Source file intervalCalculus.ml
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open Format
open Options
open Sig
open Matching_types
module Z = Numbers.Z
module Q = Numbers.Q
module L = Xliteral
module Sy = Symbols
module I = Intervals
module OracleContainer =
(val (Inequalities.get_current ()) : Inequalities.Container_SIG)
module X = Shostak.Combine
module P = Shostak.Polynome
module MP0 = Map.Make(P)
module SP = Set.Make(P)
module SX = Set.Make(struct type t = X.r let compare = X.hash_cmp end)
module MX0 = Map.Make(struct type t = X.r let compare = X.hash_cmp end)
module MPL = Expr.Map
module Oracle = OracleContainer.Make(P)
module SE = Expr.Set
module ME = Expr.Map
module Ex = Explanation
module E = Expr
module EM = Matching.Make
(struct
include Uf
let are_equal env s t ~init_terms:_ =
Uf.are_equal env s t ~added_terms:false
let term_repr env t ~init_term:_ = Uf.term_repr env t
end)
type r = P.r
module LR = Uf.LX
module MR = Map.Make(
struct
type t = r L.view
let compare a b = LR.compare (LR.make a) (LR.make b)
end)
let alien_of p = Shostak.Arith.is_mine p
let poly_of r = Shostak.Arith.embed r
let is_alien x =
match P.is_monomial @@ poly_of x with
| Some(a, _, c) -> Q.equal a Q.one && Q.equal c Q.zero
| _ -> false
module SimVar = struct
type t = X.r
let compare = X.hash_cmp
let is_int r = X.type_info r == Ty.Tint
let print fmt x =
if is_alien x then fprintf fmt "%a" X.print x
else fprintf fmt "s!%d" (X.hash x)
end
module Sim = OcplibSimplex.Basic.Make(SimVar)(Numbers.Q)(Explanation)
type t = {
inequations : Oracle.t MPL.t;
monomes: (I.t * SX.t) MX0.t;
polynomes : I.t MP0.t;
known_eqs : SX.t;
improved_p : SP.t;
improved_x : SX.t;
classes : SE.t list;
size_splits : Q.t;
int_sim : Sim.Core.t;
rat_sim : Sim.Core.t;
new_uf : Uf.t;
th_axioms : (Expr.th_elt * Explanation.t) ME.t;
linear_dep : SE.t ME.t;
syntactic_matching :
(Matching_types.trigger_info * Matching_types.gsubst list) list list;
}
module Sim_Wrap = struct
let check_unsat_result simplex env =
match Sim.Result.get None simplex with
| Sim.Core.Unknown -> assert false
| Sim.Core.Unbounded _ -> assert false
| Sim.Core.Max _ -> assert false
| Sim.Core.Sat _ -> ()
| Sim.Core.Unsat ex ->
let ex = Lazy.force ex in
if debug_fm() then
fprintf fmt "[fm] simplex derived unsat: %a@." Explanation.print ex;
raise (Ex.Inconsistent (ex, env.classes))
let solve env _i =
let int_sim = Sim.Solve.solve env.int_sim in
check_unsat_result int_sim env;
let rat_sim = Sim.Solve.solve env.rat_sim in
check_unsat_result rat_sim env;
{env with int_sim; rat_sim}
let solve env i =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Simplex Timers.F_solve;
let res = solve env i in
Timers.exec_timer_pause Timers.M_Simplex Timers.F_solve;
res
with e ->
Timers.exec_timer_pause Timers.M_Simplex Timers.F_solve;
raise e
else solve env i
let i get_lb =
let func, q =
if get_lb then I.borne_inf, Q.one else I.borne_sup, Q.m_one in
try
let bnd, expl, large = func i in
Some (bnd, if large then Q.zero else q), expl
with I.No_finite_bound -> None, Explanation.empty
let same_bnds _old _new =
match _old, _new with
| None, None -> true
| None, Some _ | Some _, None -> false
| Some(s,t), Some(u, v) -> Q.equal s u && Q.equal t v
let add_if_better p _old _new simplex =
let old_mn, _old_mn_ex = extract_bound _old true in
let old_mx, _old_mx_ex = extract_bound _old false in
let new_mn, new_mn_ex = extract_bound _new true in
let new_mx, new_mx_ex = extract_bound _new false in
if same_bnds old_mn new_mn && same_bnds old_mx new_mx then simplex
else
let l, z = P.to_list p in
assert (Q.sign z = 0);
let simplex, _changed =
match l with
[] -> assert false
| [c, x] ->
assert (Q.is_one c);
Sim.Assert.var simplex x new_mn new_mn_ex new_mx new_mx_ex
| _ ->
let l = List.rev_map (fun (c, x) -> x, c) l in
Sim.Assert.poly simplex (Sim.Core.P.from_list l) (alien_of p)
new_mn new_mn_ex new_mx new_mx_ex
in
simplex
let finite_non_point_dom info =
match info.Sim.Core.mini, info.Sim.Core.maxi with
| None, _ | _, None -> None
| Some (a, b), Some(x,y) ->
assert (Q.is_zero b);
assert (Q.is_zero y);
let c = Q.compare a x in
assert (c <= 0);
if c = 0 then None else Some (Q.sub x a)
let case_split =
let gen_cs x n s orig =
if debug_fm () then
fprintf fmt "[Sim_CS-%d] %a = %a of size %a@."
