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(0, x) → x
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0) → 0
minus(x, 0) → x
minus(0, x) → 0
minus(x, s(y)) → p(minus(x, y))
isZero(0) → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0)), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
plus(s(x), 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 [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:

plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
plus, times, minus, facIter

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

(8) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
plus, times, minus, facIter

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

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

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

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(+(n5_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c6_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(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), 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:
times, minus, facIter

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

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

Proved the following rewrite lemma:
times(gen_0':s3_0(n732_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n732_0, b)), rt ∈ Ω(1 + b·n732₀ + n732₀)

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

Induction Step:
times(gen_0':s3_0(+(n732_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n732_0), gen_0':s3_0(b))) →_IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c733_0, b))) →_L^{Ω(1 + b)}
gen_0':s3_0(+(b, *(n732_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(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n732_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n732_0, b)), rt ∈ Ω(1 + b·n732₀ + n732₀)

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:
minus, facIter

They will be analysed ascendingly in the following order:
minus < facIter

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1674_0))) → *4_0, rt ∈ Ω(n1674₀)

Induction Base:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))

Induction Step:
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n1674_0, 1)))) →_R^Ω(1)
p(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1674_0)))) →_IH
p(*4_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(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n732_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n732_0, b)), rt ∈ Ω(1 + b·n732₀ + n732₀)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1674_0))) → *4_0, rt ∈ Ω(n1674₀)

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:
facIter

(18) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n732_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n732_0, b)), rt ∈ Ω(1 + b·n732₀ + n732₀)
minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1674_0))) → *4_0, rt ∈ Ω(n1674₀)

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.

(19) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), rt ∈ Ω(1 + n5₀)
times(gen_0':s3_0(n732_0), gen_0':s3_0(b)) → gen_0':s3_0(*(n732_0, b)), rt ∈ Ω(1 + b·n732₀ + n732₀)

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.

(20) Obligation:

Innermost TRS:
Rules:
plus(0', x) → x
plus(s(x), y) → s(plus(x, y))
times(0', y) → 0'
times(s(x), y) → plus(y, times(x, y))
p(s(x)) → x
p(0') → 0'
minus(x, 0') → x
minus(0', x) → 0'
minus(x, s(y)) → p(minus(x, y))
isZero(0') → true
isZero(s(x)) → false
facIter(x, y) → if(isZero(x), minus(x, s(0')), y, times(y, x))
if(true, x, y, z) → y
if(false, x, y, z) → facIter(x, z)
factorial(x) → facIter(x, s(0'))

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
minus :: 0':s → 0':s → 0':s
isZero :: 0':s → true:false
true :: true:false
false :: true:false
facIter :: 0':s → 0':s → 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
factorial :: 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n5_0, b)), 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.