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Ctrex-rulelist

Ctrex-rulelist-fix

(ctrex-rulelist-fix x) is a usual ACL2::fty list fixing function.

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

In the logic, we apply ctrex-rule-fix to each member of the x. In the execution, none of that is actually necessary and this is just an inlined identity function.

Definitions and Theorems

Function: ctrex-rulelist-fix$inline

(defun ctrex-rulelist-fix$inline (x)
  (declare (xargs :guard (ctrex-rulelist-p x)))
  (let ((__function__ 'ctrex-rulelist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (ctrex-rule-fix (car x))
                 (ctrex-rulelist-fix (cdr x))))
         :exec x)))

Theorem: ctrex-rulelist-p-of-ctrex-rulelist-fix

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

Theorem: ctrex-rulelist-fix-when-ctrex-rulelist-p

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

Function: ctrex-rulelist-equiv$inline

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

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

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

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

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

Theorem: ctrex-rulelist-fix-under-ctrex-rulelist-equiv

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

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

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

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

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

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

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

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

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

Theorem: car-of-ctrex-rulelist-fix-x-under-ctrex-rule-equiv

(defthm car-of-ctrex-rulelist-fix-x-under-ctrex-rule-equiv
  (ctrex-rule-equiv (car (ctrex-rulelist-fix x))
                    (car x)))

Theorem: car-ctrex-rulelist-equiv-congruence-on-x-under-ctrex-rule-equiv

(defthm
    car-ctrex-rulelist-equiv-congruence-on-x-under-ctrex-rule-equiv
  (implies (ctrex-rulelist-equiv x x-equiv)
           (ctrex-rule-equiv (car x)
                             (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-ctrex-rulelist-fix-x-under-ctrex-rulelist-equiv

(defthm cdr-of-ctrex-rulelist-fix-x-under-ctrex-rulelist-equiv
  (ctrex-rulelist-equiv (cdr (ctrex-rulelist-fix x))
                        (cdr x)))

Theorem: cdr-ctrex-rulelist-equiv-congruence-on-x-under-ctrex-rulelist-equiv

(defthm
 cdr-ctrex-rulelist-equiv-congruence-on-x-under-ctrex-rulelist-equiv
 (implies (ctrex-rulelist-equiv x x-equiv)
          (ctrex-rulelist-equiv (cdr x)
                                (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-ctrex-rule-fix-x-under-ctrex-rulelist-equiv

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

Theorem: cons-ctrex-rule-equiv-congruence-on-x-under-ctrex-rulelist-equiv

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

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

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

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

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

Theorem: consp-of-ctrex-rulelist-fix

(defthm consp-of-ctrex-rulelist-fix
  (equal (consp (ctrex-rulelist-fix x))
         (consp x)))

Theorem: ctrex-rulelist-fix-under-iff

(defthm ctrex-rulelist-fix-under-iff
  (iff (ctrex-rulelist-fix x) (consp x)))

Theorem: ctrex-rulelist-fix-of-cons

(defthm ctrex-rulelist-fix-of-cons
  (equal (ctrex-rulelist-fix (cons a x))
         (cons (ctrex-rule-fix a)
               (ctrex-rulelist-fix x))))

Theorem: len-of-ctrex-rulelist-fix

(defthm len-of-ctrex-rulelist-fix
  (equal (len (ctrex-rulelist-fix x))
         (len x)))

Theorem: ctrex-rulelist-fix-of-append

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

Theorem: ctrex-rulelist-fix-of-repeat

(defthm ctrex-rulelist-fix-of-repeat
  (equal (ctrex-rulelist-fix (acl2::repeat n x))
         (acl2::repeat n (ctrex-rule-fix x))))

Theorem: list-equiv-refines-ctrex-rulelist-equiv

(defthm list-equiv-refines-ctrex-rulelist-equiv
  (implies (acl2::list-equiv x y)
           (ctrex-rulelist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-ctrex-rulelist-fix

(defthm nth-of-ctrex-rulelist-fix
  (equal (nth n (ctrex-rulelist-fix x))
         (if (< (nfix n) (len x))
             (ctrex-rule-fix (nth n x))
           nil)))

Theorem: ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-append-1

(defthm ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-append-1
  (implies (ctrex-rulelist-equiv x fty::x-equiv)
           (ctrex-rulelist-equiv (append x y)
                                 (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-append-2

(defthm ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-append-2
  (implies (ctrex-rulelist-equiv y fty::y-equiv)
           (ctrex-rulelist-equiv (append x y)
                                 (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-nthcdr-2

(defthm ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-nthcdr-2
  (implies (ctrex-rulelist-equiv l l-equiv)
           (ctrex-rulelist-equiv (nthcdr n l)
                                 (nthcdr n l-equiv)))
  :rule-classes (:congruence))

Theorem: ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-take-2

(defthm ctrex-rulelist-equiv-implies-ctrex-rulelist-equiv-take-2
  (implies (ctrex-rulelist-equiv l l-equiv)
           (ctrex-rulelist-equiv (take n l)
                                 (take n l-equiv)))
  :rule-classes (:congruence))