(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)
(*i $Id$ i*)

Require Export Rdefinitions.

Axiom NRplus : R->R->R.

Grammar rnatural ident :=
  nat_id [ prim:var($id) ] -> [$id]

with rnegnumber : constr := 
  neg_expr [ "-" rnumber ($c) ] -> [ (Ropp $c) ] 

with rnumber :=

with rformula : constr :=
  form_expr [ rexpr($p) ] -> [ $p ]
| form_eq [ rexpr($p) "==" rexpr($c) ] -> [ (eqT R $p $c) ]
| form_le [ rexpr($p) "<=" rexpr($c) ] -> [ (Rle $p $c) ]
| form_lt [ rexpr($p) "<" rexpr($c) ] -> [ (Rlt $p $c) ]
| form_ge [ rexpr($p) ">=" rexpr($c) ] -> [ (Rge $p $c) ]
| form_gt [ rexpr($p) ">" rexpr($c) ] -> [ (Rgt $p $c) ]
| form_eq_eq [ rexpr($p) "==" rexpr($c) "==" rexpr($c1) ]
              -> [ (eqT R $p $c)/\(eqT R $c $c1) ]
| form_le_le [ rexpr($p) "<=" rexpr($c) "<=" rexpr($c1) ]
              -> [ (Rle $p $c)/\(Rle $c $c1) ]
| form_le_lt [ rexpr($p) "<=" rexpr($c) "<" rexpr($c1) ]
              -> [ (Rle $p $c)/\(Rlt $c $c1) ]
| form_lt_le [ rexpr($p) "<" rexpr($c) "<=" rexpr($c1) ]
              -> [ (Rlt $p $c)/\(Rle $c $c1) ]
| form_lt_lt [ rexpr($p) "<" rexpr($c) "<" rexpr($c1) ]
              -> [ (Rlt $p $c)/\(Rlt $c $c1) ]
| form_neq  [ rexpr($p) "<>" rexpr($c) ] -> [ ~(eqT R $p $c) ]

with rexpr : constr :=
  expr_plus [ rexpr($p) "+" rexpr($c) ] -> [ (Rplus $p $c) ]
| expr_minus [ rexpr($p) "-" rexpr($c) ] -> [ (Rminus $p $c) ]
| rexpr2 [ rexpr2($e) ] -> [ $e ]

with rexpr2 : constr :=
  expr_mult [ rexpr2($p) "*" rexpr2($c) ] -> [ (Rmult $p $c) ]
| rexpr0 [ rexpr0($e) ] -> [ $e ]


with rexpr0 : constr :=
  expr_id [ constr:global($c) ] -> [ $c ]
| expr_com [ "[" constr:constr($c) "]" ] -> [ $c ]
| expr_appl [ "(" rapplication($a) ")" ] -> [ $a ]
| expr_num [ rnumber($s) ] -> [ $s ]
| expr_negnum [ "-" rnegnumber($n) ] -> [ $n ]
| expr_div [ rexpr0($p) "/" rexpr0($c) ] -> [ (Rdiv $p $c) ]
| expr_opp [ "-" rexpr0($c) ] -> [ (Ropp $c) ] 
| expr_inv [ "/" rexpr0($c) ] -> [ (Rinv $c) ]
| expr_meta [ meta($m) ] -> [ $m ]

with meta : ast :=
| rimpl [ "?" ] -> [ (ISEVAR) ]
| rmeta [ "?" prim:numarg($n) ] -> [ (META $n) ]

with rapplication : constr :=
  apply [ rapplication($p) rexpr($c1) ] -> [ ($p $c1) ]
| pair [ rexpr($p) "," rexpr($c) ] -> [ ($p, $c) ]
| appl0 [ rexpr($a) ] -> [ $a ].


Grammar constr constr0 :=
  r_in_com [ "``" rnatural:rformula($c) "``" ] -> [ $c ].

Grammar constr atomic_pattern :=
  r_in_pattern [ "``" rnatural:rnumber($c) "``" ] -> [ $c ].   

(*i* pp **)
(* pb: on rajoute des () lorsque les constantes commencent par 1 lors de
l'appel avec NRplus i*)



Syntax constr
  level 0:
   Rle [ (Rle $n1 $n2) ] -> 
      [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2) "``"]]
  | Rlt [ (Rlt $n1 $n2) ] -> 
      [[<hov 0> "``" (REXPR $n1) [1 0] "< "(REXPR $n2) "``" ]]
  | Rge [ (Rge $n1 $n2) ] -> 
      [[<hov 0> "``" (REXPR $n1) [1 0] ">= "(REXPR $n2) "``" ]]
  | Rgt [ (Rgt $n1 $n2) ] -> 
      [[<hov 0> "``" (REXPR $n1) [1 0] "> "(REXPR $n2) "``" ]]
  | Req [ (eqT R $n1 $n2) ] -> 
      [[<hov 0> "``" (REXPR $n1) [1 0] "== "(REXPR $n2)"``"]]
  | Rneq [ ~(eqT R $n1 $n2) ] ->
      [[<hov 0> "``" (REXPR $n1) [1 0] "<> "(REXPR $n2) "``"]]
  | Rle_Rle [ (Rle $n1 $n2)/\(Rle $n2 $n3) ] ->
      [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2)
                                [1 0] "<= " (REXPR $n3) "``"]]
  | Rle_Rlt [ (Rle $n1 $n2)/\(Rlt $n2 $n3) ] ->
      [[<hov 0> "``" (REXPR $n1) [1 0] "<= "(REXPR $n2)
                                [1 0] "< " (REXPR $n3) "``"]]
  | Rlt_Rle [ (Rlt $n1 $n2)/\(Rle $n2 $n3) ] ->
      [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2)
                                [1 0] "<= " (REXPR $n3) "``"]]
  | Rlt_Rlt [ (Rlt $n1 $n2)/\(Rlt $n2 $n3) ] ->
      [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2)
                                [1 0] "< " (REXPR $n3) "``"]]
  | Rzero [ R0 ] -> ["``0``"]
  | Rone [ R1 ] -> ["``1``"]
  ;

