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• Bip32-ext-key

Bip32-ext-key-fix

Fixing function for bip32-ext-key structures.

Signature

(bip32-ext-key-fix x) → new-x

Arguments

x — Guard (bip32-ext-key-p x).

Returns

new-x — Type (bip32-ext-key-p new-x).

Definitions and Theorems

Function: bip32-ext-key-fix$inline

(defun bip32-ext-key-fix$inline (x)
 (declare (xargs :guard (bip32-ext-key-p x)))
 (let ((__function__ 'bip32-ext-key-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (bip32-ext-key-kind x)
    (:priv
        (b* ((get (bip32-ext-priv-key-fix (std::da-nth 0 (cdr x)))))
          (cons :priv (list get))))
    (:pub
         (b* ((get (bip32-ext-pub-key-fix (std::da-nth 0 (cdr x)))))
           (cons :pub (list get)))))
   :exec x)))

Theorem: bip32-ext-key-p-of-bip32-ext-key-fix

(defthm bip32-ext-key-p-of-bip32-ext-key-fix
  (b* ((new-x (bip32-ext-key-fix$inline x)))
    (bip32-ext-key-p new-x))
  :rule-classes :rewrite)

Theorem: bip32-ext-key-fix-when-bip32-ext-key-p

(defthm bip32-ext-key-fix-when-bip32-ext-key-p
  (implies (bip32-ext-key-p x)
           (equal (bip32-ext-key-fix x) x)))

Function: bip32-ext-key-equiv$inline

(defun bip32-ext-key-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bip32-ext-key-p acl2::x)
                              (bip32-ext-key-p acl2::y))))
  (equal (bip32-ext-key-fix acl2::x)
         (bip32-ext-key-fix acl2::y)))

Theorem: bip32-ext-key-equiv-is-an-equivalence

(defthm bip32-ext-key-equiv-is-an-equivalence
  (and (booleanp (bip32-ext-key-equiv x y))
       (bip32-ext-key-equiv x x)
       (implies (bip32-ext-key-equiv x y)
                (bip32-ext-key-equiv y x))
       (implies (and (bip32-ext-key-equiv x y)
                     (bip32-ext-key-equiv y z))
                (bip32-ext-key-equiv x z)))
  :rule-classes (:equivalence))

Theorem: bip32-ext-key-equiv-implies-equal-bip32-ext-key-fix-1

(defthm bip32-ext-key-equiv-implies-equal-bip32-ext-key-fix-1
  (implies (bip32-ext-key-equiv acl2::x x-equiv)
           (equal (bip32-ext-key-fix acl2::x)
                  (bip32-ext-key-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: bip32-ext-key-fix-under-bip32-ext-key-equiv

(defthm bip32-ext-key-fix-under-bip32-ext-key-equiv
  (bip32-ext-key-equiv (bip32-ext-key-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bip32-ext-key-fix-1-forward-to-bip32-ext-key-equiv

(defthm equal-of-bip32-ext-key-fix-1-forward-to-bip32-ext-key-equiv
  (implies (equal (bip32-ext-key-fix acl2::x)
                  acl2::y)
           (bip32-ext-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bip32-ext-key-fix-2-forward-to-bip32-ext-key-equiv

(defthm equal-of-bip32-ext-key-fix-2-forward-to-bip32-ext-key-equiv
  (implies (equal acl2::x (bip32-ext-key-fix acl2::y))
           (bip32-ext-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-ext-key-equiv-of-bip32-ext-key-fix-1-forward

(defthm bip32-ext-key-equiv-of-bip32-ext-key-fix-1-forward
  (implies (bip32-ext-key-equiv (bip32-ext-key-fix acl2::x)
                                acl2::y)
           (bip32-ext-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-ext-key-equiv-of-bip32-ext-key-fix-2-forward

(defthm bip32-ext-key-equiv-of-bip32-ext-key-fix-2-forward
  (implies (bip32-ext-key-equiv acl2::x (bip32-ext-key-fix acl2::y))
           (bip32-ext-key-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-ext-key-kind$inline-of-bip32-ext-key-fix-x

(defthm bip32-ext-key-kind$inline-of-bip32-ext-key-fix-x
  (equal (bip32-ext-key-kind$inline (bip32-ext-key-fix x))
         (bip32-ext-key-kind$inline x)))

Theorem: bip32-ext-key-kind$inline-bip32-ext-key-equiv-congruence-on-x

(defthm
      bip32-ext-key-kind$inline-bip32-ext-key-equiv-congruence-on-x
  (implies (bip32-ext-key-equiv x x-equiv)
           (equal (bip32-ext-key-kind$inline x)
                  (bip32-ext-key-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-bip32-ext-key-fix

(defthm consp-of-bip32-ext-key-fix
  (consp (bip32-ext-key-fix x))
  :rule-classes :type-prescription)