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• Modinst

Modinst-equiv

Basic equivalence relation for modinst structures.

Definitions and Theorems

Function: modinst-equiv$inline

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

Theorem: modinst-equiv-is-an-equivalence

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

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

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

Theorem: modinst-fix-under-modinst-equiv

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

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

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

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

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

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

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

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

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