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Candidate-assigns

Candidate-assigns-equiv

Basic equivalence relation for candidate-assigns structures.

Definitions and Theorems

Function: candidate-assigns-equiv$inline

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

Theorem: candidate-assigns-equiv-is-an-equivalence

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

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

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

Theorem: candidate-assigns-fix-under-candidate-assigns-equiv

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

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

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

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

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

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

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

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

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