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Secp256k1-point

Secp256k1-point-fix

Fixing function for secp256k1-point structures.

Signature
(secp256k1-point-fix p) → new-x
Arguments: p — Guard (secp256k1-pointp p).
Returns: new-x — Type (secp256k1-pointp new-x).

Definitions and Theorems

Function: secp256k1-point-fix$inline

(defun secp256k1-point-fix$inline (p)
  (declare (xargs :guard (secp256k1-pointp p)))
  (let ((acl2::__function__ 'secp256k1-point-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((x (secp256k1-field-fix (std::da-nth 0 p)))
              (y (secp256k1-field-fix (std::da-nth 1 p))))
           (list x y))
         :exec p)))

Theorem: secp256k1-pointp-of-secp256k1-point-fix

(defthm secp256k1-pointp-of-secp256k1-point-fix
  (b* ((new-x (secp256k1-point-fix$inline p)))
    (secp256k1-pointp new-x))
  :rule-classes :rewrite)

Theorem: secp256k1-point-fix-when-secp256k1-pointp

(defthm secp256k1-point-fix-when-secp256k1-pointp
  (implies (secp256k1-pointp p)
           (equal (secp256k1-point-fix p) p)))

Function: secp256k1-point-equiv$inline

(defun secp256k1-point-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (secp256k1-pointp acl2::x)
                              (secp256k1-pointp acl2::y))))
  (equal (secp256k1-point-fix acl2::x)
         (secp256k1-point-fix acl2::y)))

Theorem: secp256k1-point-equiv-is-an-equivalence

(defthm secp256k1-point-equiv-is-an-equivalence
  (and (booleanp (secp256k1-point-equiv x y))
       (secp256k1-point-equiv x x)
       (implies (secp256k1-point-equiv x y)
                (secp256k1-point-equiv y x))
       (implies (and (secp256k1-point-equiv x y)
                     (secp256k1-point-equiv y z))
                (secp256k1-point-equiv x z)))
  :rule-classes (:equivalence))

Theorem: secp256k1-point-equiv-implies-equal-secp256k1-point-fix-1

(defthm secp256k1-point-equiv-implies-equal-secp256k1-point-fix-1
  (implies (secp256k1-point-equiv acl2::x x-equiv)
           (equal (secp256k1-point-fix acl2::x)
                  (secp256k1-point-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: secp256k1-point-fix-under-secp256k1-point-equiv

(defthm secp256k1-point-fix-under-secp256k1-point-equiv
  (secp256k1-point-equiv (secp256k1-point-fix acl2::x)
                         acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-secp256k1-point-fix-1-forward-to-secp256k1-point-equiv

(defthm
    equal-of-secp256k1-point-fix-1-forward-to-secp256k1-point-equiv
  (implies (equal (secp256k1-point-fix acl2::x)
                  acl2::y)
           (secp256k1-point-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-secp256k1-point-fix-2-forward-to-secp256k1-point-equiv

(defthm
    equal-of-secp256k1-point-fix-2-forward-to-secp256k1-point-equiv
  (implies (equal acl2::x (secp256k1-point-fix acl2::y))
           (secp256k1-point-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: secp256k1-point-equiv-of-secp256k1-point-fix-1-forward

(defthm secp256k1-point-equiv-of-secp256k1-point-fix-1-forward
  (implies (secp256k1-point-equiv (secp256k1-point-fix acl2::x)
                                  acl2::y)
           (secp256k1-point-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: secp256k1-point-equiv-of-secp256k1-point-fix-2-forward

(defthm secp256k1-point-equiv-of-secp256k1-point-fix-2-forward
  (implies
       (secp256k1-point-equiv acl2::x (secp256k1-point-fix acl2::y))
       (secp256k1-point-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)