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)))))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
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)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Infered types.
Rules:
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)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_gen_nil':cons'3 :: Nat → nil':cons'
_gen_0':s'4 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
app', sum', plus'
They will be analysed ascendingly in the following order:
app' < sum'
plus' < sum'
Rules:
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)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_gen_nil':cons'3 :: Nat → nil':cons'
_gen_0':s'4 :: Nat → 0':s'
Generator Equations:
_gen_nil':cons'3(0) ⇔ nil'
_gen_nil':cons'3(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
app', sum', plus'
They will be analysed ascendingly in the following order:
app' < sum'
plus' < sum'
Proved the following rewrite lemma:
app'(_gen_nil':cons'3(_n6), _gen_nil':cons'3(b)) → _gen_nil':cons'3(+(_n6, b)), rt ∈ Ω(1 + n6)
Induction Base:
app'(_gen_nil':cons'3(0), _gen_nil':cons'3(b)) →RΩ(1)
_gen_nil':cons'3(b)
Induction Step:
app'(_gen_nil':cons'3(+(_$n7, 1)), _gen_nil':cons'3(_b199)) →RΩ(1)
cons'(0', app'(_gen_nil':cons'3(_$n7), _gen_nil':cons'3(_b199))) →IH
cons'(0', _gen_nil':cons'3(+(_$n7, _b199)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
app'(nil', k) → k
app'(l, nil') → l
app'(cons'(x, l), k) → cons'(x, app'(l, k))
sum'(cons'(x, nil')) → cons'(x, nil')
sum'(cons'(x, cons'(y, l))) → sum'(cons'(plus'(x, y), l))
sum'(app'(l, cons'(x, cons'(y, k)))) → sum'(app'(l, sum'(cons'(x, cons'(y, k)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_gen_nil':cons'3 :: Nat → nil':cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
app'(_gen_nil':cons'3(_n6), _gen_nil':cons'3(b)) → _gen_nil':cons'3(+(_n6, b)), rt ∈ Ω(1 + n6)
Generator Equations:
_gen_nil':cons'3(0) ⇔ nil'
_gen_nil':cons'3(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
plus', sum'
They will be analysed ascendingly in the following order:
plus' < sum'
Proved the following rewrite lemma:
plus'(_gen_0':s'4(_n774), _gen_0':s'4(b)) → _gen_0':s'4(+(_n774, b)), rt ∈ Ω(1 + n774)
Induction Base:
plus'(_gen_0':s'4(0), _gen_0':s'4(b)) →RΩ(1)
_gen_0':s'4(b)
Induction Step:
plus'(_gen_0':s'4(+(_$n775, 1)), _gen_0':s'4(_b997)) →RΩ(1)
s'(plus'(_gen_0':s'4(_$n775), _gen_0':s'4(_b997))) →IH
s'(_gen_0':s'4(+(_$n775, _b997)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
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)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_gen_nil':cons'3 :: Nat → nil':cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
app'(_gen_nil':cons'3(_n6), _gen_nil':cons'3(b)) → _gen_nil':cons'3(+(_n6, b)), rt ∈ Ω(1 + n6)
plus'(_gen_0':s'4(_n774), _gen_0':s'4(b)) → _gen_0':s'4(+(_n774, b)), rt ∈ Ω(1 + n774)
Generator Equations:
_gen_nil':cons'3(0) ⇔ nil'
_gen_nil':cons'3(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
sum'
Proved the following rewrite lemma:
sum'(_gen_nil':cons'3(+(1, _n1564))) → _gen_nil':cons'3(1), rt ∈ Ω(1 + n1564)
Induction Base:
sum'(_gen_nil':cons'3(+(1, 0))) →RΩ(1)
cons'(0', nil')
Induction Step:
sum'(_gen_nil':cons'3(+(1, +(_$n1565, 1)))) →RΩ(1)
sum'(cons'(plus'(0', 0'), _gen_nil':cons'3(_$n1565))) →LΩ(1)
sum'(cons'(_gen_0':s'4(+(0, 0)), _gen_nil':cons'3(_$n1565))) →IH
_gen_nil':cons'3(1)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
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)))))
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
Types:
app' :: nil':cons' → nil':cons' → nil':cons'
nil' :: nil':cons'
cons' :: 0':s' → nil':cons' → nil':cons'
sum' :: nil':cons' → nil':cons'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_nil':cons'1 :: nil':cons'
_hole_0':s'2 :: 0':s'
_gen_nil':cons'3 :: Nat → nil':cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
app'(_gen_nil':cons'3(_n6), _gen_nil':cons'3(b)) → _gen_nil':cons'3(+(_n6, b)), rt ∈ Ω(1 + n6)
plus'(_gen_0':s'4(_n774), _gen_0':s'4(b)) → _gen_0':s'4(+(_n774, b)), rt ∈ Ω(1 + n774)
sum'(_gen_nil':cons'3(+(1, _n1564))) → _gen_nil':cons'3(1), rt ∈ Ω(1 + n1564)
Generator Equations:
_gen_nil':cons'3(0) ⇔ nil'
_gen_nil':cons'3(+(x, 1)) ⇔ cons'(0', _gen_nil':cons'3(x))
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
app'(_gen_nil':cons'3(_n6), _gen_nil':cons'3(b)) → _gen_nil':cons'3(+(_n6, b)), rt ∈ Ω(1 + n6)