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• Bip32-key-trees

Bip32-index-tree

Index trees underlying key trees.

As defined later, a key tree includes a tree of indices, represented as a closed non-empty set of paths.

Extending a path of an index tree results in a index tree, because the extension preserves non-emptiness and prefix closure.

Definitions and Theorems

Function: bip32-index-treep

(defun bip32-index-treep (x)
  (declare (xargs :guard t))
  (and (bip32-path-setp x)
       (not (emptyp x))
       (bip32-path-set-closedp x)))

Theorem: booleanp-of-bip32-index-treep

(defthm booleanp-of-bip32-index-treep
  (b* ((yes/no (bip32-index-treep x)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: bip32-path-setp-when-bip32-index-treep

(defthm bip32-path-setp-when-bip32-index-treep
  (implies (bip32-index-treep x)
           (bip32-path-setp x))
  :rule-classes (:rewrite :forward-chaining))

Theorem: bip32-index-treep-of-singleton-empty-path

(defthm bip32-index-treep-of-singleton-empty-path
  (bip32-index-treep '(nil)))

Function: bip32-index-tree-fix

(defun bip32-index-tree-fix (x)
  (declare (xargs :guard (bip32-index-treep x)))
  (mbe :logic (if (bip32-index-treep x) x (list nil))
       :exec x))

Theorem: bip32-index-treep-of-bip32-index-tree-fix

(defthm bip32-index-treep-of-bip32-index-tree-fix
  (b* ((fixed-x (bip32-index-tree-fix x)))
    (bip32-index-treep fixed-x))
  :rule-classes :rewrite)

Theorem: bip32-index-tree-fix-when-bip32-index-treep

(defthm bip32-index-tree-fix-when-bip32-index-treep
  (implies (bip32-index-treep x)
           (equal (bip32-index-tree-fix x) x)))

Function: bip32-index-tree-equiv$inline

(defun bip32-index-tree-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bip32-index-treep acl2::x)
                              (bip32-index-treep acl2::y))))
  (equal (bip32-index-tree-fix acl2::x)
         (bip32-index-tree-fix acl2::y)))

Theorem: bip32-index-tree-equiv-is-an-equivalence

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

Theorem: bip32-index-tree-equiv-implies-equal-bip32-index-tree-fix-1

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

Theorem: bip32-index-tree-fix-under-bip32-index-tree-equiv

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

Theorem: equal-of-bip32-index-tree-fix-1-forward-to-bip32-index-tree-equiv

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

Theorem: equal-of-bip32-index-tree-fix-2-forward-to-bip32-index-tree-equiv

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

Theorem: bip32-index-tree-equiv-of-bip32-index-tree-fix-1-forward

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

Theorem: bip32-index-tree-equiv-of-bip32-index-tree-fix-2-forward

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

Theorem: bip32-index-treep-of-insert-of-rcons

(defthm bip32-index-treep-of-insert-of-rcons
  (implies (and (bip32-index-treep paths)
                (in path paths)
                (ubyte32p index))
           (bip32-index-treep (insert (rcons index path) paths))))

Subtopics

Bip32-index-treep

Recognizer for bip32-index-tree.

Bip32-index-tree-fix

Fixer for bip32-index-tree.