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• Svstmtlist

Svstmtlist-equiv

Basic equivalence relation for svstmtlist structures.

Definitions and Theorems

Function: svstmtlist-equiv$inline

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

Theorem: svstmtlist-equiv-is-an-equivalence

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

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

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

Theorem: svstmtlist-fix-under-svstmtlist-equiv

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

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

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

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

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

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

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

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

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