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4veclist

4veclist-equiv

Basic equivalence relation for 4veclist structures.

Definitions and Theorems

Function: 4veclist-equiv$inline

(defun 4veclist-equiv$inline (x y)
  (declare (xargs :guard (and (4veclist-p x) (4veclist-p y))))
  (equal (4veclist-fix x)
         (4veclist-fix y)))

Theorem: 4veclist-equiv-is-an-equivalence

(defthm 4veclist-equiv-is-an-equivalence
  (and (booleanp (4veclist-equiv x y))
       (4veclist-equiv x x)
       (implies (4veclist-equiv x y)
                (4veclist-equiv y x))
       (implies (and (4veclist-equiv x y)
                     (4veclist-equiv y z))
                (4veclist-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 4veclist-equiv-implies-equal-4veclist-fix-1

(defthm 4veclist-equiv-implies-equal-4veclist-fix-1
  (implies (4veclist-equiv x x-equiv)
           (equal (4veclist-fix x)
                  (4veclist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 4veclist-fix-under-4veclist-equiv

(defthm 4veclist-fix-under-4veclist-equiv
  (4veclist-equiv (4veclist-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-4veclist-fix-1-forward-to-4veclist-equiv

(defthm equal-of-4veclist-fix-1-forward-to-4veclist-equiv
  (implies (equal (4veclist-fix x) y)
           (4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-4veclist-fix-2-forward-to-4veclist-equiv

(defthm equal-of-4veclist-fix-2-forward-to-4veclist-equiv
  (implies (equal x (4veclist-fix y))
           (4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 4veclist-equiv-of-4veclist-fix-1-forward

(defthm 4veclist-equiv-of-4veclist-fix-1-forward
  (implies (4veclist-equiv (4veclist-fix x) y)
           (4veclist-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 4veclist-equiv-of-4veclist-fix-2-forward

(defthm 4veclist-equiv-of-4veclist-fix-2-forward
  (implies (4veclist-equiv x (4veclist-fix y))
           (4veclist-equiv x y))
  :rule-classes :forward-chaining)