solver.ml 23.8 KB
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(*-----------------------------------------------------------------------
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** Copyright (C) 2001, 2002 - Verimag.
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** This file may only be copied under the terms of the GNU Library General
** Public License 
**-----------------------------------------------------------------------
**
** File: solver.ml
** Main author: jahier@imag.fr
*)

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open List
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open Formula
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open Util
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open Hashtbl
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open Gne
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open Rnumsolver
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(****************************************************************************)
	  
let (formula_list_to_conj: formula list -> formula) =
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  fun fl -> 
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    (** Transform a (non-empty) list of formula to the conjunction
       made of those formula.  
    *)
    match fl with
	[] -> assert false
      | f::[] -> f
      | f1::f2::tail -> 
          List.fold_left (fun x y -> And(x, y)) (And(f1, f2)) tail

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let (lookup: env_in -> subst list -> var_name -> var_value option) = 
  fun input pre vn ->  
    try Some(Hashtbl.find input vn)
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    with Not_found -> 
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      try Some(List.assoc vn pre)
      with Not_found -> None

(****************************************************************************)

type comp = SupZero | SupEqZero | EqZero | NeqZero


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let rec (formula_to_bdd : env_in -> formula -> Bdd.t * bool) =
  fun input f ->
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    (** Returns the bdd of [f] and replaces input and pre variables
      by their values.
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      Also returns a flag that is true iff the formula depends on
      input and pre vars. If this flag is false, the formula is
      stored (cached) in a global table ([env_state.bdd_tbl_global]);
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      otherwise, it is stored in a table that is cleared at each new
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      step ([env_state.bdd_tbl]).  
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    *)
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    try (Env_state.bdd f, true)
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    with Not_found -> 
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      try (Env_state.bdd_global f, false)
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      with Not_found -> 
	let (bdd, dep) =
	  match f with 
	      Not(f1) ->
		let (bdd_not, dep) =  (formula_to_bdd input f1) in
		  (Bdd.dnot bdd_not, dep)

	    | Or(f1, f2) ->
		let (bdd1, dep1) = (formula_to_bdd input f1)
		and (bdd2, dep2) = (formula_to_bdd input f2)
		in
		  (Bdd.dor bdd1 bdd2, dep1 || dep2)

	    | And(f1, f2) -> 
		let (bdd1, dep1) = (formula_to_bdd input f1)
		and (bdd2, dep2) = (formula_to_bdd input f2)
		in
		  (Bdd.dand bdd1 bdd2, dep1 || dep2)

	    | IteB(f1, f2, f3) -> 
		let (bdd1, dep1) = (formula_to_bdd input f1)
		and (bdd2, dep2) = (formula_to_bdd input f2)
		and (bdd3, dep3) = (formula_to_bdd input f3) 
		in
		  ((Bdd.dor (Bdd.dand bdd1 bdd2) 
		      (Bdd.dand (Bdd.dnot bdd1) bdd3)),
		   dep1 || dep2 || dep3 )
		  
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	    | True ->  (Bdd.dtrue  (Env_state.bdd_manager ()), false)
	    | False -> (Bdd.dfalse (Env_state.bdd_manager ()), false)
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	    | Bvar(vn) ->    
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		( match (lookup input (Env_state.pre ()) vn) with 
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		      Some(B(bool)) -> 
			if bool 
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			then (Bdd.dtrue  (Env_state.bdd_manager ()), true)
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			else (Bdd.dfalse (Env_state.bdd_manager ()), true)
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		    | Some(_) -> 
			print_string (vn ^ " is not a boolean!\n");
			assert false
		    | None ->
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			if List.mem vn (Env_state.pre_var_names ())
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			then failwith 
			  ("*** " ^ vn ^ " is unknown at this stage.\n "
			   ^ "*** Make sure you have not used "
			   ^ "a pre on a output var at the 1st step, \n "
			   ^ "*** or a pre on a input var at the second step in "
			   ^ "your formula in the environment.\n ")
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			else (Bdd.ithvar (Env_state.bdd_manager ()) 
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				(Env_state.atomic_formula_to_index (Bv(vn))), false)
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		)
		
