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))
Renamed function symbols to avoid clashes with predefined symbol.
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))
Infered types.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
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'
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Could not prove a rewrite lemma for the defined symbol minus'.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Could not prove a rewrite lemma for the defined symbol geq'.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Proved the following rewrite lemma:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
Induction Base:
if'(_gen_0':mark':true':false':ok'3(+(1, 0)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c))
Induction Step:
if'(_gen_0':mark':true':false':ok'3(+(1, +(_$n30, 1))), _gen_0':mark':true':false':ok'3(_b1042), _gen_0':mark':true':false':ok'3(_c1043)) →RΩ(1)
mark'(if'(_gen_0':mark':true':false':ok'3(+(1, _$n30)), _gen_0':mark':true':false':ok'3(_b1042), _gen_0':mark':true':false':ok'3(_c1043))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Proved the following rewrite lemma:
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
Induction Base:
s'(_gen_0':mark':true':false':ok'3(+(1, 0)))
Induction Step:
s'(_gen_0':mark':true':false':ok'3(+(1, +(_$n2896, 1)))) →RΩ(1)
mark'(s'(_gen_0':mark':true':false':ok'3(+(1, _$n2896)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Proved the following rewrite lemma:
div'(_gen_0':mark':true':false':ok'3(+(1, _n4361)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n4361)
Induction Base:
div'(_gen_0':mark':true':false':ok'3(+(1, 0)), _gen_0':mark':true':false':ok'3(b))
Induction Step:
div'(_gen_0':mark':true':false':ok'3(+(1, +(_$n4362, 1))), _gen_0':mark':true':false':ok'3(_b5614)) →RΩ(1)
mark'(div'(_gen_0':mark':true':false':ok'3(+(1, _$n4362)), _gen_0':mark':true':false':ok'3(_b5614))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
div'(_gen_0':mark':true':false':ok'3(+(1, _n4361)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n4361)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(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'
Could not prove a rewrite lemma for the defined symbol active'.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
div'(_gen_0':mark':true':false':ok'3(+(1, _n4361)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n4361)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))
The following defined symbols remain to be analysed:
proper', top'
They will be analysed ascendingly in the following order:
proper' < top'
Could not prove a rewrite lemma for the defined symbol proper'.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
div'(_gen_0':mark':true':false':ok'3(+(1, _n4361)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n4361)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))
The following defined symbols remain to be analysed:
top'
Could not prove a rewrite lemma for the defined symbol top'.
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':ok'1 :: 0':mark':true':false':ok'
_hole_top'2 :: top'
_gen_0':mark':true':false':ok'3 :: Nat → 0':mark':true':false':ok'
Lemmas:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)
s'(_gen_0':mark':true':false':ok'3(+(1, _n2895))) → _*4, rt ∈ Ω(n2895)
div'(_gen_0':mark':true':false':ok'3(+(1, _n4361)), _gen_0':mark':true':false':ok'3(b)) → _*4, rt ∈ Ω(n4361)
Generator Equations:
_gen_0':mark':true':false':ok'3(0) ⇔ 0'
_gen_0':mark':true':false':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':true':false':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
if'(_gen_0':mark':true':false':ok'3(+(1, _n29)), _gen_0':mark':true':false':ok'3(b), _gen_0':mark':true':false':ok'3(c)) → _*4, rt ∈ Ω(n29)