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Partsum-elt

Partsum-elt-fix

(partsum-elt-fix x) is a usual fty list fixing function.

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

In the logic, we apply partsum-comp-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: partsum-elt-fix$inline

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

Theorem: partsum-elt-p-of-partsum-elt-fix

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

Theorem: partsum-elt-fix-when-partsum-elt-p

(defthm partsum-elt-fix-when-partsum-elt-p
  (implies (partsum-elt-p x)
           (equal (partsum-elt-fix x) x)))

Function: partsum-elt-equiv$inline

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

Theorem: partsum-elt-equiv-is-an-equivalence

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

Theorem: partsum-elt-equiv-implies-equal-partsum-elt-fix-1

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

Theorem: partsum-elt-fix-under-partsum-elt-equiv

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

Theorem: equal-of-partsum-elt-fix-1-forward-to-partsum-elt-equiv

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

Theorem: equal-of-partsum-elt-fix-2-forward-to-partsum-elt-equiv

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

Theorem: partsum-elt-equiv-of-partsum-elt-fix-1-forward

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

Theorem: partsum-elt-equiv-of-partsum-elt-fix-2-forward

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

Theorem: car-of-partsum-elt-fix-x-under-partsum-comp-equiv

(defthm car-of-partsum-elt-fix-x-under-partsum-comp-equiv
  (partsum-comp-equiv (car (partsum-elt-fix acl2::x))
                      (car acl2::x)))

Theorem: car-partsum-elt-equiv-congruence-on-x-under-partsum-comp-equiv

(defthm
     car-partsum-elt-equiv-congruence-on-x-under-partsum-comp-equiv
  (implies (partsum-elt-equiv acl2::x x-equiv)
           (partsum-comp-equiv (car acl2::x)
                               (car x-equiv)))
  :rule-classes :congruence)

Theorem: cdr-of-partsum-elt-fix-x-under-partsum-elt-equiv

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

Theorem: cdr-partsum-elt-equiv-congruence-on-x-under-partsum-elt-equiv

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

Theorem: cons-of-partsum-comp-fix-x-under-partsum-elt-equiv

(defthm cons-of-partsum-comp-fix-x-under-partsum-elt-equiv
  (partsum-elt-equiv (cons (partsum-comp-fix acl2::x)
                           acl2::y)
                     (cons acl2::x acl2::y)))

Theorem: cons-partsum-comp-equiv-congruence-on-x-under-partsum-elt-equiv

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

Theorem: cons-of-partsum-elt-fix-y-under-partsum-elt-equiv

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

Theorem: cons-partsum-elt-equiv-congruence-on-y-under-partsum-elt-equiv

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

Theorem: consp-of-partsum-elt-fix

(defthm consp-of-partsum-elt-fix
  (equal (consp (partsum-elt-fix acl2::x))
         (consp acl2::x)))

Theorem: partsum-elt-fix-under-iff

(defthm partsum-elt-fix-under-iff
  (iff (partsum-elt-fix acl2::x)
       (consp acl2::x)))

Theorem: partsum-elt-fix-of-cons

(defthm partsum-elt-fix-of-cons
  (equal (partsum-elt-fix (cons a x))
         (cons (partsum-comp-fix a)
               (partsum-elt-fix x))))

Theorem: len-of-partsum-elt-fix

(defthm len-of-partsum-elt-fix
  (equal (len (partsum-elt-fix acl2::x))
         (len acl2::x)))

Theorem: partsum-elt-fix-of-append

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

Theorem: partsum-elt-fix-of-repeat

(defthm partsum-elt-fix-of-repeat
  (equal (partsum-elt-fix (repeat acl2::n acl2::x))
         (repeat acl2::n (partsum-comp-fix acl2::x))))

Theorem: list-equiv-refines-partsum-elt-equiv

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

Theorem: nth-of-partsum-elt-fix

(defthm nth-of-partsum-elt-fix
  (equal (nth acl2::n (partsum-elt-fix acl2::x))
         (if (< (nfix acl2::n) (len acl2::x))
             (partsum-comp-fix (nth acl2::n acl2::x))
           nil)))

Theorem: partsum-elt-equiv-implies-partsum-elt-equiv-append-1

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

Theorem: partsum-elt-equiv-implies-partsum-elt-equiv-append-2

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

Theorem: partsum-elt-equiv-implies-partsum-elt-equiv-nthcdr-2

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

Theorem: partsum-elt-equiv-implies-partsum-elt-equiv-take-2

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