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Svtv

Svtv-equiv

Basic equivalence relation for svtv structures.

Definitions and Theorems

Function: svtv-equiv$inline

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

Theorem: svtv-equiv-is-an-equivalence

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

Theorem: svtv-equiv-implies-equal-svtv-fix-1

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

Theorem: svtv-fix-under-svtv-equiv

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

Theorem: equal-of-svtv-fix-1-forward-to-svtv-equiv

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

Theorem: equal-of-svtv-fix-2-forward-to-svtv-equiv

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

Theorem: svtv-equiv-of-svtv-fix-1-forward

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

Theorem: svtv-equiv-of-svtv-fix-2-forward

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