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Identifier

Identifier-equiv

Basic equivalence relation for identifier structures.

Definitions and Theorems

Function: identifier-equiv$inline

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

Theorem: identifier-equiv-is-an-equivalence

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

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

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

Theorem: identifier-fix-under-identifier-equiv

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

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

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

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

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

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

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

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

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