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Tree-set

Tree-set-fix

(tree-set-fix x) is a usual ACL2::fty set fixing function.

Signature
(tree-set-fix x) → *
Arguments: x — Guard (tree-setp x).

In the logic, we apply tree-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: tree-set-fix

(defun tree-set-fix (x)
  (declare (xargs :guard (tree-setp x)))
  (mbe :logic (if (tree-setp x) x nil)
       :exec x))

Theorem: tree-setp-of-tree-set-fix

(defthm tree-setp-of-tree-set-fix
  (tree-setp (tree-set-fix x)))

Theorem: tree-set-fix-when-tree-setp

(defthm tree-set-fix-when-tree-setp
  (implies (tree-setp x)
           (equal (tree-set-fix x) x)))

Theorem: emptyp-tree-set-fix

(defthm emptyp-tree-set-fix
  (implies (or (emptyp x) (not (tree-setp x)))
           (emptyp (tree-set-fix x))))

Theorem: emptyp-of-tree-set-fix

(defthm emptyp-of-tree-set-fix
  (equal (emptyp (tree-set-fix x))
         (or (not (tree-setp x)) (emptyp x))))

Function: tree-set-equiv$inline

(defun tree-set-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (tree-setp acl2::x)
                              (tree-setp acl2::y))))
  (equal (tree-set-fix acl2::x)
         (tree-set-fix acl2::y)))

Theorem: tree-set-equiv-is-an-equivalence

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

Theorem: tree-set-equiv-implies-equal-tree-set-fix-1

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

Theorem: tree-set-fix-under-tree-set-equiv

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

Theorem: equal-of-tree-set-fix-1-forward-to-tree-set-equiv

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

Theorem: equal-of-tree-set-fix-2-forward-to-tree-set-equiv

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

Theorem: tree-set-equiv-of-tree-set-fix-1-forward

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

Theorem: tree-set-equiv-of-tree-set-fix-2-forward

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