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• Sd-problemlist

Sd-problemlist-fix

(sd-problemlist-fix x) is a usual fty list fixing function.

Signature

(sd-problemlist-fix x) → fty::newx

Arguments

x — Guard (sd-problemlist-p x).

Returns

fty::newx — Type (sd-problemlist-p fty::newx).

In the logic, we apply sd-problem-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: sd-problemlist-fix$inline

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

Theorem: sd-problemlist-p-of-sd-problemlist-fix

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

Theorem: sd-problemlist-fix-when-sd-problemlist-p

(defthm sd-problemlist-fix-when-sd-problemlist-p
  (implies (sd-problemlist-p x)
           (equal (sd-problemlist-fix x) x)))

Function: sd-problemlist-equiv$inline

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

Theorem: sd-problemlist-equiv-is-an-equivalence

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

Theorem: sd-problemlist-equiv-implies-equal-sd-problemlist-fix-1

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

Theorem: sd-problemlist-fix-under-sd-problemlist-equiv

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

Theorem: equal-of-sd-problemlist-fix-1-forward-to-sd-problemlist-equiv

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

Theorem: equal-of-sd-problemlist-fix-2-forward-to-sd-problemlist-equiv

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

Theorem: sd-problemlist-equiv-of-sd-problemlist-fix-1-forward

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

Theorem: sd-problemlist-equiv-of-sd-problemlist-fix-2-forward

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

Theorem: car-of-sd-problemlist-fix-x-under-sd-problem-equiv

(defthm car-of-sd-problemlist-fix-x-under-sd-problem-equiv
  (sd-problem-equiv (car (sd-problemlist-fix acl2::x))
                    (car acl2::x)))

Theorem: car-sd-problemlist-equiv-congruence-on-x-under-sd-problem-equiv

(defthm
    car-sd-problemlist-equiv-congruence-on-x-under-sd-problem-equiv
  (implies (sd-problemlist-equiv acl2::x x-equiv)
           (sd-problem-equiv (car acl2::x)
                             (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-sd-problemlist-fix-x-under-sd-problemlist-equiv

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

Theorem: cdr-sd-problemlist-equiv-congruence-on-x-under-sd-problemlist-equiv

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

Theorem: cons-of-sd-problem-fix-x-under-sd-problemlist-equiv

(defthm cons-of-sd-problem-fix-x-under-sd-problemlist-equiv
  (sd-problemlist-equiv (cons (sd-problem-fix acl2::x) acl2::y)
                        (cons acl2::x acl2::y)))

Theorem: cons-sd-problem-equiv-congruence-on-x-under-sd-problemlist-equiv

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

Theorem: cons-of-sd-problemlist-fix-y-under-sd-problemlist-equiv

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

Theorem: cons-sd-problemlist-equiv-congruence-on-y-under-sd-problemlist-equiv

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

Theorem: consp-of-sd-problemlist-fix

(defthm consp-of-sd-problemlist-fix
  (equal (consp (sd-problemlist-fix acl2::x))
         (consp acl2::x)))

Theorem: sd-problemlist-fix-of-cons

(defthm sd-problemlist-fix-of-cons
  (equal (sd-problemlist-fix (cons a x))
         (cons (sd-problem-fix a)
               (sd-problemlist-fix x))))

Theorem: len-of-sd-problemlist-fix

(defthm len-of-sd-problemlist-fix
  (equal (len (sd-problemlist-fix acl2::x))
         (len acl2::x)))

Theorem: sd-problemlist-fix-of-append

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

Theorem: sd-problemlist-fix-of-repeat

(defthm sd-problemlist-fix-of-repeat
  (equal (sd-problemlist-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (sd-problem-fix acl2::x))))

Theorem: nth-of-sd-problemlist-fix

(defthm nth-of-sd-problemlist-fix
  (equal (nth acl2::n (sd-problemlist-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (sd-problem-fix (nth acl2::n acl2::x))
           nil)))

Theorem: sd-problemlist-equiv-implies-sd-problemlist-equiv-append-1

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

Theorem: sd-problemlist-equiv-implies-sd-problemlist-equiv-append-2

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

Theorem: sd-problemlist-equiv-implies-sd-problemlist-equiv-nthcdr-2

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

Theorem: sd-problemlist-equiv-implies-sd-problemlist-equiv-take-2

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