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:

a__minus(0, Y) → 0
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0) → true
a__geq(0, s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0, s(Y)) → 0
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

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:

a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

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

Heuristically decided to analyse the following defined symbols:
a__minus, a__geq, mark

They will be analysed ascendingly in the following order:
a__minus < mark
a__geq < mark

(6) Obligation:

Innermost TRS:
Rules:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

The following defined symbols remain to be analysed:
a__minus, a__geq, mark

They will be analysed ascendingly in the following order:
a__minus < mark
a__geq < mark

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

Proved the following rewrite lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) →_R^Ω(1)
0'

Induction Step:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n4_0, 1))) →_R^Ω(1)
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) →_IH
gen_0':s:true:false:minus:div:geq:if2_0(0)

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:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

The following defined symbols remain to be analysed:
a__geq, mark

They will be analysed ascendingly in the following order:
a__geq < mark

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

Proved the following rewrite lemma:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) → true, rt ∈ Ω(1 + n764₀)

Induction Base:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(0), gen_0':s:true:false:minus:div:geq:if2_0(0)) →_R^Ω(1)
true

Induction Step:
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(+(n764_0, 1)), gen_0':s:true:false:minus:div:geq:if2_0(+(n764_0, 1))) →_R^Ω(1)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) →_IH
true

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:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) → true, rt ∈ Ω(1 + n764₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_0':s:true:false:minus:div:geq:if2_0(n1629_0)) → gen_0':s:true:false:minus:div:geq:if2_0(n1629_0), rt ∈ Ω(1 + n1629₀)

Induction Base:
mark(gen_0':s:true:false:minus:div:geq:if2_0(0)) →_R^Ω(1)
0'

Induction Step:
mark(gen_0':s:true:false:minus:div:geq:if2_0(+(n1629_0, 1))) →_R^Ω(1)
s(mark(gen_0':s:true:false:minus:div:geq:if2_0(n1629_0))) →_IH
s(gen_0':s:true:false:minus:div:geq:if2_0(c1630_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:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) → true, rt ∈ Ω(1 + n764₀)
mark(gen_0':s:true:false:minus:div:geq:if2_0(n1629_0)) → gen_0':s:true:false:minus:div:geq:if2_0(n1629_0), rt ∈ Ω(1 + n1629₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) → true, rt ∈ Ω(1 + n764₀)
mark(gen_0':s:true:false:minus:div:geq:if2_0(n1629_0)) → gen_0':s:true:false:minus:div:geq:if2_0(n1629_0), rt ∈ Ω(1 + n1629₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

Innermost TRS:
Rules:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)
a__geq(gen_0':s:true:false:minus:div:geq:if2_0(n764_0), gen_0':s:true:false:minus:div:geq:if2_0(n764_0)) → true, rt ∈ Ω(1 + n764₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
a__minus(0', Y) → 0'
a__minus(s(X), s(Y)) → a__minus(X, Y)
a__geq(X, 0') → true
a__geq(0', s(Y)) → false
a__geq(s(X), s(Y)) → a__geq(X, Y)
a__div(0', s(Y)) → 0'
a__div(s(X), s(Y)) → a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0')
a__if(true, X, Y) → mark(X)
a__if(false, X, Y) → mark(Y)
mark(minus(X1, X2)) → a__minus(X1, X2)
mark(geq(X1, X2)) → a__geq(X1, X2)
mark(div(X1, X2)) → a__div(mark(X1), X2)
mark(if(X1, X2, X3)) → a__if(mark(X1), X2, X3)
mark(0') → 0'
mark(s(X)) → s(mark(X))
mark(true) → true
mark(false) → false
a__minus(X1, X2) → minus(X1, X2)
a__geq(X1, X2) → geq(X1, X2)
a__div(X1, X2) → div(X1, X2)
a__if(X1, X2, X3) → if(X1, X2, X3)

Types:
a__minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
0' :: 0':s:true:false:minus:div:geq:if
s :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
true :: 0':s:true:false:minus:div:geq:if
false :: 0':s:true:false:minus:div:geq:if
a__div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
a__if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
div :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
minus :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
mark :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
geq :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
if :: 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if → 0':s:true:false:minus:div:geq:if
hole_0':s:true:false:minus:div:geq:if1_0 :: 0':s:true:false:minus:div:geq:if
gen_0':s:true:false:minus:div:geq:if2_0 :: Nat → 0':s:true:false:minus:div:geq:if

Lemmas:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':s:true:false:minus:div:geq:if2_0(0) ⇔ 0'
gen_0':s:true:false:minus:div:geq:if2_0(+(x, 1)) ⇔ s(gen_0':s:true:false:minus:div:geq:if2_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
a__minus(gen_0':s:true:false:minus:div:geq:if2_0(n4_0), gen_0':s:true:false:minus:div:geq:if2_0(n4_0)) → gen_0':s:true:false:minus:div:geq:if2_0(0), rt ∈ Ω(1 + n4₀)

(26) BOUNDS(n^1, INF)