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Type-recursion

Type-recursion-fix

Fixing function for type-recursion structures.

Signature
(type-recursion-fix x) → new-x
Arguments: x — Guard (type-recursionp x).
Returns: new-x — Type (type-recursionp new-x).

Definitions and Theorems

Function: type-recursion-fix$inline

(defun type-recursion-fix$inline (x)
 (declare (xargs :guard (type-recursionp x)))
 (let ((__function__ 'type-recursion-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((definitions
                (type-definition-list-fix (cdr (std::da-nth 0 x)))))
        (list (cons 'definitions definitions)))
      :exec x)))

Theorem: type-recursionp-of-type-recursion-fix

(defthm type-recursionp-of-type-recursion-fix
  (b* ((new-x (type-recursion-fix$inline x)))
    (type-recursionp new-x))
  :rule-classes :rewrite)

Theorem: type-recursion-fix-when-type-recursionp

(defthm type-recursion-fix-when-type-recursionp
  (implies (type-recursionp x)
           (equal (type-recursion-fix x) x)))

Function: type-recursion-equiv$inline

(defun type-recursion-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (type-recursionp acl2::x)
                              (type-recursionp acl2::y))))
  (equal (type-recursion-fix acl2::x)
         (type-recursion-fix acl2::y)))

Theorem: type-recursion-equiv-is-an-equivalence

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

Theorem: type-recursion-equiv-implies-equal-type-recursion-fix-1

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

Theorem: type-recursion-fix-under-type-recursion-equiv

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

Theorem: equal-of-type-recursion-fix-1-forward-to-type-recursion-equiv

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

Theorem: equal-of-type-recursion-fix-2-forward-to-type-recursion-equiv

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

Theorem: type-recursion-equiv-of-type-recursion-fix-1-forward

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

Theorem: type-recursion-equiv-of-type-recursion-fix-2-forward

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