• Top
  • Documentation
  • Books
  • Boolean-reasoning
  • Projects
    • Apt
    • Zfc
    • Acre
    • Milawa
    • Smtlink
    • Abnf
      • Deftreeops
      • Defdefparse
      • Defgrammar
      • Tree-utilities
      • Notation
        • Syntax-abstraction
        • Semantics
          • Tree-terminatedp
          • Tree->string
          • String-has-finite-parse-trees-p
          • Parse-trees-of-string-p
          • Tree-match-element-p
          • Parse-treep
          • Symbol
          • String-unambiguousp
          • Tree-match-num-val-p
          • Nat-match-insensitive-char-p
          • Nats-match-insensitive-chars-p
          • Tree-option
          • String-parsablep
          • Lookup-rulename
          • Nats-match-sensitive-chars-p
          • Numrep-match-repeat-range-p
          • Tree-match-char-val-p
          • Tree-list-match-repetition-p
          • String-ambiguousp
          • Parse
          • Tree-match-prose-val-p
          • Nat-match-sensitive-char-p
          • Theorems-about-terminated-trees-matching-elements
          • Tree-option-result
          • Tree-list-result
          • Tree-list-list-result
          • Tree-result
          • Tree-list-list-match-concatenation-p
          • Languagep
          • Terminal-string-for-rules-p
          • Tree-list-list-match-alternation-p
          • Tree-list-match-element-p
          • Parse!
          • String
          • Tree-set
          • Trees
        • Abstract-syntax
        • Core-rules
        • Concrete-syntax
      • Grammar-parser
      • Meta-circular-validation
      • Parsing-primitives-defresult
      • Parsing-primitives-seq
      • Operations
      • Examples
      • Differences-with-paper
      • Constructor-utilities
      • Grammar-printer
      • Parsing-tools
    • 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
    • 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

• Semantics

Parse-trees-of-string-p

Check if a finite set of trees is the set of all and only the parse trees of a string.

If this is true, then the string has a finite number of parse trees.

Definitions and Theorems

Theorem: parse-trees-of-string-p-necc0

(defthm parse-trees-of-string-p-necc0
  (implies (parse-trees-of-string-p trees string rulename rules)
           (iff (in tree (tree-set-fix trees))
                (parse-treep tree string rulename rules))))

Theorem: booleanp-of-parse-trees-of-string-p

(defthm booleanp-of-parse-trees-of-string-p
  (b*
    ((yes/no (parse-trees-of-string-p trees string rulename rules)))
    (booleanp yes/no))
  :rule-classes :rewrite)

Theorem: parse-trees-of-string-p-necc

(defthm parse-trees-of-string-p-necc
 (implies (and (parse-trees-of-string-p trees string rulename rules)
               (tree-setp trees))
          (iff (in tree trees)
               (parse-treep tree string rulename rules))))

Theorem: parse-trees-of-string-p-of-tree-set-fix-trees

(defthm parse-trees-of-string-p-of-tree-set-fix-trees
  (equal (parse-trees-of-string-p (tree-set-fix trees)
                                  string rulename rules)
         (parse-trees-of-string-p trees string rulename rules)))

Theorem: parse-trees-of-string-p-tree-set-equiv-congruence-on-trees

(defthm parse-trees-of-string-p-tree-set-equiv-congruence-on-trees
 (implies
  (tree-set-equiv trees trees-equiv)
  (equal
       (parse-trees-of-string-p trees string rulename rules)
       (parse-trees-of-string-p trees-equiv string rulename rules)))
 :rule-classes :congruence)

Theorem: parse-trees-of-string-p-of-string-fix-string

(defthm parse-trees-of-string-p-of-string-fix-string
  (equal (parse-trees-of-string-p trees (string-fix string)
                                  rulename rules)
         (parse-trees-of-string-p trees string rulename rules)))

Theorem: parse-trees-of-string-p-string-equiv-congruence-on-string

(defthm parse-trees-of-string-p-string-equiv-congruence-on-string
 (implies
  (string-equiv string string-equiv)
  (equal
       (parse-trees-of-string-p trees string rulename rules)
       (parse-trees-of-string-p trees string-equiv rulename rules)))
 :rule-classes :congruence)

Theorem: parse-trees-of-string-p-of-rulename-fix-rulename

(defthm parse-trees-of-string-p-of-rulename-fix-rulename
  (equal
       (parse-trees-of-string-p trees string (rulename-fix rulename)
                                rules)
       (parse-trees-of-string-p trees string rulename rules)))

Theorem: parse-trees-of-string-p-rulename-equiv-congruence-on-rulename

(defthm
      parse-trees-of-string-p-rulename-equiv-congruence-on-rulename
 (implies
  (rulename-equiv rulename rulename-equiv)
  (equal
       (parse-trees-of-string-p trees string rulename rules)
       (parse-trees-of-string-p trees string rulename-equiv rules)))
 :rule-classes :congruence)

Theorem: parse-trees-of-string-p-of-rulelist-fix-rules

(defthm parse-trees-of-string-p-of-rulelist-fix-rules
 (equal
      (parse-trees-of-string-p trees
                               string rulename (rulelist-fix rules))
      (parse-trees-of-string-p trees string rulename rules)))

Theorem: parse-trees-of-string-p-rulelist-equiv-congruence-on-rules

(defthm parse-trees-of-string-p-rulelist-equiv-congruence-on-rules
 (implies
  (rulelist-equiv rules rules-equiv)
  (equal
       (parse-trees-of-string-p trees string rulename rules)
       (parse-trees-of-string-p trees string rulename rules-equiv)))
 :rule-classes :congruence)

Theorem: at-most-one-parse-tree-set-of-string

(defthm at-most-one-parse-tree-set-of-string
  (implies
       (and (tree-setp trees1)
            (tree-setp trees2)
            (parse-trees-of-string-p trees1 string rulename rules)
            (parse-trees-of-string-p trees2 string rulename rules))
       (equal trees1 trees2))
  :rule-classes nil)

Theorem: parse-trees-of-string-p-when-not-string-parsablep

(defthm parse-trees-of-string-p-when-not-string-parsablep
  (implies
       (and (not (string-parsablep string rulename rules))
            (tree-setp trees))
       (equal (parse-trees-of-string-p trees string rulename rules)
              (equal trees nil))))

Theorem: not-string-parsablep-when-parse-trees-of-string-p-of-nil

(defthm not-string-parsablep-when-parse-trees-of-string-p-of-nil
  (implies (parse-trees-of-string-p nil string rulename rules)
           (not (string-parsablep string rulename rules))))

Theorem: parse-trees-of-string-p-when-string-unambiguousp

(defthm parse-trees-of-string-p-when-string-unambiguousp
 (implies
  (and (string-unambiguousp string rulename rules)
       (tree-setp trees))
  (equal
     (parse-trees-of-string-p trees string rulename rules)
     (equal trees
            (insert (string-parsablep-witness string rulename rules)
                    nil)))))

Theorem: string-unambiguousp-when-parse-trees-of-string-p-of-one

(defthm string-unambiguousp-when-parse-trees-of-string-p-of-one
  (implies (and (parse-trees-of-string-p (insert tree nil)
                                         string rulename rules)
                (treep tree))
           (string-unambiguousp string rulename rules)))