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• Modalist

Modalist-equiv

Basic equivalence relation for modalist structures.

Definitions and Theorems

Function: modalist-equiv$inline

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

Theorem: modalist-equiv-is-an-equivalence

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

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

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

Theorem: modalist-fix-under-modalist-equiv

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

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

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

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

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

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

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

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

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