active(zeros) → mark(cons(0, zeros))
active(and(tt, X)) → mark(X)
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(s(length(L)))
active(take(0, IL)) → mark(nil)
active(take(s(M), cons(N, IL))) → mark(cons(N, take(M, IL)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(and(X1, X2)) → and(active(X1), X2)
active(length(X)) → length(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
length(mark(X)) → mark(length(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
length(ok(X)) → ok(length(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(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'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Heuristically decided to analyse the following defined symbols:
active', cons', s', length', take', and', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
s' < active'
length' < active'
take' < active'
and' < active'
active' < top'
cons' < proper'
s' < proper'
length' < proper'
take' < proper'
and' < proper'
proper' < top'
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', s', length', take', and', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
s' < active'
length' < active'
take' < active'
and' < active'
active' < top'
cons' < proper'
s' < proper'
length' < proper'
take' < proper'
and' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))
Induction Step:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n6, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n6)), _gen_zeros':0':mark':tt':nil':ok'3(_b610))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
The following defined symbols remain to be analysed:
s', active', length', take', and', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
length' < active'
take' < active'
and' < active'
active' < top'
s' < proper'
length' < proper'
take' < proper'
and' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
Induction Base:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))
Induction Step:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1971, 1)))) →RΩ(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1971)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
The following defined symbols remain to be analysed:
length', active', take', and', proper', top'
They will be analysed ascendingly in the following order:
length' < active'
take' < active'
and' < active'
active' < top'
length' < proper'
take' < proper'
and' < proper'
proper' < top'
Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
Induction Base:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)))
Induction Step:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n3349, 1)))) →RΩ(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n3349)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
The following defined symbols remain to be analysed:
take', active', and', proper', top'
They will be analysed ascendingly in the following order:
take' < active'
and' < active'
active' < top'
take' < proper'
and' < proper'
proper' < top'
Proved the following rewrite lemma:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
Induction Base:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))
Induction Step:
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n4851, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b6427)) →RΩ(1)
mark'(take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n4851)), _gen_zeros':0':mark':tt':nil':ok'3(_b6427))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
The following defined symbols remain to be analysed:
and', active', proper', top'
They will be analysed ascendingly in the following order:
and' < active'
active' < top'
and' < proper'
proper' < top'
Proved the following rewrite lemma:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7884)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n7884)
Induction Base:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))
Induction Step:
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n7885, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b9569)) →RΩ(1)
mark'(and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n7885)), _gen_zeros':0':mark':tt':nil':ok'3(_b9569))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7884)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n7884)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':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'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7884)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n7884)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':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'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7884)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n7884)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':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'(zeros') → mark'(cons'(0', zeros'))
active'(and'(tt', X)) → mark'(X)
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(s'(length'(L)))
active'(take'(0', IL)) → mark'(nil')
active'(take'(s'(M), cons'(N, IL))) → mark'(cons'(N, take'(M, IL)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(and'(X1, X2)) → and'(active'(X1), X2)
active'(length'(X)) → length'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(take'(X1, X2)) → take'(active'(X1), X2)
active'(take'(X1, X2)) → take'(X1, active'(X2))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
length'(mark'(X)) → mark'(length'(X))
s'(mark'(X)) → mark'(s'(X))
take'(mark'(X1), X2) → mark'(take'(X1, X2))
take'(X1, mark'(X2)) → mark'(take'(X1, X2))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(take'(X1, X2)) → take'(proper'(X1), proper'(X2))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
length'(ok'(X)) → ok'(length'(X))
s'(ok'(X)) → ok'(s'(X))
take'(ok'(X1), ok'(X2)) → ok'(take'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
zeros' :: zeros':0':mark':tt':nil':ok'
mark' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
cons' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
0' :: zeros':0':mark':tt':nil':ok'
and' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
tt' :: zeros':0':mark':tt':nil':ok'
length' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
nil' :: zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
take' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
proper' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
ok' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
top' :: zeros':0':mark':tt':nil':ok' → top'
_hole_zeros':0':mark':tt':nil':ok'1 :: zeros':0':mark':tt':nil':ok'
_hole_top'2 :: top'
_gen_zeros':0':mark':tt':nil':ok'3 :: Nat → zeros':0':mark':tt':nil':ok'
Lemmas:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1970))) → _*4, rt ∈ Ω(n1970)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3348))) → _*4, rt ∈ Ω(n3348)
take'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n4850)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n4850)
and'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n7884)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n7884)
Generator Equations:
_gen_zeros':0':mark':tt':nil':ok'3(0) ⇔ zeros'
_gen_zeros':0':mark':tt':nil':ok'3(+(x, 1)) ⇔ mark'(_gen_zeros':0':mark':tt':nil':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n5)