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• Bip32-ext-priv-key

Bip32-ext-priv-key-equiv

Basic equivalence relation for bip32-ext-priv-key structures.

Definitions and Theorems

Function: bip32-ext-priv-key-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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