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• Montgomery-curve

Montgomery-curve-equiv

Basic equivalence relation for montgomery-curve structures.

Definitions and Theorems

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)