	    | Eq(e1, e2) -> 
		let (gne, dep) = expr_to_gne (Diff(e1, e2)) input in 
		  (gne_to_bdd gne EqZero, dep)
		  
	    | Neq(e1, e2) -> 
		let (gne, dep) = expr_to_gne (Diff(e1, e2)) input in
		  (gne_to_bdd gne NeqZero, dep)
		  
	    | SupEq(e1, e2) ->
		let (gne, dep) = expr_to_gne (Diff(e1, e2)) input in
		  (gne_to_bdd gne SupEqZero, dep)
		  
	    | Sup(e1, e2)   ->
		let (gne, dep) = expr_to_gne (Diff(e1, e2)) input in
		  (gne_to_bdd gne SupZero, dep)
		  
	    | InfEq(e1, e2) ->  
		let (gne, dep) = expr_to_gne (Diff(e2, e1)) input in
		  (gne_to_bdd gne SupEqZero, dep)
		  
	    | Inf(e1, e2)   ->  
		let (gne, dep) =  expr_to_gne (Diff(e2, e1)) input in
		  (gne_to_bdd gne SupZero, dep)
		  
	in
	  if dep
	  then 
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	    ( Env_state.set_bdd f bdd;	
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	      match f with 
		  Not(nf) -> () (* Already in the tbl thanks to the rec call *)
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		| _  -> Env_state.set_bdd (Not(f)) (Bdd.dnot bdd) 
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	    ) 
	  else 
	    (* [f] does not depend on pre nor input vars *)
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	    ( Env_state.set_bdd_global f bdd ;	
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	      match f with 
		  Not(nf) -> () (* Already in the table thanks to the rec call *)
		| _  -> 
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		    Env_state.set_bdd_global (Not(f)) (Bdd.dnot bdd)
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	    );

	  (bdd, dep)
and
  (expr_to_gne: expr -> env_in -> Gne.gn_expr * bool) =
  fun e input -> 
    (** Evaluates pre and input vars appearing in [e] and tranlates
      it into a so-called garded normal form. Also returns a flag
      that is true iff [e] depends on pre or input vars. *)
    let (gne, dep) =
      match e with  
	  Sum(e1, e2) ->
	    let (gne1, dep1) = (expr_to_gne e1 input)
	    and (gne2, dep2) = (expr_to_gne e2 input) 
	    in
	      (Gne.add  gne1 gne2, dep1 || dep2)

	| Diff(e1, e2) -> 
	    let (gne1, dep1) = (expr_to_gne e1 input)
	    and (gne2, dep2) = (expr_to_gne e2 input) 
	    in
	      (Gne.diff gne1 gne2, dep1 || dep2)

	| Prod(e1, e2) -> 
	    let (gne1, dep1) = (expr_to_gne e1 input)
	    and (gne2, dep2) = (expr_to_gne e2 input) 
	    in
	      (Gne.mult gne1 gne2, dep1 || dep2)

	| Quot(e1, e2) -> 
	    let (gne1, dep1) = (expr_to_gne e1 input)
	    and (gne2, dep2) = (expr_to_gne e2 input) 
	    in
	      (Gne.quot gne1 gne2, dep1 || dep2)

	| Mod(e1, e2)  -> 
	    let (gne1, dep1) = (expr_to_gne e1 input)
	    and (gne2, dep2) = (expr_to_gne e2 input) 
	    in
	      (Gne.modulo gne1 gne2, dep1 || dep2)

