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    • Bitsize

    Bitsize-fix

    Fixing function for bitsize structures.

    Signature
    (bitsize-fix x) → new-x
    Arguments
    x — Guard (bitsizep x).
    Returns
    new-x — Type (bitsizep new-x).

    Definitions and Theorems

    Function: bitsize-fix$inline

    (defun bitsize-fix$inline (x)
      (declare (xargs :guard (bitsizep x)))
      (let ((__function__ 'bitsize-fix))
        (declare (ignorable __function__))
        (mbe :logic
             (case (bitsize-kind x)
               (:|8| (cons :|8| (list)))
               (:|16| (cons :|16| (list)))
               (:|32| (cons :|32| (list)))
               (:|64| (cons :|64| (list)))
               (:|128| (cons :|128| (list))))
             :exec x)))

    Theorem: bitsizep-of-bitsize-fix

    (defthm bitsizep-of-bitsize-fix
      (b* ((new-x (bitsize-fix$inline x)))
        (bitsizep new-x))
      :rule-classes :rewrite)

    Theorem: bitsize-fix-when-bitsizep

    (defthm bitsize-fix-when-bitsizep
      (implies (bitsizep x)
               (equal (bitsize-fix x) x)))

    Function: bitsize-equiv$inline

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

    Theorem: bitsize-equiv-is-an-equivalence

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

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

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

    Theorem: bitsize-fix-under-bitsize-equiv

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

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

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

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

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

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

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

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

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

    Theorem: bitsize-kind$inline-of-bitsize-fix-x

    (defthm bitsize-kind$inline-of-bitsize-fix-x
      (equal (bitsize-kind$inline (bitsize-fix x))
             (bitsize-kind$inline x)))

    Theorem: bitsize-kind$inline-bitsize-equiv-congruence-on-x

    (defthm bitsize-kind$inline-bitsize-equiv-congruence-on-x
      (implies (bitsize-equiv x x-equiv)
               (equal (bitsize-kind$inline x)
                      (bitsize-kind$inline x-equiv)))
      :rule-classes :congruence)

    Theorem: consp-of-bitsize-fix

    (defthm consp-of-bitsize-fix
      (consp (bitsize-fix x))
      :rule-classes :type-prescription)