(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(x, false) → false
and(x, not(false)) → x
not(not(x)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
implies(not(x), not(y)) → implies(y, and(x, y))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(and(x1, x2)) = 2·x1 + x2
POL(false) = 2
POL(implies(x1, x2)) = 2 + 2·x1 + 2·x2
POL(not(x1)) = 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
and(x, not(false)) → x
implies(false, y) → not(false)
implies(x, false) → not(x)
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(x, false) → false
not(not(x)) → x
implies(not(x), not(y)) → implies(y, and(x, y))
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(and(x1, x2)) = x1 + x2
POL(false) = 1
POL(implies(x1, x2)) = x1 + 2·x2
POL(not(x1)) = 1 + 2·x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
not(not(x)) → x
implies(not(x), not(y)) → implies(y, and(x, y))
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
and(x, false) → false
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
false > and2
and weight map:
false=1
and_2=0
The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
and(x, false) → false
(6) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(7) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(8) YES