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• Cgraph

Cgraph-equiv

Basic equivalence relation for cgraph structures.

Definitions and Theorems

Function: cgraph-equiv$inline

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

Theorem: cgraph-equiv-is-an-equivalence

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

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

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

Theorem: cgraph-fix-under-cgraph-equiv

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

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

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

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

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

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

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

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

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