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Valid-ord-scope

Valid-ord-scopep

Signature
(valid-ord-scopep x) → *

Definitions and Theorems

Function: valid-ord-scopep

(defun valid-ord-scopep (x)
  (declare (xargs :guard t))
  (if (atom x)
      (eq x nil)
    (and (consp (car x))
         (identp (caar x))
         (valid-ord-infop (cdar x))
         (valid-ord-scopep (cdr x)))))

Theorem: valid-ord-scopep-of-revappend

(defthm valid-ord-scopep-of-revappend
  (equal (valid-ord-scopep (revappend acl2::x acl2::y))
         (and (valid-ord-scopep (list-fix acl2::x))
              (valid-ord-scopep acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-remove

(defthm valid-ord-scopep-of-remove
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-last

(defthm valid-ord-scopep-of-last
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (valid-ord-scopep (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-nthcdr

(defthm valid-ord-scopep-of-nthcdr
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (valid-ord-scopep (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-butlast

(defthm valid-ord-scopep-of-butlast
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (valid-ord-scopep (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-update-nth

(defthm valid-ord-scopep-of-update-nth
  (implies
       (valid-ord-scopep (double-rewrite acl2::x))
       (iff (valid-ord-scopep (update-nth acl2::n acl2::y acl2::x))
            (and (and (consp acl2::y)
                      (identp (car acl2::y))
                      (valid-ord-infop (cdr acl2::y)))
                 (or (<= (nfix acl2::n) (len acl2::x))
                     (and (consp nil)
                          (identp (car nil))
                          (valid-ord-infop (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-repeat

(defthm valid-ord-scopep-of-repeat
  (iff (valid-ord-scopep (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (identp (car acl2::x))
                (valid-ord-infop (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-take

(defthm valid-ord-scopep-of-take
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (iff (valid-ord-scopep (take acl2::n acl2::x))
                (or (and (consp nil)
                         (identp (car nil))
                         (valid-ord-infop (cdr nil)))
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-union-equal

(defthm valid-ord-scopep-of-union-equal
  (equal (valid-ord-scopep (union-equal acl2::x acl2::y))
         (and (valid-ord-scopep (list-fix acl2::x))
              (valid-ord-scopep (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-intersection-equal-2

(defthm valid-ord-scopep-of-intersection-equal-2
  (implies (valid-ord-scopep (double-rewrite acl2::y))
           (valid-ord-scopep (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-intersection-equal-1

(defthm valid-ord-scopep-of-intersection-equal-1
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (valid-ord-scopep (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-set-difference-equal

(defthm valid-ord-scopep-of-set-difference-equal
 (implies (valid-ord-scopep acl2::x)
          (valid-ord-scopep (set-difference-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-when-subsetp-equal

(defthm valid-ord-scopep-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (valid-ord-scopep acl2::y))
                (equal (valid-ord-scopep acl2::x)
                       (true-listp acl2::x)))
       (implies (and (valid-ord-scopep acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (valid-ord-scopep acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-rcons

(defthm valid-ord-scopep-of-rcons
  (iff (valid-ord-scopep (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (identp (car acl2::a))
                 (valid-ord-infop (cdr acl2::a)))
            (valid-ord-scopep (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-append

(defthm valid-ord-scopep-of-append
  (equal (valid-ord-scopep (append acl2::a acl2::b))
         (and (valid-ord-scopep (list-fix acl2::a))
              (valid-ord-scopep acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-rev

(defthm valid-ord-scopep-of-rev
  (equal (valid-ord-scopep (rev acl2::x))
         (valid-ord-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-duplicated-members

(defthm valid-ord-scopep-of-duplicated-members
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-difference

(defthm valid-ord-scopep-of-difference
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-intersect-2

(defthm valid-ord-scopep-of-intersect-2
  (implies (valid-ord-scopep acl2::y)
           (valid-ord-scopep (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-intersect-1

(defthm valid-ord-scopep-of-intersect-1
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-union

