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+Require Coq.Strings.String.
+
+Set Implicit Arguments.
+
+Axiom falso : False.
+Ltac admit := destruct falso.
+
+Reserved Notation "[ x ]".
+
+Record string_like (CharType : Type) :=
+  {
+    String :> Type;
+    Singleton : CharType -> String where "[ x ]" := (Singleton x);
+    Empty : String;
+    Concat : String -> String -> String where "x ++ y" := (Concat x y);
+    bool_eq : String -> String -> bool;
+    bool_eq_correct : forall x y : String, bool_eq x y = true <-> x = y;
+    Length : String -> nat
+  }.
+
+Delimit Scope string_like_scope with string_like.
+Bind Scope string_like_scope with String.
+Arguments Length {_%type_scope _} _%string_like.
+Infix "++" := (@Concat _ _) : string_like_scope.
+
+Definition str_le {CharType} {String : string_like CharType} (s1 s2 : String)
+  := Length s1 < Length s2 \/ s1 = s2.
+Infix "≤s" := str_le (at level 70, right associativity).
+
+Module Export ContextFreeGrammar.
+  Import Coq.Strings.String.
+  Import Coq.Lists.List.
+
+  Section cfg.
+    Variable CharType : Type.
+
+    Section definitions.
+
+      Inductive item :=
+    | NonTerminal (name : string).
+
+      Definition production := list item.
+      Definition productions := list production.
+
+      Record grammar :=
+        {
+          Start_symbol :> string;
+          Lookup :> string -> productions
+        }.
+    End definitions.
+
+    Section parse.
+      Variable String : string_like CharType.
+      Variable G : grammar.
+
+      Inductive parse_of : String -> productions -> Type :=
+      | ParseHead : forall str pat pats, parse_of_production str pat
+                                         -> parse_of str (pat::pats)
+      | ParseTail : forall str pat pats, parse_of str pats
+                                         -> parse_of str (pat::pats)
+      with parse_of_production : String -> production -> Type :=
+      | ParseProductionCons : forall str pat strs pats,
+                                parse_of_item str pat
+                                -> parse_of_production strs pats
+                                -> parse_of_production (str ++ strs) (pat::pats)
+      with parse_of_item : String -> item -> Type :=
+      | ParseNonTerminal : forall name str, parse_of str (Lookup G name)
+                                            -> parse_of_item str (NonTerminal 
+name).
+    End parse.
+  End cfg.
+
+End ContextFreeGrammar.
+Module Export ContextFreeGrammarProperties.
+
+  Section cfg.
+    Context CharType (String : string_like CharType) (G : grammar)
+            (P : String.string -> Type).
+
+    Fixpoint Forall_parse_of {str pats} (p : parse_of String G str pats)
+      := match p with
+           | @ParseHead _ _ _ str pat pats p'
+             => Forall_parse_of_production p'
+           | @ParseTail _ _ _ _ _ _ p'
+             => Forall_parse_of p'
+         end
+    with Forall_parse_of_production {str pat} (p : parse_of_production String G 
+str pat)
+         := let Forall_parse_of_item {str it} (p : parse_of_item String G str 
+it)
+                := match p return Type with
+                     | @ParseNonTerminal _ _ _ name str p'
+                       => (P name * Forall_parse_of p')%type
+                   end in
+            match p return Type with
+              | @ParseProductionCons _ _ _ str pat strs pats p' p''
+                => (Forall_parse_of_item p' * Forall_parse_of_production 
+p'')%type
+            end.
+
+    Definition Forall_parse_of_item {str it} (p : parse_of_item String G str it)
+      := match p return Type with
+           | @ParseNonTerminal _ _ _ name str p'
+             => (P name * Forall_parse_of p')%type
+         end.
+  End cfg.
+
+End ContextFreeGrammarProperties.
+
+Module Export DependentlyTyped.
+  Import Coq.Strings.String.
