(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(f(X1, X2, X3)) → a__f(X1, mark(X2), X3)
mark(b) → a__b
mark(a) → a
a__f(X1, X2, X3) → f(X1, X2, X3)
a__b → b
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(a__b) = 1
POL(a__f(x1, x2, x3)) = 2 + 2·x1 + 2·x2 + 2·x3
POL(b) = 0
POL(f(x1, x2, x3)) = 2 + x1 + 2·x2 + x3
POL(mark(x1)) = 1 + 2·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, X3)) → a__f(X1, mark(X2), X3)
mark(a) → a
a__b → b
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(b) → a__b
a__f(X1, X2, X3) → f(X1, X2, X3)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 0
POL(a__b) = 0
POL(a__f(x1, x2, x3)) = 2 + x1 + 2·x2 + x3
POL(b) = 0
POL(f(x1, x2, x3)) = 1 + x1 + x2 + x3
POL(mark(x1)) = 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, X3) → f(X1, X2, X3)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
mark(b) → a__b
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 1
POL(a__b) = 1
POL(a__f(x1, x2, x3)) = 2·x1 + 2·x2 + x3
POL(b) = 0
POL(mark(x1)) = 2 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(b) → a__b
(6) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__f(a, X, X) → a__f(X, a__b, b)
a__b → a
Q is empty.
(7) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a) = 2
POL(a__b) = 2
POL(a__f(x1, x2, x3)) = 2·x1 + x2 + x3
POL(b) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__f(a, X, X) → a__f(X, a__b, b)
(8) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__b → a
Q is empty.
(9) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
ab > a
and weight map:
a__b=1
a=1
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
a__b → a
(10) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(11) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(12) YES