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:

active(minus(0, Y)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0)) → mark(true)
active(geq(0, s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0, s(Y))) → mark(0)
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
s(mark(X)) →⁺ mark(s(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
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:

active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

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

Heuristically decided to analyse the following defined symbols:
active, minus, geq, if, s, div, proper, top

They will be analysed ascendingly in the following order:
minus < active
geq < active
if < active
s < active
div < active
active < top
minus < proper
geq < proper
if < proper
s < proper
div < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
minus, active, geq, if, s, div, proper, top

They will be analysed ascendingly in the following order:
minus < active
geq < active
if < active
s < active
div < active
active < top
minus < proper
geq < proper
if < proper
s < proper
div < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
geq, active, if, s, div, proper, top

They will be analysed ascendingly in the following order:
geq < active
if < active
s < active
div < active
active < top
geq < proper
if < proper
s < proper
div < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
if, active, s, div, proper, top

They will be analysed ascendingly in the following order:
if < active
s < active
div < active
active < top
if < proper
s < proper
div < proper
proper < top

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

Proved the following rewrite lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

Induction Base:
if(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))

Induction Step:
if(gen_0':mark:true:false:ok3_0(+(1, +(n19_0, 1))), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) →_R^Ω(1)
mark(if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))) →_IH
mark(*4_0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, div, proper, top

They will be analysed ascendingly in the following order:
s < active
div < active
active < top
s < proper
div < proper
proper < top

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)

Induction Base:
s(gen_0':mark:true:false:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:true:false:ok3_0(+(1, +(n1560_0, 1)))) →_R^Ω(1)
mark(s(gen_0':mark:true:false:ok3_0(+(1, n1560_0)))) →_IH
mark(*4_0)

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

(17) Complex Obligation (BEST)

(18) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
div, active, proper, top

They will be analysed ascendingly in the following order:
div < active
active < top
div < proper
proper < top

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Induction Base:
div(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))

Induction Step:
div(gen_0':mark:true:false:ok3_0(+(1, +(n2116_0, 1))), gen_0':mark:true:false:ok3_0(b)) →_R^Ω(1)
mark(div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b))) →_IH
mark(*4_0)

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

(20) Complex Obligation (BEST)

(21) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top

(22) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(23) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
proper, top

They will be analysed ascendingly in the following order:
proper < top

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

The following defined symbols remain to be analysed:
top

(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(27) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)
div(gen_0':mark:true:false:ok3_0(+(1, n2116_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n2116₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

(32) BOUNDS(n^1, INF)

(33) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)
s(gen_0':mark:true:false:ok3_0(+(1, n1560_0))) → *4_0, rt ∈ Ω(n1560₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

(35) BOUNDS(n^1, INF)

(36) Obligation:

TRS:
Rules:
active(minus(0', Y)) → mark(0')
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(geq(X, 0')) → mark(true)
active(geq(0', s(Y))) → mark(false)
active(geq(s(X), s(Y))) → mark(geq(X, Y))
active(div(0', s(Y))) → mark(0')
active(div(s(X), s(Y))) → mark(if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0'))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(s(X)) → s(active(X))
active(div(X1, X2)) → div(active(X1), X2)
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
s(mark(X)) → mark(s(X))
div(mark(X1), X2) → mark(div(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(geq(X1, X2)) → geq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(div(X1, X2)) → div(proper(X1), proper(X2))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
s(ok(X)) → ok(s(X))
geq(ok(X1), ok(X2)) → ok(geq(X1, X2))
div(ok(X1), ok(X2)) → ok(div(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
minus :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
geq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
div :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok

Lemmas:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
if(gen_0':mark:true:false:ok3_0(+(1, n19_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n19₀)

(38) BOUNDS(n^1, INF)