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• Cgraph-equivlist

Cgraph-equivlist-equiv

Basic equivalence relation for cgraph-equivlist structures.

Definitions and Theorems

Function: cgraph-equivlist-equiv$inline

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

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

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

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

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

Theorem: cgraph-equivlist-fix-under-cgraph-equivlist-equiv

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

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

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

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

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

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

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

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

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