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• Boolean-result

Boolean-result-equiv

Basic equivalence relation for boolean-result structures.

Definitions and Theorems

Function: boolean-result-equiv$inline

(defun boolean-result-equiv$inline (x y)
  (declare (xargs :guard (and (boolean-resultp x)
                              (boolean-resultp y))))
  (equal (boolean-result-fix x)
         (boolean-result-fix y)))

Theorem: boolean-result-equiv-is-an-equivalence

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

Theorem: boolean-result-equiv-implies-equal-boolean-result-fix-1

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

Theorem: boolean-result-fix-under-boolean-result-equiv

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

Theorem: equal-of-boolean-result-fix-1-forward-to-boolean-result-equiv

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

Theorem: equal-of-boolean-result-fix-2-forward-to-boolean-result-equiv

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

Theorem: boolean-result-equiv-of-boolean-result-fix-1-forward

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

Theorem: boolean-result-equiv-of-boolean-result-fix-2-forward

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