active(f(0)) → mark(cons(0, f(s(0))))
active(f(s(0))) → mark(f(p(s(0))))
active(p(s(X))) → mark(X)
active(f(X)) → f(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(p(X)) → p(active(X))
f(mark(X)) → mark(f(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
p(mark(X)) → mark(p(X))
proper(f(X)) → f(proper(X))
proper(0) → ok(0)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(p(X)) → p(proper(X))
f(ok(X)) → ok(f(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
p(ok(X)) → ok(p(X))
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'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Heuristically decided to analyse the following defined symbols:
active', cons', f', s', p', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
f' < active'
s' < active'
p' < active'
active' < top'
cons' < proper'
f' < proper'
s' < proper'
p' < proper'
proper' < top'
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', f', s', p', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
f' < active'
s' < active'
p' < active'
active' < top'
cons' < proper'
f' < proper'
s' < proper'
p' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
cons'(_gen_0':mark':ok'3(+(1, 0)), _gen_0':mark':ok'3(b))
Induction Step:
cons'(_gen_0':mark':ok'3(+(1, +(_$n6, 1))), _gen_0':mark':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_0':mark':ok'3(+(1, _$n6)), _gen_0':mark':ok'3(_b610))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))
The following defined symbols remain to be analysed:
f', active', s', p', proper', top'
They will be analysed ascendingly in the following order:
f' < active'
s' < active'
p' < active'
active' < top'
f' < proper'
s' < proper'
p' < proper'
proper' < top'
Proved the following rewrite lemma:
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
Induction Base:
f'(_gen_0':mark':ok'3(+(1, 0)))
Induction Step:
f'(_gen_0':mark':ok'3(+(1, +(_$n1365, 1)))) →RΩ(1)
mark'(f'(_gen_0':mark':ok'3(+(1, _$n1365)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))
The following defined symbols remain to be analysed:
s', active', p', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
p' < active'
active' < top'
s' < proper'
p' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
Induction Base:
s'(_gen_0':mark':ok'3(+(1, 0)))
Induction Step:
s'(_gen_0':mark':ok'3(+(1, +(_$n2341, 1)))) →RΩ(1)
mark'(s'(_gen_0':mark':ok'3(+(1, _$n2341)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))
The following defined symbols remain to be analysed:
p', active', proper', top'
They will be analysed ascendingly in the following order:
p' < active'
active' < top'
p' < proper'
proper' < top'
Proved the following rewrite lemma:
p'(_gen_0':mark':ok'3(+(1, _n3440))) → _*4, rt ∈ Ω(n3440)
Induction Base:
p'(_gen_0':mark':ok'3(+(1, 0)))
Induction Step:
p'(_gen_0':mark':ok'3(+(1, +(_$n3441, 1)))) →RΩ(1)
mark'(p'(_gen_0':mark':ok'3(+(1, _$n3441)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(f'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
p'(_gen_0':mark':ok'3(+(1, _n3440))) → _*4, rt ∈ Ω(n3440)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':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'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
p'(_gen_0':mark':ok'3(+(1, _n3440))) → _*4, rt ∈ Ω(n3440)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':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'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
p'(_gen_0':mark':ok'3(+(1, _n3440))) → _*4, rt ∈ Ω(n3440)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':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'(0')) → mark'(cons'(0', f'(s'(0'))))
active'(f'(s'(0'))) → mark'(f'(p'(s'(0'))))
active'(p'(s'(X))) → mark'(X)
active'(f'(X)) → f'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(p'(X)) → p'(active'(X))
f'(mark'(X)) → mark'(f'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
p'(mark'(X)) → mark'(p'(X))
proper'(f'(X)) → f'(proper'(X))
proper'(0') → ok'(0')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(p'(X)) → p'(proper'(X))
f'(ok'(X)) → ok'(f'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
p'(ok'(X)) → ok'(p'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':mark':ok' → 0':mark':ok'
f' :: 0':mark':ok' → 0':mark':ok'
0' :: 0':mark':ok'
mark' :: 0':mark':ok' → 0':mark':ok'
cons' :: 0':mark':ok' → 0':mark':ok' → 0':mark':ok'
s' :: 0':mark':ok' → 0':mark':ok'
p' :: 0':mark':ok' → 0':mark':ok'
proper' :: 0':mark':ok' → 0':mark':ok'
ok' :: 0':mark':ok' → 0':mark':ok'
top' :: 0':mark':ok' → top'
_hole_0':mark':ok'1 :: 0':mark':ok'
_hole_top'2 :: top'
_gen_0':mark':ok'3 :: Nat → 0':mark':ok'
Lemmas:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
f'(_gen_0':mark':ok'3(+(1, _n1364))) → _*4, rt ∈ Ω(n1364)
s'(_gen_0':mark':ok'3(+(1, _n2340))) → _*4, rt ∈ Ω(n2340)
p'(_gen_0':mark':ok'3(+(1, _n3440))) → _*4, rt ∈ Ω(n3440)
Generator Equations:
_gen_0':mark':ok'3(0) ⇔ 0'
_gen_0':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':mark':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_0':mark':ok'3(+(1, _n5)), _gen_0':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)