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• Fraig-output-map

Fraig-output-map-fix

(fraig-output-map-fix x) is a usual ACL2::fty list fixing function.

Signature

(fraig-output-map-fix x) → fty::newx

Arguments

x — Guard (fraig-output-map-p x).

Returns

fty::newx — Type (fraig-output-map-p fty::newx).

In the logic, we apply fraig-output-map-entry-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: fraig-output-map-fix$inline

(defun fraig-output-map-fix$inline (x)
  (declare (xargs :guard (fraig-output-map-p x)))
  (let ((__function__ 'fraig-output-map-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (fraig-output-map-entry-fix (car x))
                 (fraig-output-map-fix (cdr x))))
         :exec x)))

Theorem: fraig-output-map-p-of-fraig-output-map-fix

(defthm fraig-output-map-p-of-fraig-output-map-fix
  (b* ((fty::newx (fraig-output-map-fix$inline x)))
    (fraig-output-map-p fty::newx))
  :rule-classes :rewrite)

Theorem: fraig-output-map-fix-when-fraig-output-map-p

(defthm fraig-output-map-fix-when-fraig-output-map-p
  (implies (fraig-output-map-p x)
           (equal (fraig-output-map-fix x) x)))

Function: fraig-output-map-equiv$inline

(defun fraig-output-map-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (fraig-output-map-p x)
                              (fraig-output-map-p acl2::y))))
  (equal (fraig-output-map-fix x)
         (fraig-output-map-fix acl2::y)))

Theorem: fraig-output-map-equiv-is-an-equivalence

(defthm fraig-output-map-equiv-is-an-equivalence
  (and (booleanp (fraig-output-map-equiv x y))
       (fraig-output-map-equiv x x)
       (implies (fraig-output-map-equiv x y)
                (fraig-output-map-equiv y x))
       (implies (and (fraig-output-map-equiv x y)
                     (fraig-output-map-equiv y z))
                (fraig-output-map-equiv x z)))
  :rule-classes (:equivalence))

Theorem: fraig-output-map-equiv-implies-equal-fraig-output-map-fix-1

(defthm fraig-output-map-equiv-implies-equal-fraig-output-map-fix-1
  (implies (fraig-output-map-equiv x x-equiv)
           (equal (fraig-output-map-fix x)
                  (fraig-output-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: fraig-output-map-fix-under-fraig-output-map-equiv

(defthm fraig-output-map-fix-under-fraig-output-map-equiv
  (fraig-output-map-equiv (fraig-output-map-fix x)
                          x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-fraig-output-map-fix-1-forward-to-fraig-output-map-equiv

(defthm
  equal-of-fraig-output-map-fix-1-forward-to-fraig-output-map-equiv
  (implies (equal (fraig-output-map-fix x) acl2::y)
           (fraig-output-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-fraig-output-map-fix-2-forward-to-fraig-output-map-equiv

(defthm
  equal-of-fraig-output-map-fix-2-forward-to-fraig-output-map-equiv
  (implies (equal x (fraig-output-map-fix acl2::y))
           (fraig-output-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: fraig-output-map-equiv-of-fraig-output-map-fix-1-forward

(defthm fraig-output-map-equiv-of-fraig-output-map-fix-1-forward
  (implies (fraig-output-map-equiv (fraig-output-map-fix x)
                                   acl2::y)
           (fraig-output-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: fraig-output-map-equiv-of-fraig-output-map-fix-2-forward

(defthm fraig-output-map-equiv-of-fraig-output-map-fix-2-forward
  (implies (fraig-output-map-equiv x (fraig-output-map-fix acl2::y))
           (fraig-output-map-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: car-of-fraig-output-map-fix-x-under-fraig-output-map-entry-equiv

(defthm
   car-of-fraig-output-map-fix-x-under-fraig-output-map-entry-equiv
  (fraig-output-map-entry-equiv (car (fraig-output-map-fix x))
                                (car x)))

Theorem: car-fraig-output-map-equiv-congruence-on-x-under-fraig-output-map-entry-equiv

(defthm
 car-fraig-output-map-equiv-congruence-on-x-under-fraig-output-map-entry-equiv
 (implies (fraig-output-map-equiv x x-equiv)
          (fraig-output-map-entry-equiv (car x)
                                        (car x-equiv)))
 :rule-classes :congruence)

Theorem: cdr-of-fraig-output-map-fix-x-under-fraig-output-map-equiv

(defthm cdr-of-fraig-output-map-fix-x-under-fraig-output-map-equiv
  (fraig-output-map-equiv (cdr (fraig-output-map-fix x))
                          (cdr x)))

Theorem: cdr-fraig-output-map-equiv-congruence-on-x-under-fraig-output-map-equiv

(defthm
 cdr-fraig-output-map-equiv-congruence-on-x-under-fraig-output-map-equiv
 (implies (fraig-output-map-equiv x x-equiv)
          (fraig-output-map-equiv (cdr x)
                                  (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-fraig-output-map-entry-fix-x-under-fraig-output-map-equiv

