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• 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)))