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Decdig/uscore

Decdig/uscore-fix

Fixing function for decdig/uscore structures.

Signature
(decdig/uscore-fix x) → new-x
Arguments: x — Guard (decdig/uscore-p x).
Returns: new-x — Type (decdig/uscore-p new-x).

Definitions and Theorems

Function: decdig/uscore-fix$inline

(defun decdig/uscore-fix$inline (x)
 (declare (xargs :guard (decdig/uscore-p x)))
 (let ((__function__ 'decdig/uscore-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (case (decdig/uscore-kind x)
         (:digit (b* ((get (dec-digit-fix (std::da-nth 0 (cdr x)))))
                   (cons :digit (list get))))
         (:underscore (cons :underscore (list))))
       :exec x)))

Theorem: decdig/uscore-p-of-decdig/uscore-fix

(defthm decdig/uscore-p-of-decdig/uscore-fix
  (b* ((new-x (decdig/uscore-fix$inline x)))
    (decdig/uscore-p new-x))
  :rule-classes :rewrite)

Theorem: decdig/uscore-fix-when-decdig/uscore-p

(defthm decdig/uscore-fix-when-decdig/uscore-p
  (implies (decdig/uscore-p x)
           (equal (decdig/uscore-fix x) x)))

Function: decdig/uscore-equiv$inline

(defun decdig/uscore-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (decdig/uscore-p acl2::x)
                              (decdig/uscore-p acl2::y))))
  (equal (decdig/uscore-fix acl2::x)
         (decdig/uscore-fix acl2::y)))

Theorem: decdig/uscore-equiv-is-an-equivalence

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

Theorem: decdig/uscore-equiv-implies-equal-decdig/uscore-fix-1

(defthm decdig/uscore-equiv-implies-equal-decdig/uscore-fix-1
  (implies (decdig/uscore-equiv acl2::x x-equiv)
           (equal (decdig/uscore-fix acl2::x)
                  (decdig/uscore-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: decdig/uscore-fix-under-decdig/uscore-equiv

(defthm decdig/uscore-fix-under-decdig/uscore-equiv
  (decdig/uscore-equiv (decdig/uscore-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-decdig/uscore-fix-1-forward-to-decdig/uscore-equiv

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

Theorem: equal-of-decdig/uscore-fix-2-forward-to-decdig/uscore-equiv

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

Theorem: decdig/uscore-equiv-of-decdig/uscore-fix-1-forward

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

Theorem: decdig/uscore-equiv-of-decdig/uscore-fix-2-forward

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

Theorem: decdig/uscore-kind$inline-of-decdig/uscore-fix-x

(defthm decdig/uscore-kind$inline-of-decdig/uscore-fix-x
  (equal (decdig/uscore-kind$inline (decdig/uscore-fix x))
         (decdig/uscore-kind$inline x)))

Theorem: decdig/uscore-kind$inline-decdig/uscore-equiv-congruence-on-x

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

Theorem: consp-of-decdig/uscore-fix

(defthm consp-of-decdig/uscore-fix
  (consp (decdig/uscore-fix x))
  :rule-classes :type-prescription)