path: root/plugins/omega/omega.ml

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)
(**************************************************************************)
(*                                                                        *)
(* Omega: a solver of quantifier-free problems in Presburger Arithmetic   *)
(*                                                                        *)
(* Pierre Crégut (CNET, Lannion, France)                                  *)
(*                                                                        *)
(* 13/10/2002 : modified to cope with an external numbering of equations  *)
(*   and hypothesis. Its use for Omega is not more complex and it makes   *)
(*   things much simpler for the reflexive version where we should limit  *)
(*   the number of source of numbering.                                   *)
(**************************************************************************)

module type INT = sig
  type bigint
  val equal : bigint -> bigint -> bool
  val less_than : bigint -> bigint -> bool
  val add : bigint -> bigint -> bigint
  val sub : bigint -> bigint -> bigint
  val mult : bigint -> bigint -> bigint
  val euclid : bigint -> bigint -> bigint * bigint
  val neg : bigint -> bigint
  val zero : bigint
  val one : bigint
  val to_string : bigint -> string
end

let debug = ref false

module MakeOmegaSolver (I:INT) = struct

type bigint = I.bigint
let (=?) = I.equal
let (<?) = I.less_than
let (<=?) x y = I.less_than x y || x = y
let (>?) x y = I.less_than y x
let (>=?) x y = I.less_than y x || x = y
let (+) = I.add
let (-) = I.sub
let ( * ) = I.mult
let (/) x y = fst (I.euclid x y)
let (mod) x y = snd (I.euclid x y)
let zero = I.zero
let one = I.one
let two = one + one
let negone = I.neg one
let abs x = if I.less_than x zero then I.neg x else x
let string_of_bigint = I.to_string
let neg = I.neg

(* To ensure that polymorphic (<) is not used mistakenly on big integers *)
(* Warning: do not use (=) either on big int *)
let (<) = ((<) : int -> int -> bool)
let (>) = ((>) : int -> int -> bool)
let (<=) = ((<=) : int -> int -> bool)
let (>=) = ((>=) : int -> int -> bool)

let pp i = print_int i; print_newline (); flush stdout

let push v l = l := v :: !l

let rec pgcd x y = if y =? zero then x else pgcd y (x mod y)

let pgcd_l = function
  | [] -> failwith "pgcd_l"
  | x :: l -> List.fold_left pgcd x l

let floor_div a b =
  match a >=? zero , b >? zero with
    | true,true -> a / b
    | false,false -> a / b
    | true, false -> (a-one) / b - one
    | false,true  -> (a+one) / b - one

type coeff = {c: bigint ; v: int}

type linear = coeff list

type eqn_kind = EQUA | INEQ | DISE

type afine = {
  (* a number uniquely identifying the equation *)
  id: int ;
  (* a boolean true for an eq, false for an ineq (Sigma a_i x_i >= 0) *)
  kind: eqn_kind;
  (* the variables and their coefficient *)
  body: coeff list;
  (* a constant *)
  constant: bigint }

type state_action = {
  st_new_eq : afine;
  st_def    : afine; (* /!\ this represents [st_def = st_var] *)
  st_orig   : afine;
  st_coef   : bigint;
  st_var    : int }

type action =
  | DIVIDE_AND_APPROX of afine * afine * bigint * bigint
  | NOT_EXACT_DIVIDE of afine * bigint
  | FORGET_C of int
  | EXACT_DIVIDE of afine * bigint
  | SUM of int * (bigint * afine) * (bigint * afine)
  | STATE of state_action
  | HYP of afine
  | FORGET   of int * int
  | FORGET_I of int * int
  | CONTRADICTION of afine * afine
  | NEGATE_CONTRADICT of afine * afine * bool
  | MERGE_EQ of int * afine * int
  | CONSTANT_NOT_NUL of int * bigint
  | CONSTANT_NUL of int
  | CONSTANT_NEG of int * bigint
  | SPLIT_INEQ of afine * (int * action list) * (int * action list)
  | WEAKEN of int * bigint

