|
(************************************************************************)
(* * 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
¤ Dauer der Verarbeitung: 0.23 Sekunden
(vorverarbeitet)
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