The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, 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)
fac(x) → help(x, 0)
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0)

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)
fac(x) → help(x, 0')
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(s(c), help(x, s(c)))
if(false, x, c) → s(0')

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
times/0

(4) 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)
fac(x) → help(x, 0')
help(x, c) → if(lt(c, x), x, c)
if(true, x, c) → times(help(x, s(c)))
if(false, x, c) → s(0')

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

Types:
lt :: 0':s:times → 0':s:times → true:false
0' :: 0':s:times
s :: 0':s:times → 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times → 0':s:times
help :: 0':s:times → 0':s:times → 0':s:times
if :: true:false → 0':s:times → 0':s:times → 0':s:times
times :: 0':s:times → 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat → 0':s:times

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

Heuristically decided to analyse the following defined symbols:
lt, help

They will be analysed ascendingly in the following order:
lt < help

(8) Obligation:

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

Types:
lt :: 0':s:times → 0':s:times → true:false
0' :: 0':s:times
s :: 0':s:times → 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times → 0':s:times
help :: 0':s:times → 0':s:times → 0':s:times
if :: true:false → 0':s:times → 0':s:times → 0':s:times
times :: 0':s:times → 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat → 0':s:times

Generator Equations:
gen_0':s:times3_0(0) ⇔ 0'
gen_0':s:times3_0(+(x, 1)) ⇔ s(gen_0':s:times3_0(x))

The following defined symbols remain to be analysed:
lt, help

They will be analysed ascendingly in the following order:
lt < help

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

Proved the following rewrite lemma:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
lt(gen_0':s:times3_0(0), gen_0':s:times3_0(+(1, 0))) →_R^Ω(1)
true

Induction Step:
lt(gen_0':s:times3_0(+(n5_0, 1)), gen_0':s:times3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) →_IH
true

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

Types:
lt :: 0':s:times → 0':s:times → true:false
0' :: 0':s:times
s :: 0':s:times → 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times → 0':s:times
help :: 0':s:times → 0':s:times → 0':s:times
if :: true:false → 0':s:times → 0':s:times → 0':s:times
times :: 0':s:times → 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat → 0':s:times

Lemmas:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:times3_0(0) ⇔ 0'
gen_0':s:times3_0(+(x, 1)) ⇔ s(gen_0':s:times3_0(x))

The following defined symbols remain to be analysed:
help

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

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

Types:
lt :: 0':s:times → 0':s:times → true:false
0' :: 0':s:times
s :: 0':s:times → 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times → 0':s:times
help :: 0':s:times → 0':s:times → 0':s:times
if :: true:false → 0':s:times → 0':s:times → 0':s:times
times :: 0':s:times → 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat → 0':s:times

Lemmas:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:times3_0(0) ⇔ 0'
gen_0':s:times3_0(+(x, 1)) ⇔ s(gen_0':s:times3_0(x))

No more defined symbols left to analyse.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

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

Types:
lt :: 0':s:times → 0':s:times → true:false
0' :: 0':s:times
s :: 0':s:times → 0':s:times
true :: true:false
false :: true:false
fac :: 0':s:times → 0':s:times
help :: 0':s:times → 0':s:times → 0':s:times
if :: true:false → 0':s:times → 0':s:times → 0':s:times
times :: 0':s:times → 0':s:times
hole_true:false1_0 :: true:false
hole_0':s:times2_0 :: 0':s:times
gen_0':s:times3_0 :: Nat → 0':s:times

Lemmas:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:times3_0(0) ⇔ 0'
gen_0':s:times3_0(+(x, 1)) ⇔ s(gen_0':s:times3_0(x))

No more defined symbols left to analyse.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
lt(gen_0':s:times3_0(n5_0), gen_0':s:times3_0(+(1, n5_0))) → true, rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)