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Short-weierstrass

Short-weierstrass-fix

Fixing function for short-weierstrass structures.

Signature
(short-weierstrass-fix x) → new-x
Arguments: x — Guard (short-weierstrass-p x).
Returns: new-x — Type (short-weierstrass-p new-x).

Definitions and Theorems

Function: short-weierstrass-fix$inline

(defun short-weierstrass-fix$inline (x)
 (declare (xargs :guard (short-weierstrass-p x)))
 (let ((acl2::__function__ 'short-weierstrass-fix))
  (declare (ignorable acl2::__function__))
  (mbe
      :logic
      (b* ((p (nfix (cdr (std::da-nth 0 x))))
           (a (nfix (cdr (std::da-nth 1 x))))
           (b (nfix (cdr (std::da-nth 2 x)))))
        (let ((p (if (and (> p 3)
                          (fep a p)
                          (fep b p)
                          (posp (mod (+ (* 4 a a a) (* 27 b b)) p)))
                     p
                   5))
              (a (if (and (> p 3)
                          (fep a p)
                          (fep b p)
                          (posp (mod (+ (* 4 a a a) (* 27 b b)) p)))
                     a
                   0))
              (b (if (and (> p 3)
                          (fep a p)
                          (fep b p)
                          (posp (mod (+ (* 4 a a a) (* 27 b b)) p)))
                     b
                   1)))
          (list (cons 'p p)
                (cons 'a a)
                (cons 'b b))))
      :exec x)))

Theorem: short-weierstrass-p-of-short-weierstrass-fix

(defthm short-weierstrass-p-of-short-weierstrass-fix
  (b* ((new-x (short-weierstrass-fix$inline x)))
    (short-weierstrass-p new-x))
  :rule-classes :rewrite)

Theorem: short-weierstrass-fix-when-short-weierstrass-p

(defthm short-weierstrass-fix-when-short-weierstrass-p
  (implies (short-weierstrass-p x)
           (equal (short-weierstrass-fix x) x)))

Function: short-weierstrass-equiv$inline

(defun short-weierstrass-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (short-weierstrass-p acl2::x)
                              (short-weierstrass-p acl2::y))))
  (equal (short-weierstrass-fix acl2::x)
         (short-weierstrass-fix acl2::y)))

Theorem: short-weierstrass-equiv-is-an-equivalence

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

Theorem: short-weierstrass-equiv-implies-equal-short-weierstrass-fix-1

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

Theorem: short-weierstrass-fix-under-short-weierstrass-equiv

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

Theorem: equal-of-short-weierstrass-fix-1-forward-to-short-weierstrass-equiv

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

Theorem: equal-of-short-weierstrass-fix-2-forward-to-short-weierstrass-equiv

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

Theorem: short-weierstrass-equiv-of-short-weierstrass-fix-1-forward

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

Theorem: short-weierstrass-equiv-of-short-weierstrass-fix-2-forward

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