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• Cgraph-derivstate

Cgraph-derivstate-equiv

Basic equivalence relation for cgraph-derivstate structures.

Definitions and Theorems

Function: cgraph-derivstate-equiv$inline

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

Theorem: cgraph-derivstate-equiv-is-an-equivalence

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

Theorem: cgraph-derivstate-equiv-implies-equal-cgraph-derivstate-fix-1

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

Theorem: cgraph-derivstate-fix-under-cgraph-derivstate-equiv

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

Theorem: equal-of-cgraph-derivstate-fix-1-forward-to-cgraph-derivstate-equiv

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

Theorem: equal-of-cgraph-derivstate-fix-2-forward-to-cgraph-derivstate-equiv

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

Theorem: cgraph-derivstate-equiv-of-cgraph-derivstate-fix-1-forward

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

Theorem: cgraph-derivstate-equiv-of-cgraph-derivstate-fix-2-forward

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