• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
  • Debugging
  • Std
  • Community
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
    • Kestrel-books
      • Crypto-hdwallet
      • Apt
      • Error-checking
      • Fty-extensions
      • Isar
      • Kestrel-utilities
      • Set
      • Soft
      • C
      • Bv
      • Imp-language
      • Event-macros
      • Java
      • Bitcoin
      • Ethereum
      • Yul
        • Transformations
        • Language
          • Abstract-syntax
            • Escape
            • Swcase-list->value-list
            • Hex-digit-list->chars
            • Fundef-list->name-list
            • Literal
            • Path
            • Hex-string-rest-element
            • Plain-string
            • String-element
            • Hex-string-content-option
            • Hex-string-content
            • Identifier
            • Funcall-option
            • Expression-option
            • Statement-option
            • Literal-option
            • Identifier-option
            • Hex-string
            • Hex-quad
            • Hex-digit
            • Hex-pair
            • Data-value
            • Data-item
            • Statements-to-fundefs
            • String-element-list-result
            • Identifier-identifier-map-result
            • Swcase-result
            • String-element-result
            • Statement-result
            • Literal-result
            • Identifier-set-result
            • Identifier-result
            • Identifier-list-result
            • Fundef-result
            • Funcall-result
            • Expression-result
            • Escape-result
            • Path-result
            • Block-result
            • Objects
            • Statements/blocks/cases/fundefs
            • Identifier-list
            • Identifier-set
            • Identifier-identifier-map
            • Identifier-identifier-alist
              • Identifier-identifier-alistp
              • Identifier-identifier-alist-fix
              • Identifier-identifier-alist-equiv
            • Hex-string-rest-element-list
            • String-element-list
            • Path-list
            • Hex-digit-list
            • Literal-list
            • Fundef-list
            • Expressions/funcalls
            • Abstract-syntax-induction-schemas
          • Dynamic-semantics
          • Concrete-syntax
          • Static-soundness
          • Static-semantics
          • Errors
        • Yul-json
      • Zcash
      • ACL2-programming-language
      • Prime-fields
      • Json
      • Syntheto
      • File-io-light
      • Cryptography
      • Number-theory
      • Lists-light
      • Axe
      • Builtins
      • Solidity
      • Helpers
      • Htclient
      • Typed-lists-light
      • Arithmetic-light
    • X86isa
    • Axe
    • Execloader
  • Math
  • Testing-utilities

• Identifier-identifier-alist

Identifier-identifier-alistp

Recognizer for identifier-identifier-alist.

Signature

(identifier-identifier-alistp x) → *

Definitions and Theorems

Function: identifier-identifier-alistp

(defun identifier-identifier-alistp (x)
  (declare (xargs :guard t))
  (let ((__function__ 'identifier-identifier-alistp))
    (declare (ignorable __function__))
    (if (atom x)
        (eq x nil)
      (and (consp (car x))
           (identifierp (caar x))
           (identifierp (cdar x))
           (identifier-identifier-alistp (cdr x))))))

Theorem: identifier-identifier-alistp-of-revappend

(defthm identifier-identifier-alistp-of-revappend
  (equal (identifier-identifier-alistp (revappend acl2::x acl2::y))
         (and (identifier-identifier-alistp (list-fix acl2::x))
              (identifier-identifier-alistp acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-remove

(defthm identifier-identifier-alistp-of-remove
  (implies (identifier-identifier-alistp acl2::x)
           (identifier-identifier-alistp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-last

(defthm identifier-identifier-alistp-of-last
  (implies (identifier-identifier-alistp (double-rewrite acl2::x))
           (identifier-identifier-alistp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-nthcdr

(defthm identifier-identifier-alistp-of-nthcdr
  (implies (identifier-identifier-alistp (double-rewrite acl2::x))
           (identifier-identifier-alistp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-butlast

(defthm identifier-identifier-alistp-of-butlast
  (implies (identifier-identifier-alistp (double-rewrite acl2::x))
           (identifier-identifier-alistp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-update-nth

