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Signed-format

Signed-format-fix

Fixing function for signed-format structures.

Signature
(signed-format-fix x) → new-x
Arguments: x — Guard (signed-formatp x).
Returns: new-x — Type (signed-formatp new-x).

Definitions and Theorems

Function: signed-format-fix$inline

(defun signed-format-fix$inline (x)
  (declare (xargs :guard (signed-formatp x)))
  (let ((__function__ 'signed-format-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (signed-format-kind x)
           (:sign-magnitude (cons :sign-magnitude (list)))
           (:ones-complement (cons :ones-complement (list)))
           (:twos-complement (cons :twos-complement (list))))
         :exec x)))

Theorem: signed-formatp-of-signed-format-fix

(defthm signed-formatp-of-signed-format-fix
  (b* ((new-x (signed-format-fix$inline x)))
    (signed-formatp new-x))
  :rule-classes :rewrite)

Theorem: signed-format-fix-when-signed-formatp

(defthm signed-format-fix-when-signed-formatp
  (implies (signed-formatp x)
           (equal (signed-format-fix x) x)))

Function: signed-format-equiv$inline

(defun signed-format-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (signed-formatp acl2::x)
                              (signed-formatp acl2::y))))
  (equal (signed-format-fix acl2::x)
         (signed-format-fix acl2::y)))

Theorem: signed-format-equiv-is-an-equivalence

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

Theorem: signed-format-equiv-implies-equal-signed-format-fix-1

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

Theorem: signed-format-fix-under-signed-format-equiv

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

Theorem: equal-of-signed-format-fix-1-forward-to-signed-format-equiv

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

Theorem: equal-of-signed-format-fix-2-forward-to-signed-format-equiv

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

Theorem: signed-format-equiv-of-signed-format-fix-1-forward

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

Theorem: signed-format-equiv-of-signed-format-fix-2-forward

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

Theorem: signed-format-kind$inline-of-signed-format-fix-x

(defthm signed-format-kind$inline-of-signed-format-fix-x
  (equal (signed-format-kind$inline (signed-format-fix x))
         (signed-format-kind$inline x)))

Theorem: signed-format-kind$inline-signed-format-equiv-congruence-on-x

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

Theorem: consp-of-signed-format-fix

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