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Neteval-ordering

Neteval-ordering-equiv

Basic equivalence relation for neteval-ordering structures.

Definitions and Theorems

Function: neteval-ordering-equiv$inline

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

Theorem: neteval-ordering-equiv-is-an-equivalence

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

Theorem: neteval-ordering-equiv-implies-equal-neteval-ordering-fix-1

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

Theorem: neteval-ordering-fix-under-neteval-ordering-equiv

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

Theorem: equal-of-neteval-ordering-fix-1-forward-to-neteval-ordering-equiv

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

Theorem: equal-of-neteval-ordering-fix-2-forward-to-neteval-ordering-equiv

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

Theorem: neteval-ordering-equiv-of-neteval-ordering-fix-1-forward

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

Theorem: neteval-ordering-equiv-of-neteval-ordering-fix-2-forward

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