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Binding

Binding-fix

Fixing function for binding structures.

Signature
(binding-fix x) → new-x
Arguments: x — Guard (binding-p x).
Returns: new-x — Type (binding-p new-x).

Definitions and Theorems

Function: binding-fix$inline

(defun binding-fix$inline (x)
  (declare (xargs :guard (binding-p x)))
  (let ((acl2::__function__ 'binding-fix))
    (declare (ignorable acl2::__function__))
    (mbe :logic
         (b* ((var (symbol-fix (cdr (std::da-nth 0 x))))
              (expr (pseudo-term-fix (cdr (std::da-nth 1 x))))
              (type (symbol-fix (cdr (std::da-nth 2 x)))))
           (list (cons 'var var)
                 (cons 'expr expr)
                 (cons 'type type)))
         :exec x)))

Theorem: binding-p-of-binding-fix

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

Theorem: binding-fix-when-binding-p

(defthm binding-fix-when-binding-p
  (implies (binding-p x)
           (equal (binding-fix x) x)))

Function: binding-equiv$inline

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

Theorem: binding-equiv-is-an-equivalence

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

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

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

Theorem: binding-fix-under-binding-equiv

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

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

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

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

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

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

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

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

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