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:

division(x, y) → div(x, y, 0)
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0) → s(0)
inc(s(x)) → s(inc(x))

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:

division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
div, lt, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

(6) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 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:
lt, div, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

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

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

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

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:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, 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:
inc, div, minus

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

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

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

Induction Base:
inc(gen_0':s3_0(0)) →_R^Ω(1)
s(0')

Induction Step:
inc(gen_0':s3_0(+(n330_0, 1))) →_R^Ω(1)
s(inc(gen_0':s3_0(n330_0))) →_IH
s(gen_0':s3_0(+(1, 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:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
inc(gen_0':s3_0(n330_0)) → gen_0':s3_0(+(1, n330_0)), 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:
minus, div

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n572_0), gen_0':s3_0(n572_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n572₀)

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

Induction Step:
minus(gen_0':s3_0(+(n572_0, 1)), gen_0':s3_0(+(n572_0, 1))) →_R^Ω(1)
minus(gen_0':s3_0(n572_0), gen_0':s3_0(n572_0)) →_IH
gen_0':s3_0(0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
inc(gen_0':s3_0(n330_0)) → gen_0':s3_0(+(1, n330_0)), rt ∈ Ω(1 + n330₀)
minus(gen_0':s3_0(n572_0), gen_0':s3_0(n572_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n572₀)

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:
div

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

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

(17) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
inc(gen_0':s3_0(n330_0)) → gen_0':s3_0(+(1, n330_0)), rt ∈ Ω(1 + n330₀)
minus(gen_0':s3_0(n572_0), gen_0':s3_0(n572_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n572₀)

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

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
inc(gen_0':s3_0(n330_0)) → gen_0':s3_0(+(1, n330_0)), rt ∈ Ω(1 + n330₀)
minus(gen_0':s3_0(n572_0), gen_0':s3_0(n572_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n572₀)

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

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
inc(gen_0':s3_0(n330_0)) → gen_0':s3_0(+(1, n330_0)), 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(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
division(x, y) → div(x, y, 0')
div(x, y, z) → if(lt(x, y), x, y, inc(z))
if(true, x, y, z) → z
if(false, x, s(y), z) → div(minus(x, s(y)), s(y), z)
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
lt(x, 0') → false
lt(0', s(y)) → true
lt(s(x), s(y)) → lt(x, y)
inc(0') → s(0')
inc(s(x)) → s(inc(x))

Types:
division :: 0':s → 0':s → 0':s
div :: 0':s → 0':s → 0':s → 0':s
0' :: 0':s
if :: true:false → 0':s → 0':s → 0':s → 0':s
lt :: 0':s → 0':s → true:false
inc :: 0':s → 0':s
true :: true:false
false :: true:false
s :: 0':s → 0':s
minus :: 0':s → 0':s → 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat → 0':s

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

(28) BOUNDS(n^1, INF)