The Runtime Complexity (full) 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: FULL

(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: FULL

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

Infered types.

(6) Obligation:

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:

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:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

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

Induction Step:
times(gen_0':s2_0(+(n475_0, 1)), gen_0':s2_0(b)) →_R^Ω(1)
plus(times(gen_0':s2_0(n475_0), gen_0':s2_0(b)), gen_0':s2_0(b)) →_IH
plus(gen_0':s2_0(*(c476_0, b)), gen_0':s2_0(b)) →_L^{Ω(1 + b)}
gen_0':s2_0(+(b, *(n475_0, b)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

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, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n1092₀)

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

Induction Step:
p(gen_0':s2_0(+(1, +(n1092_0, 1)))) →_R^Ω(1)
s(p(s(gen_0':s2_0(n1092_0)))) →_IH
s(gen_0':s2_0(c1093_0))

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

(16) Complex Obligation (BEST)

(17) Obligation:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n1092₀)

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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n1092₀)

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.

(20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

(21) BOUNDS(n^2, INF)

(22) Obligation:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)
p(gen_0':s2_0(+(1, n1092_0))) → gen_0':s2_0(n1092_0), rt ∈ Ω(1 + n1092₀)

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.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

(24) BOUNDS(n^2, INF)

(25) Obligation:

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(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

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.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s2_0(n475_0), gen_0':s2_0(b)) → gen_0':s2_0(*(n475_0, b)), rt ∈ Ω(1 + b·n475₀ + n475₀)

(27) BOUNDS(n^2, INF)

(28) Obligation:

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.

(29) 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)

(30) BOUNDS(n^1, INF)