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• Field-div-checked

Field-div-checked-lifting

Lifting of the circuit to a predicate.

Definitions and Theorems

Theorem: field-div-checked-pred-suff

(defthm field-div-checked-pred-suff
  (implies (and (pfield::fep w prime)
                (and (field-inv-checked-pred y w prime)
                     (field-mul-pred x w z prime)))
           (field-div-checked-pred x y z prime)))

Theorem: definition-satp-to-field-div-checked-pred

(defthm definition-satp-to-field-div-checked-pred
 (implies
  (and
   (equal (pfcs::lookup-definition '(:simple "field_div_checked")
                                   pfcs::defs)
          '(:definition
                (name :simple "field_div_checked")
                (pfcs::para (:simple "x")
                            (:simple "y")
                            (:simple "z"))
                (pfcs::body (:relation (:simple "field_inv_checked")
                                       ((:var (:simple "y"))
                                        (:var (:simple "w"))))
                            (:relation (:simple "field_mul")
                                       ((:var (:simple "x"))
                                        (:var (:simple "w"))
                                        (:var (:simple "z")))))))
   (equal
       (pfcs::lookup-definition '(:simple "field_inv_checked")
                                pfcs::defs)
       '(:definition (name :simple "field_inv_checked")
                     (pfcs::para (:simple "x") (:simple "y"))
                     (pfcs::body (:equal (:mul (:var (:simple "x"))
                                               (:var (:simple "y")))
                                         (:const 1)))))
   (equal
       (pfcs::lookup-definition '(:simple "field_mul")
                                pfcs::defs)
       '(:definition (name :simple "field_mul")
                     (pfcs::para (:simple "x")
                                 (:simple "y")
                                 (:simple "z"))
                     (pfcs::body (:equal (:mul (:var (:simple "x"))
                                               (:var (:simple "y")))
                                         (:var (:simple "z"))))))
   (pfield::fep x prime)
   (pfield::fep y prime)
   (pfield::fep z prime)
   (primep prime))
  (equal (pfcs::definition-satp '(:simple "field_div_checked")
                                pfcs::defs (list x y z)
                                prime)
         (field-div-checked-pred x y z prime))))