cond(true, x) → cond(and(even(x), gr(x, 0)), p(x))
and(x, false) → false
and(false, x) → false
and(true, true) → true
even(0) → true
even(s(0)) → false
even(s(s(x))) → even(x)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Infered types.
Rules:
cond'(true', x) → cond'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s':y' → cond'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
even' :: 0':s':y' → true':false'
gr' :: 0':s':y' → 0':s':y' → true':false'
0' :: 0':s':y'
p' :: 0':s':y' → 0':s':y'
false' :: true':false'
s' :: 0':s':y' → 0':s':y'
y' :: 0':s':y'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s':y'3 :: 0':s':y'
_gen_0':s':y'4 :: Nat → 0':s':y'
Heuristically decided to analyse the following defined symbols:
cond', even', gr'
They will be analysed ascendingly in the following order:
even' < cond'
gr' < cond'
Rules:
cond'(true', x) → cond'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s':y' → cond'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
even' :: 0':s':y' → true':false'
gr' :: 0':s':y' → 0':s':y' → true':false'
0' :: 0':s':y'
p' :: 0':s':y' → 0':s':y'
false' :: true':false'
s' :: 0':s':y' → 0':s':y'
y' :: 0':s':y'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s':y'3 :: 0':s':y'
_gen_0':s':y'4 :: Nat → 0':s':y'
Generator Equations:
_gen_0':s':y'4(0) ⇔ 0'
_gen_0':s':y'4(+(x, 1)) ⇔ s'(_gen_0':s':y'4(x))
The following defined symbols remain to be analysed:
even', cond', gr'
They will be analysed ascendingly in the following order:
even' < cond'
gr' < cond'
Proved the following rewrite lemma:
even'(_gen_0':s':y'4(*(2, _n6))) → true', rt ∈ Ω(1 + n6)
Induction Base:
even'(_gen_0':s':y'4(*(2, 0))) →RΩ(1)
true'
Induction Step:
even'(_gen_0':s':y'4(*(2, +(_$n7, 1)))) →RΩ(1)
even'(_gen_0':s':y'4(*(2, _$n7))) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
cond'(true', x) → cond'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s':y' → cond'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
even' :: 0':s':y' → true':false'
gr' :: 0':s':y' → 0':s':y' → true':false'
0' :: 0':s':y'
p' :: 0':s':y' → 0':s':y'
false' :: true':false'
s' :: 0':s':y' → 0':s':y'
y' :: 0':s':y'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s':y'3 :: 0':s':y'
_gen_0':s':y'4 :: Nat → 0':s':y'
Lemmas:
even'(_gen_0':s':y'4(*(2, _n6))) → true', rt ∈ Ω(1 + n6)
Generator Equations:
_gen_0':s':y'4(0) ⇔ 0'
_gen_0':s':y'4(+(x, 1)) ⇔ s'(_gen_0':s':y'4(x))
The following defined symbols remain to be analysed:
gr', cond'
They will be analysed ascendingly in the following order:
gr' < cond'
Could not prove a rewrite lemma for the defined symbol gr'.
Rules:
cond'(true', x) → cond'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s':y' → cond'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
even' :: 0':s':y' → true':false'
gr' :: 0':s':y' → 0':s':y' → true':false'
0' :: 0':s':y'
p' :: 0':s':y' → 0':s':y'
false' :: true':false'
s' :: 0':s':y' → 0':s':y'
y' :: 0':s':y'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s':y'3 :: 0':s':y'
_gen_0':s':y'4 :: Nat → 0':s':y'
Lemmas:
even'(_gen_0':s':y'4(*(2, _n6))) → true', rt ∈ Ω(1 + n6)
Generator Equations:
_gen_0':s':y'4(0) ⇔ 0'
_gen_0':s':y'4(+(x, 1)) ⇔ s'(_gen_0':s':y'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'(and'(even'(x), gr'(x, 0')), p'(x))
and'(x, false') → false'
and'(false', x) → false'
and'(true', true') → true'
even'(0') → true'
even'(s'(0')) → false'
even'(s'(s'(x))) → even'(x)
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y')) → gr'(x, y')
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → 0':s':y' → cond'
true' :: true':false'
and' :: true':false' → true':false' → true':false'
even' :: 0':s':y' → true':false'
gr' :: 0':s':y' → 0':s':y' → true':false'
0' :: 0':s':y'
p' :: 0':s':y' → 0':s':y'
false' :: true':false'
s' :: 0':s':y' → 0':s':y'
y' :: 0':s':y'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_0':s':y'3 :: 0':s':y'
_gen_0':s':y'4 :: Nat → 0':s':y'
Lemmas:
even'(_gen_0':s':y'4(*(2, _n6))) → true', rt ∈ Ω(1 + n6)
Generator Equations:
_gen_0':s':y'4(0) ⇔ 0'
_gen_0':s':y'4(+(x, 1)) ⇔ s'(_gen_0':s':y'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
even'(_gen_0':s':y'4(*(2, _n6))) → true', rt ∈ Ω(1 + n6)