active(first(0, X)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(from(X)) → from(active(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
from(mark(X)) → mark(from(X))
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(ok(X)) → ok(from(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'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Heuristically decided to analyse the following defined symbols:
active', cons', first', from', s', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
first' < active'
from' < active'
s' < active'
active' < top'
cons' < proper'
first' < proper'
from' < proper'
s' < proper'
proper' < top'
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':mark':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', first', from', s', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
first' < active'
from' < active'
s' < active'
active' < top'
cons' < proper'
first' < proper'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
cons'(_gen_0':nil':mark':ok'3(+(1, 0)), _gen_0':nil':mark':ok'3(b))
Induction Step:
cons'(_gen_0':nil':mark':ok'3(+(1, +(_$n6, 1))), _gen_0':nil':mark':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_0':nil':mark':ok'3(+(1, _$n6)), _gen_0':nil':mark':ok'3(_b610))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':mark':ok'3(x))
The following defined symbols remain to be analysed:
first', active', from', s', proper', top'
They will be analysed ascendingly in the following order:
first' < active'
from' < active'
s' < active'
active' < top'
first' < proper'
from' < proper'
s' < proper'
proper' < top'
Proved the following rewrite lemma:
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
Induction Base:
first'(_gen_0':nil':mark':ok'3(+(1, 0)), _gen_0':nil':mark':ok'3(b))
Induction Step:
first'(_gen_0':nil':mark':ok'3(+(1, +(_$n1653, 1))), _gen_0':nil':mark':ok'3(_b2797)) →RΩ(1)
mark'(first'(_gen_0':nil':mark':ok'3(+(1, _$n1653)), _gen_0':nil':mark':ok'3(_b2797))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':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_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
Induction Base:
from'(_gen_0':nil':mark':ok'3(+(1, 0)))
Induction Step:
from'(_gen_0':nil':mark':ok'3(+(1, +(_$n3887, 1)))) →RΩ(1)
mark'(from'(_gen_0':nil':mark':ok'3(+(1, _$n3887)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
from'(_gen_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':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_0':nil':mark':ok'3(+(1, _n5251))) → _*4, rt ∈ Ω(n5251)
Induction Base:
s'(_gen_0':nil':mark':ok'3(+(1, 0)))
Induction Step:
s'(_gen_0':nil':mark':ok'3(+(1, +(_$n5252, 1)))) →RΩ(1)
mark'(s'(_gen_0':nil':mark':ok'3(+(1, _$n5252)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
from'(_gen_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
s'(_gen_0':nil':mark':ok'3(+(1, _n5251))) → _*4, rt ∈ Ω(n5251)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':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'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
from'(_gen_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
s'(_gen_0':nil':mark':ok'3(+(1, _n5251))) → _*4, rt ∈ Ω(n5251)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':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'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
from'(_gen_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
s'(_gen_0':nil':mark':ok'3(+(1, _n5251))) → _*4, rt ∈ Ω(n5251)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':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'(first'(0', X)) → mark'(nil')
active'(first'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, first'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(first'(X1, X2)) → first'(active'(X1), X2)
active'(first'(X1, X2)) → first'(X1, active'(X2))
active'(s'(X)) → s'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(from'(X)) → from'(active'(X))
first'(mark'(X1), X2) → mark'(first'(X1, X2))
first'(X1, mark'(X2)) → mark'(first'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
proper'(first'(X1, X2)) → first'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(nil') → ok'(nil')
proper'(s'(X)) → s'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
first'(ok'(X1), ok'(X2)) → ok'(first'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
first' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
0' :: 0':nil':mark':ok'
mark' :: 0':nil':mark':ok' → 0':nil':mark':ok'
nil' :: 0':nil':mark':ok'
s' :: 0':nil':mark':ok' → 0':nil':mark':ok'
cons' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
from' :: 0':nil':mark':ok' → 0':nil':mark':ok'
proper' :: 0':nil':mark':ok' → 0':nil':mark':ok'
ok' :: 0':nil':mark':ok' → 0':nil':mark':ok'
top' :: 0':nil':mark':ok' → top'
_hole_0':nil':mark':ok'1 :: 0':nil':mark':ok'
_hole_top'2 :: top'
_gen_0':nil':mark':ok'3 :: Nat → 0':nil':mark':ok'
Lemmas:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)
first'(_gen_0':nil':mark':ok'3(+(1, _n1652)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n1652)
from'(_gen_0':nil':mark':ok'3(+(1, _n3886))) → _*4, rt ∈ Ω(n3886)
s'(_gen_0':nil':mark':ok'3(+(1, _n5251))) → _*4, rt ∈ Ω(n5251)
Generator Equations:
_gen_0':nil':mark':ok'3(0) ⇔ 0'
_gen_0':nil':mark':ok'3(+(x, 1)) ⇔ mark'(_gen_0':nil':mark':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_0':nil':mark':ok'3(+(1, _n5)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n5)