• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
    • Std/lists
      • Std/lists/abstract
      • Rev
      • Defsort
      • List-fix
      • Std/lists/nth
      • Hons-remove-duplicates
      • Std/lists/update-nth
      • Set-equiv
      • Duplicity
      • Prefixp
      • Std/lists/take
      • Std/lists/intersection$
      • Nats-equiv
      • Repeat
      • Index-of
      • All-equalp
      • Sublistp
      • Std/lists/nthcdr
      • Listpos
      • List-equiv
      • Final-cdr
      • Std/lists/append
      • Std/lists/remove
      • Subseq-list
      • Rcons
      • Std/lists/revappend
      • Std/lists/remove-duplicates-equal
      • Std/lists/reverse
      • Std/lists/last
      • Std/lists/resize-list
      • Flatten
      • Suffixp
      • Std/lists/butlast
      • Std/lists/set-difference
      • Std/lists/len
      • Std/lists/intersectp
      • Std/lists/true-listp
      • Intersectp-witness
      • Subsetp-witness
      • Std/lists/remove1-equal
      • Rest-n
      • First-n
      • Std/lists/union
      • Append-without-guard
      • Std/lists/subsetp
      • Std/lists/member
    • Std/alists
    • Obags
    • Std/util
    • Std/strings
    • Std/io
    • Std/osets
    • Std/basic
    • Std/system
    • Std/typed-lists
    • Std/bitsets
    • Std/testing
    • Std/typed-alists
    • Std/stobjs
    • Std-extensions
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Math
  • Testing-utilities

• Std/lists
• Set-difference$

Std/lists/set-difference

Lemmas about set-difference$ available in the std/lists library.

Definitions and Theorems

Theorem: set-difference-equal-when-atom

(defthm set-difference-equal-when-atom
  (implies (atom x)
           (equal (set-difference-equal x y) nil)))

Theorem: set-difference-equal-of-cons

(defthm set-difference-equal-of-cons
  (equal (set-difference-equal (cons a x) y)
         (if (member-equal a y)
             (set-difference-equal x y)
           (cons a (set-difference-equal x y)))))

Theorem: set-difference-equal-when-subsetp-equal

(defthm set-difference-equal-when-subsetp-equal
  (implies (subsetp-equal x y)
           (equal (set-difference-equal x y) nil)))

Theorem: set-difference-equal-of-self

(defthm set-difference-equal-of-self
  (equal (set-difference-equal x x) nil))

Theorem: empty-intersect-with-difference-of-self

(defthm empty-intersect-with-difference-of-self
  (not (intersectp-equal a (set-difference-equal b a))))

Theorem: no-duplicatesp-of-set-difference-equal

(defthm no-duplicatesp-of-set-difference-equal
  (implies (no-duplicatesp-equal l1)
           (no-duplicatesp-equal (set-difference-equal l1 l2))))