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Reserr

Reserr-equiv

Basic equivalence relation for reserr structures.

Definitions and Theorems

Function: reserr-equiv$inline

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

Theorem: reserr-equiv-is-an-equivalence

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

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

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

Theorem: reserr-fix-under-reserr-equiv

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

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

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

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

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

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

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

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

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