cond(true, x, y) → cond(gr(x, y), x, add(x, y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
add(0, x) → x
add(s(x), y) → s(add(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Infered types.
Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 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', gr', add'
They will be analysed ascendingly in the following order:
gr' < cond'
add' < cond'
Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 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:
gr', cond', add'
They will be analysed ascendingly in the following order:
gr' < cond'
add' < cond'
Proved the following rewrite lemma:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
Induction Base:
gr'(_gen_0':s'4(0), _gen_0':s'4(0)) →RΩ(1)
false'
Induction Step:
gr'(_gen_0':s'4(+(_$n7, 1)), _gen_0':s'4(+(_$n7, 1))) →RΩ(1)
gr'(_gen_0':s'4(_$n7), _gen_0':s'4(_$n7)) →IH
false'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 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:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_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:
add', cond'
They will be analysed ascendingly in the following order:
add' < cond'
Proved the following rewrite lemma:
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)
Induction Base:
add'(_gen_0':s'4(0), _gen_0':s'4(b)) →RΩ(1)
_gen_0':s'4(b)
Induction Step:
add'(_gen_0':s'4(+(_$n355, 1)), _gen_0':s'4(_b487)) →RΩ(1)
s'(add'(_gen_0':s'4(_$n355), _gen_0':s'4(_b487))) →IH
s'(_gen_0':s'4(+(_$n355, _b487)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
cond'(true', x, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 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:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)
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, y) → cond'(gr'(x, y), x, add'(x, y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
add'(0', x) → x
add'(s'(x), y) → s'(add'(x, y))
Types:
cond' :: true':false' → 0':s' → 0':s' → cond'
true' :: true':false'
gr' :: 0':s' → 0':s' → true':false'
add' :: 0':s' → 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:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)
add'(_gen_0':s'4(_n354), _gen_0':s'4(b)) → _gen_0':s'4(+(_n354, b)), rt ∈ Ω(1 + n354)
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:
gr'(_gen_0':s'4(_n6), _gen_0':s'4(_n6)) → false', rt ∈ Ω(1 + n6)