(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)
Rewrite Strategy: INNERMOST
(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)
S is empty.
Rewrite Strategy: INNERMOST
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
Innermost TRS:
Rules:
f(a) → b
f(c) → d
f(g(x, y)) → g(f(x), f(y))
f(h(x, y)) → g(h(y, f(x)), h(x, f(y)))
g(x, x) → h(e, x)
Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
f
(6) Obligation:
Innermost TRS:
Rules:
f(
a) →
bf(
c) →
df(
g(
x,
y)) →
g(
f(
x),
f(
y))
f(
h(
x,
y)) →
g(
h(
y,
f(
x)),
h(
x,
f(
y)))
g(
x,
x) →
h(
e,
x)
Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e
Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_0(x))
The following defined symbols remain to be analysed:
f
(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
f(
gen_a:b:c:d:h:e2_0(
n4_0)) →
*3_0, rt ∈ Ω(n4
0)
Induction Base:
f(gen_a:b:c:d:h:e2_0(0))
Induction Step:
f(gen_a:b:c:d:h:e2_0(+(n4_0, 1))) →RΩ(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), f(a)), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) →RΩ(1)
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, f(gen_a:b:c:d:h:e2_0(n4_0)))) →IH
g(h(gen_a:b:c:d:h:e2_0(n4_0), b), h(a, *3_0))
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
Innermost TRS:
Rules:
f(
a) →
bf(
c) →
df(
g(
x,
y)) →
g(
f(
x),
f(
y))
f(
h(
x,
y)) →
g(
h(
y,
f(
x)),
h(
x,
f(
y)))
g(
x,
x) →
h(
e,
x)
Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e
Lemmas:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_0(x))
No more defined symbols left to analyse.
(10) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
(11) BOUNDS(n^1, INF)
(12) Obligation:
Innermost TRS:
Rules:
f(
a) →
bf(
c) →
df(
g(
x,
y)) →
g(
f(
x),
f(
y))
f(
h(
x,
y)) →
g(
h(
y,
f(
x)),
h(
x,
f(
y)))
g(
x,
x) →
h(
e,
x)
Types:
f :: a:b:c:d:h:e → a:b:c:d:h:e
a :: a:b:c:d:h:e
b :: a:b:c:d:h:e
c :: a:b:c:d:h:e
d :: a:b:c:d:h:e
g :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
h :: a:b:c:d:h:e → a:b:c:d:h:e → a:b:c:d:h:e
e :: a:b:c:d:h:e
hole_a:b:c:d:h:e1_0 :: a:b:c:d:h:e
gen_a:b:c:d:h:e2_0 :: Nat → a:b:c:d:h:e
Lemmas:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_a:b:c:d:h:e2_0(0) ⇔ a
gen_a:b:c:d:h:e2_0(+(x, 1)) ⇔ h(a, gen_a:b:c:d:h:e2_0(x))
No more defined symbols left to analyse.
(13) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
f(gen_a:b:c:d:h:e2_0(n4_0)) → *3_0, rt ∈ Ω(n40)
(14) BOUNDS(n^1, INF)