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Ident

Ident-fix

Fixing function for ident structures.

Signature
(ident-fix x) → new-x
Arguments: x — Guard (identp x).
Returns: new-x — Type (identp new-x).

Definitions and Theorems

Function: ident-fix$inline

(defun ident-fix$inline (x)
  (declare (xargs :guard (identp x)))
  (let ((__function__ 'ident-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (b* ((name (str-fix (cdr (std::da-nth 0 (cdr x))))))
           (cons :ident (list (cons 'name name))))
         :exec x)))

Theorem: identp-of-ident-fix

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

Theorem: ident-fix-when-identp

(defthm ident-fix-when-identp
  (implies (identp x)
           (equal (ident-fix x) x)))

Function: ident-equiv$inline

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

Theorem: ident-equiv-is-an-equivalence

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

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

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

Theorem: ident-fix-under-ident-equiv

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

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

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

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

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

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

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

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

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