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Alternative

Alternative-fix

Fixing function for alternative structures.

Signature
(alternative-fix x) → new-x
Arguments: x — Guard (alternativep x).
Returns: new-x — Type (alternativep new-x).

Definitions and Theorems

Function: alternative-fix$inline

(defun alternative-fix$inline (x)
  (declare (xargs :guard (alternativep x)))
  (let ((__function__ 'alternative-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((name (identifier-fix (cdr (std::da-nth 0 x))))
              (product (type-product-fix (cdr (std::da-nth 1 x)))))
           (list (cons 'name name)
                 (cons 'product product)))
         :exec x)))

Theorem: alternativep-of-alternative-fix

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

Theorem: alternative-fix-when-alternativep

(defthm alternative-fix-when-alternativep
  (implies (alternativep x)
           (equal (alternative-fix x) x)))

Function: alternative-equiv$inline

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

Theorem: alternative-equiv-is-an-equivalence

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

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

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

Theorem: alternative-fix-under-alternative-equiv

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

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

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

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

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

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

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

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

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