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• Svar-widths

Svar-widths-equiv

Basic equivalence relation for svar-widths structures.

Definitions and Theorems

Function: svar-widths-equiv$inline

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

Theorem: svar-widths-equiv-is-an-equivalence

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

Theorem: svar-widths-equiv-implies-equal-svar-widths-fix-1

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

Theorem: svar-widths-fix-under-svar-widths-equiv

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

Theorem: equal-of-svar-widths-fix-1-forward-to-svar-widths-equiv

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

Theorem: equal-of-svar-widths-fix-2-forward-to-svar-widths-equiv

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

Theorem: svar-widths-equiv-of-svar-widths-fix-1-forward

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

Theorem: svar-widths-equiv-of-svar-widths-fix-2-forward

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