active(from(X)) → mark(cons(X, from(s(X))))
active(after(0, XS)) → mark(XS)
active(after(s(N), cons(X, XS))) → mark(after(N, XS))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(after(X1, X2)) → after(active(X1), X2)
active(after(X1, X2)) → after(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
after(mark(X1), X2) → mark(after(X1, X2))
after(X1, mark(X2)) → mark(after(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(after(X1, X2)) → after(proper(X1), proper(X2))
proper(0) → ok(0)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
after(ok(X1), ok(X2)) → ok(after(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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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', from', s', after', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
after' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
after' < proper'
proper' < top'
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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', from', s', after', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
after' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
after' < 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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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:
from', active', s', after', proper', top'
They will be analysed ascendingly in the following order:
from' < active'
s' < active'
after' < active'
active' < top'
from' < proper'
s' < proper'
after' < proper'
proper' < top'
Proved the following rewrite lemma:
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
Induction Base:
from'(_gen_mark':0':ok'3(+(1, 0)))
Induction Step:
from'(_gen_mark':0':ok'3(+(1, +(_$n1641, 1)))) →RΩ(1)
mark'(from'(_gen_mark':0':ok'3(+(1, _$n1641)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
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', after', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
after' < active'
active' < top'
s' < proper'
after' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
Induction Base:
s'(_gen_mark':0':ok'3(+(1, 0)))
Induction Step:
s'(_gen_mark':0':ok'3(+(1, +(_$n2809, 1)))) →RΩ(1)
mark'(s'(_gen_mark':0':ok'3(+(1, _$n2809)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
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:
after', active', proper', top'
They will be analysed ascendingly in the following order:
after' < active'
active' < top'
after' < proper'
proper' < top'
Proved the following rewrite lemma:
after'(_gen_mark':0':ok'3(+(1, _n4100)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4100)
Induction Base:
after'(_gen_mark':0':ok'3(+(1, 0)), _gen_mark':0':ok'3(b))
Induction Step:
after'(_gen_mark':0':ok'3(+(1, +(_$n4101, 1))), _gen_mark':0':ok'3(_b5677)) →RΩ(1)
mark'(after'(_gen_mark':0':ok'3(+(1, _$n4101)), _gen_mark':0':ok'3(_b5677))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
after'(_gen_mark':0':ok'3(+(1, _n4100)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4100)
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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
after'(_gen_mark':0':ok'3(+(1, _n4100)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4100)
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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
after'(_gen_mark':0':ok'3(+(1, _n4100)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4100)
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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(after'(0', XS)) → mark'(XS)
active'(after'(s'(N), cons'(X, XS))) → mark'(after'(N, XS))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(after'(X1, X2)) → after'(active'(X1), X2)
active'(after'(X1, X2)) → after'(X1, active'(X2))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
after'(mark'(X1), X2) → mark'(after'(X1, X2))
after'(X1, mark'(X2)) → mark'(after'(X1, X2))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(after'(X1, X2)) → after'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
after'(ok'(X1), ok'(X2)) → ok'(after'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':0':ok' → mark':0':ok'
from' :: mark':0':ok' → mark':0':ok'
mark' :: mark':0':ok' → mark':0':ok'
cons' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
s' :: mark':0':ok' → mark':0':ok'
after' :: mark':0':ok' → mark':0':ok' → mark':0':ok'
0' :: 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)
from'(_gen_mark':0':ok'3(+(1, _n1640))) → _*4, rt ∈ Ω(n1640)
s'(_gen_mark':0':ok'3(+(1, _n2808))) → _*4, rt ∈ Ω(n2808)
after'(_gen_mark':0':ok'3(+(1, _n4100)), _gen_mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4100)
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)