The Runtime Complexity (full) 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: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
minus(s(x), s(y)) →⁺ minus(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
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:

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: FULL

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

Infered types.

(6) Obligation:

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

(7) 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

(8) Obligation:

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

(9) 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).

(10) Complex Obligation (BEST)

(11) Obligation:

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

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

Proved the following rewrite lemma:
inc(gen_0':s3_0(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)

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

Induction Step:
inc(gen_0':s3_0(+(n282_0, 1))) →_R^Ω(1)
s(inc(gen_0':s3_0(n282_0))) →_IH
s(gen_0':s3_0(+(1, c283_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

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(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n492_0), gen_0':s3_0(n492_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n492₀)

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(+(n492_0, 1)), gen_0':s3_0(+(n492_0, 1))) →_R^Ω(1)
minus(gen_0':s3_0(n492_0), gen_0':s3_0(n492_0)) →_IH
gen_0':s3_0(0)

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

(16) Complex Obligation (BEST)

(17) Obligation:

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(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)
minus(gen_0':s3_0(n492_0), gen_0':s3_0(n492_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n492₀)

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

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

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(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)
minus(gen_0':s3_0(n492_0), gen_0':s3_0(n492_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n492₀)

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) 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)

(21) BOUNDS(n^1, INF)

(22) Obligation:

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(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)
minus(gen_0':s3_0(n492_0), gen_0':s3_0(n492_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n492₀)

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.

(23) 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)

(24) BOUNDS(n^1, INF)

(25) Obligation:

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(n282_0)) → gen_0':s3_0(+(1, n282_0)), rt ∈ Ω(1 + n282₀)

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.

(26) 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)

(27) BOUNDS(n^1, INF)

(28) Obligation:

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.

(29) 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)

(30) BOUNDS(n^1, INF)