app(x, y) → helpa(0, plus(length(x), length(y)), x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
length(nil) → 0
length(cons(x, y)) → s(length(y))
helpa(c, l, ys, zs) → if(ge(c, l), c, l, ys, zs)
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
if(true, c, l, ys, zs) → nil
if(false, c, l, ys, zs) → helpb(c, l, greater(ys, zs), smaller(ys, zs))
greater(ys, zs) → helpc(ge(length(ys), length(zs)), ys, zs)
smaller(ys, zs) → helpc(ge(length(ys), length(zs)), zs, ys)
helpc(true, ys, zs) → ys
helpc(false, ys, zs) → zs
helpb(c, l, cons(y, ys), zs) → cons(y, helpa(s(c), l, ys, zs))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(x, y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(y, ys), zs) → cons'(y, helpa'(s'(c), l, ys, zs))
Sliced the following arguments:
cons'/0
Runtime Complexity TRS:
The TRS R consists of the following rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Infered types.
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
helpa', plus', length', ge', helpb'
They will be analysed ascendingly in the following order:
ge' < helpa'
helpa' = helpb'
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
The following defined symbols remain to be analysed:
plus', helpa', length', ge', helpb'
They will be analysed ascendingly in the following order:
ge' < helpa'
helpa' = helpb'
Proved the following rewrite lemma:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
Induction Base:
plus'(_gen_0':s'5(a), _gen_0':s'5(0)) →RΩ(1)
_gen_0':s'5(a)
Induction Step:
plus'(_gen_0':s'5(_a230), _gen_0':s'5(+(_$n8, 1))) →RΩ(1)
s'(plus'(_gen_0':s'5(_a230), _gen_0':s'5(_$n8))) →IH
s'(_gen_0':s'5(+(_$n8, _a230)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
The following defined symbols remain to be analysed:
length', helpa', ge', helpb'
They will be analysed ascendingly in the following order:
ge' < helpa'
helpa' = helpb'
Proved the following rewrite lemma:
length'(_gen_nil':cons'4(_n1242)) → _gen_0':s'5(_n1242), rt ∈ Ω(1 + n1242)
Induction Base:
length'(_gen_nil':cons'4(0)) →RΩ(1)
0'
Induction Step:
length'(_gen_nil':cons'4(+(_$n1243, 1))) →RΩ(1)
s'(length'(_gen_nil':cons'4(_$n1243))) →IH
s'(_gen_0':s'5(_$n1243))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
length'(_gen_nil':cons'4(_n1242)) → _gen_0':s'5(_n1242), rt ∈ Ω(1 + n1242)
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
The following defined symbols remain to be analysed:
ge', helpa', helpb'
They will be analysed ascendingly in the following order:
ge' < helpa'
helpa' = helpb'
Proved the following rewrite lemma:
ge'(_gen_0':s'5(_n1990), _gen_0':s'5(_n1990)) → true', rt ∈ Ω(1 + n1990)
Induction Base:
ge'(_gen_0':s'5(0), _gen_0':s'5(0)) →RΩ(1)
true'
Induction Step:
ge'(_gen_0':s'5(+(_$n1991, 1)), _gen_0':s'5(+(_$n1991, 1))) →RΩ(1)
ge'(_gen_0':s'5(_$n1991), _gen_0':s'5(_$n1991)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
length'(_gen_nil':cons'4(_n1242)) → _gen_0':s'5(_n1242), rt ∈ Ω(1 + n1242)
ge'(_gen_0':s'5(_n1990), _gen_0':s'5(_n1990)) → true', rt ∈ Ω(1 + n1990)
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
The following defined symbols remain to be analysed:
helpb', helpa'
They will be analysed ascendingly in the following order:
helpa' = helpb'
Could not prove a rewrite lemma for the defined symbol helpb'.
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
length'(_gen_nil':cons'4(_n1242)) → _gen_0':s'5(_n1242), rt ∈ Ω(1 + n1242)
ge'(_gen_0':s'5(_n1990), _gen_0':s'5(_n1990)) → true', rt ∈ Ω(1 + n1990)
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
The following defined symbols remain to be analysed:
helpa'
They will be analysed ascendingly in the following order:
helpa' = helpb'
Could not prove a rewrite lemma for the defined symbol helpa'.
Rules:
app'(x, y) → helpa'(0', plus'(length'(x), length'(y)), x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
length'(nil') → 0'
length'(cons'(y)) → s'(length'(y))
helpa'(c, l, ys, zs) → if'(ge'(c, l), c, l, ys, zs)
ge'(x, 0') → true'
ge'(0', s'(x)) → false'
ge'(s'(x), s'(y)) → ge'(x, y)
if'(true', c, l, ys, zs) → nil'
if'(false', c, l, ys, zs) → helpb'(c, l, greater'(ys, zs), smaller'(ys, zs))
greater'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), ys, zs)
smaller'(ys, zs) → helpc'(ge'(length'(ys), length'(zs)), zs, ys)
helpc'(true', ys, zs) → ys
helpc'(false', ys, zs) → zs
helpb'(c, l, cons'(ys), zs) → cons'(helpa'(s'(c), l, ys, zs))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
helpa' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
0' :: 0':s'
plus' :: 0':s' → 0':s' → 0':s'
length' :: nil':cons' → 0':s'
s' :: 0':s' → 0':s'
nil' :: nil':cons'
cons' :: nil':cons' → nil':cons'
if' :: true':false' → 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
ge' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
helpb' :: 0':s' → 0':s' → nil':cons' → nil':cons' → nil':cons'
greater' :: nil':cons' → nil':cons' → nil':cons'
smaller' :: nil':cons' → nil':cons' → nil':cons'
helpc' :: true':false' → nil':cons' → nil':cons' → nil':cons'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_hole_true':false'3 :: true':false'
_gen_nil':cons'4 :: Nat → nil':cons'
_gen_0':s'5 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)
length'(_gen_nil':cons'4(_n1242)) → _gen_0':s'5(_n1242), rt ∈ Ω(1 + n1242)
ge'(_gen_0':s'5(_n1990), _gen_0':s'5(_n1990)) → true', rt ∈ Ω(1 + n1990)
Generator Equations:
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(_gen_nil':cons'4(x))
_gen_0':s'5(0) ⇔ 0'
_gen_0':s'5(+(x, 1)) ⇔ s'(_gen_0':s'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
plus'(_gen_0':s'5(a), _gen_0':s'5(_n7)) → _gen_0':s'5(+(_n7, a)), rt ∈ Ω(1 + n7)