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• Matchstatelist

Matchstatelist-fix

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

Signature

(matchstatelist-fix x) → fty::newx

Arguments

x — Guard (matchstatelist-p x).

Returns

fty::newx — Type (matchstatelist-p fty::newx).

In the logic, we apply matchstate-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: matchstatelist-fix$inline

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

Theorem: matchstatelist-p-of-matchstatelist-fix

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

Theorem: matchstatelist-fix-when-matchstatelist-p

(defthm matchstatelist-fix-when-matchstatelist-p
  (implies (matchstatelist-p x)
           (equal (matchstatelist-fix x) x)))

Function: matchstatelist-equiv$inline

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

Theorem: matchstatelist-equiv-is-an-equivalence

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

Theorem: matchstatelist-equiv-implies-equal-matchstatelist-fix-1

(defthm matchstatelist-equiv-implies-equal-matchstatelist-fix-1
  (implies (matchstatelist-equiv acl2::x x-equiv)
           (equal (matchstatelist-fix acl2::x)
                  (matchstatelist-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: matchstatelist-fix-under-matchstatelist-equiv

(defthm matchstatelist-fix-under-matchstatelist-equiv
  (matchstatelist-equiv (matchstatelist-fix acl2::x)
                        acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-matchstatelist-fix-1-forward-to-matchstatelist-equiv

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

Theorem: equal-of-matchstatelist-fix-2-forward-to-matchstatelist-equiv

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

Theorem: matchstatelist-equiv-of-matchstatelist-fix-1-forward

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

Theorem: matchstatelist-equiv-of-matchstatelist-fix-2-forward

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

Theorem: car-of-matchstatelist-fix-x-under-matchstate-equiv

(defthm car-of-matchstatelist-fix-x-under-matchstate-equiv
  (matchstate-equiv (car (matchstatelist-fix acl2::x))
                    (car acl2::x)))

Theorem: car-matchstatelist-equiv-congruence-on-x-under-matchstate-equiv

(defthm
    car-matchstatelist-equiv-congruence-on-x-under-matchstate-equiv
  (implies (matchstatelist-equiv acl2::x x-equiv)
           (matchstate-equiv (car acl2::x)
                             (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-matchstatelist-fix-x-under-matchstatelist-equiv

(defthm cdr-of-matchstatelist-fix-x-under-matchstatelist-equiv
  (matchstatelist-equiv (cdr (matchstatelist-fix acl2::x))
                        (cdr acl2::x)))

Theorem: cdr-matchstatelist-equiv-congruence-on-x-under-matchstatelist-equiv

(defthm
 cdr-matchstatelist-equiv-congruence-on-x-under-matchstatelist-equiv
 (implies (matchstatelist-equiv acl2::x x-equiv)
          (matchstatelist-equiv (cdr acl2::x)
                                (cdr x-equiv)))
 :rule-classes :congruence)

Theorem: cons-of-matchstate-fix-x-under-matchstatelist-equiv

(defthm cons-of-matchstate-fix-x-under-matchstatelist-equiv
  (matchstatelist-equiv (cons (matchstate-fix acl2::x) acl2::y)
                        (cons acl2::x acl2::y)))

Theorem: cons-matchstate-equiv-congruence-on-x-under-matchstatelist-equiv

(defthm
   cons-matchstate-equiv-congruence-on-x-under-matchstatelist-equiv
  (implies (matchstate-equiv acl2::x x-equiv)
           (matchstatelist-equiv (cons acl2::x acl2::y)
                                 (cons x-equiv acl2::y)))
  :rule-classes :congruence)

Theorem: cons-of-matchstatelist-fix-y-under-matchstatelist-equiv

(defthm cons-of-matchstatelist-fix-y-under-matchstatelist-equiv
  (matchstatelist-equiv (cons acl2::x (matchstatelist-fix acl2::y))
                        (cons acl2::x acl2::y)))

Theorem: cons-matchstatelist-equiv-congruence-on-y-under-matchstatelist-equiv

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

Theorem: consp-of-matchstatelist-fix

(defthm consp-of-matchstatelist-fix
  (equal (consp (matchstatelist-fix acl2::x))
         (consp acl2::x)))

Theorem: matchstatelist-fix-under-iff

(defthm matchstatelist-fix-under-iff
  (iff (matchstatelist-fix acl2::x)
       (consp acl2::x)))

Theorem: matchstatelist-fix-of-cons

(defthm matchstatelist-fix-of-cons
  (equal (matchstatelist-fix (cons a x))
         (cons (matchstate-fix a)
               (matchstatelist-fix x))))

Theorem: len-of-matchstatelist-fix

(defthm len-of-matchstatelist-fix
  (equal (len (matchstatelist-fix acl2::x))
         (len acl2::x)))

Theorem: matchstatelist-fix-of-append

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

Theorem: matchstatelist-fix-of-repeat

(defthm matchstatelist-fix-of-repeat
  (equal (matchstatelist-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (matchstate-fix acl2::x))))

Theorem: list-equiv-refines-matchstatelist-equiv

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

Theorem: nth-of-matchstatelist-fix

(defthm nth-of-matchstatelist-fix
  (equal (nth acl2::n (matchstatelist-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (matchstate-fix (nth acl2::n acl2::x))
           nil)))

Theorem: matchstatelist-equiv-implies-matchstatelist-equiv-append-1

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

Theorem: matchstatelist-equiv-implies-matchstatelist-equiv-append-2

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

Theorem: matchstatelist-equiv-implies-matchstatelist-equiv-nthcdr-2

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

Theorem: matchstatelist-equiv-implies-matchstatelist-equiv-take-2

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