(defthm identifier-identifier-alistp-of-update-nth
  (implies (identifier-identifier-alistp (double-rewrite acl2::x))
           (iff (identifier-identifier-alistp
                     (update-nth acl2::n acl2::y acl2::x))
                (and (and (consp acl2::y)
                          (identifierp (car acl2::y))
                          (identifierp (cdr acl2::y)))
                     (or (<= (nfix acl2::n) (len acl2::x))
                         (and (consp nil)
                              (identifierp (car nil))
                              (identifierp (cdr nil)))))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-repeat

(defthm identifier-identifier-alistp-of-repeat
  (iff (identifier-identifier-alistp (repeat acl2::n acl2::x))
       (or (and (consp acl2::x)
                (identifierp (car acl2::x))
                (identifierp (cdr acl2::x)))
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-take

(defthm identifier-identifier-alistp-of-take
 (implies (identifier-identifier-alistp (double-rewrite acl2::x))
          (iff (identifier-identifier-alistp (take acl2::n acl2::x))
               (or (and (consp nil)
                        (identifierp (car nil))
                        (identifierp (cdr nil)))
                   (<= (nfix acl2::n) (len acl2::x)))))
 :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-union-equal

(defthm identifier-identifier-alistp-of-union-equal
 (equal
      (identifier-identifier-alistp (union-equal acl2::x acl2::y))
      (and (identifier-identifier-alistp (list-fix acl2::x))
           (identifier-identifier-alistp (double-rewrite acl2::y))))
 :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-intersection-equal-2

(defthm identifier-identifier-alistp-of-intersection-equal-2
  (implies (identifier-identifier-alistp (double-rewrite acl2::y))
           (identifier-identifier-alistp
                (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-intersection-equal-1

(defthm identifier-identifier-alistp-of-intersection-equal-1
  (implies (identifier-identifier-alistp (double-rewrite acl2::x))
           (identifier-identifier-alistp
                (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-set-difference-equal

(defthm identifier-identifier-alistp-of-set-difference-equal
  (implies (identifier-identifier-alistp acl2::x)
           (identifier-identifier-alistp
                (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-when-subsetp-equal

(defthm identifier-identifier-alistp-when-subsetp-equal
  (and (implies (and (subsetp-equal acl2::x acl2::y)
                     (identifier-identifier-alistp acl2::y))
                (equal (identifier-identifier-alistp acl2::x)
                       (true-listp acl2::x)))
       (implies (and (identifier-identifier-alistp acl2::y)
                     (subsetp-equal acl2::x acl2::y))
                (equal (identifier-identifier-alistp acl2::x)
                       (true-listp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-rcons

(defthm identifier-identifier-alistp-of-rcons
  (iff (identifier-identifier-alistp (rcons acl2::a acl2::x))
       (and (and (consp acl2::a)
                 (identifierp (car acl2::a))
                 (identifierp (cdr acl2::a)))
            (identifier-identifier-alistp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-append

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

Theorem: identifier-identifier-alistp-of-rev

(defthm identifier-identifier-alistp-of-rev
  (equal (identifier-identifier-alistp (rev acl2::x))
         (identifier-identifier-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-duplicated-members

(defthm identifier-identifier-alistp-of-duplicated-members
  (implies
       (identifier-identifier-alistp acl2::x)
       (identifier-identifier-alistp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-difference

(defthm identifier-identifier-alistp-of-difference
  (implies
       (identifier-identifier-alistp acl2::x)
       (identifier-identifier-alistp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-intersect-2

(defthm identifier-identifier-alistp-of-intersect-2
  (implies
       (identifier-identifier-alistp acl2::y)
       (identifier-identifier-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-intersect-1

(defthm identifier-identifier-alistp-of-intersect-1
  (implies
       (identifier-identifier-alistp acl2::x)
       (identifier-identifier-alistp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-union

(defthm identifier-identifier-alistp-of-union
  (iff (identifier-identifier-alistp (union acl2::x acl2::y))
       (and (identifier-identifier-alistp (sfix acl2::x))
            (identifier-identifier-alistp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-mergesort

(defthm identifier-identifier-alistp-of-mergesort
  (iff (identifier-identifier-alistp (mergesort acl2::x))
       (identifier-identifier-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-delete

(defthm identifier-identifier-alistp-of-delete
  (implies (identifier-identifier-alistp acl2::x)
           (identifier-identifier-alistp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-insert

