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Coordinate

Coordinate-fix

Fixing function for coordinate structures.

Signature
(coordinate-fix x) → new-x
Arguments: x — Guard (coordinatep x).
Returns: new-x — Type (coordinatep new-x).

Definitions and Theorems

Function: coordinate-fix$inline

(defun coordinate-fix$inline (x)
  (declare (xargs :guard (coordinatep x)))
  (let ((__function__ 'coordinate-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (coordinate-kind x)
           (:explicit (b* ((get (ifix (std::da-nth 0 (cdr x)))))
                        (cons :explicit (list get))))
           (:sign-high (cons :sign-high (list)))
           (:sign-low (cons :sign-low (list)))
           (:inferred (cons :inferred (list))))
         :exec x)))

Theorem: coordinatep-of-coordinate-fix

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

Theorem: coordinate-fix-when-coordinatep

(defthm coordinate-fix-when-coordinatep
  (implies (coordinatep x)
           (equal (coordinate-fix x) x)))

Function: coordinate-equiv$inline

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

Theorem: coordinate-equiv-is-an-equivalence

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

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

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

Theorem: coordinate-fix-under-coordinate-equiv

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

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

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

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

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

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

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

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

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

Theorem: coordinate-kind$inline-of-coordinate-fix-x

(defthm coordinate-kind$inline-of-coordinate-fix-x
  (equal (coordinate-kind$inline (coordinate-fix x))
         (coordinate-kind$inline x)))

Theorem: coordinate-kind$inline-coordinate-equiv-congruence-on-x

(defthm coordinate-kind$inline-coordinate-equiv-congruence-on-x
  (implies (coordinate-equiv x x-equiv)
           (equal (coordinate-kind$inline x)
                  (coordinate-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-coordinate-fix

(defthm consp-of-coordinate-fix
  (consp (coordinate-fix x))
  :rule-classes :type-prescription)