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:

fac(0) → 1
fac(s(x)) → *(s(x), fac(x))
floop(0, y) → y
floop(s(x), y) → floop(x, *(s(x), y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
1 → s(0)
fac(0) → s(0)

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)) →⁺ *(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 [ ].

(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:

fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 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:
fac, *', floop, +'

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

(8) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 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:
+', fac, *', floop

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

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

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

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

Induction Step:
+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
s(+'(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:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(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:
*', fac, floop

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

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

Proved the following rewrite lemma:
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

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

Induction Step:
*'(gen_0':s2_0(a), gen_0':s2_0(+(n511_0, 1))) →_R^Ω(1)
+'(*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)), gen_0':s2_0(a)) →_IH
+'(gen_0':s2_0(*(c512_0, a)), gen_0':s2_0(a)) →_L^{Ω(1 + a)}
gen_0':s2_0(+(a, *(n511_0, a)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

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, floop

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

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:
floop

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

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.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

(20) BOUNDS(n^2, INF)

(21) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) → gen_0':s2_0(+(n4_0, a)), rt ∈ Ω(1 + n4₀)
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

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.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) → gen_0':s2_0(*(n511_0, a)), rt ∈ Ω(1 + a·n511₀ + n511₀)

(23) BOUNDS(n^2, INF)

(24) Obligation:

TRS:
Rules:
fac(0') → 1'
fac(s(x)) → *'(s(x), fac(x))
floop(0', y) → y
floop(s(x), y) → floop(x, *'(s(x), y))
*'(x, 0') → 0'
*'(x, s(y)) → +'(*'(x, y), x)
+'(x, 0') → x
+'(x, s(y)) → s(+'(x, y))
1' → s(0')
fac(0') → s(0')

Types:
fac :: 0':s → 0':s
0' :: 0':s
1' :: 0':s
s :: 0':s → 0':s
*' :: 0':s → 0':s → 0':s
floop :: 0':s → 0':s → 0':s
+' :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
+'(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.

(25) LowerBoundsProof (EQUIVALENT transformation)

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

(26) BOUNDS(n^1, INF)