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Address-pos-map

Address-pos-map-fix

(address-pos-map-fix x) is a usual ACL2::fty omap fixing function.

Signature
(address-pos-map-fix x) → *
Arguments: x — Guard (address-pos-mapp x).

Definitions and Theorems

Function: address-pos-map-fix

(defun address-pos-map-fix (x)
  (declare (xargs :guard (address-pos-mapp x)))
  (mbe :logic (if (address-pos-mapp x) x nil)
       :exec x))

Theorem: address-pos-mapp-of-address-pos-map-fix

(defthm address-pos-mapp-of-address-pos-map-fix
  (address-pos-mapp (address-pos-map-fix x)))

Theorem: address-pos-map-fix-when-address-pos-mapp

(defthm address-pos-map-fix-when-address-pos-mapp
  (implies (address-pos-mapp x)
           (equal (address-pos-map-fix x) x)))

Theorem: emptyp-address-pos-map-fix

(defthm emptyp-address-pos-map-fix
  (implies (or (omap::emptyp x)
               (not (address-pos-mapp x)))
           (omap::emptyp (address-pos-map-fix x))))

Theorem: emptyp-of-address-pos-map-fix-to-not-address-pos-map-or-emptyp

(defthm
     emptyp-of-address-pos-map-fix-to-not-address-pos-map-or-emptyp
  (equal (omap::emptyp (address-pos-map-fix x))
         (or (not (address-pos-mapp x))
             (omap::emptyp x))))

Function: address-pos-map-equiv$inline

(defun address-pos-map-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (address-pos-mapp acl2::x)
                              (address-pos-mapp acl2::y))))
  (equal (address-pos-map-fix acl2::x)
         (address-pos-map-fix acl2::y)))

Theorem: address-pos-map-equiv-is-an-equivalence

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

Theorem: address-pos-map-equiv-implies-equal-address-pos-map-fix-1

(defthm address-pos-map-equiv-implies-equal-address-pos-map-fix-1
  (implies (address-pos-map-equiv acl2::x x-equiv)
           (equal (address-pos-map-fix acl2::x)
                  (address-pos-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: address-pos-map-fix-under-address-pos-map-equiv

(defthm address-pos-map-fix-under-address-pos-map-equiv
  (address-pos-map-equiv (address-pos-map-fix acl2::x)
                         acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-address-pos-map-fix-1-forward-to-address-pos-map-equiv

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

Theorem: equal-of-address-pos-map-fix-2-forward-to-address-pos-map-equiv

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

Theorem: address-pos-map-equiv-of-address-pos-map-fix-1-forward

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

Theorem: address-pos-map-equiv-of-address-pos-map-fix-2-forward

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