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Input-title

Input-title-fix

Fixing function for input-title structures.

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

Definitions and Theorems

Function: input-title-fix$inline

(defun input-title-fix$inline (x)
 (declare (xargs :guard (input-titlep x)))
 (let ((__function__ 'input-title-fix))
  (declare (ignorable __function__))
  (mbe
     :logic
     (b* ((sort (var/const-sort-fix (cdr (std::da-nth 0 (cdr x))))))
       (cons :input-title (list (cons 'sort sort))))
     :exec x)))

Theorem: input-titlep-of-input-title-fix

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

Theorem: input-title-fix-when-input-titlep

(defthm input-title-fix-when-input-titlep
  (implies (input-titlep x)
           (equal (input-title-fix x) x)))

Function: input-title-equiv$inline

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

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

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

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

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

Theorem: input-title-fix-under-input-title-equiv

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

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

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

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

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

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

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

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

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