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Address+pos

Address+pos-fix

Fixing function for address+pos structures.

Signature
(address+pos-fix x) → new-x
Arguments: x — Guard (address+posp x).
Returns: new-x — Type (address+posp new-x).

Definitions and Theorems

Function: address+pos-fix$inline

(defun address+pos-fix$inline (x)
  (declare (xargs :guard (address+posp x)))
  (let ((__function__ 'address+pos-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((address (address-fix (cdr (std::da-nth 0 x))))
              (pos (pos-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'address address)
                 (cons 'pos pos)))
         :exec x)))

Theorem: address+posp-of-address+pos-fix

(defthm address+posp-of-address+pos-fix
  (b* ((new-x (address+pos-fix$inline x)))
    (address+posp new-x))
  :rule-classes :rewrite)

Theorem: address+pos-fix-when-address+posp

(defthm address+pos-fix-when-address+posp
  (implies (address+posp x)
           (equal (address+pos-fix x) x)))

Function: address+pos-equiv$inline

(defun address+pos-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (address+posp acl2::x)
                              (address+posp acl2::y))))
  (equal (address+pos-fix acl2::x)
         (address+pos-fix acl2::y)))

Theorem: address+pos-equiv-is-an-equivalence

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

Theorem: address+pos-equiv-implies-equal-address+pos-fix-1

(defthm address+pos-equiv-implies-equal-address+pos-fix-1
  (implies (address+pos-equiv acl2::x x-equiv)
           (equal (address+pos-fix acl2::x)
                  (address+pos-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: address+pos-fix-under-address+pos-equiv

(defthm address+pos-fix-under-address+pos-equiv
  (address+pos-equiv (address+pos-fix acl2::x)
                     acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-address+pos-fix-1-forward-to-address+pos-equiv

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

Theorem: equal-of-address+pos-fix-2-forward-to-address+pos-equiv

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

Theorem: address+pos-equiv-of-address+pos-fix-1-forward

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

Theorem: address+pos-equiv-of-address+pos-fix-2-forward

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