	| Ivar(str) ->
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	    ( match (lookup input (Env_state.pre ()) str) with 
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		  Some(N(I(i))) ->
		    ( (GneMap.add 
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			 (NeMap.add "" (I(i)) NeMap.empty) 
			 (Bdd.dtrue (Env_state.bdd_manager ()))
			 GneMap.empty),
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		      true
		    )
		| None ->
		    ( (GneMap.add 
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			 (NeMap.add str (I(1)) NeMap.empty) 
			 (Bdd.dtrue (Env_state.bdd_manager ())) 
			 GneMap.empty),
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		      false
		    )     
		| Some(N(F(f))) -> 
		    print_string ((string_of_float f) 
				  ^ "is a float, but an int is expected.\n");
		    assert false
		| Some(B(f)) -> 
		    print_string ((string_of_bool f) 
				  ^ "is a bool, but an int is expected.\n");
		    assert false
	    )

	| Fvar(str) ->
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	    ( match (lookup input (Env_state.pre ()) str) with 
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		  Some(N(F(f))) ->
		    ( (GneMap.add 
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			 (NeMap.add "" (F(f)) NeMap.empty) 
			 (Bdd.dtrue (Env_state.bdd_manager ())) 
			 GneMap.empty),
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		      true
		    )
		| None ->
		    ( (GneMap.add 
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			 (NeMap.add str (F(1.)) NeMap.empty) 
			 (Bdd.dtrue (Env_state.bdd_manager ())) 
			 GneMap.empty),
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		      false
		    )
		| Some(N(I(i))) -> 
		    print_string ((string_of_int i) 
				  ^ "is an int, but a float is expected.\n");
		    assert false
		| Some(B(f)) -> 
		    print_string ((string_of_bool f) 
				  ^ "is a bool, not a float is expected.\n");
		    assert false
	    )

	| Ival(i) ->  
	    ( (GneMap.add 
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		 (NeMap.add "" (I(i)) NeMap.empty) 
		 (Bdd.dtrue (Env_state.bdd_manager ())) 
		 GneMap.empty),
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	      false
	    )

	| Fval(f) -> 
	    ( (GneMap.add 
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		 (NeMap.add "" (F(f)) NeMap.empty) 
		 (Bdd.dtrue (Env_state.bdd_manager ()))
		 GneMap.empty),
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	      false
	    )

