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Defsvtv-args

Defsvtv-args-equiv

Basic equivalence relation for defsvtv-args structures.

Definitions and Theorems

Function: defsvtv-args-equiv$inline

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

Theorem: defsvtv-args-equiv-is-an-equivalence

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

Theorem: defsvtv-args-equiv-implies-equal-defsvtv-args-fix-1

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

Theorem: defsvtv-args-fix-under-defsvtv-args-equiv

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

Theorem: equal-of-defsvtv-args-fix-1-forward-to-defsvtv-args-equiv

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

Theorem: equal-of-defsvtv-args-fix-2-forward-to-defsvtv-args-equiv

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

Theorem: defsvtv-args-equiv-of-defsvtv-args-fix-1-forward

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

Theorem: defsvtv-args-equiv-of-defsvtv-args-fix-2-forward

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