orig X.print x Q.print n Q.print s;
let ty = X.type_info x in
let r1 = x in
let r2 = alien_of (P.create [] n ty) in
[LR.mkv_eq r1 r2, true, Th_util.CS (Th_util.Th_arith, s)]
in
let aux_1 uf x (info,_) acc =
assert (X.type_info x == Ty.Tint);
match finite_non_point_dom info with
| Some s when
(Sim.Core.equals_optimum info.Sim.Core.value info.Sim.Core.mini ||
Sim.Core.equals_optimum info.Sim.Core.value info.Sim.Core.maxi)
&& Uf.is_normalized uf x ->
let v, _ = info.Sim.Core.value in
assert (Q.is_int v);
begin
match acc with
| Some (_,_,s') when Q.compare s' s <= 0 -> acc
| _ -> Some (x,v, s)
end
| _ -> acc
in
let aux_2 env uf x (info,_) acc =
let v, _ = info.Sim.Core.value in
assert (X.type_info x == Ty.Tint);
match finite_non_point_dom info with
| Some s when Q.is_int v && Uf.is_normalized uf x ->
let fnd1, cont1 =
try true, I.contains (fst (MX0.find x env.monomes)) v
with Not_found -> false, true
in
let fnd2, cont2 =
try true, I.contains (MP0.find (poly_of x) env.polynomes) v
with Not_found -> false, true
in
if (fnd1 || fnd2) && cont1 && cont2 then
match acc with
| Some (_,_,s') when Q.compare s' s <= 0 -> acc
| _ -> Some (x,v, s)
else
acc
| _ -> acc
in
fun env uf ->
let int_sim = env.int_sim in
assert (int_sim.Sim.Core.status == Sim.Core.SAT);
let acc =
Sim.Core.MX.fold (aux_1 uf) int_sim.Sim.Core.non_basic None
in
let acc =
Sim.Core.MX.fold (aux_1 uf) int_sim.Sim.Core.basic acc
in
match acc with
| Some (x, n, s) -> gen_cs x n s 1
| None ->
let acc =
Sim.Core.MX.fold (aux_2 env uf) int_sim.Sim.Core.non_basic None
in
let acc =
Sim.Core.MX.fold (aux_2 env uf) int_sim.Sim.Core.basic acc
in
match acc with
| Some (x, n, s) -> gen_cs x n s 2
| None -> []
let infer_best_bounds env p =
let best_bnd lp ~upper sim i set_new_bound =
let q = if upper then Q.one else Q.m_one in
let lp = List.rev_map (fun (c, x) -> x, Q.mult q c) lp in
let sim, mx_res = Sim.Solve.maximize sim (Sim.Core.P.from_list lp) in
match Sim.Result.get mx_res sim with
| Sim.Core.Unknown -> assert false
| Sim.Core.Sat _ -> assert false
| Sim.Core.Unsat _ -> assert false
| Sim.Core.Unbounded _ -> i
| Sim.Core.Max(mx,_sol) ->
let {Sim.Core.max_v; is_le; reason} = Lazy.force mx in
set_new_bound reason (Q.mult q max_v) ~is_le:is_le i
in
if debug_fpa()>=2 then fprintf fmt "#infer bounds for %a@." P.print p;
let ty = P.type_info p in
let sim = if ty == Ty.Tint then env.int_sim else env.rat_sim in
let i = I.undefined ty in
let lp, c = P.to_list p in
assert (Q.is_zero c);
assert (sim.Sim.Core.status == Sim.Core.SAT);
let i = best_bnd lp ~upper:true sim i I.new_borne_sup in
let i = best_bnd lp ~upper:false sim i I.new_borne_inf in
if debug_fpa () >= 2 then
fprintf fmt "## inferred bounds for %a: %a@." P.print p I.print i;
i
end
module MP = struct
include MP0
let assert_normalized_poly p =
assert
(let _p0, c0, d0 = P.normal_form_pos p in
let b = Q.is_zero c0 && Q.is_one d0 in
begin
if not b then
fprintf fmt "[IC.assert_normalized_poly] %a is not normalized@."
P.print p
end;
b)
let n_add p i old ({ polynomes; _ } as env) =
assert_normalized_poly p;
if I.is_strict_smaller i old || not (MP0.mem p polynomes) then
let ty = P.type_info p in
let polynomes = MP0.add p i polynomes in
let improved_p = SP.add p env.improved_p in
if ty == Ty.Tint then
{env with
polynomes; improved_p;
int_sim = Sim_Wrap.add_if_better p old i env.int_sim}
else
{env with
polynomes; improved_p;
rat_sim = Sim_Wrap.add_if_better p old i env.rat_sim}
else
let () = assert (I.equal i old) in
env
let n_find p mp =
assert_normalized_poly p;
MP0.find p mp
[@@@ocaml.warning "-32"]
let find (_ : unit) (_ : unit) = assert false
let add (_ : unit) (_ : unit) (_ : unit) = assert false
[@@@ocaml.warning "+32"]
end
module MX = struct
include MX0
let assert_is_alien x =
assert (
let b = is_alien x in
begin
if not b then
fprintf fmt "[IC.assert_is_alien] %a is not an alien@." X.print x
end;
b
)
let n_add x ((i,_) as e) old ({ monomes; _ } as env) =
assert_is_alien x;
if I.is_strict_smaller i old || not (MX0.mem x monomes) then
let ty = X.type_info x in
let monomes = MX0.add x e monomes in
let improved_x = SX.add x env.improved_x in
if ty == Ty.Tint then
{env with
monomes; improved_x;
int_sim = Sim_Wrap.add_if_better (poly_of x) old i env.int_sim}
else
{env with
monomes; improved_x;
rat_sim = Sim_Wrap.add_if_better (poly_of x) old i env.rat_sim}
else
let () = assert (I.equal i old) in
{env with monomes = MX0.add x e monomes}
let n_find x mp =
assert_is_alien x;
MX0.find x mp
[@@@ocaml.warning "-32"]
let find (_ : unit) (_ : unit) = assert false
let add (_ : unit) (_ : unit) (_ : unit) = assert false
[@@@ocaml.warning "+32"]
end
let generic_find xp env =
let is_mon = is_alien xp in
if debug_fm () then
fprintf fmt "generic_find: %a ... " X.print xp;
try
if not is_mon then raise Not_found;
let i, use = MX.n_find xp env.monomes in
if debug_fm () then fprintf fmt "found@.";
i, use, is_mon
with Not_found ->
if debug_fm () then
fprintf fmt "not_found@.";
let p0 = poly_of xp in
let p, c = P.separate_constant p0 in
let p, c0, d = P.normal_form_pos p in
if debug_fm() then
fprintf fmt "normalized into %a + %a * %a@."
