(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) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
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)
Types:
a__f :: g:f → g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f
(7) 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
(8) Obligation:
TRS:
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)
Types:
a__f :: g:f → g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g: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
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol mark.
(10) Obligation:
TRS:
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)
Types:
a__f :: g:f → g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g: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:
a__f
They will be analysed ascendingly in the following order:
a__f = mark
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol a__f.
(12) Obligation:
TRS:
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)
Types:
a__f :: g:f → g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g: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))
No more defined symbols left to analyse.