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Montgomery-curve

Montgomery-curve-fix

Fixing function for montgomery-curve structures.

Signature
(montgomery-curve-fix x) → new-x
Arguments: x — Guard (montgomery-curvep x).
Returns: new-x — Type (montgomery-curvep new-x).

Definitions and Theorems

Function: montgomery-curve-fix$inline

(defun montgomery-curve-fix$inline (x)
  (declare (xargs :guard (montgomery-curvep x)))
  (let ((acl2::__function__ 'montgomery-curve-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((p (cdr (std::da-nth 0 x)))
              (a (cdr (std::da-nth 1 x)))
              (b (cdr (std::da-nth 2 x))))
           (let ((p (if (and (dm::primep p) (> p 2)) p 3))
                 (a (if (and (dm::primep p)
                             (> p 2)
                             (fep a p)
                             (not (equal a 2))
                             (not (equal a (mod -2 p))))
                        a
                      0))
                 (b (if (and (dm::primep p)
                             (fep b p)
                             (not (equal b 0)))
                        b
                      1)))
             (list (cons 'p p)
                   (cons 'a a)
                   (cons 'b b))))
         :exec x)))

Theorem: montgomery-curvep-of-montgomery-curve-fix

(defthm montgomery-curvep-of-montgomery-curve-fix
  (b* ((new-x (montgomery-curve-fix$inline x)))
    (montgomery-curvep new-x))
  :rule-classes :rewrite)

Theorem: montgomery-curve-fix-when-montgomery-curvep

(defthm montgomery-curve-fix-when-montgomery-curvep
  (implies (montgomery-curvep x)
           (equal (montgomery-curve-fix x) x)))

Function: montgomery-curve-equiv$inline

(defun montgomery-curve-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (montgomery-curvep acl2::x)
                              (montgomery-curvep acl2::y))))
  (equal (montgomery-curve-fix acl2::x)
         (montgomery-curve-fix acl2::y)))

Theorem: montgomery-curve-equiv-is-an-equivalence

(defthm montgomery-curve-equiv-is-an-equivalence
  (and (booleanp (montgomery-curve-equiv x y))
       (montgomery-curve-equiv x x)
       (implies (montgomery-curve-equiv x y)
                (montgomery-curve-equiv y x))
       (implies (and (montgomery-curve-equiv x y)
                     (montgomery-curve-equiv y z))
                (montgomery-curve-equiv x z)))
  :rule-classes (:equivalence))

Theorem: montgomery-curve-equiv-implies-equal-montgomery-curve-fix-1

(defthm montgomery-curve-equiv-implies-equal-montgomery-curve-fix-1
  (implies (montgomery-curve-equiv acl2::x x-equiv)
           (equal (montgomery-curve-fix acl2::x)
                  (montgomery-curve-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: montgomery-curve-fix-under-montgomery-curve-equiv

(defthm montgomery-curve-fix-under-montgomery-curve-equiv
  (montgomery-curve-equiv (montgomery-curve-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-montgomery-curve-fix-1-forward-to-montgomery-curve-equiv

(defthm
  equal-of-montgomery-curve-fix-1-forward-to-montgomery-curve-equiv
  (implies (equal (montgomery-curve-fix acl2::x)
                  acl2::y)
           (montgomery-curve-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-montgomery-curve-fix-2-forward-to-montgomery-curve-equiv

(defthm
  equal-of-montgomery-curve-fix-2-forward-to-montgomery-curve-equiv
  (implies (equal acl2::x (montgomery-curve-fix acl2::y))
           (montgomery-curve-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: montgomery-curve-equiv-of-montgomery-curve-fix-1-forward

(defthm montgomery-curve-equiv-of-montgomery-curve-fix-1-forward
  (implies (montgomery-curve-equiv (montgomery-curve-fix acl2::x)
                                   acl2::y)
           (montgomery-curve-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: montgomery-curve-equiv-of-montgomery-curve-fix-2-forward

(defthm montgomery-curve-equiv-of-montgomery-curve-fix-2-forward
 (implies
     (montgomery-curve-equiv acl2::x (montgomery-curve-fix acl2::y))
     (montgomery-curve-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)