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Sandwich

Sandwich-fix

Fixing function for sandwich structures.

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

Definitions and Theorems

Function: sandwich-fix$inline

(defun sandwich-fix$inline (x)
  (declare (xargs :guard (sandwich-p x)))
  (let ((acl2::__function__ 'sandwich-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((bread (ifix (cdr (std::da-nth 0 x))))
              (fillings (symbol-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'bread bread)
                 (cons 'fillings fillings)))
         :exec x)))

Theorem: sandwich-p-of-sandwich-fix

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

Theorem: sandwich-fix-when-sandwich-p

(defthm sandwich-fix-when-sandwich-p
  (implies (sandwich-p x)
           (equal (sandwich-fix x) x)))

Function: sandwich-equiv$inline

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

Theorem: sandwich-equiv-is-an-equivalence

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

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

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

Theorem: sandwich-fix-under-sandwich-equiv

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

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

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

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

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

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

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

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

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