exception UNSOLVABLE

exception NO_CONTRADICTION

let display_eq print_var (l,e) =
  let _ =
    List.fold_left
      (fun not_first f ->
	 print_string
	   (if f.c <? zero then "- " else if not_first then "+ " else "");
	 let c = abs f.c in
	 if c =? one then
	   Printf.printf "%s " (print_var f.v)
	 else
	   Printf.printf "%s %s " (string_of_bigint c) (print_var f.v);
	 true)
      false l
  in
  if e >? zero then
    Printf.printf "+ %s " (string_of_bigint e)
  else if e <? zero then
    Printf.printf "- %s " (string_of_bigint (abs e))

let rec trace_length l =
  let action_length accu = function
    | SPLIT_INEQ (_,(_,l1),(_,l2)) ->
	accu + one + trace_length l1 + trace_length l2
    | _ -> accu + one in
  List.fold_left action_length zero l

let operator_of_eq = function
  | EQUA -> "=" | DISE -> "!=" | INEQ -> ">="

let kind_of = function
  | EQUA -> "equation" | DISE -> "disequation" | INEQ -> "inequation"

let display_system print_var l =
  List.iter
    (fun { kind=b; body=e; constant=c; id=id} ->
       Printf.printf "E%d: " id;
       display_eq print_var (e,c);
       Printf.printf "%s 0\n" (operator_of_eq b))
    l;
  print_string "------------------------\n\n"

let display_inequations print_var l =
  List.iter (fun e -> display_eq print_var e;print_string ">= 0\n") l;
  print_string "------------------------\n\n"

let sbi = string_of_bigint

let rec display_action print_var = function
  | act :: l -> begin match act with
      | DIVIDE_AND_APPROX (e1,e2,k,d) ->
          Printf.printf
            "Inequation E%d is divided by %s and the constant coefficient is \
            rounded by subtracting %s.\n" e1.id (sbi k) (sbi d)
      | NOT_EXACT_DIVIDE (e,k) ->
          Printf.printf
            "Constant in equation E%d is not divisible by the pgcd \
            %s of its other coefficients.\n" e.id (sbi k)
      | EXACT_DIVIDE (e,k) ->
          Printf.printf
            "Equation E%d is divided by the pgcd \
            %s of its coefficients.\n" e.id (sbi k)
      | WEAKEN (e,k) ->
          Printf.printf
            "To ensure a solution in the dark shadow \
            the equation E%d is weakened by %s.\n" e (sbi k)
      | SUM (e,(c1,e1),(c2,e2)) ->
          Printf.printf
            "We state %s E%d = %s %s E%d + %s %s E%d.\n"
            (kind_of e1.kind) e (sbi c1) (kind_of e1.kind) e1.id (sbi c2)
            (kind_of e2.kind) e2.id
      | STATE { st_new_eq = e } ->
          Printf.printf "We define a new equation E%d: " e.id;
          display_eq print_var (e.body,e.constant);
          print_string (operator_of_eq e.kind); print_string " 0"
      | HYP e ->
          Printf.printf "We define E%d: " e.id;
          display_eq print_var (e.body,e.constant);
          print_string (operator_of_eq e.kind); print_string " 0\n"
      | FORGET_C e ->  Printf.printf "E%d is trivially satisfiable.\n" e
      | FORGET (e1,e2) -> Printf.printf "E%d subsumes E%d.\n" e1 e2
      | FORGET_I (e1,e2) -> Printf.printf "E%d subsumes E%d.\n" e1 e2
      | MERGE_EQ (e,e1,e2) ->
         Printf.printf "E%d and E%d can be merged into E%d.\n" e1.id e2 e
      | CONTRADICTION (e1,e2) ->
          Printf.printf
            "Equations E%d and E%d imply a contradiction on their \
            constant factors.\n" e1.id e2.id
      | NEGATE_CONTRADICT(e1,e2,b) ->
          Printf.printf
            "Equations E%d and E%d state that their body is at the same time \
            equal and different\n" e1.id e2.id
      | CONSTANT_NOT_NUL (e,k) ->
          Printf.printf "Equation E%d states %s = 0.\n" e (sbi k)
      | CONSTANT_NEG(e,k) ->
          Printf.printf "Equation E%d states %s >= 0.\n" e (sbi k)
      | CONSTANT_NUL e ->
          Printf.printf "Inequation E%d states 0 != 0.\n" e
      | SPLIT_INEQ (e,(e1,l1),(e2,l2)) ->
          Printf.printf "Equation E%d is split in E%d and E%d\n\n" e.id e1 e2;
          display_action print_var l1;
          print_newline ();
          display_action print_var l2;
          print_newline ()
    end; display_action print_var l
  | [] ->
      flush stdout

let default_print_var v = Printf.sprintf "X%d" v (* For debugging *)

