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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)