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• Flatnorm-setup

Flatnorm-setup-equiv

Basic equivalence relation for flatnorm-setup structures.

Definitions and Theorems

Function: flatnorm-setup-equiv$inline

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

Theorem: flatnorm-setup-equiv-is-an-equivalence

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

Theorem: flatnorm-setup-equiv-implies-equal-flatnorm-setup-fix-1

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

Theorem: flatnorm-setup-fix-under-flatnorm-setup-equiv

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

Theorem: equal-of-flatnorm-setup-fix-1-forward-to-flatnorm-setup-equiv

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

Theorem: equal-of-flatnorm-setup-fix-2-forward-to-flatnorm-setup-equiv

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

Theorem: flatnorm-setup-equiv-of-flatnorm-setup-fix-1-forward

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

Theorem: flatnorm-setup-equiv-of-flatnorm-setup-fix-2-forward

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