Runtime Complexity TRS:
The TRS R consists of the following rules:
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
int(x, y) → if(le(x, y), x, y)
if(true, x, y) → cons(x, int(s(x), y))
if(false, x, y) → nil
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(x, int'(s'(x), y))
if'(false', x, y) → nil'
Sliced the following arguments:
cons'/0
Runtime Complexity TRS:
The TRS R consists of the following rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(int'(s'(x), y))
if'(false', x, y) → nil'
Infered types.
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(int'(s'(x), y))
if'(false', x, y) → nil'
Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
int' :: 0':s' → 0':s' → cons':nil'
if' :: true':false' → 0':s' → 0':s' → cons':nil'
cons' :: cons':nil' → cons':nil'
nil' :: cons':nil'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_cons':nil'3 :: cons':nil'
_gen_0':s'4 :: Nat → 0':s'
_gen_cons':nil'5 :: Nat → cons':nil'
Heuristically decided to analyse the following defined symbols:
le', int'
They will be analysed ascendingly in the following order:
le' < int'
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(int'(s'(x), y))
if'(false', x, y) → nil'
Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
int' :: 0':s' → 0':s' → cons':nil'
if' :: true':false' → 0':s' → 0':s' → cons':nil'
cons' :: cons':nil' → cons':nil'
nil' :: cons':nil'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_cons':nil'3 :: cons':nil'
_gen_0':s'4 :: Nat → 0':s'
_gen_cons':nil'5 :: Nat → cons':nil'
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
_gen_cons':nil'5(0) ⇔ nil'
_gen_cons':nil'5(+(x, 1)) ⇔ cons'(_gen_cons':nil'5(x))
The following defined symbols remain to be analysed:
le', int'
They will be analysed ascendingly in the following order:
le' < int'
Proved the following rewrite lemma:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)
Induction Base:
le'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s'4(+(_$n8, 1)), _gen_0':s'4(+(_$n8, 1))) →RΩ(1)
le'(_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:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(int'(s'(x), y))
if'(false', x, y) → nil'
Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
int' :: 0':s' → 0':s' → cons':nil'
if' :: true':false' → 0':s' → 0':s' → cons':nil'
cons' :: cons':nil' → cons':nil'
nil' :: cons':nil'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_cons':nil'3 :: cons':nil'
_gen_0':s'4 :: Nat → 0':s'
_gen_cons':nil'5 :: Nat → cons':nil'
Lemmas:
le'(_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_cons':nil'5(0) ⇔ nil'
_gen_cons':nil'5(+(x, 1)) ⇔ cons'(_gen_cons':nil'5(x))
The following defined symbols remain to be analysed:
int'
Could not prove a rewrite lemma for the defined symbol int'.
Rules:
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
int'(x, y) → if'(le'(x, y), x, y)
if'(true', x, y) → cons'(int'(s'(x), y))
if'(false', x, y) → nil'
Types:
le' :: 0':s' → 0':s' → true':false'
0' :: 0':s'
true' :: true':false'
s' :: 0':s' → 0':s'
false' :: true':false'
int' :: 0':s' → 0':s' → cons':nil'
if' :: true':false' → 0':s' → 0':s' → cons':nil'
cons' :: cons':nil' → cons':nil'
nil' :: cons':nil'
_hole_true':false'1 :: true':false'
_hole_0':s'2 :: 0':s'
_hole_cons':nil'3 :: cons':nil'
_gen_0':s'4 :: Nat → 0':s'
_gen_cons':nil'5 :: Nat → cons':nil'
Lemmas:
le'(_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_cons':nil'5(0) ⇔ nil'
_gen_cons':nil'5(+(x, 1)) ⇔ cons'(_gen_cons':nil'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
le'(_gen_0':s'4(_n7), _gen_0':s'4(_n7)) → true', rt ∈ Ω(1 + n7)