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Field-neq

Field-neq-lifting

Lifting of the circuit to a predicate.

Definitions and Theorems

Theorem: field-neq-pred-suff

(defthm field-neq-pred-suff
 (implies
   (and (pfield::fep w prime)
        (and (equal (pfield::mul (pfield::sub x y prime)
                                 w prime)
                    z)
             (equal (pfield::mul (pfield::sub x y prime)
                                 (pfield::sub (mod 1 prime) z prime)
                                 prime)
                    (mod 0 prime))))
   (field-neq-pred x y z prime)))

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

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