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Tree-list-tuple3

Tree-list-tuple3-fix

Fixing function for tree-list-tuple3 structures.

Signature
(tree-list-tuple3-fix x) → new-x
Arguments: x — Guard (tree-list-tuple3p x).
Returns: new-x — Type (tree-list-tuple3p new-x).

Definitions and Theorems

Function: tree-list-tuple3-fix$inline

(defun tree-list-tuple3-fix$inline (x)
  (declare (xargs :guard (tree-list-tuple3p x)))
  (let ((__function__ 'tree-list-tuple3-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((1st (tree-list-fix (cdr (std::da-nth 0 x))))
              (2nd (tree-list-fix (cdr (std::da-nth 1 x))))
              (3rd (tree-list-fix (cdr (std::da-nth 2 x)))))
           (list (cons '1st 1st)
                 (cons '2nd 2nd)
                 (cons '3rd 3rd)))
         :exec x)))

Theorem: tree-list-tuple3p-of-tree-list-tuple3-fix

(defthm tree-list-tuple3p-of-tree-list-tuple3-fix
  (b* ((new-x (tree-list-tuple3-fix$inline x)))
    (tree-list-tuple3p new-x))
  :rule-classes :rewrite)

Theorem: tree-list-tuple3-fix-when-tree-list-tuple3p

(defthm tree-list-tuple3-fix-when-tree-list-tuple3p
  (implies (tree-list-tuple3p x)
           (equal (tree-list-tuple3-fix x) x)))

Function: tree-list-tuple3-equiv$inline

(defun tree-list-tuple3-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (tree-list-tuple3p acl2::x)
                              (tree-list-tuple3p acl2::y))))
  (equal (tree-list-tuple3-fix acl2::x)
         (tree-list-tuple3-fix acl2::y)))

Theorem: tree-list-tuple3-equiv-is-an-equivalence

(defthm tree-list-tuple3-equiv-is-an-equivalence
  (and (booleanp (tree-list-tuple3-equiv x y))
       (tree-list-tuple3-equiv x x)
       (implies (tree-list-tuple3-equiv x y)
                (tree-list-tuple3-equiv y x))
       (implies (and (tree-list-tuple3-equiv x y)
                     (tree-list-tuple3-equiv y z))
                (tree-list-tuple3-equiv x z)))
  :rule-classes (:equivalence))

Theorem: tree-list-tuple3-equiv-implies-equal-tree-list-tuple3-fix-1

(defthm tree-list-tuple3-equiv-implies-equal-tree-list-tuple3-fix-1
  (implies (tree-list-tuple3-equiv acl2::x x-equiv)
           (equal (tree-list-tuple3-fix acl2::x)
                  (tree-list-tuple3-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: tree-list-tuple3-fix-under-tree-list-tuple3-equiv

(defthm tree-list-tuple3-fix-under-tree-list-tuple3-equiv
  (tree-list-tuple3-equiv (tree-list-tuple3-fix acl2::x)
                          acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-tree-list-tuple3-fix-1-forward-to-tree-list-tuple3-equiv

(defthm
  equal-of-tree-list-tuple3-fix-1-forward-to-tree-list-tuple3-equiv
  (implies (equal (tree-list-tuple3-fix acl2::x)
                  acl2::y)
           (tree-list-tuple3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-tree-list-tuple3-fix-2-forward-to-tree-list-tuple3-equiv

(defthm
  equal-of-tree-list-tuple3-fix-2-forward-to-tree-list-tuple3-equiv
  (implies (equal acl2::x (tree-list-tuple3-fix acl2::y))
           (tree-list-tuple3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: tree-list-tuple3-equiv-of-tree-list-tuple3-fix-1-forward

(defthm tree-list-tuple3-equiv-of-tree-list-tuple3-fix-1-forward
  (implies (tree-list-tuple3-equiv (tree-list-tuple3-fix acl2::x)
                                   acl2::y)
           (tree-list-tuple3-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: tree-list-tuple3-equiv-of-tree-list-tuple3-fix-2-forward

(defthm tree-list-tuple3-equiv-of-tree-list-tuple3-fix-2-forward
 (implies
     (tree-list-tuple3-equiv acl2::x (tree-list-tuple3-fix acl2::y))
     (tree-list-tuple3-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)