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Sinteger-format

Sinteger-format-fix

Fixing function for sinteger-format structures.

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

Definitions and Theorems

Function: sinteger-format-fix$inline

(defun sinteger-format-fix$inline (x)
 (declare (xargs :guard (sinteger-formatp x)))
 (let ((__function__ 'sinteger-format-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b* ((bits (sinteger-bit-role-list-fix (cdr (std::da-nth 0 x))))
         (signed (signed-format-fix (cdr (std::da-nth 1 x))))
         (traps (cdr (std::da-nth 2 x))))
      (let ((bits (if (sinteger-bit-roles-wfp bits)
                      bits
                    (list (sinteger-bit-role-sign)
                          (sinteger-bit-role-value 0)))))
        (list (cons 'bits bits)
              (cons 'signed signed)
              (cons 'traps traps))))
    :exec x)))

Theorem: sinteger-formatp-of-sinteger-format-fix

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

Theorem: sinteger-format-fix-when-sinteger-formatp

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

Function: sinteger-format-equiv$inline

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

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

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

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

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

Theorem: sinteger-format-fix-under-sinteger-format-equiv

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

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

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

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

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

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

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

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

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