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• Twisted-edwards-curve

Twisted-edwards-curve-equiv

Basic equivalence relation for twisted-edwards-curve structures.

Definitions and Theorems

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)