(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
not(x) → if(x, false, true)
and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
=(x, x) → true
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, true))
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(=(x1, x2)) = 2 + x1 + 2·x2
POL(and(x1, x2)) = 2 + 2·x1 + 2·x2
POL(false) = 1
POL(if(x1, x2, x3)) = x1 + x2 + x3
POL(implies(x1, x2)) = 1 + x1 + 2·x2
POL(not(x1)) = 2 + x1
POL(or(x1, x2)) = 1 + 2·x1 + 2·x2
POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
not(x) → if(x, false, true)
and(x, y) → if(x, y, false)
or(x, y) → if(x, true, y)
implies(x, y) → if(x, y, true)
=(x, x) → true
if(false, x, y) → y
if(x, x, if(x, false, true)) → true
=(x, y) → if(x, y, if(y, false, true))
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(=(x1, x2)) = 2 + 2·x1 + 2·x2
POL(if(x1, x2, x3)) = x1 + x2 + x3
POL(not(x1)) = x1
POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
=(x, y) → if(x, y, not(y))
if(true, x, y) → x
(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