Q.print c Q.print d P.print p;
assert (Q.sign d <> 0 && Q.sign c0 = 0);
let ty = P.type_info p0 in
let ip =
try MP.n_find p env.polynomes with Not_found -> I.undefined ty
in
if debug_fm() then
fprintf fmt "scaling %a by %a@." I.print ip Q.print d;
let ip = I.affine_scale ~const:c ~coef:d ip in
ip, SX.empty, is_mon
let generic_add x j use is_mon env =
let ty = X.type_info x in
if is_mon then
try MX.n_add x (j,use) (fst (MX.n_find x env.monomes)) env
with Not_found -> MX.n_add x (j, use) (I.undefined ty) env
else
let p0 = poly_of x in
let p, c = P.separate_constant p0 in
let p, c0, d = P.normal_form_pos p in
assert (Q.sign d <> 0 && Q.sign c0 = 0);
let j = I.add j (I.point (Q.minus c) ty Explanation.empty) in
let j = I.scale (Q.inv d) j in
try MP.n_add p j (MP.n_find p env.polynomes) env
with Not_found -> MP.n_add p j (I.undefined ty) env
module Debug = struct
let assume a expl =
if debug_fm () then begin
fprintf fmt "[fm] We assume: %a@." LR.print (LR.make a);
fprintf fmt "explanations: %a@." Explanation.print expl
end
let print_use fmt use =
SX.iter (fprintf fmt "%a, " X.print) use
let env env =
if debug_fm () then begin
fprintf fmt "------------ FM: inequations-------------------------@.";
MPL.iter
(fun a { Oracle.ple0 = p; is_le = is_le; _ } ->
fprintf fmt "%a%s0 | %a@."
P.print p (if is_le then "<=" else "<") E.print a
)env.inequations;
fprintf fmt "------------ FM: monomes ----------------------------@.";
MX.iter
(fun x (i, use) ->
fprintf fmt "%a : %a |-use-> {%a}@."
X.print x I.print i print_use use)
env.monomes;
fprintf fmt "------------ FM: polynomes---------------------------@.";
MP.iter
(fun p i ->
fprintf fmt "%a : %a@."
P.print p I.print i)
env.polynomes;
fprintf fmt "-----------------------------------------------------@."
end
let implied_equalities l =
if debug_fm () then
begin
fprintf fmt "[fm] %d implied equalities@." (List.length l);
List.iter
(fun (ra, _, ex, _) ->
fprintf fmt " %a %a@."
LR.print (LR.make ra) Explanation.print ex) l
end
let case_split r1 r2 =
if debug_fm () then
fprintf fmt "[case-split] %a = %a@." X.print r1 X.print r2
let no_case_split s =
if debug_fm () then fprintf fmt "[case-split] %s : nothing@." s
let inconsistent_interval expl =
if debug_fm () then
fprintf fmt "interval inconsistent %a@." Explanation.print expl
let added_inequation kind ineq =
if debug_fm () then begin
fprintf fmt "[fm] I derived the (%s) inequality: %a %s 0@."
kind P.print ineq.Oracle.ple0
(if ineq.Oracle.is_le then "<=" else "<");
fprintf fmt "from the following combination:@.";
Util.MI.iter
(fun _a (coef, ple0, is_le) ->
fprintf fmt "\t%a * (%a %s 0) + @."
Q.print coef P.print ple0 (if is_le then "<=" else "<")
)ineq.Oracle.dep;
fprintf fmt "\t0@.@."
end
let tighten_interval_modulo_eq_begin p1 p2 =
if debug_fm () then begin
fprintf fmt "@.[fm] tighten intervals modulo eq: %a = %a@."
