min(0, y) → 0
min(s(x), 0) → 0
min(s(x), s(y)) → min(x, y)
len(nil) → 0
len(cons(x, xs)) → s(len(xs))
sum(x, 0) → x
sum(x, s(y)) → s(sum(x, y))
le(0, x) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
take(0, cons(y, ys)) → y
take(s(x), cons(y, ys)) → take(x, ys)
addList(x, y) → if(le(0, min(len(x), len(y))), 0, x, y, nil)
if(false, c, x, y, z) → z
if(true, c, xs, ys, z) → if(le(s(c), min(len(xs), len(ys))), s(c), xs, ys, cons(sum(take(c, xs), take(c, ys)), z))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Infered types.
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Heuristically decided to analyse the following defined symbols:
min', len', sum', le', take', if'
They will be analysed ascendingly in the following order:
min' < if'
len' < if'
sum' < if'
le' < if'
take' < if'
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
min', len', sum', le', take', if'
They will be analysed ascendingly in the following order:
min' < if'
len' < if'
sum' < if'
le' < if'
take' < if'
Proved the following rewrite lemma:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
Induction Base:
min'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
0'
Induction Step:
min'(_gen_0':s'4(+(_$n8, 1)), _gen_0':s'4(+(_$n8, 1))) →RΩ(1)
min'(_gen_0':s'4(_$n8), _gen_0':s'4(_$n8)) →IH
_gen_0':s'4(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
len', sum', le', take', if'
They will be analysed ascendingly in the following order:
len' < if'
sum' < if'
le' < if'
take' < if'
Proved the following rewrite lemma:
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
Induction Base:
len'(_gen_nil':cons'5(0)) →RΩ(1)
0'
Induction Step:
len'(_gen_nil':cons'5(+(_$n1045, 1))) →RΩ(1)
s'(len'(_gen_nil':cons'5(_$n1045))) →IH
s'(_gen_0':s'4(_$n1045))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
sum', le', take', if'
They will be analysed ascendingly in the following order:
sum' < if'
le' < if'
take' < if'
Proved the following rewrite lemma:
sum'(_gen_0':s'4(a), _gen_0':s'4(_n1656)) → _gen_0':s'4(+(_n1656, a)), rt ∈ Ω(1 + n1656)
Induction Base:
sum'(_gen_0':s'4(a), _gen_0':s'4(0)) →RΩ(1)
_gen_0':s'4(a)
Induction Step:
sum'(_gen_0':s'4(_a1825), _gen_0':s'4(+(_$n1657, 1))) →RΩ(1)
s'(sum'(_gen_0':s'4(_a1825), _gen_0':s'4(_$n1657))) →IH
s'(_gen_0':s'4(+(_$n1657, _a1825)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
sum'(_gen_0':s'4(a), _gen_0':s'4(_n1656)) → _gen_0':s'4(+(_n1656, a)), rt ∈ Ω(1 + n1656)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
le', take', if'
They will be analysed ascendingly in the following order:
le' < if'
take' < if'
Proved the following rewrite lemma:
le'(_gen_0':s'4(_n2796), _gen_0':s'4(_n2796)) → true', rt ∈ Ω(1 + n2796)
Induction Base:
le'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s'4(+(_$n2797, 1)), _gen_0':s'4(+(_$n2797, 1))) →RΩ(1)
le'(_gen_0':s'4(_$n2797), _gen_0':s'4(_$n2797)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
sum'(_gen_0':s'4(a), _gen_0':s'4(_n1656)) → _gen_0':s'4(+(_n1656, a)), rt ∈ Ω(1 + n1656)
le'(_gen_0':s'4(_n2796), _gen_0':s'4(_n2796)) → true', rt ∈ Ω(1 + n2796)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
take', if'
They will be analysed ascendingly in the following order:
take' < if'
Proved the following rewrite lemma:
take'(_gen_0':s'4(_n3745), _gen_nil':cons'5(+(1, _n3745))) → _gen_0':s'4(0), rt ∈ Ω(1 + n3745)
Induction Base:
take'(_gen_0':s'4(0), _gen_nil':cons'5(+(1, 0))) →RΩ(1)
0'
Induction Step:
take'(_gen_0':s'4(+(_$n3746, 1)), _gen_nil':cons'5(+(1, +(_$n3746, 1)))) →RΩ(1)
take'(_gen_0':s'4(_$n3746), _gen_nil':cons'5(+(1, _$n3746))) →IH
_gen_0':s'4(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
sum'(_gen_0':s'4(a), _gen_0':s'4(_n1656)) → _gen_0':s'4(+(_n1656, a)), rt ∈ Ω(1 + n1656)
le'(_gen_0':s'4(_n2796), _gen_0':s'4(_n2796)) → true', rt ∈ Ω(1 + n2796)
take'(_gen_0':s'4(_n3745), _gen_nil':cons'5(+(1, _n3745))) → _gen_0':s'4(0), rt ∈ Ω(1 + n3745)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
The following defined symbols remain to be analysed:
if'
Could not prove a rewrite lemma for the defined symbol if'.
Rules:
min'(0', y) → 0'
min'(s'(x), 0') → 0'
min'(s'(x), s'(y)) → min'(x, y)
len'(nil') → 0'
len'(cons'(x, xs)) → s'(len'(xs))
sum'(x, 0') → x
sum'(x, s'(y)) → s'(sum'(x, y))
le'(0', x) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
take'(0', cons'(y, ys)) → y
take'(s'(x), cons'(y, ys)) → take'(x, ys)
addList'(x, y) → if'(le'(0', min'(len'(x), len'(y))), 0', x, y, nil')
if'(false', c, x, y, z) → z
if'(true', c, xs, ys, z) → if'(le'(s'(c), min'(len'(xs), len'(ys))), s'(c), xs, ys, cons'(sum'(take'(c, xs), take'(c, ys)), z))
Types:
min' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
len' :: nil':cons' → 0':s'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: 0':s' → 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
take' :: 0':s' → nil':cons' → 0':s'
addList' :: nil':cons' → nil':cons' → nil':cons'
if' :: true':false' → 0':s' → nil':cons' → nil':cons' → nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_hole_true':false'3 :: true':false'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':cons'5 :: Nat → nil':cons'
Lemmas:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)
len'(_gen_nil':cons'5(_n1044)) → _gen_0':s'4(_n1044), rt ∈ Ω(1 + n1044)
sum'(_gen_0':s'4(a), _gen_0':s'4(_n1656)) → _gen_0':s'4(+(_n1656, a)), rt ∈ Ω(1 + n1656)
le'(_gen_0':s'4(_n2796), _gen_0':s'4(_n2796)) → true', rt ∈ Ω(1 + n2796)
take'(_gen_0':s'4(_n3745), _gen_nil':cons'5(+(1, _n3745))) → _gen_0':s'4(0), rt ∈ Ω(1 + n3745)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':cons'5(0) ⇔ nil'
_gen_nil':cons'5(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
min'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → _gen_0':s'4(0), rt ∈ Ω(1 + n7)