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Twisted-edwards-curve

Twisted-edwards-curve-fix

Fixing function for twisted-edwards-curve structures.

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

Definitions and Theorems

Function: twisted-edwards-curve-fix$inline

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

Theorem: twisted-edwards-curvep-of-twisted-edwards-curve-fix

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

Theorem: twisted-edwards-curve-fix-when-twisted-edwards-curvep

(defthm twisted-edwards-curve-fix-when-twisted-edwards-curvep
  (implies (twisted-edwards-curvep x)
           (equal (twisted-edwards-curve-fix x)
                  x)))

Function: twisted-edwards-curve-equiv$inline

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

Theorem: twisted-edwards-curve-equiv-is-an-equivalence

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

Theorem: twisted-edwards-curve-equiv-implies-equal-twisted-edwards-curve-fix-1

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

Theorem: twisted-edwards-curve-fix-under-twisted-edwards-curve-equiv

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

Theorem: equal-of-twisted-edwards-curve-fix-1-forward-to-twisted-edwards-curve-equiv

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

Theorem: equal-of-twisted-edwards-curve-fix-2-forward-to-twisted-edwards-curve-equiv

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

Theorem: twisted-edwards-curve-equiv-of-twisted-edwards-curve-fix-1-forward

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

Theorem: twisted-edwards-curve-equiv-of-twisted-edwards-curve-fix-2-forward

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