• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Community
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
    • Gl
    • Esim
    • Vl2014
      • Warnings
      • Primitives
      • Use-set
      • Syntax
      • Getting-started
      • Utilities
      • Loader
      • Transforms
      • Lint
      • Mlib
        • Scopestack
        • Filtering-by-name
        • Vl-namefactory
        • Substitution
        • Allexprs
        • Hid-tools
        • Vl-consteval
        • Range-tools
        • Lvalexprs
        • Hierarchy
        • Finding-by-name
        • Expr-tools
        • Expr-slicing
        • Stripping-functions
        • Stmt-tools
        • Modnamespace
        • Vl-parse-expr-from-str
        • Welltyped
        • Reordering-by-name
        • Flat-warnings
        • Genblob
        • Expr-building
        • Datatype-tools
        • Syscalls
        • Relocate
        • Expr-cleaning
        • Namemangle
        • Caremask
        • Port-tools
          • Port-expressions
          • Vl-directionlist
            • Vl-directionlist-fix
            • Vl-directionlist-equiv
            • Vl-directionlist-p
              • Vl-directionlist-p-basics
        • Lvalues
      • Server
      • Kit
      • Printer
      • Esim-vl
      • Well-formedness
    • Sv
    • Fgl
    • Vwsim
    • Vl
    • X86isa
    • Svl
    • Rtl
  • Software-verification
  • Math
  • Testing-utilities

• Vl-directionlist-p

Vl-directionlist-p-basics

Basic theorems about vl-directionlist-p, generated by deflist.

Definitions and Theorems

Theorem: vl-directionlist-p-of-cons

(defthm vl-directionlist-p-of-cons
  (equal (vl-directionlist-p (cons acl2::a acl2::x))
         (and (vl-direction-p acl2::a)
              (vl-directionlist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-cdr-when-vl-directionlist-p

(defthm vl-directionlist-p-of-cdr-when-vl-directionlist-p
  (implies (vl-directionlist-p (double-rewrite acl2::x))
           (vl-directionlist-p (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-when-not-consp

(defthm vl-directionlist-p-when-not-consp
  (implies (not (consp acl2::x))
           (vl-directionlist-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-direction-p-of-car-when-vl-directionlist-p

(defthm vl-direction-p-of-car-when-vl-directionlist-p
  (implies (vl-directionlist-p acl2::x)
           (iff (vl-direction-p (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-append

(defthm vl-directionlist-p-of-append
  (equal (vl-directionlist-p (append acl2::a acl2::b))
         (and (vl-directionlist-p acl2::a)
              (vl-directionlist-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-list-fix

(defthm vl-directionlist-p-of-list-fix
  (equal (vl-directionlist-p (list-fix acl2::x))
         (vl-directionlist-p acl2::x))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-sfix

(defthm vl-directionlist-p-of-sfix
  (iff (vl-directionlist-p (sfix acl2::x))
       (or (vl-directionlist-p acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-insert

(defthm vl-directionlist-p-of-insert
  (iff (vl-directionlist-p (insert acl2::a acl2::x))
       (and (vl-directionlist-p (sfix acl2::x))
            (vl-direction-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-delete

(defthm vl-directionlist-p-of-delete
  (implies (vl-directionlist-p acl2::x)
           (vl-directionlist-p (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-mergesort

(defthm vl-directionlist-p-of-mergesort
  (iff (vl-directionlist-p (mergesort acl2::x))
       (vl-directionlist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-union

(defthm vl-directionlist-p-of-union
  (iff (vl-directionlist-p (union acl2::x acl2::y))
       (and (vl-directionlist-p (sfix acl2::x))
            (vl-directionlist-p (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-intersect-1

(defthm vl-directionlist-p-of-intersect-1
  (implies (vl-directionlist-p acl2::x)
           (vl-directionlist-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-intersect-2

(defthm vl-directionlist-p-of-intersect-2
  (implies (vl-directionlist-p acl2::y)
           (vl-directionlist-p (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-difference

(defthm vl-directionlist-p-of-difference
  (implies (vl-directionlist-p acl2::x)
           (vl-directionlist-p (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-duplicated-members

(defthm vl-directionlist-p-of-duplicated-members
  (implies (vl-directionlist-p acl2::x)
           (vl-directionlist-p (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-rev

