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Let-binding

Let-binding-fix

Fixing function for let-binding structures.

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

Definitions and Theorems

Function: let-binding-fix$inline

(defun let-binding-fix$inline (x)
 (declare (xargs :guard (let-binding-p x)))
 (let ((acl2::__function__ 'let-binding-fix))
  (declare (ignorable acl2::__function__))
  (mbe
     :logic
     (b* ((bindings (binding-list-fix (cdr (std::da-nth 0 x))))
          (hypotheses (hint-pair-list-fix (cdr (std::da-nth 1 x)))))
       (list (cons 'bindings bindings)
             (cons 'hypotheses hypotheses)))
     :exec x)))

Theorem: let-binding-p-of-let-binding-fix

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

Theorem: let-binding-fix-when-let-binding-p

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

Function: let-binding-equiv$inline

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

Theorem: let-binding-equiv-is-an-equivalence

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

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

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

Theorem: let-binding-fix-under-let-binding-equiv

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

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

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

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

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

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

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

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

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