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• Eprefix

Eprefix-equiv

Basic equivalence relation for eprefix structures.

Definitions and Theorems

Function: eprefix-equiv$inline

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

Theorem: eprefix-equiv-is-an-equivalence

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

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

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

Theorem: eprefix-fix-under-eprefix-equiv

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

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

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

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

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

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

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

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

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