(*""*)
let add_event, history, clear_history =
  let accu = ref [] in
  (fun (v:action) -> if !debug then display_action default_print_var [v]; push v accu),
  (fun () -> !accu),
  (fun () -> accu := [])

let nf_linear = List.sort (fun x y -> Pervasives.(-) y.v x.v)

let nf ((b : bool),(e,(x : int))) = (b,(nf_linear e,x))

let map_eq_linear f =
  let rec loop = function
    | x :: l -> let c = f x.c in if c=?zero then loop l else {v=x.v; c=c} :: loop l
    | [] -> []
  in
  loop

let map_eq_afine f e =
  { id = e.id; kind = e.kind; body = map_eq_linear f e.body;
    constant = f e.constant }

let negate_eq = map_eq_afine (fun x -> neg x)

let rec sum p0 p1 = match (p0,p1) with
  | ([], l) -> l | (l, []) -> l
  | (((x1::l1) as l1'), ((x2::l2) as l2')) ->
      if x1.v = x2.v then
	let c = x1.c + x2.c in
	if c =? zero then sum l1 l2 else {v=x1.v;c=c} :: sum l1 l2
      else if x1.v > x2.v then
	x1 :: sum l1 l2'
      else
	x2 :: sum l1' l2

let sum_afine new_eq_id eq1 eq2 =
  { kind = eq1.kind; id = new_eq_id ();
    body = sum eq1.body eq2.body; constant = eq1.constant + eq2.constant }

exception FACTOR1

let rec chop_factor_1 = function
  | x :: l ->
    if abs x.c =? one then x,l else let (c',l') = chop_factor_1 l in (c',x::l')
  | [] -> raise FACTOR1

exception CHOPVAR

let rec chop_var v = function
  | f :: l -> if f.v = v then f,l else let (f',l') = chop_var v l in (f',f::l')
  | [] -> raise CHOPVAR

let normalize ({id=id; kind=eq_flag; body=e; constant =x} as eq) =
  if e = [] then begin
    match eq_flag with
      | EQUA ->
          if x =? zero then [] else begin
            add_event (CONSTANT_NOT_NUL(id,x)); raise UNSOLVABLE
         end
      | DISE ->
          if x <> zero then [] else begin
            add_event (CONSTANT_NUL id); raise UNSOLVABLE
          end
      | INEQ ->
          if x >=? zero then [] else begin
            add_event (CONSTANT_NEG(id,x)); raise UNSOLVABLE
          end
  end else
    let gcd = pgcd_l (List.map (fun f -> abs f.c) e) in
    if eq_flag=EQUA && x mod gcd <> zero then begin
      add_event (NOT_EXACT_DIVIDE (eq,gcd)); raise UNSOLVABLE
    end else if eq_flag=DISE && x mod gcd <> zero then begin
      add_event (FORGET_C eq.id); []
    end else if gcd <> one then begin
      let c = floor_div x gcd in
      let d = x - c * gcd in
      let new_eq = {id=id; kind=eq_flag; constant=c;
                    body=map_eq_linear (fun c -> c / gcd) e} in
      add_event (if eq_flag=EQUA || eq_flag = DISE then EXACT_DIVIDE(eq,gcd)
                 else DIVIDE_AND_APPROX(eq,new_eq,gcd,d));
      [new_eq]
    end else [eq]

let eliminate_with_in new_eq_id {v=v;c=c_unite} eq2
                        ({body=e1; constant=c1} as eq1) =
  try
    let (f,_) = chop_var v e1 in
    let coeff = if c_unite=?one then neg f.c else if c_unite=? negone then f.c
                else failwith "eliminate_with_in" in
    let res = sum_afine new_eq_id eq1 (map_eq_afine (fun c -> c * coeff) eq2) in
    add_event (SUM (res.id,(one,eq1),(coeff,eq2))); res
  with CHOPVAR -> eq1

