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• Uinteger+sinteger-format

Uinteger+sinteger-format-fix

Fixing function for uinteger+sinteger-format structures.

Signature

(uinteger+sinteger-format-fix x) → new-x

Arguments

x — Guard (uinteger+sinteger-formatp x).

Returns

new-x — Type (uinteger+sinteger-formatp new-x).

Definitions and Theorems

Function: uinteger+sinteger-format-fix$inline

(defun uinteger+sinteger-format-fix$inline (x)
  (declare (xargs :guard (uinteger+sinteger-formatp x)))
  (mbe :logic
       (b* ((unsigned (uinteger-format-fix (cdr (std::da-nth 0 x))))
            (signed (sinteger-format-fix (cdr (std::da-nth 1 x)))))
         (list (cons 'unsigned unsigned)
               (cons 'signed signed)))
       :exec x))

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

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

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

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

Function: uinteger+sinteger-format-equiv$inline

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Theorem: uinteger+sinteger-format-fix$inline-of-uinteger+sinteger-format-fix-x

(defthm
 uinteger+sinteger-format-fix$inline-of-uinteger+sinteger-format-fix-x
 (equal (uinteger+sinteger-format-fix$inline
             (uinteger+sinteger-format-fix x))
        (uinteger+sinteger-format-fix$inline x)))

Theorem: uinteger+sinteger-format-fix$inline-uinteger+sinteger-format-equiv-congruence-on-x

(defthm
 uinteger+sinteger-format-fix$inline-uinteger+sinteger-format-equiv-congruence-on-x
 (implies (uinteger+sinteger-format-equiv x x-equiv)
          (equal (uinteger+sinteger-format-fix$inline x)
                 (uinteger+sinteger-format-fix$inline x-equiv)))
 :rule-classes :congruence)