(defthm
  cons-of-fraig-output-map-entry-fix-x-under-fraig-output-map-equiv
  (fraig-output-map-equiv (cons (fraig-output-map-entry-fix x)
                                acl2::y)
                          (cons x acl2::y)))

Theorem: cons-fraig-output-map-entry-equiv-congruence-on-x-under-fraig-output-map-equiv

(defthm
 cons-fraig-output-map-entry-equiv-congruence-on-x-under-fraig-output-map-equiv
 (implies (fraig-output-map-entry-equiv x x-equiv)
          (fraig-output-map-equiv (cons x acl2::y)
                                  (cons x-equiv acl2::y)))
 :rule-classes :congruence)

Theorem: cons-of-fraig-output-map-fix-y-under-fraig-output-map-equiv

(defthm cons-of-fraig-output-map-fix-y-under-fraig-output-map-equiv
  (fraig-output-map-equiv (cons x (fraig-output-map-fix acl2::y))
                          (cons x acl2::y)))

Theorem: cons-fraig-output-map-equiv-congruence-on-y-under-fraig-output-map-equiv

(defthm
 cons-fraig-output-map-equiv-congruence-on-y-under-fraig-output-map-equiv
 (implies (fraig-output-map-equiv acl2::y y-equiv)
          (fraig-output-map-equiv (cons x acl2::y)
                                  (cons x y-equiv)))
 :rule-classes :congruence)

Theorem: consp-of-fraig-output-map-fix

(defthm consp-of-fraig-output-map-fix
  (equal (consp (fraig-output-map-fix x))
         (consp x)))

Theorem: fraig-output-map-fix-under-iff

(defthm fraig-output-map-fix-under-iff
  (iff (fraig-output-map-fix x)
       (consp x)))

Theorem: fraig-output-map-fix-of-cons

(defthm fraig-output-map-fix-of-cons
  (equal (fraig-output-map-fix (cons a x))
         (cons (fraig-output-map-entry-fix a)
               (fraig-output-map-fix x))))

Theorem: len-of-fraig-output-map-fix

(defthm len-of-fraig-output-map-fix
  (equal (len (fraig-output-map-fix x))
         (len x)))

Theorem: fraig-output-map-fix-of-append

(defthm fraig-output-map-fix-of-append
  (equal (fraig-output-map-fix (append std::a std::b))
         (append (fraig-output-map-fix std::a)
                 (fraig-output-map-fix std::b))))

Theorem: fraig-output-map-fix-of-repeat

(defthm fraig-output-map-fix-of-repeat
  (equal (fraig-output-map-fix (acl2::repeat acl2::n x))
         (acl2::repeat acl2::n
                       (fraig-output-map-entry-fix x))))

Theorem: list-equiv-refines-fraig-output-map-equiv

(defthm list-equiv-refines-fraig-output-map-equiv
  (implies (list-equiv x acl2::y)
           (fraig-output-map-equiv x acl2::y))
  :rule-classes :refinement)

Theorem: nth-of-fraig-output-map-fix

(defthm nth-of-fraig-output-map-fix
  (equal (nth acl2::n (fraig-output-map-fix x))
         (if (< (nfix acl2::n) (len x))
             (fraig-output-map-entry-fix (nth acl2::n x))
           nil)))

Theorem: fraig-output-map-equiv-implies-fraig-output-map-equiv-append-1

(defthm
     fraig-output-map-equiv-implies-fraig-output-map-equiv-append-1
  (implies (fraig-output-map-equiv x fty::x-equiv)
           (fraig-output-map-equiv (append x acl2::y)
                                   (append fty::x-equiv acl2::y)))
  :rule-classes (:congruence))

Theorem: fraig-output-map-equiv-implies-fraig-output-map-equiv-append-2

(defthm
     fraig-output-map-equiv-implies-fraig-output-map-equiv-append-2
  (implies (fraig-output-map-equiv acl2::y fty::y-equiv)
           (fraig-output-map-equiv (append x acl2::y)
                                   (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: fraig-output-map-equiv-implies-fraig-output-map-equiv-nthcdr-2

(defthm
     fraig-output-map-equiv-implies-fraig-output-map-equiv-nthcdr-2
  (implies (fraig-output-map-equiv acl2::l l-equiv)
           (fraig-output-map-equiv (nthcdr acl2::n acl2::l)
                                   (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: fraig-output-map-equiv-implies-fraig-output-map-equiv-take-2

(defthm fraig-output-map-equiv-implies-fraig-output-map-equiv-take-2
  (implies (fraig-output-map-equiv acl2::l l-equiv)
           (fraig-output-map-equiv (take acl2::n acl2::l)
                                   (take acl2::n l-equiv)))
  :rule-classes (:congruence))