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Bip39

Bip39-entropy

Entropy values.

The entropy is a sequence of bits whose size is one of the allowed ones (see bip39-entropy-size-p).

We introduce a fixtype for the possible values of the entropy.

Definitions and Theorems

Function: bip39-entropyp

(defun bip39-entropyp (x)
  (declare (xargs :guard t))
  (and (bit-listp x)
       (bip39-entropy-size-p (len x))
       t))

Theorem: booleanp-of-bip39-entropyp

(defthm booleanp-of-bip39-entropyp
  (b* ((yes/no (bip39-entropyp x)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Function: bip39-entropy-fix

(defun bip39-entropy-fix (x)
  (declare (xargs :guard (bip39-entropyp x)))
  (mbe :logic (if (bip39-entropyp x) x (repeat 128 0))
       :exec x))

Theorem: bip39-entropyp-of-bip39-entropy-fix

(defthm bip39-entropyp-of-bip39-entropy-fix
  (b* ((fixed-x (bip39-entropy-fix x)))
    (bip39-entropyp fixed-x))
  :rule-classes :rewrite)

Theorem: bip39-entropy-fix-when-bip39-entropyp

(defthm bip39-entropy-fix-when-bip39-entropyp
  (implies (bip39-entropyp x)
           (equal (bip39-entropy-fix x) x)))

Function: bip39-entropy-equiv$inline

(defun bip39-entropy-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bip39-entropyp acl2::x)
                              (bip39-entropyp acl2::y))))
  (equal (bip39-entropy-fix acl2::x)
         (bip39-entropy-fix acl2::y)))

Theorem: bip39-entropy-equiv-is-an-equivalence

(defthm bip39-entropy-equiv-is-an-equivalence
  (and (booleanp (bip39-entropy-equiv x y))
       (bip39-entropy-equiv x x)
       (implies (bip39-entropy-equiv x y)
                (bip39-entropy-equiv y x))
       (implies (and (bip39-entropy-equiv x y)
                     (bip39-entropy-equiv y z))
                (bip39-entropy-equiv x z)))
  :rule-classes (:equivalence))

Theorem: bip39-entropy-equiv-implies-equal-bip39-entropy-fix-1

(defthm bip39-entropy-equiv-implies-equal-bip39-entropy-fix-1
  (implies (bip39-entropy-equiv acl2::x x-equiv)
           (equal (bip39-entropy-fix acl2::x)
                  (bip39-entropy-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: bip39-entropy-fix-under-bip39-entropy-equiv

(defthm bip39-entropy-fix-under-bip39-entropy-equiv
  (bip39-entropy-equiv (bip39-entropy-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bip39-entropy-fix-1-forward-to-bip39-entropy-equiv

(defthm equal-of-bip39-entropy-fix-1-forward-to-bip39-entropy-equiv
  (implies (equal (bip39-entropy-fix acl2::x)
                  acl2::y)
           (bip39-entropy-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bip39-entropy-fix-2-forward-to-bip39-entropy-equiv

(defthm equal-of-bip39-entropy-fix-2-forward-to-bip39-entropy-equiv
  (implies (equal acl2::x (bip39-entropy-fix acl2::y))
           (bip39-entropy-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip39-entropy-equiv-of-bip39-entropy-fix-1-forward

(defthm bip39-entropy-equiv-of-bip39-entropy-fix-1-forward
  (implies (bip39-entropy-equiv (bip39-entropy-fix acl2::x)
                                acl2::y)
           (bip39-entropy-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip39-entropy-equiv-of-bip39-entropy-fix-2-forward

(defthm bip39-entropy-equiv-of-bip39-entropy-fix-2-forward
  (implies (bip39-entropy-equiv acl2::x (bip39-entropy-fix acl2::y))
           (bip39-entropy-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Subtopics

Bip39-entropyp: Recognizer for bip39-entropy.
Bip39-entropy-fix: Fixer for bip39-entropy.