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• Svstate

Svstate-equiv

Basic equivalence relation for svstate structures.

Definitions and Theorems

Function: svstate-equiv$inline

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

Theorem: svstate-equiv-is-an-equivalence

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

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

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

Theorem: svstate-fix-under-svstate-equiv

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

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

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

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

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

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

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

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

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