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(0, Y) → 0
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0) → true
geq(0, s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0, s(Y)) → 0
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) → X
if(false, X, Y) → Y

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:

minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus, geq, div

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

(8) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus, geq, div

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

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)

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

Induction Step:
minus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) →_R^Ω(1)
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) →_IH
gen_0':s3_0(0)

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(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), 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:
geq, div

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

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

Proved the following rewrite lemma:
geq(gen_0':s3_0(n212_0), gen_0':s3_0(n212_0)) → true, rt ∈ Ω(1 + n212₀)

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

Induction Step:
geq(gen_0':s3_0(+(n212_0, 1)), gen_0':s3_0(+(n212_0, 1))) →_R^Ω(1)
geq(gen_0':s3_0(n212_0), gen_0':s3_0(n212_0)) →_IH
true

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)
geq(gen_0':s3_0(n212_0), gen_0':s3_0(n212_0)) → true, rt ∈ Ω(1 + n212₀)

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

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)
geq(gen_0':s3_0(n212_0), gen_0':s3_0(n212_0)) → true, rt ∈ Ω(1 + n212₀)

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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)

(18) BOUNDS(n^1, INF)

(19) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)
geq(gen_0':s3_0(n212_0), gen_0':s3_0(n212_0)) → true, rt ∈ Ω(1 + n212₀)

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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)

(21) BOUNDS(n^1, INF)

(22) Obligation:

TRS:
Rules:
minus(0', Y) → 0'
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0') → true
geq(0', s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0', s(Y)) → 0'
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
if(true, X, Y) → X
if(false, X, Y) → Y

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
geq :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
div :: 0':s → 0':s → 0':s
if :: true:false → 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:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), 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.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) → gen_0':s3_0(0), rt ∈ Ω(1 + n5₀)

(24) BOUNDS(n^1, INF)