active(from(X)) → mark(cons(X, from(s(X))))
active(length(nil)) → mark(0)
active(length(cons(X, Y))) → mark(s(length1(Y)))
active(length1(X)) → mark(length(X))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(length(X)) → length(proper(X))
proper(nil) → ok(nil)
proper(0) → ok(0)
proper(length1(X)) → length1(proper(X))
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
length(ok(X)) → ok(length(X))
length1(ok(X)) → ok(length1(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'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Heuristically decided to analyse the following defined symbols:
active', cons', from', s', length1', length', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
length1' < active'
length' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
length1' < proper'
length' < proper'
proper' < top'
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
cons', active', from', s', length1', length', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
from' < active'
s' < active'
length1' < active'
length' < active'
active' < top'
cons' < proper'
from' < proper'
s' < proper'
length1' < proper'
length' < proper'
proper' < top'
Proved the following rewrite lemma:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
cons'(_gen_mark':nil':0':ok'3(+(1, 0)), _gen_mark':nil':0':ok'3(b))
Induction Step:
cons'(_gen_mark':nil':0':ok'3(+(1, +(_$n6, 1))), _gen_mark':nil':0':ok'3(_b610)) →RΩ(1)
mark'(cons'(_gen_mark':nil':0':ok'3(+(1, _$n6)), _gen_mark':nil':0':ok'3(_b610))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
from', active', s', length1', length', proper', top'
They will be analysed ascendingly in the following order:
from' < active'
s' < active'
length1' < active'
length' < active'
active' < top'
from' < proper'
s' < proper'
length1' < proper'
length' < proper'
proper' < top'
Proved the following rewrite lemma:
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
Induction Base:
from'(_gen_mark':nil':0':ok'3(+(1, 0)))
Induction Step:
from'(_gen_mark':nil':0':ok'3(+(1, +(_$n1452, 1)))) →RΩ(1)
mark'(from'(_gen_mark':nil':0':ok'3(+(1, _$n1452)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
s', active', length1', length', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
length1' < active'
length' < active'
active' < top'
s' < proper'
length1' < proper'
length' < proper'
proper' < top'
Proved the following rewrite lemma:
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Induction Base:
s'(_gen_mark':nil':0':ok'3(+(1, 0)))
Induction Step:
s'(_gen_mark':nil':0':ok'3(+(1, +(_$n2485, 1)))) →RΩ(1)
mark'(s'(_gen_mark':nil':0':ok'3(+(1, _$n2485)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
length1', active', length', proper', top'
They will be analysed ascendingly in the following order:
length1' < active'
length' < active'
active' < top'
length1' < proper'
length' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol length1'.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
length', active', proper', top'
They will be analysed ascendingly in the following order:
length' < active'
active' < top'
length' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol length'.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
active', proper', top'
They will be analysed ascendingly in the following order:
active' < top'
proper' < top'
Could not prove a rewrite lemma for the defined symbol active'.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
proper', top'
They will be analysed ascendingly in the following order:
proper' < top'
Could not prove a rewrite lemma for the defined symbol proper'.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
The following defined symbols remain to be analysed:
top'
Could not prove a rewrite lemma for the defined symbol top'.
Rules:
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(length'(nil')) → mark'(0')
active'(length'(cons'(X, Y))) → mark'(s'(length1'(Y)))
active'(length1'(X)) → mark'(length'(X))
active'(from'(X)) → from'(active'(X))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(s'(X)) → s'(active'(X))
from'(mark'(X)) → mark'(from'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
s'(mark'(X)) → mark'(s'(X))
proper'(from'(X)) → from'(proper'(X))
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(s'(X)) → s'(proper'(X))
proper'(length'(X)) → length'(proper'(X))
proper'(nil') → ok'(nil')
proper'(0') → ok'(0')
proper'(length1'(X)) → length1'(proper'(X))
from'(ok'(X)) → ok'(from'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
s'(ok'(X)) → ok'(s'(X))
length'(ok'(X)) → ok'(length'(X))
length1'(ok'(X)) → ok'(length1'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: mark':nil':0':ok' → mark':nil':0':ok'
from' :: mark':nil':0':ok' → mark':nil':0':ok'
mark' :: mark':nil':0':ok' → mark':nil':0':ok'
cons' :: mark':nil':0':ok' → mark':nil':0':ok' → mark':nil':0':ok'
s' :: mark':nil':0':ok' → mark':nil':0':ok'
length' :: mark':nil':0':ok' → mark':nil':0':ok'
nil' :: mark':nil':0':ok'
0' :: mark':nil':0':ok'
length1' :: mark':nil':0':ok' → mark':nil':0':ok'
proper' :: mark':nil':0':ok' → mark':nil':0':ok'
ok' :: mark':nil':0':ok' → mark':nil':0':ok'
top' :: mark':nil':0':ok' → top'
_hole_mark':nil':0':ok'1 :: mark':nil':0':ok'
_hole_top'2 :: top'
_gen_mark':nil':0':ok'3 :: Nat → mark':nil':0':ok'
Lemmas:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)
from'(_gen_mark':nil':0':ok'3(+(1, _n1451))) → _*4, rt ∈ Ω(n1451)
s'(_gen_mark':nil':0':ok'3(+(1, _n2484))) → _*4, rt ∈ Ω(n2484)
Generator Equations:
_gen_mark':nil':0':ok'3(0) ⇔ nil'
_gen_mark':nil':0':ok'3(+(x, 1)) ⇔ mark'(_gen_mark':nil':0':ok'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
cons'(_gen_mark':nil':0':ok'3(+(1, _n5)), _gen_mark':nil':0':ok'3(b)) → _*4, rt ∈ Ω(n5)