let omega_mod a b = a - b * floor_div (two * a + b) (two * b)
let banerjee_step (new_eq_id,new_var_id,print_var) original l1 l2 =
  let e = original.body in
  let sigma = new_var_id () in
  if e == [] then begin
    display_system print_var [original] ; failwith "TL"
    end;
  let smallest,var =
    List.fold_left (fun (v,p) c ->  if v >? (abs c.c) then abs c.c,c.v else (v,p))
      (abs (List.hd e).c, (List.hd e).v) (List.tl e)
  in
  let m = smallest + one in
  let new_eq =
    { constant = omega_mod original.constant m;
      body = {c= neg m;v=sigma} ::
             map_eq_linear (fun a -> omega_mod a m) original.body;
      id = new_eq_id (); kind = EQUA } in
  let definition =
    { constant = neg (floor_div (two * original.constant + m) (two * m));
      body = map_eq_linear (fun a -> neg (floor_div (two * a + m) (two * m)))
                             original.body;
      id = new_eq_id (); kind = EQUA } in
  add_event (STATE {st_new_eq = new_eq; st_def = definition;
		    st_orig = original; st_coef = m; st_var = sigma});
  let new_eq = List.hd (normalize new_eq) in
  let eliminated_var, def = chop_var var new_eq.body in
  let other_equations =
    Util.List.map_append
      (fun e ->
        normalize (eliminate_with_in new_eq_id eliminated_var new_eq e)) l1 in
  let inequations =
    Util.List.map_append
      (fun e ->
        normalize (eliminate_with_in new_eq_id eliminated_var new_eq e)) l2 in
  let original' = eliminate_with_in new_eq_id eliminated_var new_eq original in
  let mod_original = map_eq_afine (fun c -> c / m) original' in
  add_event (EXACT_DIVIDE (original',m));
  List.hd (normalize mod_original),other_equations,inequations

let rec eliminate_one_equation ((new_eq_id,new_var_id,print_var) as new_ids) (e,other,ineqs) =
  if !debug then display_system print_var (e::other);
  try
    let v,def = chop_factor_1 e.body in
    (Util.List.map_append
      (fun e' -> normalize (eliminate_with_in new_eq_id v e e')) other,
     Util.List.map_append
       (fun e' -> normalize (eliminate_with_in new_eq_id v e e')) ineqs)
  with FACTOR1 ->
    eliminate_one_equation new_ids (banerjee_step new_ids e other ineqs)

let rec banerjee ((_,_,print_var) as new_ids) (sys_eq,sys_ineq) =
  let rec fst_eq_1 = function
     (eq::l) ->
        if List.exists (fun x -> abs x.c =? one) eq.body then eq,l
        else let (eq',l') = fst_eq_1 l in (eq',eq::l')
   | [] -> raise Not_found in
  match sys_eq with
     [] -> if !debug then display_system print_var sys_ineq; sys_ineq
   | (e1::rest) ->
       let eq,other = try fst_eq_1 sys_eq with Not_found -> (e1,rest) in
       if eq.body = [] then
         if eq.constant =? zero then begin
           add_event (FORGET_C eq.id); banerjee new_ids (other,sys_ineq)
         end else begin
           add_event (CONSTANT_NOT_NUL(eq.id,eq.constant)); raise UNSOLVABLE
         end
       else
	 banerjee new_ids
	   (eliminate_one_equation new_ids (eq,other,sys_ineq))

