• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
    • Apt
    • Zfc
    • Acre
    • Milawa
    • Smtlink
    • Abnf
    • Vwsim
    • Isar
    • Wp-gen
    • Dimacs-reader
    • Pfcs
    • Legacy-defrstobj
    • C
    • Proof-checker-array
    • Soft
    • Farray
    • Rp-rewriter
    • Instant-runoff-voting
    • Imp-language
    • Sidekick
    • Ethereum
    • Leftist-trees
    • Java
    • Riscv
    • Taspi
    • Bitcoin
    • Zcash
    • Des
    • X86isa
    • Sha-2
    • Yul
    • Proof-checker-itp13
    • Regex
    • ACL2-programming-language
    • Json
    • Jfkr
    • Equational
    • Cryptography
    • Axe
    • Poseidon
    • Where-do-i-place-my-book
    • Aleo
      • Aleobft
      • Aleovm
      • Leo
        • Grammar
        • Early-version
          • Json2ast
          • Testing
          • Definition
            • Flattening
            • Abstract-syntax
              • Expression
              • Syntax-abstraction
              • Statement
              • Files
              • Input-files
                • Input-section-list->title-list
                • Input-title-option
                • Input-section-option
                • Input-item
                • Input-type
                • Input-section
                • Input-expression
                • Input-title
                • Input-file
                • Input-section-option-result
                • Input-section-list-result
                • Input-type-result
                • Input-title-result
                • Input-title-option-result
                • Input-section-result
                • Input-item-result
                • Input-item-list-result
                • Input-file-result
                • Input-expression-result
                • Input-section-list
                • Input-item-list
                • Input-title-list
                  • Input-title-list-fix
                  • Input-title-list-equiv
                  • Input-title-listp
                    • Input-title-listp-basics
              • Identifiers
              • Types
              • Struct-init
              • Branch
              • Statements
              • Format-strings
              • Input-syntax-abstraction
              • Expressions
              • Output-files
              • Addresses
              • Literals
              • Characters
              • Expression-list
              • Statement-list
              • Output-syntax-abstraction
              • Struct-init-list
              • Branch-list
              • Annotations
              • Abstract-syntax-trees
              • Symbols
              • Keywords
              • Programs
              • Packages
              • Bit-sizes
            • Dynamic-semantics
            • Compilation
            • Static-semantics
            • Concrete-syntax
    • Bigmems
    • Builtins
    • Execloader
    • Solidity
    • Paco
    • Concurrent-programs
    • Bls12-377-curves
  • Debugging
  • Community
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Math
  • Testing-utilities

• Input-title-listp

Input-title-listp-basics

Basic theorems about input-title-listp, generated by std::deflist.

Definitions and Theorems

Theorem: input-title-listp-of-cons

(defthm input-title-listp-of-cons
  (equal (input-title-listp (cons acl2::a acl2::x))
         (and (input-titlep acl2::a)
              (input-title-listp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-cdr-when-input-title-listp

(defthm input-title-listp-of-cdr-when-input-title-listp
  (implies (input-title-listp (double-rewrite acl2::x))
           (input-title-listp (cdr acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-when-not-consp

(defthm input-title-listp-when-not-consp
  (implies (not (consp acl2::x))
           (equal (input-title-listp acl2::x)
                  (not acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-titlep-of-car-when-input-title-listp

(defthm input-titlep-of-car-when-input-title-listp
  (implies (input-title-listp acl2::x)
           (iff (input-titlep (car acl2::x))
                (consp acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: true-listp-when-input-title-listp-compound-recognizer

(defthm true-listp-when-input-title-listp-compound-recognizer
  (implies (input-title-listp acl2::x)
           (true-listp acl2::x))
  :rule-classes :compound-recognizer)

Theorem: input-title-listp-of-list-fix

(defthm input-title-listp-of-list-fix
  (implies (input-title-listp acl2::x)
           (input-title-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-sfix

(defthm input-title-listp-of-sfix
  (iff (input-title-listp (sfix acl2::x))
       (or (input-title-listp acl2::x)
           (not (setp acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-insert

(defthm input-title-listp-of-insert
  (iff (input-title-listp (insert acl2::a acl2::x))
       (and (input-title-listp (sfix acl2::x))
            (input-titlep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-delete

(defthm input-title-listp-of-delete
  (implies (input-title-listp acl2::x)
           (input-title-listp (delete acl2::k acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-mergesort

(defthm input-title-listp-of-mergesort
  (iff (input-title-listp (mergesort acl2::x))
       (input-title-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-union

