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Theorem

Theorem-fix

Fixing function for theorem structures.

Signature
(theorem-fix x) → new-x
Arguments: x — Guard (theoremp x).
Returns: new-x — Type (theoremp new-x).

Definitions and Theorems

Function: theorem-fix$inline

(defun theorem-fix$inline (x)
 (declare (xargs :guard (theoremp x)))
 (let ((__function__ 'theorem-fix))
  (declare (ignorable __function__))
  (mbe
    :logic
    (b*
      ((name (identifier-fix (cdr (std::da-nth 0 x))))
       (variables (typed-variable-list-fix (cdr (std::da-nth 1 x))))
       (formula (expression-fix (cdr (std::da-nth 2 x)))))
      (list (cons 'name name)
            (cons 'variables variables)
            (cons 'formula formula)))
    :exec x)))

Theorem: theoremp-of-theorem-fix

(defthm theoremp-of-theorem-fix
  (b* ((new-x (theorem-fix$inline x)))
    (theoremp new-x))
  :rule-classes :rewrite)

Theorem: theorem-fix-when-theoremp

(defthm theorem-fix-when-theoremp
  (implies (theoremp x)
           (equal (theorem-fix x) x)))

Function: theorem-equiv$inline

(defun theorem-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (theoremp acl2::x)
                              (theoremp acl2::y))))
  (equal (theorem-fix acl2::x)
         (theorem-fix acl2::y)))

Theorem: theorem-equiv-is-an-equivalence

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

Theorem: theorem-equiv-implies-equal-theorem-fix-1

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

Theorem: theorem-fix-under-theorem-equiv

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

Theorem: equal-of-theorem-fix-1-forward-to-theorem-equiv

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

Theorem: equal-of-theorem-fix-2-forward-to-theorem-equiv

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

Theorem: theorem-equiv-of-theorem-fix-1-forward

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

Theorem: theorem-equiv-of-theorem-fix-2-forward

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