exp(x, 0) → s(0)
exp(x, s(y)) → *(x, exp(x, y))
*(0, y) → 0
*(s(x), y) → +(y, *(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(y, *'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Sliced the following arguments:
+'/0
Runtime Complexity TRS:
The TRS R consists of the following rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Infered types.
Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Types:
exp' :: 0':s':+' → 0':s':+' → 0':s':+'
0' :: 0':s':+'
s' :: 0':s':+' → 0':s':+'
*' :: 0':s':+' → 0':s':+' → 0':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:
exp', *', -'
They will be analysed ascendingly in the following order:
*' < exp'
Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Types:
exp' :: 0':s':+' → 0':s':+' → 0':s':+'
0' :: 0':s':+'
s' :: 0':s':+' → 0':s':+'
*' :: 0':s':+' → 0':s':+' → 0':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:
*', exp', -'
They will be analysed ascendingly in the following order:
*' < exp'
Proved the following rewrite lemma:
*'(_gen_0':s':+'2(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)
Induction Base:
*'(_gen_0':s':+'2(+(1, 0)), _gen_0':s':+'2(b))
Induction Step:
*'(_gen_0':s':+'2(+(1, +(_$n5, 1))), _gen_0':s':+'2(_b711)) →RΩ(1)
+'(*'(_gen_0':s':+'2(+(1, _$n5)), _gen_0':s':+'2(_b711))) →IH
+'(_*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Types:
exp' :: 0':s':+' → 0':s':+' → 0':s':+'
0' :: 0':s':+'
s' :: 0':s':+' → 0':s':+'
*' :: 0':s':+' → 0':s':+' → 0':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(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(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:
exp', -'
Proved the following rewrite lemma:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)
Induction Base:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(0))
Induction Step:
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(+(_$n1019, 1))) →RΩ(1)
*'(_gen_0':s':+'2(0), exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_$n1019))) →IH
*'(_gen_0':s':+'2(0), _*3)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Types:
exp' :: 0':s':+' → 0':s':+' → 0':s':+'
0' :: 0':s':+'
s' :: 0':s':+' → 0':s':+'
*' :: 0':s':+' → 0':s':+' → 0':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(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)
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:
-'
Proved the following rewrite lemma:
-'(_gen_0':s':+'2(_n2763), _gen_0':s':+'2(_n2763)) → _gen_0':s':+'2(0), rt ∈ Ω(1 + n2763)
Induction Base:
-'(_gen_0':s':+'2(0), _gen_0':s':+'2(0)) →RΩ(1)
0'
Induction Step:
-'(_gen_0':s':+'2(+(_$n2764, 1)), _gen_0':s':+'2(+(_$n2764, 1))) →RΩ(1)
-'(_gen_0':s':+'2(_$n2764), _gen_0':s':+'2(_$n2764)) →IH
_gen_0':s':+'2(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
exp'(x, 0') → s'(0')
exp'(x, s'(y)) → *'(x, exp'(x, y))
*'(0', y) → 0'
*'(s'(x), y) → +'(*'(x, y))
-'(0', y) → 0'
-'(x, 0') → x
-'(s'(x), s'(y)) → -'(x, y)
Types:
exp' :: 0':s':+' → 0':s':+' → 0':s':+'
0' :: 0':s':+'
s' :: 0':s':+' → 0':s':+'
*' :: 0':s':+' → 0':s':+' → 0':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(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)
exp'(_gen_0':s':+'2(0), _gen_0':s':+'2(_n1018)) → _*3, rt ∈ Ω(n1018)
-'(_gen_0':s':+'2(_n2763), _gen_0':s':+'2(_n2763)) → _gen_0':s':+'2(0), rt ∈ Ω(1 + n2763)
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(+(1, _n4)), _gen_0':s':+'2(b)) → _*3, rt ∈ Ω(n4)