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Uid-equiv

Basic equivalence relation for uid structures.

Definitions and Theorems

Function: uid-equiv$inline

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

Theorem: uid-equiv-is-an-equivalence

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

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

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

Theorem: uid-fix-under-uid-equiv

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

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

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

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

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

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

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

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

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