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Num-val

Num-val-fix

Fixing function for num-val structures.

Signature
(num-val-fix x) → new-x
Arguments: x — Guard (num-val-p x).
Returns: new-x — Type (num-val-p new-x).

Definitions and Theorems

Function: num-val-fix$inline

(defun num-val-fix$inline (x)
 (declare (xargs :guard (num-val-p x)))
 (let ((__function__ 'num-val-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (case (num-val-kind x)
         (:direct (b* ((base (num-base-fix (std::da-nth 0 (cdr x))))
                       (get (nat-list-fix (std::da-nth 1 (cdr x)))))
                    (cons :direct (list base get))))
         (:range (b* ((base (num-base-fix (std::da-nth 0 (cdr x))))
                      (min (nfix (std::da-nth 1 (cdr x))))
                      (max (nfix (std::da-nth 2 (cdr x)))))
                   (cons :range (list base min max)))))
       :exec x)))

Theorem: num-val-p-of-num-val-fix

(defthm num-val-p-of-num-val-fix
  (b* ((new-x (num-val-fix$inline x)))
    (num-val-p new-x))
  :rule-classes :rewrite)

Theorem: num-val-fix-when-num-val-p

(defthm num-val-fix-when-num-val-p
  (implies (num-val-p x)
           (equal (num-val-fix x) x)))

Function: num-val-equiv$inline

(defun num-val-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (num-val-p acl2::x)
                              (num-val-p acl2::y))))
  (equal (num-val-fix acl2::x)
         (num-val-fix acl2::y)))

Theorem: num-val-equiv-is-an-equivalence

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

Theorem: num-val-equiv-implies-equal-num-val-fix-1

(defthm num-val-equiv-implies-equal-num-val-fix-1
  (implies (num-val-equiv acl2::x x-equiv)
           (equal (num-val-fix acl2::x)
                  (num-val-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: num-val-fix-under-num-val-equiv

(defthm num-val-fix-under-num-val-equiv
  (num-val-equiv (num-val-fix acl2::x)
                 acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-num-val-fix-1-forward-to-num-val-equiv

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

Theorem: equal-of-num-val-fix-2-forward-to-num-val-equiv

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

Theorem: num-val-equiv-of-num-val-fix-1-forward

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

Theorem: num-val-equiv-of-num-val-fix-2-forward

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

Theorem: num-val-kind$inline-of-num-val-fix-x

(defthm num-val-kind$inline-of-num-val-fix-x
  (equal (num-val-kind$inline (num-val-fix x))
         (num-val-kind$inline x)))

Theorem: num-val-kind$inline-num-val-equiv-congruence-on-x

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

Theorem: consp-of-num-val-fix

(defthm consp-of-num-val-fix
  (consp (num-val-fix x))
  :rule-classes :type-prescription)