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• Boolean-assert-neq

Boolean-assert-neq-lifting

Lifting of the circuit to a predicate.

Definitions and Theorems

Theorem: boolean-assert-neq-pred-suff

(defthm boolean-assert-neq-pred-suff
  (implies (and (pfield::fep z prime)
                (and (boolean-neq-pred x y z prime)
                     (boolean-assert-true-pred z prime)))
           (boolean-assert-neq-pred x y prime)))

Theorem: definition-satp-to-boolean-assert-neq-pred

(defthm definition-satp-to-boolean-assert-neq-pred
 (implies
  (and
   (equal
        (pfcs::lookup-definition '(:simple "boolean_assert_neq")
                                 pfcs::defs)
        '(:definition
              (name :simple "boolean_assert_neq")
              (pfcs::para (:simple "x") (:simple "y"))
              (pfcs::body (:relation (:simple "boolean_neq")
                                     ((:var (:simple "x"))
                                      (:var (:simple "y"))
                                      (:var (:simple "z"))))
                          (:relation (:simple "boolean_assert_true")
                                     ((:var (:simple "z")))))))
   (equal
    (pfcs::lookup-definition '(:simple "boolean_assert_true")
                             pfcs::defs)
    '(:definition
          (name :simple "boolean_assert_true")
          (pfcs::para (:simple "x"))
          (pfcs::body (:equal (:mul (:var (:simple "x")) (:const 1))
                              (:const 1)))))
   (equal
     (pfcs::lookup-definition '(:simple "boolean_neq")
                              pfcs::defs)
     '(:definition (name :simple "boolean_neq")
                   (pfcs::para (:simple "x")
                               (:simple "y")
                               (:simple "z"))
                   (pfcs::body (:relation (:simple "boolean_xor")
                                          ((:var (:simple "x"))
                                           (:var (:simple "y"))
                                           (:var (:simple "z")))))))
   (equal
     (pfcs::lookup-definition '(:simple "boolean_xor")
                              pfcs::defs)
     '(:definition
           (name :simple "boolean_xor")
           (pfcs::para (:simple "x")
                       (:simple "y")
                       (:simple "z"))
           (pfcs::body
                (:equal (:mul (:mul (:const 2) (:var (:simple "x")))
                              (:var (:simple "y")))
                        (:sub (:add (:var (:simple "x"))
                                    (:var (:simple "y")))
                              (:var (:simple "z")))))))
   (pfield::fep x prime)
   (pfield::fep y prime)
   (primep prime))
  (equal (pfcs::definition-satp '(:simple "boolean_assert_neq")
                                pfcs::defs (list x y)
                                prime)
         (boolean-assert-neq-pred x y prime))))