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• Semantics

String-has-finite-parse-trees-p

Check if a string has a finite number of parse trees.

If this is not true, the string is ambiguous, because it has an infinite number of parse trees.

Definitions and Theorems

Theorem: string-has-finite-parse-trees-p-suff

(defthm string-has-finite-parse-trees-p-suff
  (implies
       (and (tree-setp trees)
            (parse-trees-of-string-p trees string rulename rules))
       (string-has-finite-parse-trees-p string rulename rules)))

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

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

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

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

Theorem: string-has-finite-parse-trees-p-string-equiv-congruence-on-string

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

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

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

Theorem: string-has-finite-parse-trees-p-rulename-equiv-congruence-on-rulename

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

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

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

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

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

Theorem: tree-setp-of-witness-when-string-has-finite-parse-trees-p

(defthm tree-setp-of-witness-when-string-has-finite-parse-trees-p
  (implies (string-has-finite-parse-trees-p string rulename rules)
           (tree-setp (string-has-finite-parse-trees-p-witness
                           string rulename rules))))

Theorem: string-has-finite-parse-trees-p-when-not-string-parsablep

(defthm string-has-finite-parse-trees-p-when-not-string-parsablep
  (implies (not (string-parsablep string rulename rules))
           (string-has-finite-parse-trees-p string rulename rules)))

Theorem: string-has-finite-parse-trees-p-when-string-unambiguousp

(defthm string-has-finite-parse-trees-p-when-string-unambiguousp
  (implies (string-unambiguousp string rulename rules)
           (string-has-finite-parse-trees-p string rulename rules)))

Theorem: string-has-finite-parse-trees-p-witness-when-not-string-parsablep

(defthm
  string-has-finite-parse-trees-p-witness-when-not-string-parsablep
 (implies
  (not (string-parsablep string rulename rules))
  (equal
     (string-has-finite-parse-trees-p-witness string rulename rules)
     nil)))

Theorem: string-has-finite-parse-trees-p-witness-when-string-unambiguousp

(defthm
   string-has-finite-parse-trees-p-witness-when-string-unambiguousp
 (implies
  (string-unambiguousp string rulename rules)
  (equal
     (string-has-finite-parse-trees-p-witness string rulename rules)
     (insert (string-parsablep-witness string rulename rules)
             nil))))

Theorem: not-string-parsablep-when-string-has-finite-parse-trees-p-nil

(defthm
      not-string-parsablep-when-string-has-finite-parse-trees-p-nil
 (implies
  (and
   (string-has-finite-parse-trees-p string rulename rules)
   (equal
     (string-has-finite-parse-trees-p-witness string rulename rules)
     nil))
  (not (string-parsablep string rulename rules))))

Theorem: string-unambiguousp-when-string-has-finite-parse-trees-p-one

(defthm string-unambiguousp-when-string-has-finite-parse-trees-p-one
 (implies
  (and
   (string-has-finite-parse-trees-p string rulename rules)
   (equal
     (string-has-finite-parse-trees-p-witness string rulename rules)
     (insert (string-parsablep-witness string rulename rules)
             nil)))
  (string-unambiguousp string rulename rules)))

Theorem: string-ambiguousp-when-infinite-parse-trees

(defthm string-ambiguousp-when-infinite-parse-trees
  (implies
       (not (string-has-finite-parse-trees-p string rulename rules))
       (string-ambiguousp string rulename rules)))