(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
a__b → a
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__b → b
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(a__b) = 1
POL(a__f(x1, x2)) = x1 + x2
POL(b) = 0
POL(f(x1, x2)) = x1 + x2
POL(mark(x1)) = 2 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__b → a
mark(b) → a__b
mark(a) → a
a__b → b
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
mark(f(X1, X2)) → a__f(mark(X1), X2)
a__f(X1, X2) → f(X1, X2)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(a__f(x1, x2)) = 2 + x1 + x2
POL(b) = 0
POL(f(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__f(X1, X2) → f(X1, X2)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
mark(f(X1, X2)) → a__f(mark(X1), X2)
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(a__f(x1, x2)) = x1 + x2
POL(b) = 0
POL(f(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(f(X1, X2)) → a__f(mark(X1), X2)
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
Q is empty.
(7) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is none
The TRS R 2 is
a__f(X, X) → a__f(a, b)
The signature Sigma is {
a__f}
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
The set Q consists of the following terms:
a__f(x0, x0)
(9) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A__F(X, X) → A__F(a, b)
The TRS R consists of the following rules:
a__f(X, X) → a__f(a, b)
The set Q consists of the following terms:
a__f(x0, x0)
We have to consider all minimal (P,Q,R)-chains.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(12) TRUE