P.print p1 P.print p2;
end
let tighten_interval_modulo_eq_middle p1 p2 i1 i2 j =
if debug_fm () then begin
fprintf fmt " %a has interval %a@." P.print p1 I.print i1;
fprintf fmt " %a has interval %a@." P.print p2 I.print i2;
fprintf fmt " intersection is %a@." I.print j;
end
let tighten_interval_modulo_eq_end p1 p2 b1 b2 =
if debug_fm () then begin
if b1 then fprintf fmt " > improve interval of %a@.@." P.print p1;
if b2 then fprintf fmt " > improve interval of %a@.@." P.print p2;
if not b1 && not b2 then fprintf fmt " > no improvement@.@."
end
end
let empty classes = {
inequations = MPL.empty;
monomes = MX.empty ;
polynomes = MP.empty ;
known_eqs = SX.empty ;
improved_p = SP.empty ;
improved_x = SX.empty ;
classes = classes;
size_splits = Q.one;
new_uf = Uf.empty ();
rat_sim =
Sim.Solve.solve
(Sim.Core.empty ~is_int:false ~check_invs:false ~debug:0);
int_sim =
Sim.Solve.solve
(Sim.Core.empty ~is_int:true ~check_invs:false ~debug:0);
th_axioms = ME.empty;
linear_dep = ME.empty;
syntactic_matching = [];
}
(** computes an interval for vars = x_1^n_1 * ..... * x_i^n_i
(1) if some var is not in monomes, then return undefined
(2) check that all vars are in monomes before doing interval ops **)
let mult_bornes_vars vars env ty =
try
let l =
List.rev_map (fun (y,n) ->
let i, _, _ = generic_find y env in i, n
) vars
in
List.fold_left
(fun ui (yi,n) -> I.mult ui (I.power n yi))
(I.point Q.one ty Explanation.empty) l
with Not_found -> I.undefined ty
(** computes the interval of a polynome from those of its monomes.
The monomes are supposed to be already added in env **)
let intervals_from_monomes ?(monomes_inited=true) env p =
let pl, v = P.to_list p in
List.fold_left
(fun i (a, x) ->
let i_x, _ =
try MX.n_find x env.monomes
with Not_found ->
if monomes_inited then assert false;
I.undefined (X.type_info x), SX.empty
in
I.add (I.scale a i_x) i
) (I.point v (P.type_info p) Explanation.empty) pl
let cannot_be_equal_to_zero env p ip =
try
let z = alien_of (P.create [] Q.zero (P.type_info p)) in
match X.solve (alien_of p) z with
| [] -> None
| _ -> I.doesnt_contain_0 ip
with Util.Unsolvable -> Some (Explanation.empty, env.classes)
let rec init_monomes_of_poly are_eq env p use_p expl =
List.fold_left
(fun env (_, x) ->
try
let u, old_use_x = MX.n_find x env.monomes in
MX.n_add x (u, SX.union old_use_x use_p) u env
with Not_found ->
update_monome are_eq expl use_p env x
) env (fst (P.to_list p))
and init_alien are_eq expl p (normal_p, c, d) ty use_x env =
let env = init_monomes_of_poly are_eq env p use_x expl in
let i = intervals_from_monomes env p in
let i =
try
let old_i = MP.n_find normal_p env.polynomes in
let old_i = I.scale d
(I.add old_i (I.point c ty Explanation.empty)) in
I.intersect i old_i
with Not_found -> i
in
env, i
and update_monome are_eq expl use_x env x =
let ty = X.type_info x in
let ui, env = match X.ac_extract x with
| Some { h; l; _ }
when Symbols.equal h (Symbols.Op Symbols.Mult) ->
let use_x = SX.singleton x in
let env =
List.fold_left
(fun env (r,_) ->
let rp, _, _ = poly_of r |> P.normal_form_pos in
match P.is_monomial rp with
| Some (a,y,b) when Q.equal a Q.one && Q.sign b = 0 ->
update_monome are_eq expl use_x env y
| _ -> env
) env l in
let m = mult_bornes_vars l env ty in
m, env
| _ ->
match X.term_extract x with
| Some t, _ ->
let use_x = SX.singleton x in
begin
match E.term_view t with
| E.Not_a_term _ -> assert false
| E.Term { E.f = (Sy.Op Sy.Div); xs = [a; b]; _ } ->
let ra, ea =
let (ra, _) as e = Uf.find env.new_uf a in
if List.filter (X.equal x) (X.leaves ra) == [] then e
else fst (X.make a), Explanation.empty
in
let rb, eb =
let (rb, _) as e = Uf.find env.new_uf b in
if List.filter (X.equal x) (X.leaves rb) == [] then e
else fst (X.make b), Explanation.empty
in
let expl = Explanation.union expl (Explanation.union ea eb) in
let pa = poly_of ra in
let pb = poly_of rb in
let (pa', ca, da) as npa = P.normal_form_pos pa in
let (pb', cb, db) as npb = P.normal_form_pos pb in
let env, ia =
init_alien are_eq expl pa npa ty use_x env in
let ia = I.add_explanation ia ea in
let env, ib =
init_alien are_eq expl pb npb ty use_x env in
let ib = I.add_explanation ib eb in
let ia, ib = match cannot_be_equal_to_zero env pb ib with