(defthm input-title-listp-of-union
  (iff (input-title-listp (union acl2::x acl2::y))
       (and (input-title-listp (sfix acl2::x))
            (input-title-listp (sfix acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-intersect-1

(defthm input-title-listp-of-intersect-1
  (implies (input-title-listp acl2::x)
           (input-title-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-intersect-2

(defthm input-title-listp-of-intersect-2
  (implies (input-title-listp acl2::y)
           (input-title-listp (intersect acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-difference

(defthm input-title-listp-of-difference
  (implies (input-title-listp acl2::x)
           (input-title-listp (difference acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-duplicated-members

(defthm input-title-listp-of-duplicated-members
  (implies (input-title-listp acl2::x)
           (input-title-listp (duplicated-members acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-rev

(defthm input-title-listp-of-rev
  (equal (input-title-listp (rev acl2::x))
         (input-title-listp (list-fix acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-append

(defthm input-title-listp-of-append
  (equal (input-title-listp (append acl2::a acl2::b))
         (and (input-title-listp (list-fix acl2::a))
              (input-title-listp acl2::b)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-rcons

(defthm input-title-listp-of-rcons
  (iff (input-title-listp (rcons acl2::a acl2::x))
       (and (input-titlep acl2::a)
            (input-title-listp (list-fix acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: input-titlep-when-member-equal-of-input-title-listp

(defthm input-titlep-when-member-equal-of-input-title-listp
  (and (implies (and (member-equal acl2::a acl2::x)
                     (input-title-listp acl2::x))
                (input-titlep acl2::a))
       (implies (and (input-title-listp acl2::x)
                     (member-equal acl2::a acl2::x))
                (input-titlep acl2::a)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-when-subsetp-equal

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

Theorem: input-title-listp-of-set-difference-equal

(defthm input-title-listp-of-set-difference-equal
  (implies
       (input-title-listp acl2::x)
       (input-title-listp (set-difference-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-intersection-equal-1

(defthm input-title-listp-of-intersection-equal-1
  (implies (input-title-listp (double-rewrite acl2::x))
           (input-title-listp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-intersection-equal-2

(defthm input-title-listp-of-intersection-equal-2
  (implies (input-title-listp (double-rewrite acl2::y))
           (input-title-listp (intersection-equal acl2::x acl2::y)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-union-equal

(defthm input-title-listp-of-union-equal
  (equal (input-title-listp (union-equal acl2::x acl2::y))
         (and (input-title-listp (list-fix acl2::x))
              (input-title-listp (double-rewrite acl2::y))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-take

(defthm input-title-listp-of-take
  (implies (input-title-listp (double-rewrite acl2::x))
           (iff (input-title-listp (take acl2::n acl2::x))
                (or (input-titlep nil)
                    (<= (nfix acl2::n) (len acl2::x)))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-repeat

(defthm input-title-listp-of-repeat
  (iff (input-title-listp (repeat acl2::n acl2::x))
       (or (input-titlep acl2::x)
           (zp acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: input-titlep-of-nth-when-input-title-listp

(defthm input-titlep-of-nth-when-input-title-listp
  (implies (input-title-listp acl2::x)
           (iff (input-titlep (nth acl2::n acl2::x))
                (< (nfix acl2::n) (len acl2::x))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-update-nth

(defthm input-title-listp-of-update-nth
  (implies
       (input-title-listp (double-rewrite acl2::x))
       (iff (input-title-listp (update-nth acl2::n acl2::y acl2::x))
            (and (input-titlep acl2::y)
                 (or (<= (nfix acl2::n) (len acl2::x))
                     (input-titlep nil)))))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-butlast

(defthm input-title-listp-of-butlast
  (implies (input-title-listp (double-rewrite acl2::x))
           (input-title-listp (butlast acl2::x acl2::n)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-nthcdr

(defthm input-title-listp-of-nthcdr
  (implies (input-title-listp (double-rewrite acl2::x))
           (input-title-listp (nthcdr acl2::n acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-last

(defthm input-title-listp-of-last
  (implies (input-title-listp (double-rewrite acl2::x))
           (input-title-listp (last acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-remove

(defthm input-title-listp-of-remove
  (implies (input-title-listp acl2::x)
           (input-title-listp (remove acl2::a acl2::x)))
  :rule-classes ((:rewrite)))

Theorem: input-title-listp-of-revappend

(defthm input-title-listp-of-revappend
  (equal (input-title-listp (revappend acl2::x acl2::y))
         (and (input-title-listp (list-fix acl2::x))
              (input-title-listp acl2::y)))
  :rule-classes ((:rewrite)))