  level 7:
    Rplus [ (Rplus $n1 $n2) ]
      -> [ [<hov 0> "``"(REXPR $n1):E "+"  [0 0] (REXPR $n2):L "``"] ]
   | Rconst [(Rplus R1 $r)] -> [$r:"r_printer_outside"]
   | Rminus [ (Rminus $n1 $n2) ]
      -> [ [<hov 0> "``"(REXPR $n1):E "-" [0 0] (REXPR $n2):L "``"] ]
  ;

  level 6:
    Rmult [ (Rmult $n1 $n2) ]
      -> [ [<hov 0> "``"(REXPR $n1):E "*" [0 0] (REXPR $n2):L "``"] ]
    |Rdiv [ (Rdiv $n1 $n2) ]
      -> [ [<hov 0> "``"(REXPR $n1):E "/" [0 0] (REXPR $n2):L "``"] ]
  ;

  level 8:
    Ropp [(Ropp $n1)] -> [ [<hov 0> "``" "-"(REXPR $n1):E "``"] ]
    |Rinv [(Rinv $n1)] -> [ [<hov 0> "``" "/"(REXPR $n1):E "``"] ]
  ;

  level 0:
    rescape_inside [<< (REXPR $r) >>] -> [ "[" $r:E "]" ]
  ;

  level 4:
    Rappl_inside [<<(REXPR (APPLIST $h ($LIST $t)))>>]
 -> [ [<hov 0> "("(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E ")"] ]
  | Rappl_inside_tail [<<(RAPPLINSIDETAIL $h ($LIST $t))>>]
      -> [(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E] 
  | Rappl_inside_one [<<(RAPPLINSIDETAIL $e)>>] ->[(REXPR $e):E]
  | rpair_inside [<<(REXPR <<(pair $s1 $s2 $r1 $r2)>>)>>] 
      -> [ [<hov 0> "("(REXPR $r1):E "," [1 0] (REXPR $r2):E ")"] ]
  ;

 level 3:
    rvar_inside [<<(REXPR ($VAR $i))>>] -> [$i]
  | rsecvar_inside [<<(REXPR (SECVAR $i))>>] -> [(SECVAR $i)]
  | rconst_inside [<<(REXPR (CONST $c))>>] -> [(CONST $c)]
  | rmutind_inside [<<(REXPR (MUTIND $i $n))>>] 
      -> [(MUTIND $i $n)]
  | rmutconstruct_inside [<<(REXPR (MUTCONSTRUCT $c1 $c2 $c3))>>]
      -> [ (MUTCONSTRUCT $c1 $c2 $c3) ]
  | rimplicit_head_inside [<<(REXPR (XTRA "!" $c))>>] -> [ $c ]
  | rimplicit_arg_inside  [<<(REXPR (XTRA "!" $n $c))>>] -> [ ]

  ;

  
  level 7:
   Rplus_inside
      [<<(REXPR <<(Rplus $n1 $n2)>>)>>]
      	 -> [ (REXPR $n1):E "+" [0 0] (REXPR $n2):L ]
  | Rminus_inside
      [<<(REXPR <<(Rminus $n1 $n2)>>)>>]
      	 -> [ (REXPR $n1):E "-" [0 0] (REXPR $n2):L ]
  | NRplus_inside
      [<<(REXPR <<(NRplus $n1 $n2)>>)>>]
      	 -> ["(" (REXPR $n1):E "+" [0 0] (REXPR $n2):L ")"]
  ;

  level 6:
    Rmult_inside
      [<<(REXPR <<(Rmult $n1 $n2)>>)>>]
      	 -> [ (REXPR $n1):E "*" [0 0] (REXPR $n2):L ]
  ;

  level 5:
 Ropp_inside [<<(REXPR <<(Ropp $n1)>>)>>] -> [ " -" (REXPR $n1):E  ]
|Rinv_inside [<<(REXPR <<(Rinv $n1)>>)>>] -> [ "/" (REXPR $n1):E  ]
|Rdiv_inside
      [<<(REXPR <<(Rdiv $n1 $n2)>>)>>]
      	 -> [ (REXPR $n1):E "/" [0 0] (REXPR $n2):L ]
  ;  

  level 0:
    Rzero_inside [<<(REXPR <<R0>>)>>] -> ["0"]
  | Rone_inside [<<(REXPR <<R1>>)>>] -> ["1"]
  | Rconst_inside [<<(REXPR <<(Rplus R1 $r)>>)>>] -> [$r:"r_printer"].