active(fst(0, Z)) → mark(nil)
active(fst(s(X), cons(Y, Z))) → mark(cons(Y, fst(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(len(nil)) → mark(0)
active(len(cons(X, Z))) → mark(s(len(Z)))
active(cons(X1, X2)) → cons(active(X1), X2)
active(fst(X1, X2)) → fst(active(X1), X2)
active(fst(X1, X2)) → fst(X1, active(X2))
active(from(X)) → from(active(X))
active(add(X1, X2)) → add(active(X1), X2)
active(add(X1, X2)) → add(X1, active(X2))
active(len(X)) → len(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
fst(mark(X1), X2) → mark(fst(X1, X2))
fst(X1, mark(X2)) → mark(fst(X1, X2))
from(mark(X)) → mark(from(X))
add(mark(X1), X2) → mark(add(X1, X2))
add(X1, mark(X2)) → mark(add(X1, X2))
len(mark(X)) → mark(len(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(fst(X1, X2)) → fst(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(add(X1, X2)) → add(proper(X1), proper(X2))
proper(len(X)) → len(proper(X))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
fst(ok(X1), ok(X2)) → ok(fst(X1, X2))
from(ok(X)) → ok(from(X))
add(ok(X1), ok(X2)) → ok(add(X1, X2))
len(ok(X)) → ok(len(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'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Infered types.
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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', fst', from', s', add', len', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
fst' < active'
from' < active'
s' < active'
add' < active'
len' < active'
active' < top'
cons' < proper'
fst' < proper'
from' < proper'
s' < proper'
add' < proper'
len' < proper'
proper' < top'
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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', fst', from', s', add', len', proper', top'
They will be analysed ascendingly in the following order:
cons' < active'
fst' < active'
from' < active'
s' < active'
add' < active'
len' < active'
active' < top'
cons' < proper'
fst' < proper'
from' < proper'
s' < proper'
add' < proper'
len' < 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'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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:
fst', active', from', s', add', len', proper', top'
They will be analysed ascendingly in the following order:
fst' < active'
from' < active'
s' < active'
add' < active'
len' < active'
active' < top'
fst' < proper'
from' < proper'
s' < proper'
add' < proper'
len' < proper'
proper' < top'
Proved the following rewrite lemma:
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
Induction Base:
fst'(_gen_0':nil':mark':ok'3(+(1, 0)), _gen_0':nil':mark':ok'3(b))
Induction Step:
fst'(_gen_0':nil':mark':ok'3(+(1, +(_$n2196, 1))), _gen_0':nil':mark':ok'3(_b3340)) →RΩ(1)
mark'(fst'(_gen_0':nil':mark':ok'3(+(1, _$n2196)), _gen_0':nil':mark':ok'3(_b3340))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
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', add', len', proper', top'
They will be analysed ascendingly in the following order:
from' < active'
s' < active'
add' < active'
len' < active'
active' < top'
from' < proper'
s' < proper'
add' < proper'
len' < proper'
proper' < top'
Proved the following rewrite lemma:
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
Induction Base:
from'(_gen_0':nil':mark':ok'3(+(1, 0)))
Induction Step:
from'(_gen_0':nil':mark':ok'3(+(1, +(_$n4973, 1)))) →RΩ(1)
mark'(from'(_gen_0':nil':mark':ok'3(+(1, _$n4973)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
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', add', len', proper', top'
They will be analysed ascendingly in the following order:
s' < active'
add' < active'
len' < active'
active' < top'
s' < proper'
add' < proper'
len' < proper'
proper' < top'
Could not prove a rewrite lemma for the defined symbol s'.
