active(zeros) → mark(cons(0, zeros))
active(U11(tt, L)) → mark(U12(tt, L))
active(U12(tt, L)) → mark(s(length(L)))
active(length(nil)) → mark(0)
active(length(cons(N, L))) → mark(U11(tt, L))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X1, X2)) → U12(active(X1), X2)
active(s(X)) → s(active(X))
active(length(X)) → length(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X1), X2) → mark(U12(X1, X2))
s(mark(X)) → mark(s(X))
length(mark(X)) → mark(length(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(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'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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', U12', s', length', U11', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
U12' < active'
s' < active'
length' < active'
U11' < active'
active' < top'
cons' < proper'
U12' < proper'
s' < proper'
length' < proper'
U11' < proper'
proper' < top'
Rules:
active'(zeros') → mark'(cons'(0', zeros'))
active'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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', U12', s', length', U11', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
U12' < active'
s' < active'
length' < active'
U11' < active'
active' < top'
cons' < proper'
U12' < proper'
s' < proper'
length' < proper'
U11' < 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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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:
U12', active', s', length', U11', proper', top'
They will be analysed ascendingly in the following order:
U12' < active'
s' < active'
length' < active'
U11' < active'
active' < top'
U12' < proper'
s' < proper'
length' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
Induction Base:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))
Induction Step:
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n1788, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b2716)) →RΩ(1)
mark'(U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n1788)), _gen_zeros':0':mark':tt':nil':ok'3(_b2716))) →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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
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', U11', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
length' < active'
U11' < active'
active' < top'
s' < proper'
length' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
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, +(_$n3938, 1)))) →RΩ(1)
mark'(s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n3938)))) →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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
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', U11', proper', top'
They will be analysed ascendingly in the following order:
length' < active'
U11' < active'
active' < top'
length' < proper'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
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, +(_$n5384, 1)))) →RΩ(1)
mark'(length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n5384)))) →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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
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:
U11', active', proper', top'
They will be analysed ascendingly in the following order:
U11' < active'
active' < top'
U11' < proper'
proper' < top'
Proved the following rewrite lemma:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n6953)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n6953)
Induction Base:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, 0)), _gen_zeros':0':mark':tt':nil':ok'3(b))
Induction Step:
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, +(_$n6954, 1))), _gen_zeros':0':mark':tt':nil':ok'3(_b8638)) →RΩ(1)
mark'(U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _$n6954)), _gen_zeros':0':mark':tt':nil':ok'3(_b8638))) →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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n6953)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n6953)
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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n6953)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n6953)
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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n6953)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n6953)
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'(U11'(tt', L)) → mark'(U12'(tt', L))
active'(U12'(tt', L)) → mark'(s'(length'(L)))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(N, L))) → mark'(U11'(tt', L))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
active'(length'(X)) → length'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
length'(mark'(X)) → mark'(length'(X))
proper'(zeros') → ok'(zeros')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
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'
U11' :: 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'
U12' :: zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok' → zeros':0':mark':tt':nil':ok'
s' :: zeros':0':mark':tt':nil':ok' → 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'
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)
U12'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n1787)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n1787)
s'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n3937))) → _*4, rt ∈ Ω(n3937)
length'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n5383))) → _*4, rt ∈ Ω(n5383)
U11'(_gen_zeros':0':mark':tt':nil':ok'3(+(1, _n6953)), _gen_zeros':0':mark':tt':nil':ok'3(b)) → _*4, rt ∈ Ω(n6953)
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)