	| Ite(f, e1, e2) -> 
	    let (add_formula_to_gne_acc : Bdd.t -> n_expr -> Bdd.t -> 
		   Gne.gn_expr -> Gne.gn_expr) = 
	      fun bdd nexpr c acc -> 
		(* Used (by a GneMap.fold) to add the condition [c] to every
		   condition of a garded expression. *)
		let _ = assert (
		  try 
		    let _ = GneMap.find nexpr acc in
		      false
		  with Not_found -> true
		) 
		in
		let new_bdd = (Bdd.dand bdd c) in
		  if Bdd.is_false new_bdd
		  then acc
		  else GneMap.add nexpr new_bdd acc
	    in
	    let (bdd, dep1) = formula_to_bdd input f in
	    let bdd_not = Bdd.dnot bdd
	    and (gne_t, dep2) = (expr_to_gne e1 input)
	    and (gne_e, dep3) = (expr_to_gne e2 input) in
	    let gne1 = GneMap.fold (add_formula_to_gne_acc bdd) gne_t GneMap.empty in
	    let gne  = GneMap.fold (add_formula_to_gne_acc bdd_not) gne_e gne1 in
	      (gne, dep1 || dep2 || dep3)
    in
      (gne, dep)
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and
  (gne_to_bdd : Gne.gn_expr -> comp -> Bdd.t) =
  fun gne cmp -> 
    (** Use [cmp] to compare [gne] with 0 and returns the
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      corresponding formula.  E.g., if [gne] is bounded to
      [e1 -> c1; e2 -> c2], then [gne_to_bdd gne SupZero] returns
      (the bdd corresponding to) the formula [(c1 and (e1 > 0)) or
      (c2 and (e2 > 0))] *)
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    match cmp with
	SupZero ->
	  ( GneMap.fold 
	      (fun nexpr c acc -> 
		 let bdd = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i > 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f > 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (GZ(nexpr))) 
		 in
		   Bdd.dor (Bdd.dand c bdd) acc
	      )
	      gne 
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	      (Bdd.dfalse (Env_state.bdd_manager ()))
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	  )
      | SupEqZero ->
	  ( GneMap.fold 
	      (fun nexpr c acc -> 
		 let bdd = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i >= 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f >= 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (GeqZ(nexpr))) 
		 in
		   Bdd.dor (Bdd.dand c bdd) acc
	      )
	      gne 
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	      (Bdd.dfalse (Env_state.bdd_manager ()))
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	  )
      | EqZero -> 
	  ( GneMap.fold 
	      (fun nexpr c acc -> 
		 let bdd1 = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i >= 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f >= 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (GeqZ(nexpr))) 
		 in
		 let bdd2 = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i <= 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f <= 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (SeqZ(nexpr))) 
		 in
		 let bdd = Bdd.dand bdd1 bdd2 in 
		   (* We transform [e1 = e2] into [e1 <= e2 ^ e1 >= e2] as the 
		      numeric solver can not handle equalities *)
		   Bdd.dor (Bdd.dand c bdd) acc
	      )
	      gne 
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	      (Bdd.dfalse (Env_state.bdd_manager ()))
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	  )
      | NeqZero -> 
	  ( GneMap.fold 
	      (fun nexpr c acc -> 
		 let bdd1 = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i > 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f > 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (GZ(nexpr))) 
		 in
		 let bdd2 = 
		   if is_n_expr_a_constant nexpr
		   then 
		     let cst = NeMap.find "" nexpr in
		       match cst with
			   I(i) -> 
			     if i < 0 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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			 | F(f) -> 
			     if f < 0. 
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			     then (Bdd.dtrue (Env_state.bdd_manager ()))
			     else (Bdd.dfalse (Env_state.bdd_manager ()))
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		   else 
		     Bdd.ithvar 
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		       (Env_state.bdd_manager ()) 
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		       (Env_state.atomic_formula_to_index (SZ(nexpr))) 
		 in
		 let bdd = Bdd.dor bdd1 bdd2 in 
		   (* We transform [e1 <> e2] into [e1 < e2 or e1 > e2] as the 
		      numeric solver can not handle disequalities *)
		   Bdd.dor (Bdd.dand c bdd) acc
	      )
	      gne 
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	      (Bdd.dfalse (Env_state.bdd_manager ()))
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	  )
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(****************************************************************************)
(****************************************************************************)


(* Exported *)
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let rec (is_satisfiable: env_in -> formula list -> bool) = 
  fun input fl -> 
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    let f = formula_list_to_conj fl in
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    let (bdd, _) = formula_to_bdd input f in
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      not (Bdd.is_false bdd) &&
      ( 
	try 
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	  let (n, m) = Env_state.sol_number bdd in 
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	    not ((zero_sol, zero_sol) = (n, m))
	with Not_found -> true
      )
      


(****************************************************************************)
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(****************************************************************************)

type var = int

(** In the following, we call a comb the bdd of a conjunction of
 litterals (var). They provide the ordering in which litterals
 appear in the bdds we manipulate.
*)



let rec (build_sol_nb_table: Bdd.t -> Bdd.t -> sol_nb * sol_nb) =
  fun bdd comb -> 
    (** Returns the relative (to which bbd points to it) number of
      solutions of [bdd] and the one of its negation. Also udpates
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      the solution number table for [bdd] and its negation, and
      recursively for all its sub-bdds.

