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• Plexeme

Plexeme-equiv

Basic equivalence relation for plexeme structures.

Definitions and Theorems

Function: plexeme-equiv$inline

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

Theorem: plexeme-equiv-is-an-equivalence

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

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

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

Theorem: plexeme-fix-under-plexeme-equiv

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

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

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

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

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

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

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

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

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