(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(f(g(X2938_0), X2)) →+ a__f(mark(mark(X2938_0)), f(g(mark(X2938_0)), X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X2938_0 / f(g(X2938_0), X2)].
The result substitution is [ ].
The rewrite sequence
mark(f(g(X2938_0), X2)) →+ a__f(mark(mark(X2938_0)), f(g(mark(X2938_0)), X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X2938_0 / f(g(X2938_0), X2)].
The result substitution is [ ].
(2) BOUNDS(2^n, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)
S is empty.
Rewrite Strategy: FULL
(5) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
a__f/1
f/1
(6) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)
S is empty.
Rewrite Strategy: FULL
(7) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(8) Obligation:
TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)
Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
(9) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
a__f,
markThey will be analysed ascendingly in the following order:
a__f = mark
(10) Obligation:
TRS:
Rules:
a__f(
g(
X)) →
a__f(
mark(
X))
mark(
f(
X1)) →
a__f(
mark(
X1))
mark(
g(
X)) →
g(
mark(
X))
a__f(
X1) →
f(
X1)
Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))
The following defined symbols remain to be analysed:
mark, a__f
They will be analysed ascendingly in the following order:
a__f = mark
(11) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
mark(
gen_g:f2_0(
+(
1,
n4_0))) →
*3_0, rt ∈ Ω(n4
0)
Induction Base:
mark(gen_g:f2_0(+(1, 0)))
Induction Step:
mark(gen_g:f2_0(+(1, +(n4_0, 1)))) →RΩ(1)
g(mark(gen_g:f2_0(+(1, n4_0)))) →IH
g(*3_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(12) Complex Obligation (BEST)
(13) Obligation:
TRS:
Rules:
a__f(
g(
X)) →
a__f(
mark(
X))
mark(
f(
X1)) →
a__f(
mark(
X1))
mark(
g(
X)) →
g(
mark(
X))
a__f(
X1) →
f(
X1)
Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))
The following defined symbols remain to be analysed:
a__f
They will be analysed ascendingly in the following order:
a__f = mark
(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__f.
(15) Obligation:
TRS:
Rules:
a__f(
g(
X)) →
a__f(
mark(
X))
mark(
f(
X1)) →
a__f(
mark(
X1))
mark(
g(
X)) →
g(
mark(
X))
a__f(
X1) →
f(
X1)
Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))
No more defined symbols left to analyse.
(16) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(17) BOUNDS(n^1, INF)
(18) Obligation:
TRS:
Rules:
a__f(
g(
X)) →
a__f(
mark(
X))
mark(
f(
X1)) →
a__f(
mark(
X1))
mark(
g(
X)) →
g(
mark(
X))
a__f(
X1) →
f(
X1)
Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))
No more defined symbols left to analyse.
(19) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
(20) BOUNDS(n^1, INF)