      [comb] is a positive cube that ougth to contain the indexes of
      boolean vars that are still to be generated, and the numerical
      indexes that appears in [bdd].  
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    *)
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    let _ = assert (not (Bdd.is_cst bdd) 
		    && (Bdd.topvar comb) = (Bdd.topvar bdd)) 
    in
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    let bdd_not = (Bdd.dnot bdd) in
    let (sol_nb, sol_nb_not) =
      try
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	let (nt, ne) = Env_state.sol_number bdd 
	and (not_nt, not_ne) = Env_state.sol_number bdd_not in
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	  (* solutions numbers in the table are absolute *)
	  ((add_sol_nb nt ne), (add_sol_nb not_nt not_ne))
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      with Not_found ->
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	let (nt, not_nt) = compute_absolute_sol_nb (Bdd.dthen bdd) comb in
	let (ne, not_ne) = compute_absolute_sol_nb (Bdd.delse bdd) comb in
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	  Env_state.set_sol_number bdd (nt, ne) ;
	  Env_state.set_sol_number bdd_not (not_nt, not_ne) ;
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	  ((add_sol_nb nt ne), (add_sol_nb not_nt not_ne))
    in
      (sol_nb, sol_nb_not)
and 
  (compute_absolute_sol_nb: Bdd.t -> Bdd.t -> sol_nb * sol_nb) =
  fun sub_bdd comb -> 
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    (* Returns the absolute number of solutions of [sub_bdd] (and its
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       negation) w.r.t. [comb], where [comb] is the comb of the
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       father of [sub_bdd].

       The [comb] is used to know which output boolean variables are
       unconstraint along a path in the bdd. Indeed, the comb is made
       of all the boolean output var indexes plus the num contraints
       indexes that appears in the bdd; hence, if the topvar of the
       bdd is different from the topvar of the comb, it means that
       the topvar of the comb is unsconstraint and we need to
       multiply the number of solution of the branch by 2.
    *)
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    if Bdd.is_cst sub_bdd 
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    then
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      let sol_nb = 
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	if Bdd.is_true comb
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	then one_sol
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	else (two_power_of (List.length (Bdd.list_of_support (Bdd.dthen comb)))) 
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      in
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	if Bdd.is_true sub_bdd
	then (sol_nb, zero_sol) 
	else (zero_sol, sol_nb)
    else 
      let topvar = Bdd.topvar sub_bdd in
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      let rec
	(count_missing_vars: Bdd.t -> var -> int -> Bdd.t * int) =
	fun comb var cpt -> 
	  (* Returns [cpt] + the number of variables occurring in [comb]
	     before reaching [var] ([var] excluded). Also returns the comb
	     whch topvar is [var]. *)
	  let _ = assert (not (Bdd.is_cst comb)) in
	  let combvar = Bdd.topvar comb in
	    if var = combvar
	    then (comb, cpt)
	    else count_missing_vars (Bdd.dthen comb) var (cpt+1)
      in
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      let (sub_comb, missing_vars_nb) = 
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	count_missing_vars (Bdd.dthen comb) topvar 0
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      in
      let (n0, not_n0) = build_sol_nb_table sub_bdd sub_comb in
      let factor = (two_power_of missing_vars_nb) in
	(mult_sol_nb n0 factor, mult_sol_nb not_n0 factor)
	
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(****************************************************************************)
(****************************************************************************)

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let (toss_up_one_var: var -> subst option) =
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  fun var -> 
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    (* if [var] is a index that corresponds to a boolean variable,
       this fonction performs a toss and returns a substitution for
       the corresponding boolean variable. It returns [None]
       otherwise.

       Indeed, if it happens that a numerical constraint does not
       appear along a path, we simply ignore it and hence it will not
       be added to the store.
    *)
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    let af = Env_state.index_to_atomic_formula var in
      match af with 
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          Bv(vn) -> 
	    let ran = Random.float 1. in
	      if (ran < 0.5) 
	      then Some(vn, Formula.B(true)) 
	      else Some(vn, Formula.B(false))
	| _  -> None
     
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let (is_a_numeric_constraint : atomic_formula -> bool) =
  fun af -> 
    match af with
	Bv(_) -> false
      | GZ(_)   -> true 
      | GeqZ(_) -> true
      | SZ(_)   -> true
      | SeqZ(_) -> true
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(* exported *)
exception No_numeric_solution

