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Assigns

Assigns-equiv

Basic equivalence relation for assigns structures.

Definitions and Theorems

Function: assigns-equiv$inline

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

Theorem: assigns-equiv-is-an-equivalence

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

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

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

Theorem: assigns-fix-under-assigns-equiv

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

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

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

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

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

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

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

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

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