(defthm valid-ord-scopep-of-union
  (iff (valid-ord-scopep (union acl2::x acl2::y))
       (and (valid-ord-scopep (sfix acl2::x))
            (valid-ord-scopep (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-mergesort

(defthm valid-ord-scopep-of-mergesort
  (iff (valid-ord-scopep (mergesort acl2::x))
       (valid-ord-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-delete

(defthm valid-ord-scopep-of-delete
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-insert

(defthm valid-ord-scopep-of-insert
  (iff (valid-ord-scopep (insert acl2::a acl2::x))
       (and (valid-ord-scopep (sfix acl2::x))
            (and (consp acl2::a)
                 (identp (car acl2::a))
                 (valid-ord-infop (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-sfix

(defthm valid-ord-scopep-of-sfix
  (iff (valid-ord-scopep (sfix acl2::x))
       (or (valid-ord-scopep acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-list-fix

(defthm valid-ord-scopep-of-list-fix
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-valid-ord-scopep-compound-recognizer

(defthm true-listp-when-valid-ord-scopep-compound-recognizer
  (implies (valid-ord-scopep acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: valid-ord-scopep-when-not-consp

(defthm valid-ord-scopep-when-not-consp
  (implies (not (consp acl2::x))
           (equal (valid-ord-scopep acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-cdr-when-valid-ord-scopep

(defthm valid-ord-scopep-of-cdr-when-valid-ord-scopep
  (implies (valid-ord-scopep (double-rewrite acl2::x))
           (valid-ord-scopep (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-cons

(defthm valid-ord-scopep-of-cons
  (equal (valid-ord-scopep (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (identp (car acl2::a))
                   (valid-ord-infop (cdr acl2::a)))
              (valid-ord-scopep acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-remove-assoc

(defthm valid-ord-scopep-of-remove-assoc
  (implies
       (valid-ord-scopep acl2::x)
       (valid-ord-scopep (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-put-assoc

(defthm valid-ord-scopep-of-put-assoc
 (implies
  (and (valid-ord-scopep acl2::x))
  (iff
   (valid-ord-scopep (put-assoc-equal acl2::name acl2::val acl2::x))
   (and (identp acl2::name)
        (valid-ord-infop acl2::val))))
 :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-fast-alist-clean

(defthm valid-ord-scopep-of-fast-alist-clean
  (implies (valid-ord-scopep acl2::x)
           (valid-ord-scopep (fast-alist-clean acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-hons-shrink-alist

(defthm valid-ord-scopep-of-hons-shrink-alist
  (implies (and (valid-ord-scopep acl2::x)
                (valid-ord-scopep acl2::y))
           (valid-ord-scopep (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-scopep-of-hons-acons

(defthm valid-ord-scopep-of-hons-acons
  (equal (valid-ord-scopep (hons-acons acl2::a acl2::n acl2::x))
         (and (identp acl2::a)
              (valid-ord-infop acl2::n)
              (valid-ord-scopep acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: valid-ord-infop-of-cdr-of-hons-assoc-equal-when-valid-ord-scopep

(defthm
   valid-ord-infop-of-cdr-of-hons-assoc-equal-when-valid-ord-scopep
 (implies
     (valid-ord-scopep acl2::x)
     (iff (valid-ord-infop (cdr (hons-assoc-equal acl2::k acl2::x)))
          (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-valid-ord-scopep-rewrite

(defthm alistp-when-valid-ord-scopep-rewrite
  (implies (valid-ord-scopep acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-valid-ord-scopep

(defthm alistp-when-valid-ord-scopep
  (implies (valid-ord-scopep acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: valid-ord-infop-of-cdar-when-valid-ord-scopep

(defthm valid-ord-infop-of-cdar-when-valid-ord-scopep
  (implies (valid-ord-scopep acl2::x)
           (iff (valid-ord-infop (cdar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identp-of-caar-when-valid-ord-scopep

(defthm identp-of-caar-when-valid-ord-scopep
  (implies (valid-ord-scopep acl2::x)
           (iff (identp (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))