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• Lexeme

Lexeme-equiv

Basic equivalence relation for lexeme structures.

Definitions and Theorems

Function: lexeme-equiv$inline

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

Theorem: lexeme-equiv-is-an-equivalence

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

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

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

Theorem: lexeme-fix-under-lexeme-equiv

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

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

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

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

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

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

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

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

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