| Some (ex, _) when Q.equal ca cb
&& P.compare pa' pb' = 0 ->
let expl = Explanation.union ex expl in
I.point da ty expl, I.point db ty expl
| Some (ex, _) ->
begin
match are_eq a b with
| Some (ex_eq, _) ->
let expl = Explanation.union ex expl in
let expl = Explanation.union ex_eq expl in
I.point Q.one ty expl,
I.point Q.one ty expl
| None -> ia, ib
end
| None -> ia, ib
in
I.div ia ib, env
| _ -> I.undefined ty, env
end
| _ -> I.undefined ty, env
in
let u, use_x' =
try MX.n_find x env.monomes
with Not_found -> I.undefined (X.type_info x), use_x in
let ui = I.intersect ui u in
MX.n_add x (ui, (SX.union use_x use_x')) u env
let rec tighten_ac are_eq x env expl =
let ty = X.type_info x in
let u, _use_x =
try MX.n_find x env.monomes
with Not_found -> I.undefined ty, SX.empty in
try
match X.ac_extract x with
| Some { h; l = [x,n]; _ }
when Symbols.equal h (Symbols.Op Symbols.Mult) && n mod 2 = 0 ->
let env =
if is_alien x then
let u = I.root n u in
let (pu, use_px) =
try MX.n_find x env.monomes
with Not_found -> I.undefined ty, SX.empty
in
let u = I.intersect u pu in
let env = MX.n_add x (u, use_px) pu env in
tighten_non_lin are_eq x use_px env expl
else
env
in
env
| Some { h; l = [x,n]; _ } when
Symbols.equal h (Symbols.Op Symbols.Mult) && n > 2 ->
let env =
if is_alien x then
let u = I.root n u in
let pu, use_px =
try MX.n_find x env.monomes
with Not_found -> I.undefined ty, SX.empty
in
let u = I.intersect u pu in
let env = MX.n_add x (u, use_px) pu env in
tighten_non_lin are_eq x use_px env expl
else
env
in
env
| _ -> env
with Q.Not_a_float -> env
and tighten_div _ env _ =
env
and tighten_non_lin are_eq x use_x env expl =
let env' = tighten_ac are_eq x env expl in
let env' = tighten_div x env' expl in
SX.fold
(fun x acc ->
let _, use = MX.n_find x acc.monomes in
update_monome are_eq expl use acc x
)
use_x env'
let update_monomes_from_poly p i env =
let lp, _ = P.to_list p in
let ty = P.type_info p in
List.fold_left (fun env (a,x) ->
let np = P.remove x p in
let (np,c,d) = P.normal_form_pos np in
try
let inp = MP.n_find np env.polynomes in
let new_ix =
I.scale
(Q.div Q.one a)
(I.add i
(I.scale (Q.minus d)
(I.add inp
(I.point c ty Explanation.empty)))) in
let old_ix, ux = MX.n_find x env.monomes in
let ix = I.intersect old_ix new_ix in
MX.n_add x (ix, ux) old_ix env
with Not_found -> env
) env lp
let update_polynomes_intervals env =
MP.fold
(fun p ip env ->
let new_i = intervals_from_monomes env p in
let i = I.intersect new_i ip in
if I.is_strict_smaller i ip then
update_monomes_from_poly p i (MP.n_add p i ip env)
else env
) env.polynomes env
let update_non_lin_monomes_intervals are_eq env =
MX.fold
(fun x (_, use_x) env ->
tighten_non_lin are_eq x use_x env Explanation.empty
) env.monomes env
let find_one_eq x u =
match I.is_point u with
| Some (v, ex) when X.type_info x != Ty.Tint || Q.is_int v ->
let eq =
LR.mkv_eq x (alien_of (P.create [] v (X.type_info x))) in
Some (eq, None, ex, Th_util.Other)
| _ -> None
let find_eq eqs x u env =
match find_one_eq x u with
| None -> eqs
| Some eq1 ->
begin
match X.ac_extract x with
| Some { h; l = [y,_]; _ }
when Symbols.equal h (Symbols.Op Symbols.Mult) ->
let neweqs = try
let u, _, _ = generic_find y env in
match find_one_eq y u with
| None -> eq1::eqs
| Some eq2 -> eq1::eq2::eqs
with Not_found -> eq1::eqs
in neweqs
| _ -> eq1::eqs
end
type ineq_status =
| Trivial_eq
| Trivial_ineq of Q.t
| Bottom
| Monome of Q.t * P.r * Q.t
| Other
let ineq_status { Oracle.ple0 = p ; is_le; _ } =
match P.is_monomial p with
Some (a, x, v) -> Monome (a, x, v)
| None ->
if P.is_empty p then
let _, v = P.separate_constant p in
let c = Q.sign v in
if c > 0 || (c >=0 && not is_le) then Bottom
else
if c = 0 && is_le then Trivial_eq
else Trivial_ineq v
else Other
let mk_equality p =
let r1 = alien_of p in
let r2 = alien_of (P.create [] Q.zero (P.type_info p)) in
LR.mkv_eq r1 r2
let fm_equalities eqs { Oracle.dep; expl = ex; _ } =
Util.MI.fold
(fun _ (_, p, _) eqs ->
(mk_equality p, None, ex, Th_util.Other) :: eqs
) dep eqs
let update_intervals are_eq env eqs expl (a, x, v) is_le =
let (u0, use_x0) as ixx = MX.n_find x env.monomes in
let uints, use_x =
match X.ac_extract x with