type kind = INVERTED | NORMAL

let redundancy_elimination new_eq_id system =
  let normal = function
     ({body=f::_} as e) when f.c <? zero ->  negate_eq e, INVERTED
   | e -> e,NORMAL in
  let table = Hashtbl.create 7 in
  List.iter
    (fun e ->
       let ({body=ne} as nx) ,kind = normal e in
       if ne = [] then
         if nx.constant <? zero then begin
           add_event (CONSTANT_NEG(nx.id,nx.constant)); raise UNSOLVABLE
         end else add_event (FORGET_C nx.id)
       else
       try
         let (optnormal,optinvert) = Hashtbl.find table ne in
         let final =
           if kind = NORMAL then begin
             match optnormal with
                Some v ->
                  let kept =
                    if v.constant <? nx.constant
                    then begin add_event (FORGET (v.id,nx.id));v end
                    else begin add_event (FORGET (nx.id,v.id));nx end in
                  (Some(kept),optinvert)
              | None -> Some nx,optinvert
           end else begin
             match optinvert with
                Some v ->
                  let _kept =
                    if v.constant >? nx.constant
                    then begin add_event (FORGET_I (v.id,nx.id));v end
                    else begin add_event (FORGET_I (nx.id,v.id));nx end in
                  (optnormal,Some(if v.constant >? nx.constant then v else nx))
              | None -> optnormal,Some nx
           end in
         begin match final with
            (Some high, Some low) ->
              if high.constant <? low.constant then begin
                add_event(CONTRADICTION (high,negate_eq low));
                raise UNSOLVABLE
              end
          | _ -> () end;
         Hashtbl.remove table ne;
         Hashtbl.add table ne final
       with Not_found ->
         Hashtbl.add table ne
           (if kind = NORMAL then (Some nx,None) else (None,Some nx)))
    system;
  let accu_eq = ref [] in
  let accu_ineq = ref [] in
  Hashtbl.iter
    (fun p0 p1 -> match (p0,p1) with
       | (e, (Some x, Some y)) when x.constant =? y.constant ->
           let id=new_eq_id () in
           add_event (MERGE_EQ(id,x,y.id));
           push {id=id; kind=EQUA; body=x.body; constant=x.constant} accu_eq
       | (e, (optnorm,optinvert)) ->
           begin match optnorm with
               Some x -> push x accu_ineq | _ -> () end;
           begin match optinvert with
               Some x -> push (negate_eq x) accu_ineq | _ -> () end)
    table;
  !accu_eq,!accu_ineq

exception SOLVED_SYSTEM

let select_variable system =
  let table = Hashtbl.create 7 in
  let push v c=
    try let r = Hashtbl.find table v in r := max !r (abs c)
    with Not_found -> Hashtbl.add table v (ref (abs c)) in
  List.iter (fun {body=l} -> List.iter (fun f -> push f.v f.c) l) system;
  let vmin,cmin = ref (-1), ref zero in
  let var_cpt = ref 0 in
  Hashtbl.iter
    (fun v ({contents =  c}) ->
       incr var_cpt;
       if c <? !cmin || !vmin = (-1) then begin vmin := v; cmin := c end)
    table;
  if !var_cpt < 1 then raise SOLVED_SYSTEM;
  !vmin

let classify v system =
  List.fold_left
    (fun (not_occ,below,over) eq ->
       try let f,eq' = chop_var v eq.body in
       if f.c >=? zero then (not_occ,((f.c,eq) :: below),over)
       else (not_occ,below,((neg f.c,eq) :: over))
       with CHOPVAR -> (eq::not_occ,below,over))
    ([],[],[]) system

let product new_eq_id dark_shadow low high =
  List.fold_left
    (fun accu (a,eq1) ->
       List.fold_left
         (fun accu (b,eq2) ->
            let eq =
              sum_afine new_eq_id (map_eq_afine (fun c -> c * b) eq1)
                (map_eq_afine (fun c -> c * a) eq2) in
            add_event(SUM(eq.id,(b,eq1),(a,eq2)));
            match normalize eq with
	      | [eq] ->
                  let final_eq =
                    if dark_shadow then
                      let delta = (a - one) * (b - one) in
                      add_event(WEAKEN(eq.id,delta));
                      {id = eq.id; kind=INEQ; body = eq.body;
                       constant = eq.constant - delta}
                    else eq
                  in final_eq :: accu
              | (e::_) -> failwith "Product dardk"
              | [] -> accu)
         accu high)
    [] low

let fourier_motzkin (new_eq_id,_,print_var) dark_shadow system =
  let v = select_variable system in
  let (ineq_out, ineq_low,ineq_high) = classify v system in
  let expanded = ineq_out @ product new_eq_id dark_shadow ineq_low ineq_high in
  if !debug then display_system print_var expanded; expanded

