(1 + n1276₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

The following defined symbols remain to be analysed:
fac

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
fac(gen_0':s2_0(+(1, n1492_0))) → *3_0, rt ∈ Ω(n1492₀ + n1492₀²)

Induction Base:
fac(gen_0':s2_0(+(1, 0)))

Induction Step:
fac(gen_0':s2_0(+(1, +(n1492_0, 1)))) →_R^Ω(1)
times(fac(p(s(gen_0':s2_0(+(1, n1492_0))))), s(gen_0':s2_0(+(1, n1492_0)))) →_L^{Ω(2 + n1492₀)}
times(fac(gen_0':s2_0(+(1, n1492_0))), s(gen_0':s2_0(+(1, n1492_0)))) →_IH
times(*3_0, s(gen_0':s2_0(+(1, n1492_0))))

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

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)
p(gen_0':s2_0(+(1, n1276_0))) → gen_0':s2_0(n1276_0), rt ∈ Ω(1 + n1276₀)
fac(gen_0':s2_0(+(1, n1492_0))) → *3_0, rt ∈ Ω(n1492₀ + n1492₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)
p(gen_0':s2_0(+(1, n1276_0))) → gen_0':s2_0(n1276_0), rt ∈ Ω(1 + n1276₀)
fac(gen_0':s2_0(+(1, n1492_0))) → *3_0, rt ∈ Ω(n1492₀ + n1492₀²)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

(25) BOUNDS(n^2, INF)

(26) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)
p(gen_0':s2_0(+(1, n1276_0))) → gen_0':s2_0(n1276_0), rt ∈ Ω(1 + n1276₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

(28) BOUNDS(n^2, INF)

(29) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n555_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n555_0, b)), rt ∈ Ω(1 + b·n555₀ + n555₀)

(31) BOUNDS(n^2, INF)

(32) Obligation:

Innermost TRS:
Rules:
plus(x, 0') → x
plus(x, s(y)) → s(plus(x, y))
times(0', y) → 0'
times(x, 0') → 0'
times(s(x), y) → plus(times(x, y), y)
p(s(s(x))) → s(p(s(x)))
p(s(0')) → 0'
fac(s(x)) → times(fac(p(s(x))), s(x))

Types:
plus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
times :: 0':s → 0':s → 0':s
p :: 0':s → 0':s
fac :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s2_0(0) ⇔ 0'
gen_0':s2_0(+(x, 1)) ⇔ s(gen_0':s2_0(x))

No more defined symbols left to analyse.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)

(34) BOUNDS(n^1, INF)