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Svar-split

Svar-split-equiv

Basic equivalence relation for svar-split structures.

Definitions and Theorems

Function: svar-split-equiv$inline

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

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

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

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

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

Theorem: svar-split-fix-under-svar-split-equiv

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

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

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

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

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

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

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

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

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