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Natoption/natoptionlist

Natoption/natoptionlist-fix

Fixing function for natoption/natoptionlist structures.

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

Definitions and Theorems

Function: natoption/natoptionlist-fix$inline

(defun natoption/natoptionlist-fix$inline (x)
 (declare (xargs :guard (natoption/natoptionlist-p x)))
 (let ((__function__ 'natoption/natoptionlist-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (case (natoption/natoptionlist-kind x)
    (:one (b* ((get (nat-option-fix (std::da-nth 0 (cdr x)))))
            (cons :one (list get))))
    (:list (b* ((get (nat-option-list-fix (std::da-nth 0 (cdr x)))))
             (cons :list (list get)))))
   :exec x)))

Theorem: natoption/natoptionlist-p-of-natoption/natoptionlist-fix

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

Theorem: natoption/natoptionlist-fix-when-natoption/natoptionlist-p

(defthm natoption/natoptionlist-fix-when-natoption/natoptionlist-p
  (implies (natoption/natoptionlist-p x)
           (equal (natoption/natoptionlist-fix x)
                  x)))

Function: natoption/natoptionlist-equiv$inline

(defun natoption/natoptionlist-equiv$inline (x y)
  (declare (xargs :guard (and (natoption/natoptionlist-p x)
                              (natoption/natoptionlist-p y))))
  (equal (natoption/natoptionlist-fix x)
         (natoption/natoptionlist-fix y)))

Theorem: natoption/natoptionlist-equiv-is-an-equivalence

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

Theorem: natoption/natoptionlist-equiv-implies-equal-natoption/natoptionlist-fix-1

(defthm
 natoption/natoptionlist-equiv-implies-equal-natoption/natoptionlist-fix-1
 (implies (natoption/natoptionlist-equiv x x-equiv)
          (equal (natoption/natoptionlist-fix x)
                 (natoption/natoptionlist-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: natoption/natoptionlist-fix-under-natoption/natoptionlist-equiv

(defthm
    natoption/natoptionlist-fix-under-natoption/natoptionlist-equiv
  (natoption/natoptionlist-equiv (natoption/natoptionlist-fix x)
                                 x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-natoption/natoptionlist-fix-1-forward-to-natoption/natoptionlist-equiv

(defthm
 equal-of-natoption/natoptionlist-fix-1-forward-to-natoption/natoptionlist-equiv
 (implies (equal (natoption/natoptionlist-fix x)
                 y)
          (natoption/natoptionlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-natoption/natoptionlist-fix-2-forward-to-natoption/natoptionlist-equiv

(defthm
 equal-of-natoption/natoptionlist-fix-2-forward-to-natoption/natoptionlist-equiv
 (implies (equal x (natoption/natoptionlist-fix y))
          (natoption/natoptionlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: natoption/natoptionlist-equiv-of-natoption/natoptionlist-fix-1-forward

(defthm
 natoption/natoptionlist-equiv-of-natoption/natoptionlist-fix-1-forward
 (implies
      (natoption/natoptionlist-equiv (natoption/natoptionlist-fix x)
                                     y)
      (natoption/natoptionlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: natoption/natoptionlist-equiv-of-natoption/natoptionlist-fix-2-forward

(defthm
 natoption/natoptionlist-equiv-of-natoption/natoptionlist-fix-2-forward
 (implies
   (natoption/natoptionlist-equiv x (natoption/natoptionlist-fix y))
   (natoption/natoptionlist-equiv x y))
 :rule-classes :forward-chaining)

Theorem: natoption/natoptionlist-kind$inline-of-natoption/natoptionlist-fix-x

(defthm
 natoption/natoptionlist-kind$inline-of-natoption/natoptionlist-fix-x
 (equal (natoption/natoptionlist-kind$inline
             (natoption/natoptionlist-fix x))
        (natoption/natoptionlist-kind$inline x)))

Theorem: natoption/natoptionlist-kind$inline-natoption/natoptionlist-equiv-congruence-on-x

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

Theorem: consp-of-natoption/natoptionlist-fix

(defthm consp-of-natoption/natoptionlist-fix
  (consp (natoption/natoptionlist-fix x))
  :rule-classes :type-prescription)