• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Community
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
    • Kestrel-books
      • Crypto-hdwallet
      • Apt
      • Error-checking
      • Fty-extensions
      • Isar
      • Kestrel-utilities
      • Set
      • C
        • Syntax-for-tools
          • Formalized-subset
          • Mapping-to-language-definition
          • Input-files
          • Compilation-database
          • Printer
          • Output-files
          • Abstract-syntax-operations
          • Implementation-environments
          • Abstract-syntax
          • Concrete-syntax
          • Disambiguation
          • Validation
          • Gcc-builtins
          • Preprocessing
            • Preprocessor
              • Preprocessor-states
                • Plexeme
                • Ppstate
                • Pnumber
                  • Pnumber-count
                  • Pnumber-fix
                  • Pnumberp
                  • Pnumber-case
                  • Pnumber-equiv
                  • Pnumber-number-nondigit
                  • Pnumber-number-digit
                  • Pnumber-number-upcase-p-sign
                  • Pnumber-number-upcase-e-sign
                  • Pnumber-number-locase-p-sign
                  • Pnumber-number-locase-e-sign
                  • Pnumber-dot-digit
                  • Pnumber-digit
                  • Pnumber-number-dot
                  • Pnumber-kind
                • Macro-info
                • Macro-add
                • Init-ppstate
                • Plexeme-token/newline-p
                • Newline
                • Plexeme-option
                • Macro-info-option
                • Macro-table
                • Plexeme+span
                • Update-ppstate->lexemes-unread
                • Update-ppstate->lexemes-read
                • Update-ppstate->lexemes-length
                • Update-ppstate->chars-unread
                • Update-ppstate->chars-read
                • Update-ppstate->version
                • Update-ppstate->position
                • Update-ppstate->macros
                • Update-ppstate->chars-length
                • Macro-lookup
                • Update-ppstate->bytes
                • Update-ppstate->size
                • Plexeme-token/space-p
                • Plexeme-tokenp
                • Ppstate->gcc
                • Plexeme-list-token/space-p
                • Ppstate->size
                • Ppstate->lexemes-unread
                • Ppstate->lexemes-read
                • Ppstate->lexemes-length
                • Ppstate->chars-unread
                • Ppstate->bytes
                • Plexeme-list-token/newline-p
                • Plexeme-list-not-token/newline-p
                • Ident-macroinfo-alist
                • Update-ppstate->lexeme
                • Update-ppstate->char
                • Ppstate->version
                • Ppstate->macros
                • Ppstate->chars-read
                • Ppstate->chars-length
                • Plexeme-list-tokenp
                • Plexeme-list-not-tokenp
                • Macro-table-init
                • Ppstate->lexeme
                • Ppstate->char
                • Ppstate-fix
                • Plexeme+span-list
                • Plexeme-list
                • Irr-pnumber
                • Irr-plexeme
                • Irr-macro-table
                • Ppstate->position
              • Preprocessor-printer
              • Pread-token/newline
              • Pread-token
              • Preprocessor-lexer
              • Pproc-files
              • Pproc-file
              • Filepath-plexeme-list-alist-to-filepath-filedata-map
              • Filepath-plexeme-list-alist
              • Preprocessor-reader
              • Preprocessor-messages
            • External-preprocessing
          • Parsing
        • Atc
        • Transformation-tools
        • Language
        • Representation
        • Insertion-sort
        • Pack
      • Soft
      • Bv
      • Imp-language
      • Ethereum
      • Event-macros
      • Java
      • Riscv
      • Bitcoin
      • Zcash
      • Yul
      • ACL2-programming-language
      • Prime-fields
      • Json
      • Syntheto
      • File-io-light
      • Cryptography
      • Number-theory
      • Axe
      • Lists-light
      • Builtins
      • Solidity
      • Helpers
      • Htclient
      • Typed-lists-light
      • Arithmetic-light
    • X86isa
    • Axe
    • Execloader
  • Math
  • Testing-utilities

• Pnumber

Pnumber-equiv

Basic equivalence relation for pnumber structures.

Definitions and Theorems

Function: pnumber-equiv$inline

(defun pnumber-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (pnumberp acl2::x)
                              (pnumberp acl2::y))))
  (equal (pnumber-fix acl2::x)
         (pnumber-fix acl2::y)))

Theorem: pnumber-equiv-is-an-equivalence

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

Theorem: pnumber-equiv-implies-equal-pnumber-fix-1

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

Theorem: pnumber-fix-under-pnumber-equiv

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

Theorem: equal-of-pnumber-fix-1-forward-to-pnumber-equiv

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

Theorem: equal-of-pnumber-fix-2-forward-to-pnumber-equiv

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

Theorem: pnumber-equiv-of-pnumber-fix-1-forward

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

Theorem: pnumber-equiv-of-pnumber-fix-2-forward

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