eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(nil) → 0
min(add(n, x)) → minIter(add(n, x), add(n, x), 0)
minIter(nil, add(n, y), m) → minIter(add(n, y), add(n, y), s(m))
minIter(add(n, x), y, m) → if_min(le(n, m), x, y, m)
if_min(true, x, y, m) → m
if_min(false, x, y, m) → minIter(x, y, m)
head(add(n, x)) → n
tail(add(n, x)) → x
tail(nil) → nil
null(nil) → true
null(add(n, x)) → false
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Infered types.
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Heuristically decided to analyse the following defined symbols:
eq', le', app', minIter', rm', minsort'
They will be analysed ascendingly in the following order:
eq' < rm'
eq' < minsort'
le' < minIter'
app' < minsort'
rm' < minsort'
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
eq', le', app', minIter', rm', minsort'
They will be analysed ascendingly in the following order:
eq' < rm'
eq' < minsort'
le' < minIter'
app' < minsort'
rm' < minsort'
Proved the following rewrite lemma:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
Induction Base:
eq'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'
Induction Step:
eq'(_gen_0':s'4(+(_$n8, 1)), _gen_0':s'4(+(_$n8, 1))) →RΩ(1)
eq'(_gen_0':s'4(_$n8), _gen_0':s'4(_$n8)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', 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':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
le', app', minIter', rm', minsort'
They will be analysed ascendingly in the following order:
le' < minIter'
app' < minsort'
rm' < minsort'
Proved the following rewrite lemma:
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
Induction Base:
le'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s'4(+(_$n1534, 1)), _gen_0':s'4(+(_$n1534, 1))) →RΩ(1)
le'(_gen_0':s'4(_$n1534), _gen_0':s'4(_$n1534)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
app', minIter', rm', minsort'
They will be analysed ascendingly in the following order:
app' < minsort'
rm' < minsort'
Proved the following rewrite lemma:
app'(_gen_nil':add'5(_n2977), _gen_nil':add'5(b)) → _gen_nil':add'5(+(_n2977, b)), rt ∈ Ω(1 + n2977)
Induction Base:
app'(_gen_nil':add'5(0), _gen_nil':add'5(b)) →RΩ(1)
_gen_nil':add'5(b)
Induction Step:
app'(_gen_nil':add'5(+(_$n2978, 1)), _gen_nil':add'5(_b3208)) →RΩ(1)
add'(0', app'(_gen_nil':add'5(_$n2978), _gen_nil':add'5(_b3208))) →IH
add'(0', _gen_nil':add'5(+(_$n2978, _b3208)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
app'(_gen_nil':add'5(_n2977), _gen_nil':add'5(b)) → _gen_nil':add'5(+(_n2977, b)), rt ∈ Ω(1 + n2977)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
minIter', rm', minsort'
They will be analysed ascendingly in the following order:
rm' < minsort'
Could not prove a rewrite lemma for the defined symbol minIter'.
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
app'(_gen_nil':add'5(_n2977), _gen_nil':add'5(b)) → _gen_nil':add'5(+(_n2977, b)), rt ∈ Ω(1 + n2977)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
rm', minsort'
They will be analysed ascendingly in the following order:
rm' < minsort'
Proved the following rewrite lemma:
rm'(_gen_0':s'4(0), _gen_nil':add'5(_n6014)) → _gen_nil':add'5(0), rt ∈ Ω(1 + n6014)
Induction Base:
rm'(_gen_0':s'4(0), _gen_nil':add'5(0)) →RΩ(1)
nil'
Induction Step:
rm'(_gen_0':s'4(0), _gen_nil':add'5(+(_$n6015, 1))) →RΩ(1)
if_rm'(eq'(_gen_0':s'4(0), 0'), _gen_0':s'4(0), add'(0', _gen_nil':add'5(_$n6015))) →LΩ(1)
if_rm'(true', _gen_0':s'4(0), add'(0', _gen_nil':add'5(_$n6015))) →RΩ(1)
rm'(_gen_0':s'4(0), _gen_nil':add'5(_$n6015)) →IH
_gen_nil':add'5(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
app'(_gen_nil':add'5(_n2977), _gen_nil':add'5(b)) → _gen_nil':add'5(+(_n2977, b)), rt ∈ Ω(1 + n2977)
rm'(_gen_0':s'4(0), _gen_nil':add'5(_n6014)) → _gen_nil':add'5(0), rt ∈ Ω(1 + n6014)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
The following defined symbols remain to be analysed:
minsort'
Could not prove a rewrite lemma for the defined symbol minsort'.
Rules:
eq'(0', 0') → true'
eq'(0', s'(x)) → false'
eq'(s'(x), 0') → false'
eq'(s'(x), s'(y)) → eq'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
app'(nil', y) → y
app'(add'(n, x), y) → add'(n, app'(x, y))
min'(nil') → 0'
min'(add'(n, x)) → minIter'(add'(n, x), add'(n, x), 0')
minIter'(nil', add'(n, y), m) → minIter'(add'(n, y), add'(n, y), s'(m))
minIter'(add'(n, x), y, m) → if_min'(le'(n, m), x, y, m)
if_min'(true', x, y, m) → m
if_min'(false', x, y, m) → minIter'(x, y, m)
head'(add'(n, x)) → n
tail'(add'(n, x)) → x
tail'(nil') → nil'
null'(nil') → true'
null'(add'(n, x)) → false'
rm'(n, nil') → nil'
rm'(n, add'(m, x)) → if_rm'(eq'(n, m), n, add'(m, x))
if_rm'(true', n, add'(m, x)) → rm'(n, x)
if_rm'(false', n, add'(m, x)) → add'(m, rm'(n, x))
minsort'(nil', nil') → nil'
minsort'(add'(n, x), y) → if_minsort'(eq'(n, min'(add'(n, x))), add'(n, x), y)
if_minsort'(true', add'(n, x), y) → add'(n, minsort'(app'(rm'(n, x), y), nil'))
if_minsort'(false', add'(n, x), y) → minsort'(x, add'(n, y))
Types:
eq' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
le' :: 0':s' → 0':s' → true':false'
app' :: nil':add' → nil':add' → nil':add'
nil' :: nil':add'
add' :: 0':s' → nil':add' → nil':add'
min' :: nil':add' → 0':s'
minIter' :: nil':add' → nil':add' → 0':s' → 0':s'
if_min' :: true':false' → nil':add' → nil':add' → 0':s' → 0':s'
head' :: nil':add' → 0':s'
tail' :: nil':add' → nil':add'
null' :: nil':add' → true':false'
rm' :: 0':s' → nil':add' → nil':add'
if_rm' :: true':false' → 0':s' → nil':add' → nil':add'
minsort' :: nil':add' → nil':add' → nil':add'
if_minsort' :: true':false' → nil':add' → nil':add' → nil':add'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_nil':add'3 :: nil':add'
_gen_0':s'4 :: Nat → 0':s'
_gen_nil':add'5 :: Nat → nil':add'
Lemmas:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
le'(_gen_0':s'4(_n1533), _gen_0':s'4(_n1533)) → true', rt ∈ Ω(1 + n1533)
app'(_gen_nil':add'5(_n2977), _gen_nil':add'5(b)) → _gen_nil':add'5(+(_n2977, b)), rt ∈ Ω(1 + n2977)
rm'(_gen_0':s'4(0), _gen_nil':add'5(_n6014)) → _gen_nil':add'5(0), rt ∈ Ω(1 + n6014)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_nil':add'5(0) ⇔ nil'
_gen_nil':add'5(+(x, 1)) ⇔ add'(0', _gen_nil':add'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
eq'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)