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Maybe-theorem

Maybe-theorem-fix

Fixing function for maybe-theorem structures.

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

Definitions and Theorems

Function: maybe-theorem-fix$inline

(defun maybe-theorem-fix$inline (x)
  (declare (xargs :guard (maybe-theoremp x)))
  (let ((__function__ 'maybe-theorem-fix))
    (declare (ignorable __function__))
    (mbe :logic (cond ((not x) nil)
                      (t (b* ((fty::val (theorem-fix x)))
                           fty::val)))
         :exec x)))

Theorem: maybe-theoremp-of-maybe-theorem-fix

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

Theorem: maybe-theorem-fix-when-maybe-theoremp

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

Function: maybe-theorem-equiv$inline

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

Theorem: maybe-theorem-equiv-is-an-equivalence

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

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

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

Theorem: maybe-theorem-fix-under-maybe-theorem-equiv

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

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

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

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

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

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

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

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

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