(0) Obligation:
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)
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) → 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)
S is empty.
Rewrite Strategy: INNERMOST
(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(4) Obligation:
Innermost TRS:
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:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g
(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
f, e
(6) Obligation:
Innermost TRS:
Rules:
f(
a) →
f(
c(
a))
f(
c(
X)) →
Xf(
c(
a)) →
f(
d(
b))
f(
a) →
f(
d(
a))
f(
d(
X)) →
Xf(
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:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g
Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))
The following defined symbols remain to be analysed:
f, e
(7) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(8) Obligation:
Innermost TRS:
Rules:
f(
a) →
f(
c(
a))
f(
c(
X)) →
Xf(
c(
a)) →
f(
d(
b))
f(
a) →
f(
d(
a))
f(
d(
X)) →
Xf(
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:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g
Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))
The following defined symbols remain to be analysed:
e
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
e(
gen_g5_0(
+(
1,
n43_0))) →
*6_0, rt ∈ Ω(n43
0)
Induction Base:
e(gen_g5_0(+(1, 0)))
Induction Step:
e(gen_g5_0(+(1, +(n43_0, 1)))) →RΩ(1)
e(gen_g5_0(+(1, n43_0))) →IH
*6_0
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
Innermost TRS:
Rules:
f(
a) →
f(
c(
a))
f(
c(
X)) →
Xf(
c(
a)) →
f(
d(
b))
f(
a) →
f(
d(
a))
f(
d(
X)) →
Xf(
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:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g
Lemmas:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n430)
Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))
No more defined symbols left to analyse.
(12) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n430)
(13) BOUNDS(n^1, INF)
(14) Obligation:
Innermost TRS:
Rules:
f(
a) →
f(
c(
a))
f(
c(
X)) →
Xf(
c(
a)) →
f(
d(
b))
f(
a) →
f(
d(
a))
f(
d(
X)) →
Xf(
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:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat → a:c:b:d
gen_g5_0 :: Nat → g
Lemmas:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n430)
Generator Equations:
gen_a:c:b:d4_0(0) ⇔ b
gen_a:c:b:d4_0(+(x, 1)) ⇔ c(gen_a:c:b:d4_0(x))
gen_g5_0(0) ⇔ hole_g3_0
gen_g5_0(+(x, 1)) ⇔ g(gen_g5_0(x))
No more defined symbols left to analyse.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
e(gen_g5_0(+(1, n43_0))) → *6_0, rt ∈ Ω(n430)
(16) BOUNDS(n^1, INF)