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• Field-eq

Field-eq-lifting

Lifting of the circuit to a predicate.

Definitions and Theorems

Theorem: field-eq-pred-suff

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

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

(defthm definition-satp-to-field-eq-pred
 (implies
  (and
   (equal
    (pfcs::lookup-definition "field_eq" pfcs::defs)
    '(:definition
          (pfcs::name . "field_eq")
          (pfcs::para "x" "y" "z")
          (pfcs::body (:relation "field_neq"
                                 ((:var "x") (:var "y") (:var "w")))
                      (:relation "boolean_not"
                                 ((:var "w") (:var "z"))))))
   (equal
    (pfcs::lookup-definition "boolean_not" pfcs::defs)
    '(:definition
      (pfcs::name . "boolean_not")
      (pfcs::para "x" "y")
      (pfcs::body (:equal (:mul (:add (:const 1)
                                      (:mul (:const -1) (:var "x")))
                                (:const 1))
                          (:var "y")))))
   (equal
      (pfcs::lookup-definition "field_neq" pfcs::defs)
      '(:definition
            (pfcs::name . "field_neq")
            (pfcs::para "x" "y" "z")
            (pfcs::body
                 (:equal (:mul (:add (:var "x")
                                     (:mul (:const -1) (:var "y")))
                               (:var "w"))
                         (:var "z"))
                 (:equal (:mul (:add (:var "x")
                                     (:mul (:const -1) (:var "y")))
                               (:add (:const 1)
                                     (:mul (:const -1) (:var "z"))))
                         (:const 0)))))
   (pfield::fep x prime)
   (pfield::fep y prime)
   (pfield::fep z prime)
   (primep prime))
  (equal (pfcs::definition-satp "field_eq" pfcs::defs (list x y z)
                                prime)
         (field-eq-pred x y z prime))))