• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
    • Apt
    • Acre
    • Milawa
    • Smtlink
    • Aleobft
    • Abnf
      • Deftreeops
      • Defdefparse
      • Defgrammar
      • Abnf-tree-utilities
      • Notation
        • Syntax-abstraction
        • Semantics
        • Abstract-syntax
          • Convenience-constructors
          • Num-val
          • Char-val
          • Repeat-range
          • Rulename
          • Rule
          • Rulename-option
          • Num-base
          • Rule-option
          • Prose-val
          • Rulelist
          • Char-val-set
          • Rulename-set
          • Rulename-list
          • Grammar
          • Alt/conc/rep/elem
            • Element
            • Repetition
              • Repetitionp
              • Repetition-equiv
              • Make-repetition
              • Repetition->range
              • Repetition->element
              • Change-repetition
              • Repetition-fix
              • Repetition-count
            • Alternation
            • Concatenation
        • Core-rules
        • Concrete-syntax
      • Grammar-parser
      • Meta-circular-validation
      • Parsing-primitives-defresult
      • Parsing-primitives-seq
      • Operations
      • Examples
      • Differences-with-paper
      • Grammar-printer
      • Parsing-tools
    • Vwsim
    • Isar
    • Wp-gen
    • Dimacs-reader
    • Pfcs
    • Legacy-defrstobj
    • Proof-checker-array
    • Soft
    • Farray
    • Rp-rewriter
    • Instant-runoff-voting
    • Imp-language
    • Sidekick
    • Leftist-trees
    • C
    • Java
    • Taspi
    • Bitcoin
    • Des
    • Ethereum
    • X86isa
    • Sha-2
    • Yul
    • Riscv
    • Zcash
    • Proof-checker-itp13
    • Regex
    • ACL2-programming-language
    • Json
    • Jfkr
    • Equational
    • Cryptography
    • Poseidon
    • Where-do-i-place-my-book
    • Bigmems
    • Builtins
    • Axe
    • Execloader
    • Solidity
    • Leo
    • Paco
    • Concurrent-programs
  • Debugging
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Math
  • Testing-utilities

• Repetition

Repetition-equiv

Basic equivalence relation for repetition structures.

Definitions and Theorems

Function: repetition-equiv$inline

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

Theorem: repetition-equiv-is-an-equivalence

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

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

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

Theorem: repetition-fix-under-repetition-equiv

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

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

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

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

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

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

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

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

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