+
+  Section recursive_descent_parser.
+
+    Class parser_computational_predataT :=
+      { nonterminal_names_listT : Type;
+        initial_nonterminal_names_data : nonterminal_names_listT;
+        is_valid_nonterminal_name : nonterminal_names_listT -> string -> bool;
+        remove_nonterminal_name : nonterminal_names_listT -> string -> 
+nonterminal_names_listT }.
+
+  End recursive_descent_parser.
+
+End DependentlyTyped.
+Import Coq.Strings.String.
+Import Coq.Lists.List.
+
+Section cfg.
+  Context CharType (String : string_like CharType) (G : grammar).
+  Context (names_listT : Type)
+          (initial_names_data : names_listT)
+          (is_valid_name : names_listT -> string -> bool)
+          (remove_name : names_listT -> string -> names_listT).
+
+  Inductive minimal_parse_of
+  : forall (str0 : String) (valid : names_listT)
+           (str : String),
+      productions -> Type :=
+  | MinParseHead : forall str0 valid str pat pats,
+                     @minimal_parse_of_production str0 valid str pat
+                     -> @minimal_parse_of str0 valid str (pat::pats)
+  | MinParseTail : forall str0 valid str pat pats,
+                     @minimal_parse_of str0 valid str pats
+                     -> @minimal_parse_of str0 valid str (pat::pats)
+  with minimal_parse_of_production
+       : forall (str0 : String) (valid : names_listT)
+                (str : String),
+           production -> Type :=
+  | MinParseProductionNil : forall str0 valid,
+                              @minimal_parse_of_production str0 valid (Empty _) 
+nil
+  | MinParseProductionCons : forall str0 valid str strs pat pats,
+                               str ++ strs ≤s str0
+                               -> @minimal_parse_of_item str0 valid str pat
+                               -> @minimal_parse_of_production str0 valid strs 
+pats
+                               -> @minimal_parse_of_production str0 valid (str 
+++ strs) (pat::pats)
+  with minimal_parse_of_item
+       : forall (str0 : String) (valid : names_listT)
+                (str : String),
+           item -> Type :=
+  | MinParseNonTerminal
+    : forall str0 valid str name,
+        @minimal_parse_of_name str0 valid str name
+        -> @minimal_parse_of_item str0 valid str (NonTerminal name)
+  with minimal_parse_of_name
+       : forall (str0 : String) (valid : names_listT)
+                (str : String),
+           string -> Type :=
+  | MinParseNonTerminalStrLt
+    : forall str0 valid name str,
+        @minimal_parse_of str initial_names_data str (Lookup G name)
+        -> @minimal_parse_of_name str0 valid str name
+  | MinParseNonTerminalStrEq
+    : forall str valid name,
+        @minimal_parse_of str (remove_name valid name) str (Lookup G name)
+        -> @minimal_parse_of_name str valid str name.
+  Definition parse_of_item_name__of__minimal_parse_of_name
+  : forall {str0 valid str name} (p : @minimal_parse_of_name str0 valid str 
+name),
+      parse_of_item String G str (NonTerminal name).
+  Proof.
+    admit.
+  Defined.
+
+End cfg.
+
+Section recursive_descent_parser.
+  Context (CharType : Type)
+          (String : string_like CharType)
+          (G : grammar).
+  Context {premethods : parser_computational_predataT}.
+  Let P : string -> Prop.
+  Proof.
+    admit.
+  Defined.
+
+  Let mp_parse_nonterminal_name str0 valid str nonterminal_name
+    := { p' : minimal_parse_of_name String G initial_nonterminal_names_data 
+remove_nonterminal_name str0 valid str nonterminal_name & Forall_parse_of_item 
+P (parse_of_item_name__of__minimal_parse_of_name p') }.
+
+  Goal False.
+  Proof.
+    clear -mp_parse_nonterminal_name.
+    subst P.
+    simpl in *.
+    admit.
+  Qed.