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• Vl-immdeps

Vl-immdeps-equiv

Basic equivalence relation for vl-immdeps structures.

Definitions and Theorems

Function: vl-immdeps-equiv$inline

(defun vl-immdeps-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (vl-immdeps-p acl2::x)
                              (vl-immdeps-p acl2::y))))
  (equal (vl-immdeps-fix acl2::x)
         (vl-immdeps-fix acl2::y)))

Theorem: vl-immdeps-equiv-is-an-equivalence

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

Theorem: vl-immdeps-equiv-implies-equal-vl-immdeps-fix-1

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

Theorem: vl-immdeps-fix-under-vl-immdeps-equiv

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

Theorem: equal-of-vl-immdeps-fix-1-forward-to-vl-immdeps-equiv

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

Theorem: equal-of-vl-immdeps-fix-2-forward-to-vl-immdeps-equiv

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

Theorem: vl-immdeps-equiv-of-vl-immdeps-fix-1-forward

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

Theorem: vl-immdeps-equiv-of-vl-immdeps-fix-2-forward

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