(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
f(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X
Rewrite Strategy: FULL
(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(X) → g(n__h(n__f(X)))
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
g/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
f(X) → g
h(X) → n__h(X)
f(X) → n__f(X)
activate(n__h(X)) → h(activate(X))
activate(n__f(X)) → f(activate(X))
activate(X) → X
Types:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
activate
(8) Obligation:
TRS:
Rules:
f(
X) →
gh(
X) →
n__h(
X)
f(
X) →
n__f(
X)
activate(
n__h(
X)) →
h(
activate(
X))
activate(
n__f(
X)) →
f(
activate(
X))
activate(
X) →
XTypes:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f
Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_0(x))
The following defined symbols remain to be analysed:
activate
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
activate(
gen_g:n__h:n__f2_0(
+(
1,
n4_0))) →
*3_0, rt ∈ Ω(n4
0)
Induction Base:
activate(gen_g:n__h:n__f2_0(+(1, 0)))
Induction Step:
activate(gen_g:n__h:n__f2_0(+(1, +(n4_0, 1)))) →RΩ(1)
h(activate(gen_g:n__h:n__f2_0(+(1, n4_0)))) →IH
h(*3_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
f(
X) →
gh(
X) →
n__h(
X)
f(
X) →
n__f(
X)
activate(
n__h(
X)) →
h(
activate(
X))
activate(
n__f(
X)) →
f(
activate(
X))
activate(
X) →
XTypes:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f
Lemmas:
activate(gen_g:n__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_0(x))
No more defined symbols left to analyse.
(12) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_g:n__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(13) BOUNDS(n^1, INF)
(14) Obligation:
TRS:
Rules:
f(
X) →
gh(
X) →
n__h(
X)
f(
X) →
n__f(
X)
activate(
n__h(
X)) →
h(
activate(
X))
activate(
n__f(
X)) →
f(
activate(
X))
activate(
X) →
XTypes:
f :: g:n__h:n__f → g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f → g:n__h:n__f
n__h :: g:n__h:n__f → g:n__h:n__f
n__f :: g:n__h:n__f → g:n__h:n__f
activate :: g:n__h:n__f → g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat → g:n__h:n__f
Lemmas:
activate(gen_g:n__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_g:n__h:n__f2_0(0) ⇔ g
gen_g:n__h:n__f2_0(+(x, 1)) ⇔ n__h(gen_g:n__h:n__f2_0(x))
No more defined symbols left to analyse.
(15) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
activate(gen_g:n__h:n__f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(16) BOUNDS(n^1, INF)