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• Svtv-cyclephase

Svtv-cyclephase-equiv

Basic equivalence relation for svtv-cyclephase structures.

Definitions and Theorems

Function: svtv-cyclephase-equiv$inline

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

Theorem: svtv-cyclephase-equiv-is-an-equivalence

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

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

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

Theorem: svtv-cyclephase-fix-under-svtv-cyclephase-equiv

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

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

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

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

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

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

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

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

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