(0) Obligation:
Q restricted rewrite system:
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)
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))
A__F(g(X), Y) → MARK(X)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)
MARK(g(X)) → MARK(X)
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)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A__F(g(X), Y) → MARK(X)
MARK(f(X1, X2)) → A__F(mark(X1), X2)
MARK(f(X1, X2)) → MARK(X1)
MARK(g(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__F(
x0,
x1,
x2) =
A__F(
x0,
x1)
MARK(
x0,
x1) =
MARK(
x0)
Tags:
A__F has argument tags [6,5,0] and root tag 1
MARK has argument tags [5,2] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__F(
x1,
x2) =
A__F
g(
x1) =
g(
x1)
mark(
x1) =
mark(
x1)
f(
x1,
x2) =
f(
x1)
MARK(
x1) =
MARK(
x1)
a__f(
x1,
x2) =
a__f(
x1)
Recursive path order with status [RPO].
Quasi-Precedence:
[AF, g1, mark1, MARK1] > af1 > f1
Status:
AF: []
g1: [1]
mark1: [1]
f1: multiset
MARK1: [1]
af1: multiset
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))
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)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
A__F(g(X), Y) → A__F(mark(X), f(g(X), Y))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
A__F(
x0,
x1,
x2) =
A__F(
x1)
Tags:
A__F has argument tags [2,1,1] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
A__F(
x1,
x2) =
x2
g(
x1) =
g(
x1)
mark(
x1) =
x1
f(
x1,
x2) =
f
a__f(
x1,
x2) =
a__f
Recursive path order with status [RPO].
Quasi-Precedence:
[g1, f, af]
Status:
g1: multiset
f: []
af: []
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
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)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE