minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Infered types.
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Heuristically decided to analyse the following defined symbols:
minus', quot', plus', app', sum'
They will be analysed ascendingly in the following order:
minus' < quot'
plus' < minus'
plus' < sum'
app' < sum'
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
The following defined symbols remain to be analysed:
plus', minus', quot', app', sum'
They will be analysed ascendingly in the following order:
minus' < quot'
plus' < minus'
plus' < sum'
app' < sum'
Proved the following rewrite lemma:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
Induction Base:
plus'(_gen_0':s'3(0), _gen_0':s'3(b)) →RΩ(1)
_gen_0':s'3(b)
Induction Step:
plus'(_gen_0':s'3(+(_$n7, 1)), _gen_0':s'3(_b175)) →RΩ(1)
s'(plus'(_gen_0':s'3(_$n7), _gen_0':s'3(_b175))) →IH
s'(_gen_0':s'3(+(_$n7, _b175)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Lemmas:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
The following defined symbols remain to be analysed:
minus', quot', app', sum'
They will be analysed ascendingly in the following order:
minus' < quot'
app' < sum'
Proved the following rewrite lemma:
minus'(_gen_0':s'3(+(1, _n976)), _gen_0':s'3(+(1, _n976))) → _*5, rt ∈ Ω(n976)
Induction Base:
minus'(_gen_0':s'3(+(1, 0)), _gen_0':s'3(+(1, 0)))
Induction Step:
minus'(_gen_0':s'3(+(1, +(_$n977, 1))), _gen_0':s'3(+(1, +(_$n977, 1)))) →RΩ(1)
minus'(_gen_0':s'3(+(1, _$n977)), _gen_0':s'3(+(1, _$n977))) →IH
_*5
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Lemmas:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
minus'(_gen_0':s'3(+(1, _n976)), _gen_0':s'3(+(1, _n976))) → _*5, rt ∈ Ω(n976)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
The following defined symbols remain to be analysed:
quot', app', sum'
They will be analysed ascendingly in the following order:
app' < sum'
Could not prove a rewrite lemma for the defined symbol quot'.
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Lemmas:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
minus'(_gen_0':s'3(+(1, _n976)), _gen_0':s'3(+(1, _n976))) → _*5, rt ∈ Ω(n976)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
The following defined symbols remain to be analysed:
app', sum'
They will be analysed ascendingly in the following order:
app' < sum'
Proved the following rewrite lemma:
app'(_gen_nil':cons'4(_n5497), _gen_nil':cons'4(b)) → _gen_nil':cons'4(+(_n5497, b)), rt ∈ Ω(1 + n5497)
Induction Base:
app'(_gen_nil':cons'4(0), _gen_nil':cons'4(b)) →RΩ(1)
_gen_nil':cons'4(b)
Induction Step:
app'(_gen_nil':cons'4(+(_$n5498, 1)), _gen_nil':cons'4(_b5786)) →RΩ(1)
cons'(0', app'(_gen_nil':cons'4(_$n5498), _gen_nil':cons'4(_b5786))) →IH
cons'(0', _gen_nil':cons'4(+(_$n5498, _b5786)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Lemmas:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
minus'(_gen_0':s'3(+(1, _n976)), _gen_0':s'3(+(1, _n976))) → _*5, rt ∈ Ω(n976)
app'(_gen_nil':cons'4(_n5497), _gen_nil':cons'4(b)) → _gen_nil':cons'4(+(_n5497, b)), rt ∈ Ω(1 + n5497)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
The following defined symbols remain to be analysed:
sum'
Proved the following rewrite lemma:
sum'(_gen_nil':cons'4(+(1, _n6726))) → _gen_nil':cons'4(1), rt ∈ Ω(1 + n6726)
Induction Base:
sum'(_gen_nil':cons'4(+(1, 0))) →RΩ(1)
cons'(0', nil')
Induction Step:
sum'(_gen_nil':cons'4(+(1, +(_$n6727, 1)))) →RΩ(1)
sum'(cons'(plus'(0', 0'), _gen_nil':cons'4(_$n6727))) →LΩ(1)
sum'(cons'(_gen_0':s'3(+(0, 0)), _gen_nil':cons'4(_$n6727))) →IH
_gen_nil':cons'4(1)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
quot'(0', s'(y)) → 0'
quot'(s'(x), s'(y)) → s'(quot'(minus'(x, y), s'(y)))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
_hole_0':s'1 :: 0':s'
_hole_nil':cons'2 :: nil':cons'
_gen_0':s'3 :: Nat → 0':s'
_gen_nil':cons'4 :: Nat → nil':cons'
Lemmas:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)
minus'(_gen_0':s'3(+(1, _n976)), _gen_0':s'3(+(1, _n976))) → _*5, rt ∈ Ω(n976)
app'(_gen_nil':cons'4(_n5497), _gen_nil':cons'4(b)) → _gen_nil':cons'4(+(_n5497, b)), rt ∈ Ω(1 + n5497)
sum'(_gen_nil':cons'4(+(1, _n6726))) → _gen_nil':cons'4(1), rt ∈ Ω(1 + n6726)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
_gen_nil':cons'4(0) ⇔ nil'
_gen_nil':cons'4(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
plus'(_gen_0':s'3(_n6), _gen_0':s'3(b)) → _gen_0':s'3(+(_n6, b)), rt ∈ Ω(1 + n6)