active(f(X)) → mark(cons(X, f(g(X))))
active(g(0)) → mark(s(0))
active(g(s(X))) → mark(s(s(g(X))))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(g(X)) → g(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
g(mark(X)) → mark(g(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
proper(f(X)) → f(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(g(X)) → g(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
g(ok(X)) → ok(g(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
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'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Heuristically decided to analyse the following defined symbols:
active', cons', f', g', s', sel', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
f' < active'
g' < active'
s' < active'
sel' < active'
active' < top'
cons' < proper'
f' < proper'
g' < proper'
s' < proper'
sel' < proper'
proper' < top'
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', f', g', s', sel', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
f' < active'
g' < active'
s' < active'
sel' < active'
active' < top'
cons' < proper'
f' < proper'
g' < proper'
s' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
cons'(_gen_mark':0':ok'3(+(1, 0)), _gen_mark':0':ok'3(b))
Induction Step:
cons'(_gen_mark':0':ok'3(+(1, +(_$n6, 1))), _gen_mark':0':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_mark':0':ok'3(+(1, _$n6)), _gen_mark':0':ok'3(_b610))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
The following defined symbols remain to be analysed:
f', active', g', s', sel', proper', top'
They will be analysed ascendingly in the following order:
f' < active'
g' < active'
s' < active'
sel' < active'
active' < top'
f' < proper'
g' < proper'
s' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
Induction Base:
f'(_gen_mark':0':ok'3(+(1, 0)))
Induction Step:
f'(_gen_mark':0':ok'3(+(1, +(_$n1839, 1)))) →RΩ(1)
mark'(f'(_gen_mark':0':ok'3(+(1, _$n1839)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
The following defined symbols remain to be analysed:
g', active', s', sel', proper', top'
They will be analysed ascendingly in the following order:
g' < active'
s' < active'
sel' < active'
active' < top'
g' < proper'
s' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
Induction Base:
g'(_gen_mark':0':ok'3(+(1, 0)))
Induction Step:
g'(_gen_mark':0':ok'3(+(1, +(_$n3133, 1)))) →RΩ(1)
mark'(g'(_gen_mark':0':ok'3(+(1, _$n3133)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
The following defined symbols remain to be analysed:
s', active', sel', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
sel' < active'
active' < top'
s' < proper'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
Induction Base:
s'(_gen_mark':0':ok'3(+(1, 0)))
Induction Step:
s'(_gen_mark':0':ok'3(+(1, +(_$n4551, 1)))) →RΩ(1)
mark'(s'(_gen_mark':0':ok'3(+(1, _$n4551)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
The following defined symbols remain to be analysed:
sel', active', proper', top'
They will be analysed ascendingly in the following order:
sel' < active'
active' < top'
sel' < proper'
proper' < top'
Proved the following rewrite lemma:
sel'(_gen_mark':0':ok'3(+(1, _n6092)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n6092)
Induction Base:
sel'(_gen_mark':0':ok'3(+(1, 0)), _gen_mark':0':ok'3(b))
Induction Step:
sel'(_gen_mark':0':ok'3(+(1, +(_$n6093, 1))), _gen_mark':0':ok'3(_b7885)) →RΩ(1)
mark'(sel'(_gen_mark':0':ok'3(+(1, _$n6093)), _gen_mark':0':ok'3(_b7885))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
sel'(_gen_mark':0':ok'3(+(1, _n6092)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n6092)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
sel'(_gen_mark':0':ok'3(+(1, _n6092)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n6092)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
sel'(_gen_mark':0':ok'3(+(1, _n6092)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n6092)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':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'(f'(X)) → mark'(cons'(X, f'(g'(X))))
active'(g'(0')) → mark'(s'(0'))
active'(g'(s'(X))) → mark'(s'(s'(g'(X))))
active'(sel'(0', cons'(X, Y))) → mark'(X)
active'(sel'(s'(X), cons'(Y, Z))) → mark'(sel'(X, Z))
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(g'(X)) → g'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(sel'(X1, X2)) → sel'(active'(X1), X2)
active'(sel'(X1, X2)) → sel'(X1, active'(X2))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
g'(mark'(X)) → mark'(g'(X))
s'(mark'(X)) → mark'(s'(X))
sel'(mark'(X1), X2) → mark'(sel'(X1, X2))
sel'(X1, mark'(X2)) → mark'(sel'(X1, X2))
proper'(f'(X)) → f'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(g'(X)) → g'(proper'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(sel'(X1, X2)) → sel'(proper'(X1), proper'(X2))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
g'(ok'(X)) → ok'(g'(X))
s'(ok'(X)) → ok'(s'(X))
sel'(ok'(X1), ok'(X2)) → ok'(sel'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
f' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
g' :: mark':0':ok' → mark':0':ok'
0' :: mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
sel' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
proper' :: mark':0':ok' → mark':0':ok'
ok' :: mark':0':ok' → mark':0':ok'
top' :: mark':0':ok' → top'
_hole_mark':0':ok'1 :: mark':0':ok'
_hole_top'2 :: top'
_gen_mark':0':ok'3 :: Nat → mark':0':ok'
Lemmas:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_mark':0':ok'3(+(1, _n1838))) → _*4, rt ∈ Ω(n1838)
g'(_gen_mark':0':ok'3(+(1, _n3132))) → _*4, rt ∈ Ω(n3132)
s'(_gen_mark':0':ok'3(+(1, _n4550))) → _*4, rt ∈ Ω(n4550)
sel'(_gen_mark':0':ok'3(+(1, _n6092)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n6092)
Generator Equations:
_gen_mark':0':ok'3(0) ⇔ 0'
_gen_mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':0':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_mark':0':ok'3(+(1, _n5)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n5)