cond(true, x) → cond(odd(x), p(p(p(x))))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
cond'(true', x) → cond'(odd'(x), p'(p'(p'(x))))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x
Infered types.
Rules:
cond'(true', x) → cond'(odd'(x), p'(p'(p'(x))))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
cond', odd'
They will be analysed ascendingly in the following order:
odd' < cond'
Rules:
cond'(true', x) → cond'(odd'(x), p'(p'(p'(x))))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'
Generator Equations:
_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:
odd', cond'
They will be analysed ascendingly in the following order:
odd' < cond'
Proved the following rewrite lemma:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)
Induction Base:
odd'(_gen_0':s'4(*(2, 0))) →RΩ(1)
false'
Induction Step:
odd'(_gen_0':s'4(*(2, +(_$n7, 1)))) →RΩ(1)
odd'(_gen_0':s'4(*(2, _$n7))) →IH
false'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
cond'(true', x) → cond'(odd'(x), p'(p'(p'(x))))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)
Generator Equations:
_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:
cond'
Could not prove a rewrite lemma for the defined symbol cond'.
Rules:
cond'(true', x) → cond'(odd'(x), p'(p'(p'(x))))
odd'(0') → false'
odd'(s'(0')) → true'
odd'(s'(s'(x))) → odd'(x)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s' → cond'
true' :: true':false'
odd' :: 0':s' → true':false'
p' :: 0':s' → 0':s'
0' :: 0':s'
false' :: true':false'
s' :: 0':s' → 0':s'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)
Generator Equations:
_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:
odd'(_gen_0':s'4(*(2, _n6))) → false', rt ∈ Ω(1 + n6)