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:

minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0
p(0) → 0
p(s(x)) → x
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0, y) → false
gt(s(x), 0) → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0), x, y)
if1(false, x, y) → 0
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0

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(x13_3), 0) →⁺ s(minus(x13_3, 0))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x13_3 / s(x13_3)].
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:

minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

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

Heuristically decided to analyse the following defined symbols:
minus, gt, ge, div

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

(8) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
gt, minus, ge, div

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

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

Proved the following rewrite lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Induction Base:
gt(gen_s:0'3_0(0), gen_s:0'3_0(0)) →_R^Ω(1)
false

Induction Step:
gt(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) →_R^Ω(1)
gt(gen_s:0'3_0(n5_0), gen_s:0'3_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:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
minus, ge, div

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

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

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

(13) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
ge, div

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

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
ge(gen_s:0'3_0(n327_0), gen_s:0'3_0(n327_0)) → true, rt ∈ Ω(1 + n327₀)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) →_R^Ω(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n327_0, 1)), gen_s:0'3_0(+(n327_0, 1))) →_R^Ω(1)
ge(gen_s:0'3_0(n327_0), gen_s:0'3_0(n327_0)) →_IH
true

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

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
ge(gen_s:0'3_0(n327_0), gen_s:0'3_0(n327_0)) → true, rt ∈ Ω(1 + n327₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

The following defined symbols remain to be analysed:
div

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
ge(gen_s:0'3_0(n327_0), gen_s:0'3_0(n327_0)) → true, rt ∈ Ω(1 + n327₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)
ge(gen_s:0'3_0(n327_0), gen_s:0'3_0(n327_0)) → true, rt ∈ Ω(1 + n327₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
minus(x, y) → if(gt(x, y), x, y)
if(true, x, y) → s(minus(p(x), y))
if(false, x, y) → 0'
p(0') → 0'
p(s(x)) → x
ge(x, 0') → true
ge(0', s(x)) → false
ge(s(x), s(y)) → ge(x, y)
gt(0', y) → false
gt(s(x), 0') → true
gt(s(x), s(y)) → gt(x, y)
div(x, y) → if1(ge(x, y), x, y)
if1(true, x, y) → if2(gt(y, 0'), x, y)
if1(false, x, y) → 0'
if2(true, x, y) → s(div(minus(x, y), y))
if2(false, x, y) → 0'

Types:
minus :: s:0' → s:0' → s:0'
if :: true:false → s:0' → s:0' → s:0'
gt :: s:0' → s:0' → true:false
true :: true:false
s :: s:0' → s:0'
p :: s:0' → s:0'
false :: true:false
0' :: s:0'
ge :: s:0' → s:0' → true:false
div :: s:0' → s:0' → s:0'
if1 :: true:false → s:0' → s:0' → s:0'
if2 :: true:false → s:0' → s:0' → s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat → s:0'

Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_s:0'3_0(0) ⇔ 0'
gen_s:0'3_0(+(x, 1)) ⇔ s(gen_s:0'3_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) → false, rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)