Module Grammar

Require Import List.
Require Import Syntax.
Require Import Alphabet.
Require Import Orders.
Require Tuples.

The terminal non-terminal alphabets of the grammar. *

Module Type Alphs.
  Parameters terminal nonterminal : Type.
  Declare Instance TerminalAlph: Alphabet terminal.
  Declare Instance NonTerminalAlph: Alphabet nonterminal.
End Alphs.

Definition of the alphabet of symbols, given the alphabet of terminals and the alphabet of non terminals *

Module Symbol(Import A:Alphs).

  Inductive symbol :=
    | T: terminal -> symbol
    | NT: nonterminal -> symbol.

  Program Instance SymbolAlph : Alphabet symbol :=
    { AlphabetComparable := {| compare := fun x y =>
        match x, y return comparison with
          | T _, NT _ => Gt
          | NT _, T _ => Lt
          | T x, T y => compare x y
          | NT x, NT y => compare x y
        end |};
      AlphabetFinite := {| all_list :=
        map T all_list++map NT all_list |} }.

Next Obligation.

destruct x; destruct y; intuition; apply compare_antisym.
Qed.

Next Obligation.

  destruct x; destruct y; destruct z; intuition; try discriminate.
  apply (compare_trans _ t0); intuition.
  apply (compare_trans _ n0); intuition.
  Qed.

Next Obligation.

  intros x y.
  destruct x; destruct y; try discriminate; intros.
  rewrite (compare_eq t t0); intuition.
  rewrite (compare_eq n n0); intuition.
  Qed.

Next Obligation.

  rewrite in_app_iff.
  destruct x; [left | right]; apply in_map; apply all_list_forall.
  Qed.

End Symbol.

Module Type T.
Export Tuples.

Include Alphs <+ Symbol.

symbol_semantic_type maps a symbols to the type of its semantic values. *

Parameter symbol_semantic_type: symbol -> Type.

The type of productions identifiers *

Parameter production : Type.
Declare Instance ProductionAlph : Alphabet production.

Accessors for productions: left hand side, right hand side, and semantic action. The semantic actions are given in the form of curryfied functions, that take arguments in the reverse order. *

  Parameter prod_lhs: production -> nonterminal.
  Parameter prod_rhs_rev: production -> list symbol.
  Parameter prod_action:
    forall p:production,
      arrows_left
        (map symbol_semantic_type (rev (prod_rhs_rev p)))
        (symbol_semantic_type (NT (prod_lhs p))).

End T.

Module Defs(Import G:T).

A token is a terminal and a semantic value for this terminal. *

Definition token := {t:terminal & symbol_semantic_type (T t)}.

A grammar creates a relation between word of tokens and semantic values. This relation is parametrized by the head symbol. It defines the "semantics" of the grammar. This relation is defined by a notion of parse tree. *

  Inductive parse_tree:
    forall (head_symbol:symbol) (word:list token)
      (semantic_value:symbol_semantic_type head_symbol), Type :=

A single token has its semantic value as semantic value, for the corresponding terminal as head symbol. *

  | Terminal_pt:
    forall (t:terminal) (sem:symbol_semantic_type (T t)),
      parse_tree (T t)
      [existT (fun t => symbol_semantic_type (T t)) t sem] sem

Given a production, if a word has a list of semantic values for the right hand side as head symbols, then this word has the semantic value given by the semantic action of the production for the left hand side as head symbol.*

  | Non_terminal_pt:
    forall {p:production} {word:list token}
      {semantic_values:tuple (map symbol_semantic_type (rev (prod_rhs_rev p)))},
      parse_tree_list (rev (prod_rhs_rev p)) word semantic_values ->
      parse_tree (NT (prod_lhs p)) word (uncurry (prod_action p) semantic_values)

Basically the same relation as before, but for list of head symbols (ie. We are building a forest of syntax trees. It is mutually recursive with the previous relation *

  with parse_tree_list:
    forall (head_symbols:list symbol) (word:list token)
      (semantic_values:tuple (map symbol_semantic_type head_symbols)),
      Type :=

The empty word has () as semantic for [] as head symbols list *

| Nil_ptl: parse_tree_list [] [] ()

The cons of the semantic value for one head symbol and for a list of head symbols *

| Cons_ptl:

The semantic for the head *

    forall {head_symbolt:symbol} {wordt:list token}
      {semantic_valuet:symbol_semantic_type head_symbolt},
      parse_tree head_symbolt wordt semantic_valuet ->

and the semantic for the tail *

    forall {head_symbolsq:list symbol} {wordq:list token}
      {semantic_valuesq:tuple (map symbol_semantic_type head_symbolsq)},
      parse_tree_list head_symbolsq wordq semantic_valuesq ->

give the semantic of the cons *

      parse_tree_list
        (head_symbolt::head_symbolsq)
        (wordt++wordq)
        (semantic_valuet, semantic_valuesq).

  Fixpoint pt_size {head_symbol word sem} (tree:parse_tree head_symbol word sem) :=
    match tree with
      | Terminal_pt _ _ => 1
      | Non_terminal_pt l => S (ptl_size l)
    end
  with ptl_size {head_symbols word sems} (tree:parse_tree_list head_symbols word sems) :=
    match tree with
      | Nil_ptl => 0
      | Cons_ptl t q =>
         pt_size t + ptl_size q
    end.
End Defs.