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• Boolean-eq

Boolean-eq-lifting

Lifting of the circuit to a predicate.

Definitions and Theorems

Theorem: boolean-eq-pred-suff

(defthm boolean-eq-pred-suff
  (implies (and (pfield::fep w prime)
                (and (boolean-neq-pred x y w prime)
                     (boolean-not-pred w z prime)))
           (boolean-eq-pred x y z prime)))

Theorem: definition-satp-to-boolean-eq-pred

(defthm definition-satp-to-boolean-eq-pred
 (implies
  (and
   (equal
    (pfcs::lookup-definition "boolean_eq" pfcs::defs)
    '(:definition
          (pfcs::name . "boolean_eq")
          (pfcs::para "x" "y" "z")
          (pfcs::body (:relation "boolean_neq"
                                 ((:var "x") (:var "y") (:var "w")))
                      (:relation "boolean_not"
                                 ((:var "w") (:var "z"))))))
   (equal
    (pfcs::lookup-definition "boolean_neq" pfcs::defs)
    '(:definition
       (pfcs::name . "boolean_neq")
       (pfcs::para "x" "y" "z")
       (pfcs::body (:relation "boolean_xor"
                              ((:var "x") (:var "y") (:var "z"))))))
   (equal
        (pfcs::lookup-definition "boolean_xor" pfcs::defs)
        '(:definition
              (pfcs::name . "boolean_xor")
              (pfcs::para "x" "y" "z")
              (pfcs::body (:equal (:mul (:mul (:const 2) (:var "x"))
                                        (:var "y"))
                                  (:sub (:add (:var "x") (:var "y"))
                                        (:var "z"))))))
   (equal
        (pfcs::lookup-definition "boolean_not" pfcs::defs)
        '(:definition
              (pfcs::name . "boolean_not")
              (pfcs::para "x" "y")
              (pfcs::body (:equal (:mul (:sub (:const 1) (:var "x"))
                                        (:const 1))
                                  (:var "y")))))
   (pfield::fep x prime)
   (pfield::fep y prime)
   (pfield::fep z prime)
   (primep prime))
  (equal (pfcs::definition-satp "boolean_eq" pfcs::defs (list x y z)
                                prime)
         (boolean-eq-pred x y z prime))))