f(a) → f(c(a))
f(c(X)) → X
f(c(a)) → f(d(b))
f(a) → f(d(a))
f(d(X)) → X
f(c(b)) → f(d(a))
e(g(X)) → e(X)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(a') → f'(c'(a'))
f'(c'(X)) → X
f'(c'(a')) → f'(d'(b'))
f'(a') → f'(d'(a'))
f'(d'(X)) → X
f'(c'(b')) → f'(d'(a'))
e'(g'(X)) → e'(X)
Infered types.
Rules:
f'(a') → f'(c'(a'))
f'(c'(X)) → X
f'(c'(a')) → f'(d'(b'))
f'(a') → f'(d'(a'))
f'(d'(X)) → X
f'(c'(b')) → f'(d'(a'))
e'(g'(X)) → e'(X)
Types:
f' :: a':c':b':d' → a':c':b':d'
a' :: a':c':b':d'
c' :: a':c':b':d' → a':c':b':d'
d' :: a':c':b':d' → a':c':b':d'
b' :: a':c':b':d'
e' :: g' → e'
g' :: g' → g'
_hole_a':c':b':d'1 :: a':c':b':d'
_hole_e'2 :: e'
_hole_g'3 :: g'
_gen_a':c':b':d'4 :: Nat → a':c':b':d'
_gen_g'5 :: Nat → g'
Heuristically decided to analyse the following defined symbols:
f', e'
Rules:
f'(a') → f'(c'(a'))
f'(c'(X)) → X
f'(c'(a')) → f'(d'(b'))
f'(a') → f'(d'(a'))
f'(d'(X)) → X
f'(c'(b')) → f'(d'(a'))
e'(g'(X)) → e'(X)
Types:
f' :: a':c':b':d' → a':c':b':d'
a' :: a':c':b':d'
c' :: a':c':b':d' → a':c':b':d'
d' :: a':c':b':d' → a':c':b':d'
b' :: a':c':b':d'
e' :: g' → e'
g' :: g' → g'
_hole_a':c':b':d'1 :: a':c':b':d'
_hole_e'2 :: e'
_hole_g'3 :: g'
_gen_a':c':b':d'4 :: Nat → a':c':b':d'
_gen_g'5 :: Nat → g'
Generator Equations:
_gen_a':c':b':d'4(0) ⇔ b'
_gen_a':c':b':d'4(+(x, 1)) ⇔ c'(_gen_a':c':b':d'4(x))
_gen_g'5(0) ⇔ _hole_g'3
_gen_g'5(+(x, 1)) ⇔ g'(_gen_g'5(x))
The following defined symbols remain to be analysed:
f', e'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(a') → f'(c'(a'))
f'(c'(X)) → X
f'(c'(a')) → f'(d'(b'))
f'(a') → f'(d'(a'))
f'(d'(X)) → X
f'(c'(b')) → f'(d'(a'))
e'(g'(X)) → e'(X)
Types:
f' :: a':c':b':d' → a':c':b':d'
a' :: a':c':b':d'
c' :: a':c':b':d' → a':c':b':d'
d' :: a':c':b':d' → a':c':b':d'
b' :: a':c':b':d'
e' :: g' → e'
g' :: g' → g'
_hole_a':c':b':d'1 :: a':c':b':d'
_hole_e'2 :: e'
_hole_g'3 :: g'
_gen_a':c':b':d'4 :: Nat → a':c':b':d'
_gen_g'5 :: Nat → g'
Generator Equations:
_gen_a':c':b':d'4(0) ⇔ b'
_gen_a':c':b':d'4(+(x, 1)) ⇔ c'(_gen_a':c':b':d'4(x))
_gen_g'5(0) ⇔ _hole_g'3
_gen_g'5(+(x, 1)) ⇔ g'(_gen_g'5(x))
The following defined symbols remain to be analysed:
e'
Proved the following rewrite lemma:
e'(_gen_g'5(+(1, _n77))) → _*6, rt ∈ Ω(n77)
Induction Base:
e'(_gen_g'5(+(1, 0)))
Induction Step:
e'(_gen_g'5(+(1, +(_$n78, 1)))) →RΩ(1)
e'(_gen_g'5(+(1, _$n78))) →IH
_*6
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(a') → f'(c'(a'))
f'(c'(X)) → X
f'(c'(a')) → f'(d'(b'))
f'(a') → f'(d'(a'))
f'(d'(X)) → X
f'(c'(b')) → f'(d'(a'))
e'(g'(X)) → e'(X)
Types:
f' :: a':c':b':d' → a':c':b':d'
a' :: a':c':b':d'
c' :: a':c':b':d' → a':c':b':d'
d' :: a':c':b':d' → a':c':b':d'
b' :: a':c':b':d'
e' :: g' → e'
g' :: g' → g'
_hole_a':c':b':d'1 :: a':c':b':d'
_hole_e'2 :: e'
_hole_g'3 :: g'
_gen_a':c':b':d'4 :: Nat → a':c':b':d'
_gen_g'5 :: Nat → g'
Lemmas:
e'(_gen_g'5(+(1, _n77))) → _*6, rt ∈ Ω(n77)
Generator Equations:
_gen_a':c':b':d'4(0) ⇔ b'
_gen_a':c':b':d'4(+(x, 1)) ⇔ c'(_gen_a':c':b':d'4(x))
_gen_g'5(0) ⇔ _hole_g'3
_gen_g'5(+(x, 1)) ⇔ g'(_gen_g'5(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
e'(_gen_g'5(+(1, _n77))) → _*6, rt ∈ Ω(n77)