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• Sinteger-bit-role

Sinteger-bit-role-fix

Fixing function for sinteger-bit-role structures.

Signature

(sinteger-bit-role-fix x) → new-x

Arguments

x — Guard (sinteger-bit-rolep x).

Returns

new-x — Type (sinteger-bit-rolep new-x).

Definitions and Theorems

Function: sinteger-bit-role-fix$inline

(defun sinteger-bit-role-fix$inline (x)
  (declare (xargs :guard (sinteger-bit-rolep x)))
  (mbe :logic
       (case (sinteger-bit-role-kind x)
         (:sign (cons :sign (list)))
         (:value (b* ((exp (nfix (std::da-nth 0 (cdr x)))))
                   (cons :value (list exp))))
         (:padding (cons :padding (list))))
       :exec x))

Theorem: sinteger-bit-rolep-of-sinteger-bit-role-fix

(defthm sinteger-bit-rolep-of-sinteger-bit-role-fix
  (b* ((new-x (sinteger-bit-role-fix$inline x)))
    (sinteger-bit-rolep new-x))
  :rule-classes :rewrite)

Theorem: sinteger-bit-role-fix-when-sinteger-bit-rolep

(defthm sinteger-bit-role-fix-when-sinteger-bit-rolep
  (implies (sinteger-bit-rolep x)
           (equal (sinteger-bit-role-fix x) x)))

Function: sinteger-bit-role-equiv$inline

(defun sinteger-bit-role-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (sinteger-bit-rolep acl2::x)
                              (sinteger-bit-rolep acl2::y))))
  (equal (sinteger-bit-role-fix acl2::x)
         (sinteger-bit-role-fix acl2::y)))

Theorem: sinteger-bit-role-equiv-is-an-equivalence

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

Theorem: sinteger-bit-role-equiv-implies-equal-sinteger-bit-role-fix-1

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

Theorem: sinteger-bit-role-fix-under-sinteger-bit-role-equiv

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

Theorem: equal-of-sinteger-bit-role-fix-1-forward-to-sinteger-bit-role-equiv

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

Theorem: equal-of-sinteger-bit-role-fix-2-forward-to-sinteger-bit-role-equiv

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

Theorem: sinteger-bit-role-equiv-of-sinteger-bit-role-fix-1-forward

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

Theorem: sinteger-bit-role-equiv-of-sinteger-bit-role-fix-2-forward

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

Theorem: sinteger-bit-role-kind$inline-of-sinteger-bit-role-fix-x

(defthm sinteger-bit-role-kind$inline-of-sinteger-bit-role-fix-x
  (equal (sinteger-bit-role-kind$inline (sinteger-bit-role-fix x))
         (sinteger-bit-role-kind$inline x)))

Theorem: sinteger-bit-role-kind$inline-sinteger-bit-role-equiv-congruence-on-x

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

Theorem: consp-of-sinteger-bit-role-fix

(defthm consp-of-sinteger-bit-role-fix
  (consp (sinteger-bit-role-fix x))
  :rule-classes :type-prescription)

Theorem: sinteger-bit-role-fix$inline-of-sinteger-bit-role-fix-x

(defthm sinteger-bit-role-fix$inline-of-sinteger-bit-role-fix-x
  (equal (sinteger-bit-role-fix$inline (sinteger-bit-role-fix x))
         (sinteger-bit-role-fix$inline x)))

Theorem: sinteger-bit-role-fix$inline-sinteger-bit-role-equiv-congruence-on-x

(defthm
 sinteger-bit-role-fix$inline-sinteger-bit-role-equiv-congruence-on-x
 (implies (sinteger-bit-role-equiv x x-equiv)
          (equal (sinteger-bit-role-fix$inline x)
                 (sinteger-bit-role-fix$inline x-equiv)))
 :rule-classes :congruence)