| Some { h; l; _ } when Symbols.equal h (Symbols.Op Symbols.Mult) ->
let m = mult_bornes_vars l env (X.type_info x) in
I.intersect m u0, use_x0
| _ -> ixx
in
let b = Q.div (Q.mult Q.m_one v) a in
let u =
if Q.sign a > 0 then
I.new_borne_sup expl b ~is_le uints
else
I.new_borne_inf expl b ~is_le uints in
let env = MX.n_add x (u, use_x) u0 env in
let env = tighten_non_lin are_eq x use_x env expl in
env, (find_eq eqs x u env)
let update_ple0 are_eq env p0 is_le expl =
if P.is_empty p0 then env
else
let ty = P.type_info p0 in
let a, _ = P.choose p0 in
let p, change =
if Q.sign a < 0 then
P.mult_const Q.m_one p0, true
else p0, false in
let p, c, _ = P.normal_form p in
let c = Q.minus c in
let u =
if change then
I.new_borne_inf expl c ~is_le (I.undefined ty)
else
I.new_borne_sup expl c ~is_le (I.undefined ty) in
let u, pu =
try
let pu = MP.n_find p env.polynomes in
let i = I.intersect u pu in
i, pu
with Not_found -> u, I.undefined ty
in
let env =
if I.is_strict_smaller u pu then
update_monomes_from_poly p u (MP.n_add p u pu env)
else env
in
match P.to_list p0 with
| [a,x], v ->
fst(update_intervals are_eq env [] expl (a, x, v) is_le)
| _ -> env
let register_relationship c x pi expl (x_rels, p_rels) =
let x_rels =
let a = Q.minus c, expl in
let s = Q.sign c in
assert (s <> 0);
let low, up =
try MX0.find x x_rels with Not_found -> MP0.empty, MP0.empty in
let v =
if s < 0 then
MP0.add pi a low, up
else
low, MP0.add pi a up
in
MX0.add x v x_rels
in
let p_rels =
let p0, c0, d0 = P.normal_form_pos pi in
let b = c, Q.minus c0, Q.minus d0, expl in
let s = Q.sign d0 in
assert (s <> 0);
let low,up =
try MP0.find p0 p_rels with Not_found -> MX0.empty, MX0.empty in
let w =
if s < 0 then
MX0.add x b low, up
else
low, MX0.add x b up
in
MP0.add p0 w p_rels
in
x_rels, p_rels
let add_inequations are_eq acc x_opt lin =
List.fold_left
(fun ((env, eqs, rels) as acc) ineq ->
let expl = ineq.Oracle.expl in
match ineq_status ineq with
| Bottom ->
Debug.added_inequation "Bottom" ineq;
raise (Ex.Inconsistent (expl, env.classes))
| Trivial_eq ->
Debug.added_inequation "Trivial_eq" ineq;
env, fm_equalities eqs ineq, rels
| Trivial_ineq c ->
Debug.added_inequation "Trivial_ineq" ineq;
let n, pp =
Util.MI.fold
(fun _ (_, p, is_le) ((n, pp) as acc) ->
if is_le then acc else
match pp with
| Some _ -> n+1, None
| None when n=0 -> 1, Some p
| _ -> n+1, None) ineq.Oracle.dep (0,None)
in
let env =
Util.MI.fold
(fun _ (coef, p, is_le) env ->
let ty = P.type_info p in
let is_le =
match pp with
Some x -> P.compare x p = 0 | _ -> is_le && n=0
in
let p' = P.sub (P.create [] (Q.div c coef) ty) p in
update_ple0 are_eq env p' is_le expl
) ineq.Oracle.dep env
in
env, eqs, rels
| Monome (a, x, v) ->
Debug.added_inequation "Monome" ineq;
let env, eqs =
update_intervals
are_eq env eqs expl (a, x, v) ineq.Oracle.is_le
in
env, eqs, rels
| Other ->
match x_opt with
| None -> acc
| Some x ->
let ple0 = ineq.Oracle.ple0 in
let c = try P.find x ple0 with Not_found -> assert false in
let ple0 = P.remove x ple0 in
env, eqs, register_relationship c x ple0 ineq.Oracle.expl rels
) acc lin
let split_problem env ineqs aliens =
let current_age = Oracle.current_age () in
let l, all_lvs =
List.fold_left
(fun (acc, all_lvs) ({ Oracle.ple0 = p; _ } as ineq) ->
match ineq_status ineq with
| Trivial_eq | Trivial_ineq _ -> (acc, all_lvs)
| Bottom ->
raise (Ex.Inconsistent (ineq.Oracle.expl, env.classes))
| _ ->
let lvs =
List.fold_left (fun acc e -> SX.add e acc) SX.empty (aliens p)
in
([ineq], lvs) :: acc , SX.union lvs all_lvs
)([], SX.empty) ineqs
in
let ll =
SX.fold
(fun x l ->
let lx, l_nx = List.partition (fun (_,s) -> SX.mem x s) l in
match lx with
| [] -> assert false
| e:: lx ->
let elx =
List.fold_left
(fun (l, lvs) (l', lvs') ->
List.rev_append l l', SX.union lvs lvs') e lx in
elx :: l_nx
) all_lvs l
in
let ll =
List.filter
(fun (ineqs, _) ->
List.exists
(fun ineq -> Z.equal current_age ineq.Oracle.age) ineqs
)ll
in
List.fast_sort (fun (a,_) (b,_) -> List.length a - List.length b) ll
let is_normalized_poly uf p =
let p = alien_of p in
let rp, _ = Uf.find_r uf p in
if X.equal p rp then true
else begin
fprintf fmt "%a <= 0 NOT normalized@." X.print p;
