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Tyname

Tyname-fix

Fixing function for tyname structures.

Signature
(tyname-fix x) → new-x
Arguments: x — Guard (tynamep x).
Returns: new-x — Type (tynamep new-x).

Definitions and Theorems

Function: tyname-fix$inline

(defun tyname-fix$inline (x)
 (declare (xargs :guard (tynamep x)))
 (let ((__function__ 'tyname-fix))
  (declare (ignorable __function__))
  (mbe
      :logic
      (b* ((tyspec (tyspecseq-fix (cdr (std::da-nth 0 (cdr x)))))
           (declor (obj-adeclor-fix (cdr (std::da-nth 1 (cdr x))))))
        (cons :tyname (list (cons 'tyspec tyspec)
                            (cons 'declor declor))))
      :exec x)))

Theorem: tynamep-of-tyname-fix

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

Theorem: tyname-fix-when-tynamep

(defthm tyname-fix-when-tynamep
  (implies (tynamep x)
           (equal (tyname-fix x) x)))

Function: tyname-equiv$inline

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

Theorem: tyname-equiv-is-an-equivalence

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

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

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

Theorem: tyname-fix-under-tyname-equiv

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

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

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

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

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

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

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

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

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