(defthm vl-directionlist-p-of-rev
  (equal (vl-directionlist-p (rev acl2::x))
         (vl-directionlist-p (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-rcons

(defthm vl-directionlist-p-of-rcons
  (iff (vl-directionlist-p (acl2::rcons acl2::a acl2::x))
       (and (vl-direction-p acl2::a)
            (vl-directionlist-p (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-direction-p-when-member-equal-of-vl-directionlist-p

(defthm vl-direction-p-when-member-equal-of-vl-directionlist-p
  (and (implies (and (member-equal acl2::a acl2::x)
                     (vl-directionlist-p acl2::x))
                (vl-direction-p acl2::a))
       (implies (and (vl-directionlist-p acl2::x)
                     (member-equal acl2::a acl2::x))
                (vl-direction-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-when-subsetp-equal

(defthm vl-directionlist-p-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (vl-directionlist-p acl2::y))
                (vl-directionlist-p acl2::x))
       (implies (and (vl-directionlist-p acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (vl-directionlist-p acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-set-equiv-congruence

(defthm vl-directionlist-p-set-equiv-congruence
  (implies (set-equiv acl2::x acl2::y)
           (equal (vl-directionlist-p acl2::x)
                  (vl-directionlist-p acl2::y)))
  :rule-classes :congruence)

Theorem: vl-directionlist-p-of-set-difference-equal

(defthm vl-directionlist-p-of-set-difference-equal
  (implies
       (vl-directionlist-p acl2::x)
       (vl-directionlist-p (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-intersection-equal-1

(defthm vl-directionlist-p-of-intersection-equal-1
 (implies (vl-directionlist-p (double-rewrite acl2::x))
          (vl-directionlist-p (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-intersection-equal-2

(defthm vl-directionlist-p-of-intersection-equal-2
 (implies (vl-directionlist-p (double-rewrite acl2::y))
          (vl-directionlist-p (intersection-equal acl2::x acl2::y)))
 :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-union-equal

(defthm vl-directionlist-p-of-union-equal
  (equal (vl-directionlist-p (union-equal acl2::x acl2::y))
         (and (vl-directionlist-p (list-fix acl2::x))
              (vl-directionlist-p (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-take

(defthm vl-directionlist-p-of-take
  (implies (vl-directionlist-p (double-rewrite acl2::x))
           (iff (vl-directionlist-p (take acl2::n acl2::x))
                (or (vl-direction-p nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-repeat

(defthm vl-directionlist-p-of-repeat
  (iff (vl-directionlist-p (repeat acl2::n acl2::x))
       (or (vl-direction-p acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-direction-p-of-nth-when-vl-directionlist-p

(defthm vl-direction-p-of-nth-when-vl-directionlist-p
  (implies (vl-directionlist-p acl2::x)
           (iff (vl-direction-p (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-update-nth

(defthm vl-directionlist-p-of-update-nth
 (implies
      (vl-directionlist-p (double-rewrite acl2::x))
      (iff (vl-directionlist-p (update-nth acl2::n acl2::y acl2::x))
           (and (vl-direction-p acl2::y)
                (or (<= (nfix acl2::n) (len acl2::x))
                    (vl-direction-p nil)))))
 :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-butlast

(defthm vl-directionlist-p-of-butlast
  (implies (vl-directionlist-p (double-rewrite acl2::x))
           (vl-directionlist-p (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-nthcdr

(defthm vl-directionlist-p-of-nthcdr
  (implies (vl-directionlist-p (double-rewrite acl2::x))
           (vl-directionlist-p (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-last

(defthm vl-directionlist-p-of-last
  (implies (vl-directionlist-p (double-rewrite acl2::x))
           (vl-directionlist-p (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-remove

(defthm vl-directionlist-p-of-remove
  (implies (vl-directionlist-p acl2::x)
           (vl-directionlist-p (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: vl-directionlist-p-of-revappend

(defthm vl-directionlist-p-of-revappend
  (equal (vl-directionlist-p (revappend acl2::x acl2::y))
         (and (vl-directionlist-p (list-fix acl2::x))
              (vl-directionlist-p acl2::y)))
  :rule-classes ((:rewrite)))