(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)
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:
MARK(f(X)) → A__F(mark(X))
MARK(f(X)) → MARK(X)
MARK(h(X)) → MARK(X)
The TRS R consists of the following rules:
a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(h(X)) → MARK(X)
MARK(f(X)) → MARK(X)
The TRS R consists of the following rules:
a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)
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.
MARK(h(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:
MARK(
x0,
x1) =
MARK(
x1)
Tags:
MARK has argument tags [0,0] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(MARK(x1)) = 1
POL(f(x1)) = x1
POL(h(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
MARK(f(X)) → MARK(X)
The TRS R consists of the following rules:
a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
MARK(f(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:
MARK(
x0,
x1) =
MARK(
x1)
Tags:
MARK has argument tags [1,1] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:
POL(MARK(x1)) = 0
POL(f(x1)) = 1 + x1
The following usable rules [FROCOS05] were oriented:
none
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a__f(X) → g(h(f(X)))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → g(X)
mark(h(X)) → h(mark(X))
a__f(X) → f(X)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) TRUE