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Usuffix

Usuffix-equiv

Basic equivalence relation for usuffix structures.

Definitions and Theorems

Function: usuffix-equiv$inline

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

Theorem: usuffix-equiv-is-an-equivalence

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

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

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

Theorem: usuffix-fix-under-usuffix-equiv

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

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

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

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

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

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

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

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

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