active(2nd(cons1(X, cons(Y, Z)))) → mark(Y)
active(2nd(cons(X, X1))) → mark(2nd(cons1(X, X1)))
active(from(X)) → mark(cons(X, from(s(X))))
active(2nd(X)) → 2nd(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
active(s(X)) → s(active(X))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
2nd(mark(X)) → mark(2nd(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
s(mark(X)) → mark(s(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
proper(2nd(X)) → 2nd(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(s(X)) → s(proper(X))
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
2nd(ok(X)) → ok(2nd(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(X))
s(ok(X)) → ok(s(X))
cons1(ok(X1), ok(X2)) → ok(cons1(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'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Heuristically decided to analyse the following defined symbols:
active', 2nd', cons1', cons', from', s', proper', top'
They will be analysed ascendingly in the following order:
2nd' < active'
cons1' < active'
cons' < active'
from' < active'
s' < active'
active' < top'
2nd' < proper'
cons1' < proper'
cons' < proper'
from' < proper'
s' < proper'
proper' < top'
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
2nd', active', cons1', cons', from', s', proper', top'
They will be analysed ascendingly in the following order:
2nd' < active'
cons1' < active'
cons' < active'
from' < active'
s' < active'
active' < top'
2nd' < proper'
cons1' < proper'
cons' < proper'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Induction Base:
2nd'(_gen_mark':ok'3(+(1, 0)))
Induction Step:
2nd'(_gen_mark':ok'3(+(1, +(_$n6, 1)))) →RΩ(1)
mark'(2nd'(_gen_mark':ok'3(+(1, _$n6)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
cons1', active', cons', from', s', proper', top'
They will be analysed ascendingly in the following order:
cons1' < active'
cons' < active'
from' < active'
s' < active'
active' < top'
cons1' < proper'
cons' < proper'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
Induction Base:
cons1'(_gen_mark':ok'3(+(1, 0)), _gen_mark':ok'3(b))
Induction Step:
cons1'(_gen_mark':ok'3(+(1, +(_$n1076, 1))), _gen_mark':ok'3(_b2112)) →RΩ(1)
mark'(cons1'(_gen_mark':ok'3(+(1, _$n1076)), _gen_mark':ok'3(_b2112))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', from', s', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
Induction Base:
cons'(_gen_mark':ok'3(+(1, 0)), _gen_mark':ok'3(b))
Induction Step:
cons'(_gen_mark':ok'3(+(1, +(_$n3312, 1))), _gen_mark':ok'3(_b4456)) →RΩ(1)
mark'(cons'(_gen_mark':ok'3(+(1, _$n3312)), _gen_mark':ok'3(_b4456))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
from', active', s', proper', top'
They will be analysed ascendingly in the following order:
from' < active'
s' < active'
active' < top'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
Induction Base:
from'(_gen_mark':ok'3(+(1, 0)))
Induction Step:
from'(_gen_mark':ok'3(+(1, +(_$n5697, 1)))) →RΩ(1)
mark'(from'(_gen_mark':ok'3(+(1, _$n5697)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
s', active', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
active' < top'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_mark':ok'3(+(1, _n7272))) → _*4, rt ∈ Ω(n7272)
Induction Base:
s'(_gen_mark':ok'3(+(1, 0)))
Induction Step:
s'(_gen_mark':ok'3(+(1, +(_$n7273, 1)))) →RΩ(1)
mark'(s'(_gen_mark':ok'3(+(1, _$n7273)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
s'(_gen_mark':ok'3(+(1, _n7272))) → _*4, rt ∈ Ω(n7272)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
active', proper', top'
They will be analysed ascendingly in the following order:
active' < top'
proper' < top'
Could not prove a rewrite lemma for the defined symbol active'.
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
s'(_gen_mark':ok'3(+(1, _n7272))) → _*4, rt ∈ Ω(n7272)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
proper', top'
They will be analysed ascendingly in the following order:
proper' < top'
Could not prove a rewrite lemma for the defined symbol proper'.
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
s'(_gen_mark':ok'3(+(1, _n7272))) → _*4, rt ∈ Ω(n7272)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
The following defined symbols remain to be analysed:
top'
Could not prove a rewrite lemma for the defined symbol top'.
Rules:
active'(2nd'(cons1'(X, cons'(Y, Z)))) → mark'(Y)
active'(2nd'(cons'(X, X1))) → mark'(2nd'(cons1'(X, X1)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(2nd'(X)) → 2nd'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
active'(s'(X)) → s'(active'(X))
active'(cons1'(X1, X2)) → cons1'(active'(X1), X2)
active'(cons1'(X1, X2)) → cons1'(X1, active'(X2))
2nd'(mark'(X)) → mark'(2nd'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
s'(mark'(X)) → mark'(s'(X))
cons1'(mark'(X1), X2) → mark'(cons1'(X1, X2))
cons1'(X1, mark'(X2)) → mark'(cons1'(X1, X2))
proper'(2nd'(X)) → 2nd'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(s'(X)) → s'(proper'(X))
proper'(cons1'(X1, X2)) → cons1'(proper'(X1), proper'(X2))
2nd'(ok'(X)) → ok'(2nd'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
s'(ok'(X)) → ok'(s'(X))
cons1'(ok'(X1), ok'(X2)) → ok'(cons1'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':ok' → mark':ok'
2nd' :: mark':ok' → mark':ok'
cons1' :: mark':ok' → mark':ok' → mark':ok'
cons' :: mark':ok' → mark':ok' → mark':ok'
mark' :: mark':ok' → mark':ok'
from' :: mark':ok' → mark':ok'
s' :: mark':ok' → mark':ok'
proper' :: mark':ok' → mark':ok'
ok' :: mark':ok' → mark':ok'
top' :: mark':ok' → top'
_hole_mark':ok'1 :: mark':ok'
_hole_top'2 :: top'
_gen_mark':ok'3 :: Nat → mark':ok'
Lemmas:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
cons1'(_gen_mark':ok'3(+(1, _n1075)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n1075)
cons'(_gen_mark':ok'3(+(1, _n3311)), _gen_mark':ok'3(b)) → _*4, rt ∈ Ω(n3311)
from'(_gen_mark':ok'3(+(1, _n5696))) → _*4, rt ∈ Ω(n5696)
s'(_gen_mark':ok'3(+(1, _n7272))) → _*4, rt ∈ Ω(n7272)
Generator Equations:
_gen_mark':ok'3(0) ⇔ _hole_mark':ok'1
_gen_mark':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
2nd'(_gen_mark':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)