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Sig-table

Sig-table-fix

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

Signature
(sig-table-fix x) → fty::newx
Arguments: x — Guard (sig-table-p x).
Returns: fty::newx — Type (sig-table-p fty::newx).

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

Definitions and Theorems

Function: sig-table-fix$inline

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

Theorem: sig-table-p-of-sig-table-fix

(defthm sig-table-p-of-sig-table-fix
  (b* ((fty::newx (sig-table-fix$inline x)))
    (sig-table-p fty::newx))
  :rule-classes :rewrite)

Theorem: sig-table-fix-when-sig-table-p

(defthm sig-table-fix-when-sig-table-p
  (implies (sig-table-p x)
           (equal (sig-table-fix x) x)))

Function: sig-table-equiv$inline

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

Theorem: sig-table-equiv-is-an-equivalence

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

Theorem: sig-table-equiv-implies-equal-sig-table-fix-1

(defthm sig-table-equiv-implies-equal-sig-table-fix-1
  (implies (sig-table-equiv x x-equiv)
           (equal (sig-table-fix x)
                  (sig-table-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: sig-table-fix-under-sig-table-equiv

(defthm sig-table-fix-under-sig-table-equiv
  (sig-table-equiv (sig-table-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-sig-table-fix-1-forward-to-sig-table-equiv

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

Theorem: equal-of-sig-table-fix-2-forward-to-sig-table-equiv

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

Theorem: sig-table-equiv-of-sig-table-fix-1-forward

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

Theorem: sig-table-equiv-of-sig-table-fix-2-forward

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

Theorem: cons-of-fgl-object-bindingslist-fix-v-under-sig-table-equiv

(defthm cons-of-fgl-object-bindingslist-fix-v-under-sig-table-equiv
  (sig-table-equiv (cons (cons k (fgl-object-bindingslist-fix v))
                         x)
                   (cons (cons k v) x)))

Theorem: cons-fgl-object-bindingslist-equiv-congruence-on-v-under-sig-table-equiv

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

Theorem: cons-of-sig-table-fix-y-under-sig-table-equiv

(defthm cons-of-sig-table-fix-y-under-sig-table-equiv
  (sig-table-equiv (cons x (sig-table-fix y))
                   (cons x y)))

Theorem: cons-sig-table-equiv-congruence-on-y-under-sig-table-equiv

(defthm cons-sig-table-equiv-congruence-on-y-under-sig-table-equiv
  (implies (sig-table-equiv y y-equiv)
           (sig-table-equiv (cons x y)
                            (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: sig-table-fix-of-acons

(defthm sig-table-fix-of-acons
  (equal
       (sig-table-fix (cons (cons a b) x))
       (let ((rest (sig-table-fix x)))
         (if (and (fgl-objectlist-p a))
             (let ((fty::first-key a)
                   (fty::first-val (fgl-object-bindingslist-fix b)))
               (cons (cons fty::first-key fty::first-val)
                     rest))
           rest))))

Theorem: hons-assoc-equal-of-sig-table-fix

(defthm hons-assoc-equal-of-sig-table-fix
 (equal
     (hons-assoc-equal k (sig-table-fix x))
     (let ((fty::pair (hons-assoc-equal k x)))
       (and (fgl-objectlist-p k)
            fty::pair
            (cons k
                  (fgl-object-bindingslist-fix (cdr fty::pair)))))))

Theorem: sig-table-fix-of-append

(defthm sig-table-fix-of-append
  (equal (sig-table-fix (append std::a std::b))
         (append (sig-table-fix std::a)
                 (sig-table-fix std::b))))

Theorem: consp-car-of-sig-table-fix

(defthm consp-car-of-sig-table-fix
  (equal (consp (car (sig-table-fix x)))
         (consp (sig-table-fix x))))