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Fgl-object-bindings

Fgl-object-bindings-fix

(fgl-object-bindings-fix x) is an ACL2::fty alist fixing function that follows the drop-keys strategy.

Signature
(fgl-object-bindings-fix x) → fty::newx
Arguments: x — Guard (fgl-object-bindings-p x).
Returns: fty::newx — Type (fgl-object-bindings-p fty::newx).

Note that in the execution this is just an inline identity function.

Definitions and Theorems

Function: fgl-object-bindings-fix$inline

(defun fgl-object-bindings-fix$inline (x)
  (declare (xargs :guard (fgl-object-bindings-p x)))
  (let ((__function__ 'fgl-object-bindings-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (let ((rest (fgl-object-bindings-fix (cdr x))))
             (if (and (consp (car x))
                      (pseudo-var-p (caar x)))
                 (let ((fty::first-key (caar x))
                       (fty::first-val (fgl-object-fix (cdar x))))
                   (cons (cons fty::first-key fty::first-val)
                         rest))
               rest)))
         :exec x)))

Theorem: fgl-object-bindings-p-of-fgl-object-bindings-fix

(defthm fgl-object-bindings-p-of-fgl-object-bindings-fix
  (b* ((fty::newx (fgl-object-bindings-fix$inline x)))
    (fgl-object-bindings-p fty::newx))
  :rule-classes :rewrite)

Theorem: fgl-object-bindings-fix-when-fgl-object-bindings-p

(defthm fgl-object-bindings-fix-when-fgl-object-bindings-p
  (implies (fgl-object-bindings-p x)
           (equal (fgl-object-bindings-fix x) x)))

Function: fgl-object-bindings-equiv$inline

(defun fgl-object-bindings-equiv$inline (x y)
  (declare (xargs :guard (and (fgl-object-bindings-p x)
                              (fgl-object-bindings-p y))))
  (equal (fgl-object-bindings-fix x)
         (fgl-object-bindings-fix y)))

Theorem: fgl-object-bindings-equiv-is-an-equivalence

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

Theorem: fgl-object-bindings-equiv-implies-equal-fgl-object-bindings-fix-1

(defthm
  fgl-object-bindings-equiv-implies-equal-fgl-object-bindings-fix-1
  (implies (fgl-object-bindings-equiv x x-equiv)
           (equal (fgl-object-bindings-fix x)
                  (fgl-object-bindings-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: fgl-object-bindings-fix-under-fgl-object-bindings-equiv

(defthm fgl-object-bindings-fix-under-fgl-object-bindings-equiv
  (fgl-object-bindings-equiv (fgl-object-bindings-fix x)
                             x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-fgl-object-bindings-fix-1-forward-to-fgl-object-bindings-equiv

(defthm
 equal-of-fgl-object-bindings-fix-1-forward-to-fgl-object-bindings-equiv
 (implies (equal (fgl-object-bindings-fix x) y)
          (fgl-object-bindings-equiv x y))
 :rule-classes :forward-chaining)

Theorem: equal-of-fgl-object-bindings-fix-2-forward-to-fgl-object-bindings-equiv

(defthm
 equal-of-fgl-object-bindings-fix-2-forward-to-fgl-object-bindings-equiv
 (implies (equal x (fgl-object-bindings-fix y))
          (fgl-object-bindings-equiv x y))
 :rule-classes :forward-chaining)

Theorem: fgl-object-bindings-equiv-of-fgl-object-bindings-fix-1-forward

(defthm
     fgl-object-bindings-equiv-of-fgl-object-bindings-fix-1-forward
  (implies (fgl-object-bindings-equiv (fgl-object-bindings-fix x)
                                      y)
           (fgl-object-bindings-equiv x y))
  :rule-classes :forward-chaining)

Theorem: fgl-object-bindings-equiv-of-fgl-object-bindings-fix-2-forward

(defthm
     fgl-object-bindings-equiv-of-fgl-object-bindings-fix-2-forward
  (implies (fgl-object-bindings-equiv x (fgl-object-bindings-fix y))
           (fgl-object-bindings-equiv x y))
  :rule-classes :forward-chaining)

Theorem: cons-of-fgl-object-fix-v-under-fgl-object-bindings-equiv

(defthm cons-of-fgl-object-fix-v-under-fgl-object-bindings-equiv
  (fgl-object-bindings-equiv (cons (cons k (fgl-object-fix v)) x)
                             (cons (cons k v) x)))

Theorem: cons-fgl-object-equiv-congruence-on-v-under-fgl-object-bindings-equiv

(defthm
 cons-fgl-object-equiv-congruence-on-v-under-fgl-object-bindings-equiv
 (implies (fgl-object-equiv v v-equiv)
          (fgl-object-bindings-equiv (cons (cons k v) x)
                                     (cons (cons k v-equiv) x)))
 :rule-classes :congruence)

Theorem: cons-of-fgl-object-bindings-fix-y-under-fgl-object-bindings-equiv

(defthm
  cons-of-fgl-object-bindings-fix-y-under-fgl-object-bindings-equiv
  (fgl-object-bindings-equiv (cons x (fgl-object-bindings-fix y))
                             (cons x y)))

Theorem: cons-fgl-object-bindings-equiv-congruence-on-y-under-fgl-object-bindings-equiv

(defthm
 cons-fgl-object-bindings-equiv-congruence-on-y-under-fgl-object-bindings-equiv
 (implies (fgl-object-bindings-equiv y y-equiv)
          (fgl-object-bindings-equiv (cons x y)
                                     (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: fgl-object-bindings-fix-of-acons

(defthm fgl-object-bindings-fix-of-acons
  (equal (fgl-object-bindings-fix (cons (cons a b) x))
         (let ((rest (fgl-object-bindings-fix x)))
           (if (and (pseudo-var-p a))
               (let ((fty::first-key a)
                     (fty::first-val (fgl-object-fix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-fgl-object-bindings-fix

(defthm hons-assoc-equal-of-fgl-object-bindings-fix
  (equal (hons-assoc-equal k (fgl-object-bindings-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and (pseudo-var-p k)
                fty::pair
                (cons k (fgl-object-fix (cdr fty::pair)))))))

Theorem: fgl-object-bindings-fix-of-append

(defthm fgl-object-bindings-fix-of-append
  (equal (fgl-object-bindings-fix (append std::a std::b))
         (append (fgl-object-bindings-fix std::a)
                 (fgl-object-bindings-fix std::b))))

Theorem: consp-car-of-fgl-object-bindings-fix

(defthm consp-car-of-fgl-object-bindings-fix
  (equal (consp (car (fgl-object-bindings-fix x)))
         (consp (fgl-object-bindings-fix x))))