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• Database

Database-fix

(database-fix x) is a usual ACL2::fty omap fixing function.

Signature

(database-fix x) → *

Arguments

x — Guard (databasep x).

Definitions and Theorems

Function: database-fix

(defun database-fix (x)
  (declare (xargs :guard (databasep x)))
  (mbe :logic (if (databasep x) x nil)
       :exec x))

Theorem: databasep-of-database-fix

(defthm databasep-of-database-fix
  (databasep (database-fix x)))

Theorem: database-fix-when-databasep

(defthm database-fix-when-databasep
  (implies (databasep x)
           (equal (database-fix x) x)))

Theorem: emptyp-database-fix

(defthm emptyp-database-fix
  (implies (or (omap::emptyp x)
               (not (databasep x)))
           (omap::emptyp (database-fix x))))

Theorem: emptyp-of-database-fix-to-not-database-or-emptyp

(defthm emptyp-of-database-fix-to-not-database-or-emptyp
  (equal (omap::emptyp (database-fix x))
         (or (not (databasep x))
             (omap::emptyp x))))

Function: database-equiv$inline

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

Theorem: database-equiv-is-an-equivalence

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

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

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

Theorem: database-fix-under-database-equiv

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

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

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

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

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

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

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

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

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