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• Svex-s4env

Svex-s4env-equiv

Basic equivalence relation for svex-s4env structures.

Definitions and Theorems

Function: svex-s4env-equiv$inline

(defun svex-s4env-equiv$inline (x y)
  (declare (xargs :guard (and (svex-s4env-p x)
                              (svex-s4env-p y))))
  (equal (svex-s4env-fix x)
         (svex-s4env-fix y)))

Theorem: svex-s4env-equiv-is-an-equivalence

(defthm svex-s4env-equiv-is-an-equivalence
  (and (booleanp (svex-s4env-equiv x y))
       (svex-s4env-equiv x x)
       (implies (svex-s4env-equiv x y)
                (svex-s4env-equiv y x))
       (implies (and (svex-s4env-equiv x y)
                     (svex-s4env-equiv y z))
                (svex-s4env-equiv x z)))
  :rule-classes (:equivalence))

Theorem: svex-s4env-equiv-implies-equal-svex-s4env-fix-1

(defthm svex-s4env-equiv-implies-equal-svex-s4env-fix-1
  (implies (svex-s4env-equiv x x-equiv)
           (equal (svex-s4env-fix x)
                  (svex-s4env-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: svex-s4env-fix-under-svex-s4env-equiv

(defthm svex-s4env-fix-under-svex-s4env-equiv
  (svex-s4env-equiv (svex-s4env-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-svex-s4env-fix-1-forward-to-svex-s4env-equiv

(defthm equal-of-svex-s4env-fix-1-forward-to-svex-s4env-equiv
  (implies (equal (svex-s4env-fix x) y)
           (svex-s4env-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-svex-s4env-fix-2-forward-to-svex-s4env-equiv

(defthm equal-of-svex-s4env-fix-2-forward-to-svex-s4env-equiv
  (implies (equal x (svex-s4env-fix y))
           (svex-s4env-equiv x y))
  :rule-classes :forward-chaining)

Theorem: svex-s4env-equiv-of-svex-s4env-fix-1-forward

(defthm svex-s4env-equiv-of-svex-s4env-fix-1-forward
  (implies (svex-s4env-equiv (svex-s4env-fix x) y)
           (svex-s4env-equiv x y))
  :rule-classes :forward-chaining)

Theorem: svex-s4env-equiv-of-svex-s4env-fix-2-forward

(defthm svex-s4env-equiv-of-svex-s4env-fix-2-forward
  (implies (svex-s4env-equiv x (svex-s4env-fix y))
           (svex-s4env-equiv x y))
  :rule-classes :forward-chaining)