let simplify ((new_eq_id,new_var_id,print_var) as new_ids) dark_shadow system =
  if List.exists (fun e -> e.kind = DISE) system then
    failwith "disequation in simplify";
  clear_history ();
  List.iter (fun e -> add_event (HYP e)) system;
  let system = Util.List.map_append normalize system in
  let eqs,ineqs = List.partition (fun e -> e.kind=EQUA) system in
  let simp_eq,simp_ineq = redundancy_elimination new_eq_id ineqs in
  let system = (eqs @ simp_eq,simp_ineq) in
  let rec loop1a system =
    let sys_ineq = banerjee new_ids system in
    loop1b sys_ineq
  and loop1b sys_ineq =
    let simp_eq,simp_ineq = redundancy_elimination new_eq_id sys_ineq in
    if simp_eq = [] then simp_ineq else loop1a (simp_eq,simp_ineq)
  in
  let rec loop2 system =
    try
      let expanded = fourier_motzkin new_ids dark_shadow system in
      loop2 (loop1b expanded)
    with SOLVED_SYSTEM ->
      if !debug then display_system print_var system; system
  in
  loop2 (loop1a system)

let rec depend relie_on accu = function
  | act :: l ->
      begin match act with
	| DIVIDE_AND_APPROX (e,_,_,_) ->
            if Int.List.mem e.id relie_on then depend relie_on (act::accu) l
            else depend relie_on accu l
	| EXACT_DIVIDE (e,_) ->
            if Int.List.mem e.id relie_on then depend relie_on (act::accu) l
            else depend relie_on accu l
	| WEAKEN (e,_) ->
            if Int.List.mem e relie_on then depend relie_on (act::accu) l
            else depend relie_on accu l
	| SUM (e,(_,e1),(_,e2)) ->
            if Int.List.mem e relie_on then
	      depend (e1.id::e2.id::relie_on) (act::accu) l
            else
	      depend relie_on accu l
	| STATE {st_new_eq=e;st_orig=o} ->
            if Int.List.mem e.id relie_on then depend (o.id::relie_on) (act::accu) l
            else depend relie_on accu l
	| HYP e ->
            if Int.List.mem e.id relie_on then depend relie_on (act::accu) l
            else depend relie_on accu l
	| FORGET_C _ -> depend relie_on accu l
	| FORGET _ -> depend relie_on accu l
	| FORGET_I _ -> depend relie_on accu l
	| MERGE_EQ (e,e1,e2) ->
            if Int.List.mem e relie_on then
	      depend (e1.id::e2::relie_on) (act::accu) l
            else
	      depend relie_on accu l
	| NOT_EXACT_DIVIDE (e,_) -> depend (e.id::relie_on) (act::accu) l
	| CONTRADICTION (e1,e2) ->
	    depend (e1.id::e2.id::relie_on) (act::accu) l
	| CONSTANT_NOT_NUL (e,_) -> depend (e::relie_on) (act::accu) l
	| CONSTANT_NEG (e,_) -> depend (e::relie_on) (act::accu) l
	| CONSTANT_NUL e -> depend (e::relie_on) (act::accu) l
	| NEGATE_CONTRADICT (e1,e2,_) ->
            depend (e1.id::e2.id::relie_on) (act::accu) l
	| SPLIT_INEQ _ -> failwith "depend"
      end
  | [] -> relie_on, accu

let negation (eqs,ineqs) =
  let diseq,_ = List.partition (fun e -> e.kind = DISE) ineqs in
  let normal = function
    | ({body=f::_} as e) when f.c <? zero ->  negate_eq e, INVERTED
    | e -> e,NORMAL in
  let table = Hashtbl.create 7 in
  List.iter (fun e ->
               let {body=ne;constant=c} ,kind = normal e in
               Hashtbl.add table (ne,c) (kind,e)) diseq;
  List.iter (fun e ->
               assert (e.kind = EQUA);
               let {body=ne;constant=c},kind = normal e in
               try
		 let (kind',e') = Hashtbl.find table (ne,c) in
		 add_event (NEGATE_CONTRADICT (e,e',kind=kind'));
		 raise UNSOLVABLE
               with Not_found -> ()) eqs