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let rec (draw_in_bdd: subst list * store -> Bdd.t -> Bdd.t -> 
	   subst list * store) = 
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  fun (sl, store) bdd comb ->
    (** Returns [sl] appended to a draw of all the boolean variables
      bigger than the topvar of [bdd] according to the ordering
      induced by the comb [comb]. Also returns the (non empty) store
      obtained by adding to [store] all the numeric constraints that
      were encountered during this draw.
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      Raises the [No_numeric_solution] exception whenever no valid
      path in [bdd] leads to a satisfiable set of numeric
      constraints.  
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    *)
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    if Bdd.is_true bdd
    then
      (* Toss the remaining bool vars. *)
      ( (List.append sl
	   (Util.list_map_option toss_up_one_var (Bdd.list_of_support comb))),
	store )
    else
      let _ = assert (not (Bdd.is_false bdd)) in 
      let _ = assert (Env_state.sol_number_exists bdd) in
      let bddvar  = Bdd.topvar bdd in
      let af = (Env_state.index_to_atomic_formula bddvar) in 
      let top_var_is_numeric = is_a_numeric_constraint af in
	if
	  bddvar <> (Bdd.topvar comb) &&
	  not top_var_is_numeric
	then
	  let new_sl =
	    match toss_up_one_var (Bdd.topvar comb) with
		Some(s) -> s::sl
	      | None -> sl
	  in
	    draw_in_bdd (new_sl, store) bdd (Bdd.dthen comb)
	else 
	  (* bddvar = combvar xor top_var_is_numeric *) 
	  (* nb: I handle those two cases alltogether to avoid code
	     duplication (i.e., retrieving sol numbers, performing the
	     toss, the recursive call, handling the base case where a
	     dtrue bdd is reached, etc).  It makes the code a little
	     bit more obscur, but ...  *)
	  let (n, m) = Env_state.sol_number bdd in
	  let _ =
	    if ((eq_sol_nb n zero_sol) && (eq_sol_nb m zero_sol))
	    then raise No_numeric_solution ;
	  in
	  let (store_plus_af, store_plus_not_af) = 
	    (* A first trick to avoid code dup (cf nb above) *)
	    if top_var_is_numeric
	    then 
	      split_store store af
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	    else 
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	      (store, store)
	  in
	  let (swap, store1, bdd1, bool1, sol_nb1, store2, bdd2, bool2, sol_nb2) =
	    (* Depending on the result of a toss (based on the number
	       of solution in each branch), we try the [else] or the
	       [then] branch first. [swap] indicates whether or not the
	       [else] part is put before the [then] one. *)
	    let ran = Random.float 1. in
	      if ran < ((float_of_sol_nb n) /. (float_of_sol_nb (add_sol_nb n m)))
	      then
		(false, store_plus_af, (Bdd.dthen bdd), true, n,
		 store_plus_not_af, (Bdd.delse bdd), false, m )
	      else 
		(true, store_plus_not_af, (Bdd.delse bdd), false, m,
		 store_plus_af, (Bdd.dthen bdd), true, n )
	  in
	  let (sl1, sl2, new_comb) = (
	    (* A second trick to avoid code dup (cf nb above) *)
	    match af with 
		Bv(vn) -> 
		  (((vn, Formula.B(bool1))::sl), 
		   ((vn, Formula.B(bool2))::sl), 
		   (if Bdd.is_true comb then comb else Bdd.dthen comb) )
	      | _ -> 
		  (* top_var_is_numeric *)
		  (sl, sl, comb)
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	  )
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	  in
	  let res_opt =
	    (* A solution will be found in this branch iff there exists
	       at least one path in the bdd that leads to a satisfiable
	       set of numeric constraints. If it is not the case,
	       [res_opt] is bound to [None]. *)
	    if not (is_empty store1)
	    then 
	      try 
		let tail_draw1 = draw_in_bdd (sl1, store1) bdd1 new_comb in
		  Some(tail_draw1)
	      with No_numeric_solution -> 
		None
	    else
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	      None
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	  in
	    match res_opt with 
		Some(res) -> res
	      | None -> 
		  (* The second branch is now tried because no path in
		     the first bdd leaded to a satisfiable set of
		     numeric constraints. *)
		  if not (eq_sol_nb sol_nb2 zero_sol)
		  then
		    if not (is_empty store2)
		    then draw_in_bdd (sl2, store2) bdd2 new_comb 
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		    else
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		      raise No_numeric_solution
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		  else
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		    raise No_numeric_solution
