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Input-section

Input-section-fix

Fixing function for input-section structures.

Signature
(input-section-fix x) → new-x
Arguments: x — Guard (input-sectionp x).
Returns: new-x — Type (input-sectionp new-x).

Definitions and Theorems

Function: input-section-fix$inline

(defun input-section-fix$inline (x)
 (declare (xargs :guard (input-sectionp x)))
 (let ((__function__ 'input-section-fix))
  (declare (ignorable __function__))
  (mbe
   :logic
   (b* ((title (input-title-fix (cdr (std::da-nth 0 (cdr x)))))
        (items (input-item-list-fix (cdr (std::da-nth 1 (cdr x))))))
     (cons :input-section (list (cons 'title title)
                                (cons 'items items))))
   :exec x)))

Theorem: input-sectionp-of-input-section-fix

(defthm input-sectionp-of-input-section-fix
  (b* ((new-x (input-section-fix$inline x)))
    (input-sectionp new-x))
  :rule-classes :rewrite)

Theorem: input-section-fix-when-input-sectionp

(defthm input-section-fix-when-input-sectionp
  (implies (input-sectionp x)
           (equal (input-section-fix x) x)))

Function: input-section-equiv$inline

(defun input-section-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (input-sectionp acl2::x)
                              (input-sectionp acl2::y))))
  (equal (input-section-fix acl2::x)
         (input-section-fix acl2::y)))

Theorem: input-section-equiv-is-an-equivalence

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

Theorem: input-section-equiv-implies-equal-input-section-fix-1

(defthm input-section-equiv-implies-equal-input-section-fix-1
  (implies (input-section-equiv acl2::x x-equiv)
           (equal (input-section-fix acl2::x)
                  (input-section-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: input-section-fix-under-input-section-equiv

(defthm input-section-fix-under-input-section-equiv
  (input-section-equiv (input-section-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-input-section-fix-1-forward-to-input-section-equiv

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

Theorem: equal-of-input-section-fix-2-forward-to-input-section-equiv

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

Theorem: input-section-equiv-of-input-section-fix-1-forward

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

Theorem: input-section-equiv-of-input-section-fix-2-forward

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