fprintf fmt "It is equal to %a@." X.print rp;
false
end
let better_upper_bound_from_intervals env p =
let p0, c0, d0 = P.normal_form_pos p in
assert (Q.is_zero c0);
try
let i = MP.n_find p0 env.polynomes in
if Q.is_one d0 then I.borne_sup i
else
if Q.is_m_one d0 then
let bi, ex, is_large = I.borne_inf i in
Q.minus bi, ex, is_large
else assert false
with I.No_finite_bound | Not_found ->
assert false
let better_bound_from_intervals env ({ Oracle.ple0; is_le; dep; _ } as v) =
let p, c = P.separate_constant ple0 in
assert (not (P.is_empty p));
let cur_up_bnd = Q.minus c in
let i_up_bnd, expl, is_large = better_upper_bound_from_intervals env p in
let new_p = P.add_const (Q.minus i_up_bnd) p in
let a = match Util.MI.bindings dep with [a,_] -> a | _ -> assert false in
let cmp = Q.compare i_up_bnd cur_up_bnd in
assert (cmp <= 0);
if cmp = 0 then
match is_le, is_large with
| false, true -> assert false
| false, false | true, true -> v
| true , false ->
{v with
Oracle.ple0 = new_p;
expl = expl;
is_le = false;
dep = Util.MI.singleton a (Q.one, new_p, false)}
else
{v with
Oracle.ple0 = new_p;
expl = expl;
is_le = is_large;
dep = Util.MI.singleton a (Q.one, new_p, is_large)}
let args_of p = List.rev_map snd (fst (P.to_list p))
let update_linear_dep env rclass_of ineqs =
let terms =
List.fold_left
(fun st { Oracle.ple0; _ } ->
List.fold_left
(fun st (_, x) -> SE.union st (rclass_of x))
st (fst (P.to_list ple0))
)SE.empty ineqs
in
let linear_dep =
SE.fold
(fun t linear_dep -> ME.add t terms linear_dep) terms env.linear_dep
in
{env with linear_dep}
let refine_x_bounds ix env rels is_low =
MP.fold
(fun p (m_cx, ineq_ex) ix ->
try
assert (is_low == (Q.sign m_cx > 0));
let ip, _, _ = generic_find (alien_of p) env in
let b, ex_b, is_le = I.borne_inf ip in
let b = Q.div b m_cx in
let func = if is_low then I.new_borne_inf else I.new_borne_sup in
func (Explanation.union ineq_ex ex_b) b ~is_le ix
with I.No_finite_bound -> ix
)rels ix
let monomes_relational_deductions env x_rels =
MX.fold
(fun x (low, up) env ->
let ix0, use_x =
try MX.n_find x env.monomes with Not_found -> assert false in
let ix = refine_x_bounds ix0 env low true in
let ix = refine_x_bounds ix env up false in
if I.is_strict_smaller ix ix0 then MX.n_add x (ix, use_x) ix0 env
else env
)x_rels env
let refine_p_bounds ip _p env rels is_low =
MX.fold
(fun x (cx, mc0, md0, ineq_ex) ip ->
try
assert (is_low == (Q.sign md0 > 0));
let ix,_ =
try MX.n_find x env.monomes with Not_found -> raise Exit in
let bx, ex_b, is_le =
(if Q.sign cx > 0 then I.borne_inf else I.borne_sup) ix in
let b = Q.mult cx bx in
let b = Q.add (Q.div b md0) mc0 in
let func = if is_low then I.new_borne_inf else I.new_borne_sup in
func (Explanation.union ineq_ex ex_b) b ~is_le ip
with Exit | I.No_finite_bound -> ip
)rels ip
let polynomes_relational_deductions env p_rels =
MP.fold
(fun p0 (low, up) env ->
let xp = alien_of p0 in
if not (MP.mem p0 env.polynomes || MX.mem xp env.monomes) then env
else
let ip0, use, is_mon = generic_find xp env in
let ip = refine_p_bounds ip0 p0 env low true in
let ip = refine_p_bounds ip p0 env up false in
if I.is_strict_smaller ip ip0 then
if is_mon then MX.n_add xp (ip, use) ip0 env
else MP.n_add p0 ip ip0 env
else
env
)p_rels env
let fm uf are_eq rclass_of env eqs =
if debug_fm () then fprintf fmt "[fm] in fm/fm-simplex@.";
Options.tool_req 4 "TR-Arith-Fm";
let ineqs =
MPL.fold (fun _ v acc ->
if Options.enable_assertions() then
assert (is_normalized_poly uf v.Oracle.ple0);
(better_bound_from_intervals env v) :: acc
) env.inequations []
in
let pbs = split_problem env ineqs args_of in
let res = List.fold_left
(fun (env, eqs) (ineqs, _) ->
let env = update_linear_dep env rclass_of ineqs in
let mp = Oracle.MINEQS.add_to_map Oracle.MINEQS.empty ineqs in
let env, eqs, (x_rels, p_rels) =
Oracle.available add_inequations are_eq
(env, eqs, (MX.empty, MP.empty)) mp
in
let env = monomes_relational_deductions env x_rels in
let env = polynomes_relational_deductions env p_rels in
env, eqs
)(env, eqs) pbs
in
if debug_fm () then fprintf fmt "[fm] out fm/fm-simplex@.";
res
let is_num r =
let ty = X.type_info r in ty == Ty.Tint || ty == Ty.Treal
let add_disequality are_eq env eqs p expl =
