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Ctrex-ruletable

Ctrex-ruletable-fix

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

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

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

Definitions and Theorems

Function: ctrex-ruletable-fix$inline

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

Theorem: ctrex-ruletable-p-of-ctrex-ruletable-fix

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

Theorem: ctrex-ruletable-fix-when-ctrex-ruletable-p

(defthm ctrex-ruletable-fix-when-ctrex-ruletable-p
  (implies (ctrex-ruletable-p x)
           (equal (ctrex-ruletable-fix x) x)))

Function: ctrex-ruletable-equiv$inline

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

Theorem: ctrex-ruletable-equiv-is-an-equivalence

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

Theorem: ctrex-ruletable-equiv-implies-equal-ctrex-ruletable-fix-1

(defthm ctrex-ruletable-equiv-implies-equal-ctrex-ruletable-fix-1
  (implies (ctrex-ruletable-equiv x x-equiv)
           (equal (ctrex-ruletable-fix x)
                  (ctrex-ruletable-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ctrex-ruletable-fix-under-ctrex-ruletable-equiv

(defthm ctrex-ruletable-fix-under-ctrex-ruletable-equiv
  (ctrex-ruletable-equiv (ctrex-ruletable-fix x)
                         x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ctrex-ruletable-fix-1-forward-to-ctrex-ruletable-equiv

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

Theorem: equal-of-ctrex-ruletable-fix-2-forward-to-ctrex-ruletable-equiv

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

Theorem: ctrex-ruletable-equiv-of-ctrex-ruletable-fix-1-forward

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

Theorem: ctrex-ruletable-equiv-of-ctrex-ruletable-fix-2-forward

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

Theorem: cons-of-ctrex-rulelist-fix-v-under-ctrex-ruletable-equiv

(defthm cons-of-ctrex-rulelist-fix-v-under-ctrex-ruletable-equiv
  (ctrex-ruletable-equiv (cons (cons k (ctrex-rulelist-fix v)) x)
                         (cons (cons k v) x)))

Theorem: cons-ctrex-rulelist-equiv-congruence-on-v-under-ctrex-ruletable-equiv

(defthm
 cons-ctrex-rulelist-equiv-congruence-on-v-under-ctrex-ruletable-equiv
 (implies (ctrex-rulelist-equiv v v-equiv)
          (ctrex-ruletable-equiv (cons (cons k v) x)
                                 (cons (cons k v-equiv) x)))
 :rule-classes :congruence)

Theorem: cons-of-ctrex-ruletable-fix-y-under-ctrex-ruletable-equiv

(defthm cons-of-ctrex-ruletable-fix-y-under-ctrex-ruletable-equiv
  (ctrex-ruletable-equiv (cons x (ctrex-ruletable-fix y))
                         (cons x y)))

Theorem: cons-ctrex-ruletable-equiv-congruence-on-y-under-ctrex-ruletable-equiv

(defthm
 cons-ctrex-ruletable-equiv-congruence-on-y-under-ctrex-ruletable-equiv
 (implies (ctrex-ruletable-equiv y y-equiv)
          (ctrex-ruletable-equiv (cons x y)
                                 (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: ctrex-ruletable-fix-of-acons

(defthm ctrex-ruletable-fix-of-acons
  (equal (ctrex-ruletable-fix (cons (cons a b) x))
         (let ((rest (ctrex-ruletable-fix x)))
           (if (and (pseudo-fnsym-p a))
               (let ((fty::first-key a)
                     (fty::first-val (ctrex-rulelist-fix b)))
                 (cons (cons fty::first-key fty::first-val)
                       rest))
             rest))))

Theorem: hons-assoc-equal-of-ctrex-ruletable-fix

(defthm hons-assoc-equal-of-ctrex-ruletable-fix
  (equal (hons-assoc-equal k (ctrex-ruletable-fix x))
         (let ((fty::pair (hons-assoc-equal k x)))
           (and (pseudo-fnsym-p k)
                fty::pair
                (cons k
                      (ctrex-rulelist-fix (cdr fty::pair)))))))

Theorem: ctrex-ruletable-fix-of-append

(defthm ctrex-ruletable-fix-of-append
  (equal (ctrex-ruletable-fix (append std::a std::b))
         (append (ctrex-ruletable-fix std::a)
                 (ctrex-ruletable-fix std::b))))

Theorem: consp-car-of-ctrex-ruletable-fix

(defthm consp-car-of-ctrex-ruletable-fix
  (equal (consp (car (ctrex-ruletable-fix x)))
         (consp (ctrex-ruletable-fix x))))