exception FULL_SOLUTION of action list * int list

let simplify_strong ((new_eq_id,new_var_id,print_var) as new_ids) system =
  clear_history ();
  List.iter (fun e -> add_event (HYP e)) system;
  (* Initial simplification phase *)
  let rec loop1a system =
    negation system;
    let sys_ineq = banerjee new_ids system in
    loop1b sys_ineq
  and loop1b sys_ineq =
    let dise,ine = List.partition (fun e -> e.kind = DISE) sys_ineq in
    let simp_eq,simp_ineq = redundancy_elimination new_eq_id ine in
    if simp_eq = [] then dise @ simp_ineq
    else loop1a (simp_eq,dise @ simp_ineq)
  in
  let rec loop2 system =
    try
      let expanded = fourier_motzkin new_ids false system in
      loop2 (loop1b expanded)
    with SOLVED_SYSTEM -> if !debug then display_system print_var system; system
  in
  let rec explode_diseq = function
    | (de::diseq,ineqs,expl_map) ->
	let id1 = new_eq_id ()
	and id2 = new_eq_id () in
	let e1 =
          {id = id1; kind=INEQ; body = de.body; constant = de.constant -one} in
	let e2 =
	  {id = id2; kind=INEQ; body = map_eq_linear neg de.body;
           constant = neg de.constant - one} in
	let new_sys =
          List.map (fun (what,sys) -> ((de.id,id1,true)::what, e1::sys))
	    ineqs @
          List.map (fun (what,sys) -> ((de.id,id2,false)::what,e2::sys))
	    ineqs
	in
	explode_diseq (diseq,new_sys,(de.id,(de,id1,id2))::expl_map)
    | ([],ineqs,expl_map) -> ineqs,expl_map
  in
  try
    let system = Util.List.map_append normalize system in
    let eqs,ineqs = List.partition (fun e -> e.kind=EQUA) system in
    let dise,ine = List.partition (fun e -> e.kind = DISE) ineqs in
    let simp_eq,simp_ineq = redundancy_elimination new_eq_id ine in
    let system = (eqs @ simp_eq,simp_ineq @ dise) in
    let system' = loop1a system in
    let diseq,ineq = List.partition (fun e -> e.kind = DISE) system' in
    let first_segment = history () in
    let sys_exploded,explode_map = explode_diseq (diseq,[[],ineq],[]) in
    let all_solutions =
      List.map
        (fun (decomp,sys) ->
           clear_history ();
           try let _ = loop2 sys in raise NO_CONTRADICTION
           with UNSOLVABLE ->
             let relie_on,path = depend [] [] (history ()) in
             let dc,_ = List.partition (fun (_,id,_) -> Int.List.mem id relie_on) decomp in
             let red = List.map (fun (x,_,_) -> x) dc in
             (red,relie_on,decomp,path))
        sys_exploded
    in
    let max_count sys =
      let tbl = Hashtbl.create 7 in
      let augment x =
        try incr (Hashtbl.find tbl x)
	with Not_found -> Hashtbl.add tbl x (ref 1) in
      let eq = ref (-1) and c = ref 0 in
      List.iter (function
		   | ([],r_on,_,path) -> raise (FULL_SOLUTION (path,r_on))
                   | (l,_,_,_) -> List.iter augment l) sys;
      Hashtbl.iter (fun x v -> if !v > !c then begin eq := x; c := !v end) tbl;
      !eq
    in
    let rec solve systems =
      try
	let id = max_count systems in
        let rec sign = function
	  | ((id',_,b)::l) -> if id=id' then b else sign l
          | [] -> failwith "solve" in
        let s1,s2 =
          List.partition (fun (_,_,decomp,_) -> sign decomp) systems in
        let remove_int (dep,ro,dc,pa) =
          (Util.List.except Int.equal id dep,ro,dc,pa)
        in
        let s1' = List.map remove_int s1 in
        let s2' = List.map remove_int s2 in
        let (r1,relie1) = solve s1'
	and (r2,relie2) = solve s2' in
	let (eq,id1,id2) = Int.List.assoc id explode_map in
	[SPLIT_INEQ(eq,(id1,r1),(id2, r2))],
        eq.id :: Util.List.union Int.equal relie1 relie2
      with FULL_SOLUTION (x0,x1) -> (x0,x1)
    in
    let act,relie_on = solve all_solutions in
    snd(depend relie_on act first_segment)
  with UNSOLVABLE -> snd (depend [] [] (history ()))

end