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Error

Error-equiv

Basic equivalence relation for error structures.

Definitions and Theorems

Function: error-equiv$inline

(defun error-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (errorp acl2::x)
                              (errorp acl2::y))))
  (equal (error-fix acl2::x)
         (error-fix acl2::y)))

Theorem: error-equiv-is-an-equivalence

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

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

(defthm error-equiv-implies-equal-error-fix-1
  (implies (error-equiv acl2::x x-equiv)
           (equal (error-fix acl2::x)
                  (error-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: error-fix-under-error-equiv

(defthm error-fix-under-error-equiv
  (error-equiv (error-fix acl2::x)
               acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

(defthm equal-of-error-fix-1-forward-to-error-equiv
  (implies (equal (error-fix acl2::x) acl2::y)
           (error-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm equal-of-error-fix-2-forward-to-error-equiv
  (implies (equal acl2::x (error-fix acl2::y))
           (error-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm error-equiv-of-error-fix-1-forward
  (implies (error-equiv (error-fix acl2::x)
                        acl2::y)
           (error-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm error-equiv-of-error-fix-2-forward
  (implies (error-equiv acl2::x (error-fix acl2::y))
           (error-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)