• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
    • Kestrel-books
      • Crypto-hdwallet
      • Apt
      • Error-checking
      • Fty-extensions
      • Isar
      • Kestrel-utilities
      • Soft
      • Bv
      • Imp-language
      • C
      • Event-macros
      • Java
      • Bitcoin
        • Bip32
          • Bip32-wallet-structure
          • Bip32-key-trees
            • Bip32-extend-tree
            • Bip32-valid-depths-p
            • Bip32-key-tree
            • Bip32-path-set-closedp
            • Bip32-valid-keys-p
            • Bip32-index-tree
            • Bip32-path-in-tree-p
            • Bip32-ckd*
            • Bip32-get-pub-key-at-path
            • Bip32-get-priv-key-at-path
            • Bip32-ckd-priv*
            • Bip32-ckd-pub*
            • Bip32-path
            • Bip32-key-tree-priv-p
            • Bip32-path-set
              • Bip32-path-sfix
              • Bip32-path-setp
              • Bip32-path-set-equiv
          • Bip32-key-serialization
          • Bip32-key-derivation
          • Bip32-executable-attachments
          • Bip32-extended-keys
          • Bip32-master-key-generation
        • Bech32
        • Bip39
        • Bip44
        • Base58
        • Bip43
        • Bytes
        • Base58check
        • Cryptography
        • Bip-350
        • Bip-173
      • Ethereum
      • Yul
      • Zcash
      • ACL2-programming-language
      • Prime-fields
      • Json
      • Syntheto
      • File-io-light
      • Number-theory
      • Cryptography
      • Lists-light
      • Builtins
      • Axe
      • Solidity
      • Helpers
      • Htclient
      • Typed-lists-light
      • Arithmetic-light
    • X86isa
    • Axe
    • Execloader
  • Math
  • Testing-utilities

• Bip32-path-set

Bip32-path-set-equiv

Basic equivalence relation for bip32-path-set structures.

Definitions and Theorems

Function: bip32-path-set-equiv$inline

(defun bip32-path-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bip32-path-setp acl2::x)
                              (bip32-path-setp acl2::y))))
  (equal (bip32-path-sfix acl2::x)
         (bip32-path-sfix acl2::y)))

Theorem: bip32-path-set-equiv-is-an-equivalence

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

Theorem: bip32-path-set-equiv-implies-equal-bip32-path-sfix-1

(defthm bip32-path-set-equiv-implies-equal-bip32-path-sfix-1
  (implies (bip32-path-set-equiv acl2::x x-equiv)
           (equal (bip32-path-sfix acl2::x)
                  (bip32-path-sfix x-equiv)))
  :rule-classes (:congruence))

Theorem: bip32-path-sfix-under-bip32-path-set-equiv

(defthm bip32-path-sfix-under-bip32-path-set-equiv
  (bip32-path-set-equiv (bip32-path-sfix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bip32-path-sfix-1-forward-to-bip32-path-set-equiv

(defthm equal-of-bip32-path-sfix-1-forward-to-bip32-path-set-equiv
  (implies (equal (bip32-path-sfix acl2::x)
                  acl2::y)
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bip32-path-sfix-2-forward-to-bip32-path-set-equiv

(defthm equal-of-bip32-path-sfix-2-forward-to-bip32-path-set-equiv
  (implies (equal acl2::x (bip32-path-sfix acl2::y))
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-path-set-equiv-of-bip32-path-sfix-1-forward

(defthm bip32-path-set-equiv-of-bip32-path-sfix-1-forward
  (implies (bip32-path-set-equiv (bip32-path-sfix acl2::x)
                                 acl2::y)
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bip32-path-set-equiv-of-bip32-path-sfix-2-forward

(defthm bip32-path-set-equiv-of-bip32-path-sfix-2-forward
  (implies (bip32-path-set-equiv acl2::x (bip32-path-sfix acl2::y))
           (bip32-path-set-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)