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• Maybe-rational

Maybe-rational-equiv

Basic equivalence relation for maybe-rational structures.

Definitions and Theorems

Function: maybe-rational-equiv$inline

(defun maybe-rational-equiv$inline (x y)
  (declare (xargs :guard (and (maybe-rationalp x)
                              (maybe-rationalp y))))
  (equal (maybe-rational-fix x)
         (maybe-rational-fix y)))

Theorem: maybe-rational-equiv-is-an-equivalence

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

Theorem: maybe-rational-equiv-implies-equal-maybe-rational-fix-1

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

Theorem: maybe-rational-fix-under-maybe-rational-equiv

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

Theorem: equal-of-maybe-rational-fix-1-forward-to-maybe-rational-equiv

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

Theorem: equal-of-maybe-rational-fix-2-forward-to-maybe-rational-equiv

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

Theorem: maybe-rational-equiv-of-maybe-rational-fix-1-forward

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

Theorem: maybe-rational-equiv-of-maybe-rational-fix-2-forward

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