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:

lt(0, s(x)) → true
lt(x, 0) → false
lt(s(x), s(y)) → lt(x, y)
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0), s(0))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → 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:

lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

(6) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

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

Proved the following rewrite lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) →_R^Ω(1)
true

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) →_IH
true

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:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

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

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

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)

Induction Base:
plus(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n330_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(plus(gen_0':s3_0(n330_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c331_0)))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
times, loop

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
times(gen_0':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

Induction Base:
times(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
0'

Induction Step:
times(gen_0':s3_0(+(n953_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n953_0), gen_0':s3_0(b))) →_IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c954_0, b))) →_L^{Ω(1 + b)}
gen_0':s3_0(+(b, *(n953_0, b)))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)
times(gen_0':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

The following defined symbols remain to be analysed:
loop

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)
times(gen_0':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
times(gen_0':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

(19) BOUNDS(n^2, INF)

(20) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)
times(gen_0':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_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':s3_0(n953_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n953_0, b)), rt ∈ Ω(1 + b·n953₀ + n953₀)

(22) BOUNDS(n^2, INF)

(23) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)
plus(gen_0':s3_0(n330_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n330_0, b)), rt ∈ Ω(1 + n330₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
lt(0', s(x)) → true
lt(x, 0') → false
lt(s(x), s(y)) → lt(x, y)
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
fac(x) → loop(x, s(0'), s(0'))
loop(x, c, y) → if(lt(x, c), x, c, y)
if(false, x, c, y) → loop(x, s(c), times(y, s(c)))
if(true, x, c, y) → y

Types:
lt :: 0':s → 0':s → true:false
0' :: 0':s
s :: 0':s → 0':s
true :: true:false
false :: true:false
times :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
fac :: 0':s → 0':s
loop :: 0':s → 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))

No more defined symbols left to analyse.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(28) BOUNDS(n^1, INF)