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3col4vecline

3col4vecline-equiv

Basic equivalence relation for 3col4vecline structures.

Definitions and Theorems

Function: 3col4vecline-equiv$inline

(defun 3col4vecline-equiv$inline (x y)
  (declare (xargs :guard (and (3col4vecline-p x)
                              (3col4vecline-p y))))
  (equal (3col4vecline-fix x)
         (3col4vecline-fix y)))

Theorem: 3col4vecline-equiv-is-an-equivalence

(defthm 3col4vecline-equiv-is-an-equivalence
  (and (booleanp (3col4vecline-equiv x y))
       (3col4vecline-equiv x x)
       (implies (3col4vecline-equiv x y)
                (3col4vecline-equiv y x))
       (implies (and (3col4vecline-equiv x y)
                     (3col4vecline-equiv y z))
                (3col4vecline-equiv x z)))
  :rule-classes (:equivalence))

Theorem: 3col4vecline-equiv-implies-equal-3col4vecline-fix-1

(defthm 3col4vecline-equiv-implies-equal-3col4vecline-fix-1
  (implies (3col4vecline-equiv x x-equiv)
           (equal (3col4vecline-fix x)
                  (3col4vecline-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: 3col4vecline-fix-under-3col4vecline-equiv

(defthm 3col4vecline-fix-under-3col4vecline-equiv
  (3col4vecline-equiv (3col4vecline-fix x)
                      x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-3col4vecline-fix-1-forward-to-3col4vecline-equiv

(defthm equal-of-3col4vecline-fix-1-forward-to-3col4vecline-equiv
  (implies (equal (3col4vecline-fix x) y)
           (3col4vecline-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-3col4vecline-fix-2-forward-to-3col4vecline-equiv

(defthm equal-of-3col4vecline-fix-2-forward-to-3col4vecline-equiv
  (implies (equal x (3col4vecline-fix y))
           (3col4vecline-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3col4vecline-equiv-of-3col4vecline-fix-1-forward

(defthm 3col4vecline-equiv-of-3col4vecline-fix-1-forward
  (implies (3col4vecline-equiv (3col4vecline-fix x)
                               y)
           (3col4vecline-equiv x y))
  :rule-classes :forward-chaining)

Theorem: 3col4vecline-equiv-of-3col4vecline-fix-2-forward

(defthm 3col4vecline-equiv-of-3col4vecline-fix-2-forward
  (implies (3col4vecline-equiv x (3col4vecline-fix y))
           (3col4vecline-equiv x y))
  :rule-classes :forward-chaining)