(defthm identifier-identifier-alistp-of-insert
  (iff (identifier-identifier-alistp (insert acl2::a acl2::x))
       (and (identifier-identifier-alistp (sfix acl2::x))
            (and (consp acl2::a)
                 (identifierp (car acl2::a))
                 (identifierp (cdr acl2::a)))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-sfix

(defthm identifier-identifier-alistp-of-sfix
  (iff (identifier-identifier-alistp (sfix acl2::x))
       (or (identifier-identifier-alistp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-list-fix

(defthm identifier-identifier-alistp-of-list-fix
  (implies (identifier-identifier-alistp acl2::x)
           (identifier-identifier-alistp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-identifier-identifier-alistp-compound-recognizer

(defthm
   true-listp-when-identifier-identifier-alistp-compound-recognizer
  (implies (identifier-identifier-alistp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: identifier-identifier-alistp-when-not-consp

(defthm identifier-identifier-alistp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (identifier-identifier-alistp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-cdr-when-identifier-identifier-alistp

(defthm
 identifier-identifier-alistp-of-cdr-when-identifier-identifier-alistp
 (implies (identifier-identifier-alistp (double-rewrite acl2::x))
          (identifier-identifier-alistp (cdr acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-cons

(defthm identifier-identifier-alistp-of-cons
  (equal (identifier-identifier-alistp (cons acl2::a acl2::x))
         (and (and (consp acl2::a)
                   (identifierp (car acl2::a))
                   (identifierp (cdr acl2::a)))
              (identifier-identifier-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-remove-assoc

(defthm identifier-identifier-alistp-of-remove-assoc
  (implies (identifier-identifier-alistp acl2::x)
           (identifier-identifier-alistp
                (remove-assoc-equal acl2::name acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-put-assoc

(defthm identifier-identifier-alistp-of-put-assoc
  (implies (and (identifier-identifier-alistp acl2::x))
           (iff (identifier-identifier-alistp
                     (put-assoc-equal acl2::name acl2::val acl2::x))
                (and (identifierp acl2::name)
                     (identifierp acl2::val))))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-fast-alist-clean

(defthm identifier-identifier-alistp-of-fast-alist-clean
 (implies (identifier-identifier-alistp acl2::x)
          (identifier-identifier-alistp (fast-alist-clean acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-hons-shrink-alist

(defthm identifier-identifier-alistp-of-hons-shrink-alist
  (implies (and (identifier-identifier-alistp acl2::x)
                (identifier-identifier-alistp acl2::y))
           (identifier-identifier-alistp
                (hons-shrink-alist acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: identifier-identifier-alistp-of-hons-acons

(defthm identifier-identifier-alistp-of-hons-acons
  (equal (identifier-identifier-alistp
              (hons-acons acl2::a acl2::n acl2::x))
         (and (identifierp acl2::a)
              (identifierp acl2::n)
              (identifier-identifier-alistp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifierp-of-cdr-of-hons-assoc-equal-when-identifier-identifier-alistp

(defthm
 identifierp-of-cdr-of-hons-assoc-equal-when-identifier-identifier-alistp
 (implies
      (identifier-identifier-alistp acl2::x)
      (iff (identifierp (cdr (hons-assoc-equal acl2::k acl2::x)))
           (hons-assoc-equal acl2::k acl2::x)))
 :rule-classes ((:rewrite)))

Theorem: alistp-when-identifier-identifier-alistp-rewrite

(defthm alistp-when-identifier-identifier-alistp-rewrite
  (implies (identifier-identifier-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes ((:rewrite)))

Theorem: alistp-when-identifier-identifier-alistp

(defthm alistp-when-identifier-identifier-alistp
  (implies (identifier-identifier-alistp acl2::x)
           (alistp acl2::x))
  :rule-classes :tau-system)

Theorem: identifierp-of-cdar-when-identifier-identifier-alistp

(defthm identifierp-of-cdar-when-identifier-identifier-alistp
  (implies (identifier-identifier-alistp acl2::x)
           (iff (identifierp (cdar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: identifierp-of-caar-when-identifier-identifier-alistp

(defthm identifierp-of-caar-when-identifier-identifier-alistp
  (implies (identifier-identifier-alistp acl2::x)
           (iff (identifierp (caar acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))