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• Svl-env

Svl-env-equiv

Basic equivalence relation for svl-env structures.

Definitions and Theorems

Function: svl-env-equiv$inline

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

Theorem: svl-env-equiv-is-an-equivalence

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

Theorem: svl-env-equiv-implies-equal-svl-env-fix-1

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

Theorem: svl-env-fix-under-svl-env-equiv

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

Theorem: equal-of-svl-env-fix-1-forward-to-svl-env-equiv

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

Theorem: equal-of-svl-env-fix-2-forward-to-svl-env-equiv

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

Theorem: svl-env-equiv-of-svl-env-fix-1-forward

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

Theorem: svl-env-equiv-of-svl-env-fix-2-forward

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