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let (draw : vn list -> vnt list -> Bdd.t -> Bdd.t -> subst list * subst list) =
  fun bool_vars_to_gen num_vnt_to_gen comb bdd ->
    (** Draw the output and local vars to be generated by the environnent. *)
    let (bool_subst_l, store) = 
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      draw_in_bdd ([], (new_store num_vnt_to_gen)) bdd comb 
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    in
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    let num_subst_l = draw_inside store in
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    let subst_l = append bool_subst_l num_subst_l in
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    let (out_vars, _) = List.split (Env_state.output_var_names ()) 
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    in
      assert ( 
	(*  Checks that we generated all variables. *)
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	let (gen_vars, _) = List.split subst_l in
	let (num_vars_to_gen, _) = List.split num_vnt_to_gen in
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	let vars_to_gen = append bool_vars_to_gen num_vars_to_gen in
          (sort (compare) gen_vars) = (sort (compare) vars_to_gen) 
      );
      (* Splits output and local vars. *)
      List.partition 
	(fun (vn, _) -> List.mem vn out_vars) 
	subst_l
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(* Exported *)
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let (solve_formula: env_in -> int -> formula list -> vn list -> vnt list ->
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       (subst list * subst list) list) =
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  fun input p fl bool_vars_to_gen num_vars_to_gen ->
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    let f = formula_list_to_conj fl in
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    let bdd = 
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      (* The bdd of f has necessarily been computed (by is_satisfiable) *)
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      try Env_state.bdd f
      with Not_found -> Env_state.bdd_global f
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    in
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    let bool_vars_to_gen_f = 
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      List.fold_left
	(fun acc vn -> (And(Bvar(vn), acc)))
	True
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	bool_vars_to_gen
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    in
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    let (comb0, _) = formula_to_bdd input bool_vars_to_gen_f in
    let comb = 
      (* All boolean vars should appear in the comb so that when we
	 find that such a var is missing along a bdd path, we can
	 perform a (fair) toss for it. On the contrary, if a
	 numerical contraint disappear from a bdd (eg, consider [(f
	 && false) || true]), it is not important; fairly tossing a
	 (boolean) value for a num constaint [nc] and performing a
	 fair toss in the resulting domain is equivalent to directly
	 perform the toss in the (unconstraint wrt [nc]) initial
	 domain.  
      *)
      Bdd.dand (Bdd.support bdd) comb0 
    in
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    let _ =
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      if not (Env_state.sol_number_exists bdd)
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      then
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	let rec skip_unconstraint_bool_var_at_top comb v =
	  (* [build_sol_nb_table] supposes that the bdd and its comb 
	     have the same top var. 
	  *)
	  if Bdd.is_true comb then comb
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	  else 
	    let topvar = (Bdd.topvar comb) in
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	      if v = topvar then comb 
	      else skip_unconstraint_bool_var_at_top (Bdd.dthen comb) v
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	in
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	let comb2 = skip_unconstraint_bool_var_at_top comb (Bdd.topvar bdd) in 
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	let _ = build_sol_nb_table bdd comb2 in
	  ()
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    in
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      try 	
	Util.unfold (draw bool_vars_to_gen num_vars_to_gen comb) bdd p
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      with No_numeric_solution -> 
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	Env_state.set_sol_number bdd (zero_sol, zero_sol);
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	[]
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