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Points-fty

Point

Fixtype of elliptic curve points.

Definitions and Theorems

Function: point-equiv$inline

(defun point-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (pointp acl2::x)
                              (pointp acl2::y))))
  (equal (point-fix acl2::x)
         (point-fix acl2::y)))

Theorem: point-equiv-is-an-equivalence

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

Theorem: point-equiv-implies-equal-point-fix-1

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

Theorem: point-fix-under-point-equiv

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

Theorem: equal-of-point-fix-1-forward-to-point-equiv

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

Theorem: equal-of-point-fix-2-forward-to-point-equiv

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

Theorem: point-equiv-of-point-fix-1-forward

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

Theorem: point-equiv-of-point-fix-2-forward

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