• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
    • Apt
    • Zfc
    • Acre
    • Milawa
    • Smtlink
    • Abnf
    • Vwsim
    • Isar
    • Wp-gen
    • Dimacs-reader
    • Pfcs
      • Proof-support
      • Abstract-syntax
        • Syntax-abstraction
        • Expression
        • Definition
          • Definition-fix
          • Definitionp
          • Definition-equiv
          • Make-definition
          • Definition->para
          • Definition->body
          • Change-definition
          • Definition->name
        • Constraint
        • Definition-option
        • Abstract-syntax-operations
        • System
        • Convenience-constructors
        • System-result
        • Expression-result
        • Expression-list-result
        • Definition-result
        • Definition-list-result
        • Constraint-result
        • Constraint-list-result
        • Expression-list
        • Definition-list
        • Constraint-list
      • R1cs-subset
      • Semantics
      • Abstract-syntax-operations
      • Indexed-names
      • Well-formedness
      • Concrete-syntax
      • R1cs-bridge
      • Parser-interface
    • Legacy-defrstobj
    • Proof-checker-array
    • Soft
    • C
    • Farray
    • Rp-rewriter
    • Instant-runoff-voting
    • Imp-language
    • Sidekick
    • Leftist-trees
    • Java
    • Taspi
    • Bitcoin
    • Riscv
    • Des
    • Ethereum
    • X86isa
    • Sha-2
    • Yul
    • Zcash
    • Proof-checker-itp13
    • Regex
    • ACL2-programming-language
    • Json
    • Jfkr
    • Equational
    • Cryptography
    • Poseidon
    • Where-do-i-place-my-book
    • Axe
    • Bigmems
    • Builtins
    • Execloader
    • Aleo
    • Solidity
    • Paco
    • Concurrent-programs
    • Bls12-377-curves
  • Debugging
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Math
  • Testing-utilities

• Definition

Definition-equiv

Basic equivalence relation for definition structures.

Definitions and Theorems

Function: definition-equiv$inline

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

Theorem: definition-equiv-is-an-equivalence

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

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

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

Theorem: definition-fix-under-definition-equiv

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

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

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

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

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

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

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

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

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