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Elab-modlist

Elab-modlist-fix

(elab-modlist-fix x) is a usual fty list fixing function.

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

In the logic, we apply elab-mod$a-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: elab-modlist-fix$inline

(defun elab-modlist-fix$inline (x)
  (declare (xargs :guard (elab-modlist-p x)))
  (let ((__function__ 'elab-modlist-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (if (atom x)
             nil
           (cons (elab-mod$a-fix (car x))
                 (elab-modlist-fix (cdr x))))
         :exec x)))

Theorem: elab-modlist-p-of-elab-modlist-fix

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

Theorem: elab-modlist-fix-when-elab-modlist-p

(defthm elab-modlist-fix-when-elab-modlist-p
  (implies (elab-modlist-p x)
           (equal (elab-modlist-fix x) x)))

Function: elab-modlist-equiv$inline

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

Theorem: elab-modlist-equiv-is-an-equivalence

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

Theorem: elab-modlist-equiv-implies-equal-elab-modlist-fix-1

(defthm elab-modlist-equiv-implies-equal-elab-modlist-fix-1
  (implies (elab-modlist-equiv x x-equiv)
           (equal (elab-modlist-fix x)
                  (elab-modlist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: elab-modlist-fix-under-elab-modlist-equiv

(defthm elab-modlist-fix-under-elab-modlist-equiv
  (elab-modlist-equiv (elab-modlist-fix x)
                      x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-elab-modlist-fix-1-forward-to-elab-modlist-equiv

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

Theorem: equal-of-elab-modlist-fix-2-forward-to-elab-modlist-equiv

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

Theorem: elab-modlist-equiv-of-elab-modlist-fix-1-forward

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

Theorem: elab-modlist-equiv-of-elab-modlist-fix-2-forward

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

Theorem: car-of-elab-modlist-fix-x-under-elab-mod$a-equiv

(defthm car-of-elab-modlist-fix-x-under-elab-mod$a-equiv
  (elab-mod$a-equiv (car (elab-modlist-fix x))
                    (car x)))

Theorem: car-elab-modlist-equiv-congruence-on-x-under-elab-mod$a-equiv

(defthm
      car-elab-modlist-equiv-congruence-on-x-under-elab-mod$a-equiv
  (implies (elab-modlist-equiv x x-equiv)
           (elab-mod$a-equiv (car x)
                             (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-elab-modlist-fix-x-under-elab-modlist-equiv

(defthm cdr-of-elab-modlist-fix-x-under-elab-modlist-equiv
  (elab-modlist-equiv (cdr (elab-modlist-fix x))
                      (cdr x)))

Theorem: cdr-elab-modlist-equiv-congruence-on-x-under-elab-modlist-equiv

(defthm
    cdr-elab-modlist-equiv-congruence-on-x-under-elab-modlist-equiv
  (implies (elab-modlist-equiv x x-equiv)
           (elab-modlist-equiv (cdr x)
                               (cdr x-equiv)))
  :rule-classes :congruence)

Theorem: cons-of-elab-mod$a-fix-x-under-elab-modlist-equiv

(defthm cons-of-elab-mod$a-fix-x-under-elab-modlist-equiv
  (elab-modlist-equiv (cons (elab-mod$a-fix x) y)
                      (cons x y)))

Theorem: cons-elab-mod$a-equiv-congruence-on-x-under-elab-modlist-equiv

(defthm
     cons-elab-mod$a-equiv-congruence-on-x-under-elab-modlist-equiv
  (implies (elab-mod$a-equiv x x-equiv)
           (elab-modlist-equiv (cons x y)
                               (cons x-equiv y)))
  :rule-classes :congruence)

Theorem: cons-of-elab-modlist-fix-y-under-elab-modlist-equiv

(defthm cons-of-elab-modlist-fix-y-under-elab-modlist-equiv
  (elab-modlist-equiv (cons x (elab-modlist-fix y))
                      (cons x y)))

Theorem: cons-elab-modlist-equiv-congruence-on-y-under-elab-modlist-equiv

(defthm
   cons-elab-modlist-equiv-congruence-on-y-under-elab-modlist-equiv
  (implies (elab-modlist-equiv y y-equiv)
           (elab-modlist-equiv (cons x y)
                               (cons x y-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-elab-modlist-fix

(defthm consp-of-elab-modlist-fix
  (equal (consp (elab-modlist-fix x))
         (consp x)))

Theorem: elab-modlist-fix-under-iff

(defthm elab-modlist-fix-under-iff
  (iff (elab-modlist-fix x) (consp x)))

Theorem: elab-modlist-fix-of-cons

(defthm elab-modlist-fix-of-cons
  (equal (elab-modlist-fix (cons a x))
         (cons (elab-mod$a-fix a)
               (elab-modlist-fix x))))

Theorem: len-of-elab-modlist-fix

(defthm len-of-elab-modlist-fix
  (equal (len (elab-modlist-fix x))
         (len x)))

Theorem: elab-modlist-fix-of-append

(defthm elab-modlist-fix-of-append
  (equal (elab-modlist-fix (append std::a std::b))
         (append (elab-modlist-fix std::a)
                 (elab-modlist-fix std::b))))

Theorem: elab-modlist-fix-of-repeat

(defthm elab-modlist-fix-of-repeat
  (equal (elab-modlist-fix (repeat acl2::n x))
         (repeat acl2::n (elab-mod$a-fix x))))

Theorem: list-equiv-refines-elab-modlist-equiv

(defthm list-equiv-refines-elab-modlist-equiv
  (implies (list-equiv x y)
           (elab-modlist-equiv x y))
  :rule-classes :refinement)

Theorem: nth-of-elab-modlist-fix

(defthm nth-of-elab-modlist-fix
  (equal (nth acl2::n (elab-modlist-fix x))
         (if (< (nfix acl2::n) (len x))
             (elab-mod$a-fix (nth acl2::n x))
           nil)))

Theorem: elab-modlist-equiv-implies-elab-modlist-equiv-append-1

(defthm elab-modlist-equiv-implies-elab-modlist-equiv-append-1
  (implies (elab-modlist-equiv x fty::x-equiv)
           (elab-modlist-equiv (append x y)
                               (append fty::x-equiv y)))
  :rule-classes (:congruence))

Theorem: elab-modlist-equiv-implies-elab-modlist-equiv-append-2

(defthm elab-modlist-equiv-implies-elab-modlist-equiv-append-2
  (implies (elab-modlist-equiv y fty::y-equiv)
           (elab-modlist-equiv (append x y)
                               (append x fty::y-equiv)))
  :rule-classes (:congruence))

Theorem: elab-modlist-equiv-implies-elab-modlist-equiv-nthcdr-2

(defthm elab-modlist-equiv-implies-elab-modlist-equiv-nthcdr-2
  (implies (elab-modlist-equiv acl2::l l-equiv)
           (elab-modlist-equiv (nthcdr acl2::n acl2::l)
                               (nthcdr acl2::n l-equiv)))
  :rule-classes (:congruence))

Theorem: elab-modlist-equiv-implies-elab-modlist-equiv-take-2

(defthm elab-modlist-equiv-implies-elab-modlist-equiv-take-2
  (implies (elab-modlist-equiv acl2::l l-equiv)
           (elab-modlist-equiv (take acl2::n acl2::l)
                               (take acl2::n l-equiv)))
  :rule-classes (:congruence))