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
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:
add', active', len', proper', top'
They will be analysed ascendingly in the following order:
add' < active'
len' < active'
active' < top'
add' < proper'
len' < proper'
proper' < top'
Proved the following rewrite lemma:
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
Induction Base:
add'(_gen_0':nil':mark':ok'3(+(1, 0)), _gen_0':nil':mark':ok'3(b))
Induction Step:
add'(_gen_0':nil':mark':ok'3(+(1, +(_$n6727, 1))), _gen_0':nil':mark':ok'3(_b8411)) →RΩ(1)
mark'(add'(_gen_0':nil':mark':ok'3(+(1, _$n6727)), _gen_0':nil':mark':ok'3(_b8411))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
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:
len', active', proper', top'
They will be analysed ascendingly in the following order:
len' < active'
active' < top'
len' < proper'
proper' < top'
Proved the following rewrite lemma:
len'(_gen_0':nil':mark':ok'3(+(1, _n10112))) → _*4, rt ∈ Ω(n10112)
Induction Base:
len'(_gen_0':nil':mark':ok'3(+(1, 0)))
Induction Step:
len'(_gen_0':nil':mark':ok'3(+(1, +(_$n10113, 1)))) →RΩ(1)
mark'(len'(_gen_0':nil':mark':ok'3(+(1, _$n10113)))) →IH
mark'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
active'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
len'(_gen_0':nil':mark':ok'3(+(1, _n10112))) → _*4, rt ∈ Ω(n10112)
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'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
len'(_gen_0':nil':mark':ok'3(+(1, _n10112))) → _*4, rt ∈ Ω(n10112)
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'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
len'(_gen_0':nil':mark':ok'3(+(1, _n10112))) → _*4, rt ∈ Ω(n10112)
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'(fst'(0', Z)) → mark'(nil')
active'(fst'(s'(X), cons'(Y, Z))) → mark'(cons'(Y, fst'(X, Z)))
active'(from'(X)) → mark'(cons'(X, from'(s'(X))))
active'(add'(0', X)) → mark'(X)
active'(add'(s'(X), Y)) → mark'(s'(add'(X, Y)))
active'(len'(nil')) → mark'(0')
active'(len'(cons'(X, Z))) → mark'(s'(len'(Z)))
active'(cons'(X1, X2)) → cons'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(active'(X1), X2)
active'(fst'(X1, X2)) → fst'(X1, active'(X2))
active'(from'(X)) → from'(active'(X))
active'(add'(X1, X2)) → add'(active'(X1), X2)
active'(add'(X1, X2)) → add'(X1, active'(X2))
active'(len'(X)) → len'(active'(X))
cons'(mark'(X1), X2) → mark'(cons'(X1, X2))
fst'(mark'(X1), X2) → mark'(fst'(X1, X2))
fst'(X1, mark'(X2)) → mark'(fst'(X1, X2))
from'(mark'(X)) → mark'(from'(X))
add'(mark'(X1), X2) → mark'(add'(X1, X2))
add'(X1, mark'(X2)) → mark'(add'(X1, X2))
len'(mark'(X)) → mark'(len'(X))
proper'(0') → ok'(0')
proper'(s'(X)) → s'(proper'(X))
proper'(nil') → ok'(nil')
proper'(cons'(X1, X2)) → cons'(proper'(X1), proper'(X2))
proper'(fst'(X1, X2)) → fst'(proper'(X1), proper'(X2))
proper'(from'(X)) → from'(proper'(X))
proper'(add'(X1, X2)) → add'(proper'(X1), proper'(X2))
proper'(len'(X)) → len'(proper'(X))
s'(ok'(X)) → ok'(s'(X))
cons'(ok'(X1), ok'(X2)) → ok'(cons'(X1, X2))
fst'(ok'(X1), ok'(X2)) → ok'(fst'(X1, X2))
from'(ok'(X)) → ok'(from'(X))
add'(ok'(X1), ok'(X2)) → ok'(add'(X1, X2))
len'(ok'(X)) → ok'(len'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))
Types:
active' :: 0':nil':mark':ok' → 0':nil':mark':ok'
fst' :: 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'
add' :: 0':nil':mark':ok' → 0':nil':mark':ok' → 0':nil':mark':ok'
len' :: 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)
fst'(_gen_0':nil':mark':ok'3(+(1, _n2195)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n2195)
from'(_gen_0':nil':mark':ok'3(+(1, _n4972))) → _*4, rt ∈ Ω(n4972)
add'(_gen_0':nil':mark':ok'3(+(1, _n6726)), _gen_0':nil':mark':ok'3(b)) → _*4, rt ∈ Ω(n6726)
len'(_gen_0':nil':mark':ok'3(+(1, _n10112))) → _*4, rt ∈ Ω(n10112)
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)