double(0) → 0
double(s(x)) → s(s(double(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(x, y))
double(x) → +(x, x)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
+'(s'(x), y) → s'(+'(x, y))
double'(x) → +'(x, x)
Infered types.
Rules:
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
+'(s'(x), y) → s'(+'(x, y))
double'(x) → +'(x, x)
Types:
double' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
double', +'
They will be analysed ascendingly in the following order:
+' < double'
Rules:
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
+'(s'(x), y) → s'(+'(x, y))
double'(x) → +'(x, x)
Types:
double' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
+', double'
They will be analysed ascendingly in the following order:
+' < double'
Proved the following rewrite lemma:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
Induction Base:
+'(_gen_0':s'2(a), _gen_0':s'2(0)) →RΩ(1)
_gen_0':s'2(a)
Induction Step:
+'(_gen_0':s'2(_a161), _gen_0':s'2(+(_$n5, 1))) →RΩ(1)
s'(+'(_gen_0':s'2(_a161), _gen_0':s'2(_$n5))) →IH
s'(_gen_0':s'2(+(_$n5, _a161)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
+'(s'(x), y) → s'(+'(x, y))
double'(x) → +'(x, x)
Types:
double' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
double'
Proved the following rewrite lemma:
double'(_gen_0':s'2(_n466)) → _gen_0':s'2(*(2, _n466)), rt ∈ Ω(1 + n466)
Induction Base:
double'(_gen_0':s'2(0)) →RΩ(1)
0'
Induction Step:
double'(_gen_0':s'2(+(_$n467, 1))) →RΩ(1)
s'(s'(double'(_gen_0':s'2(_$n467)))) →IH
s'(s'(_gen_0':s'2(*(2, _$n467))))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
double'(0') → 0'
double'(s'(x)) → s'(s'(double'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
+'(s'(x), y) → s'(+'(x, y))
double'(x) → +'(x, x)
Types:
double' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
+' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
double'(_gen_0':s'2(_n466)) → _gen_0':s'2(*(2, _n466)), rt ∈ Ω(1 + n466)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
+'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)