• 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
      • Ethereum
      • Yul
      • Zcash
      • ACL2-programming-language
      • Prime-fields
      • Json
      • Syntheto
        • Process-syntheto-toplevel-fn
        • Translation
        • Language
          • Static-semantics
          • Abstract-syntax
            • Expression
            • Expression-sum-field-list
            • Type
            • Binary-op
            • Expression-product-field-list
            • Abstract-syntax-operations
            • Function-definition-list->header-list
            • Type-definition-list->name-list
            • Initializer-list->value-list
            • Function-header-list->name-list
            • Typed-variable-list->type-list
            • Typed-variable-list->name-list
            • Branch-list->condition-list
            • Alternative-list->name-list
            • Function-specifier
            • Expression-variable-list
            • Type-subset
            • Field-list->type-list
            • Field-list->name-list
            • Function-specification
            • Identifier
            • Toplevel
            • Function-definer
            • Function-header
            • Type-definer
            • Literal
            • Type-product
            • Function-definition
            • Type-sum
            • Maybe-expression
            • Transform-argument-value
            • Transform
            • Theorem
            • Quantifier
            • Maybe-function-specification
            • Maybe-typed-variable
            • Maybe-type-definition
            • Maybe-function-header
            • Maybe-function-definition
            • Maybe-type-sum
            • Maybe-type-subset
            • Maybe-type-product
            • Maybe-type-definer
            • Maybe-theorem
            • Maybe-type
            • Initializer
            • Type-definition
            • Alternative
            • Unary-op
            • Typed-variable
            • Branch
            • Field
              • Field-fix
              • Field-equiv
              • Make-field
              • Field->type
              • Field->name
              • Change-field
              • Fieldp
            • Transform-argument
            • Type-recursion
            • Program
            • Function-recursion
            • Typed-variable-list
            • Toplevel-name
            • Toplevel-list
            • Initializer-list
            • Expression-fixtypes
            • Toplevel-fn-names
            • Lookup-transform-argument
            • Function-definition-names
            • Type-definition-list
            • Transform-argument-list
            • Function-header-list
            • Function-definition-list
            • Alternative-list
            • Type-list
            • Identifier-set
            • Identifier-list
            • Field-list
            • Expression-list
            • Branch-list
            • Extract-default-param-alist
            • Create-arg-defaults-table
          • Outcome
          • Abstract-syntax-operations
          • Outcome-list
          • Outcomes
        • Process-syntheto-toplevel
        • Shallow-embedding
      • File-io-light
      • Number-theory
      • Cryptography
      • Lists-light
      • Builtins
      • Axe
      • Solidity
      • Helpers
      • Htclient
      • Typed-lists-light
      • Arithmetic-light
    • X86isa
    • Axe
    • Execloader
  • Math
  • Testing-utilities

• Field

Field-equiv

Basic equivalence relation for field structures.

Definitions and Theorems

Function: field-equiv$inline

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

Theorem: field-equiv-is-an-equivalence

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

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

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

Theorem: field-fix-under-field-equiv

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

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

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

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

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

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

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

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

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