• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
    • Gl
    • Esim
    • Vl2014
    • Sv
      • Svex-stvs
      • Svex-decomposition-methodology
      • Sv-versus-esim
      • Svex-decomp
      • Svex-compose-dfs
      • Svex-compilation
      • Moddb
      • Svmods
        • Address
        • Wire
        • Module
        • Lhs
          • Lhs.lisp
          • Lhs-p
            • Lhs-p-basics
          • Lhs-fix
          • Lhrange
          • Lhs-eval-zx
          • Lhs-equiv
          • Lhs-eval
          • Lhs->svex
        • Path
        • Svar-add-namespace
        • Design
        • Modinst
        • Lhs-add-namespace
        • Modalist
        • Path-add-namespace
        • Modname->submodnames
        • Name
        • Constraintlist-addr-p
        • Svex-alist-addr-p
        • Svar-map-addr-p
        • Lhspairs-addr-p
        • Modname
        • Assigns-addr-p
        • Lhs-addr-p
        • Lhatom-addr-p
        • Modhier-list-measure
        • Attributes
        • Modhier-measure
        • Modhier-list-measure-aux
        • Modhier-loopfreelist-p
        • Modhier-loopfree-p
      • Svstmt
      • Sv-tutorial
      • Expressions
      • Symbolic-test-vector
      • Vl-to-svex
    • Fgl
    • Vwsim
    • Vl
    • X86isa
    • Svl
    • Rtl
  • Software-verification
  • Math
  • Testing-utilities

• Lhs-p

Lhs-p-basics

Basic theorems about lhs-p, generated by std::deflist.

Definitions and Theorems

Theorem: lhs-p-of-cons

(defthm lhs-p-of-cons
  (equal (lhs-p (cons acl2::a x))
         (and (lhrange-p acl2::a) (lhs-p x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-cdr-when-lhs-p

(defthm lhs-p-of-cdr-when-lhs-p
  (implies (lhs-p (double-rewrite x))
           (lhs-p (cdr x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-when-not-consp

(defthm lhs-p-when-not-consp
  (implies (not (consp x))
           (equal (lhs-p x) (not x)))
  :rule-classes ((:rewrite)))

Theorem: lhrange-p-of-car-when-lhs-p

(defthm lhrange-p-of-car-when-lhs-p
  (implies (lhs-p x)
           (iff (lhrange-p (car x))
                (or (consp x) (lhrange-p nil))))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-lhs-p-compound-recognizer

(defthm true-listp-when-lhs-p-compound-recognizer
  (implies (lhs-p x) (true-listp x))
  :rule-classes :compound-recognizer)

Theorem: lhs-p-of-list-fix

(defthm lhs-p-of-list-fix
  (implies (lhs-p x) (lhs-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-rev

(defthm lhs-p-of-rev
  (equal (lhs-p (rev x))
         (lhs-p (list-fix x)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-repeat

(defthm lhs-p-of-repeat
  (iff (lhs-p (repeat acl2::n x))
       (or (lhrange-p x) (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-append

(defthm lhs-p-of-append
  (equal (lhs-p (append acl2::a acl2::b))
         (and (lhs-p (list-fix acl2::a))
              (lhs-p acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-rcons

(defthm lhs-p-of-rcons
  (iff (lhs-p (acl2::rcons acl2::a x))
       (and (lhrange-p acl2::a)
            (lhs-p (list-fix x))))
  :rule-classes ((:rewrite)))

Theorem: lhrange-p-when-member-equal-of-lhs-p

(defthm lhrange-p-when-member-equal-of-lhs-p
  (and (implies (and (member-equal acl2::a x) (lhs-p x))
                (lhrange-p acl2::a))
       (implies (and (lhs-p x) (member-equal acl2::a x))
                (lhrange-p acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-when-subsetp-equal

(defthm lhs-p-when-subsetp-equal
  (and (implies (and (subsetp-equal x y) (lhs-p y))
                (equal (lhs-p x) (true-listp x)))
       (implies (and (lhs-p y) (subsetp-equal x y))
                (equal (lhs-p x) (true-listp x))))
  :rule-classes ((:rewrite)))

Theorem: lhs-p-of-set-difference-equal

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

Theorem: lhs-p-of-intersection-equal-1

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

Theorem: lhs-p-of-intersection-equal-2

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

Theorem: lhs-p-of-union-equal

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

Theorem: lhs-p-of-update-nth

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

Theorem: lhs-p-of-butlast

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