(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(x) → xor(x, true)
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(=(x1, x2)) = 2 + x1 + x2
POL(and(x1, x2)) = x1 + x2
POL(implies(x1, x2)) = 1 + 2·x1 + 2·x2
POL(not(x1)) = 2·x1
POL(or(x1, x2)) = 1 + 2·x1 + 2·x2
POL(true) = 0
POL(xor(x1, x2)) = x1 + x2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
implies(x, y) → xor(and(x, y), xor(x, true))
or(x, y) → xor(and(x, y), xor(x, y))
=(x, y) → xor(x, xor(y, true))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(x) → xor(x, true)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Knuth-Bendix order [KBO] with precedence:
true > not1 > xor2
and weight map:
true=1
not_1=1
xor_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:
not(x) → xor(x, true)
(4) Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.
(5) RisEmptyProof (EQUIVALENT transformation)
The TRS R is empty. Hence, termination is trivially proven.
(6) YES