let ty = P.type_info p in
match P.to_list p with
| ([], v) ->
if Q.sign v = 0 then
raise (Ex.Inconsistent (expl, env.classes));
env, eqs
| ([a, x], v) ->
let b = Q.div (Q.minus v) a in
let i1 = I.point b ty expl in
let i2, use2 =
try
MX.n_find x env.monomes
with Not_found -> I.undefined ty, SX.empty
in
let i = I.exclude i1 i2 in
let env = MX.n_add x (i,use2) i2 env in
let env = tighten_non_lin are_eq x use2 env expl in
env, find_eq eqs x i env
| _ ->
let p, c, _ = P.normal_form_pos p in
let i1 = I.point (Q.minus c) ty expl in
let i2 =
try
MP.n_find p env.polynomes
with Not_found -> I.undefined ty
in
let i = I.exclude i1 i2 in
let env =
if I.is_strict_smaller i i2 then
update_monomes_from_poly p i (MP.n_add p i i2 env)
else env
in
env, eqs
let add_equality are_eq env eqs p expl =
let ty = P.type_info p in
match P.to_list p with
| ([], v) ->
if Q.sign v <> 0 then
raise (Ex.Inconsistent (expl, env.classes));
env, eqs
| ([a, x], v) ->
let b = Q.div (Q.minus v) a in
let i = I.point b ty expl in
let i2, use =
try MX.n_find x env.monomes
with Not_found -> I.undefined ty, SX.empty
in
let i = I.intersect i i2 in
let env = MX.n_add x (i, use) i2 env in
let env = tighten_non_lin are_eq x use env expl in
env, find_eq eqs x i env
| _ ->
let p, c, _ = P.normal_form_pos p in
let i = I.point (Q.minus c) ty expl in
let i, ip =
try
let ip = MP.n_find p env.polynomes in
I.intersect i ip, ip
with Not_found -> i, I.undefined ty
in
let env =
if I.is_strict_smaller i ip then
update_monomes_from_poly p i (MP.n_add p i ip env)
else env
in
let env =
{ env with
known_eqs = SX.add (alien_of p) env.known_eqs
} in
env, eqs
let normal_form a = match a with
| L.Builtin (false, L.LE, [r1; r2])
when X.type_info r1 == Ty.Tint ->
let pred_r1 = P.sub (poly_of r1) (P.create [] Q.one Ty.Tint) in
LR.mkv_builtin true L.LE [r2; alien_of pred_r1]
| L.Builtin (true, L.LT, [r1; r2]) when
X.type_info r1 == Ty.Tint ->
let pred_r2 = P.sub (poly_of r2) (P.create [] Q.one Ty.Tint) in
LR.mkv_builtin true L.LE [r1; alien_of pred_r2]
| L.Builtin (false, L.LE, [r1; r2]) ->
LR.mkv_builtin true L.LT [r2; r1]
| L.Builtin (false, L.LT, [r1; r2]) ->
LR.mkv_builtin true L.LE [r2; r1]
| _ -> a
let remove_trivial_eqs eqs la =
let la =
List.fold_left (fun m ((a, _, _, _) as e) -> MR.add a e m) MR.empty la
in
let eqs, _ =
List.fold_left
(fun ((eqs, m) as acc) ((sa, _, _, _) as e) ->
if MR.mem sa m then acc else e :: eqs, MR.add sa e m
)([], la) eqs
in
eqs
let equalities_from_polynomes env eqs =
let known, eqs =
MP.fold
(fun p i (knw, eqs) ->
let xp = alien_of p in
if SX.mem xp knw then knw, eqs
else
match I.is_point i with
| Some (num, ex) ->
let r2 = alien_of (P.create [] num (P.type_info p)) in
SX.add xp knw, (LR.mkv_eq xp r2, None, ex, Th_util.Other) :: eqs
| None -> knw, eqs
) env.polynomes (env.known_eqs, eqs)
in {env with known_eqs= known}, eqs
let equalities_from_monomes env eqs =
let known, eqs =
MX.fold
(fun x (i,_) (knw, eqs) ->
if SX.mem x knw then knw, eqs
else
match I.is_point i with
| Some (num, ex) ->
let r2 = alien_of (P.create [] num (X.type_info x)) in
SX.add x knw, (LR.mkv_eq x r2, None, ex, Th_util.Other) :: eqs
| None -> knw, eqs
) env.monomes (env.known_eqs, eqs)
in {env with known_eqs= known}, eqs
let equalities_from_intervals env eqs =
let env, eqs = equalities_from_polynomes env eqs in
equalities_from_monomes env eqs
let count_splits env la =
let nb =
List.fold_left
(fun nb (_,_,_,i) ->
match i with
| Th_util.CS (Th_util.Th_arith, n) -> Numbers.Q.mult nb n
| _ -> nb
)env.size_splits la
in
{env with size_splits = nb}
let remove_ineq a ineqs =
match a with None -> ineqs | Some a -> MPL.remove a ineqs
let add_ineq a v ineqs =
match a with None -> ineqs | Some a -> MPL.add a v ineqs
let tighten_eq_bounds env r1 r2 p1 p2 origin_eq expl =
if P.is_const p1 != None || P.is_const p2 != None then env
else
match origin_eq with
| Th_util.CS _ | Th_util.NCS _ -> env
| Th_util.Subst | Th_util.Other ->
Debug.tighten_interval_modulo_eq_begin p1 p2;
let i1, us1, is_mon_1 = generic_find r1 env in
let i2, us2, is_mon_2 = generic_find r2 env in
let j = I.add_explanation (I.intersect i1 i2) expl in
Debug.tighten_interval_modulo_eq_middle p1 p2 i1 i2 j;
let impr_i1 = I.is_strict_smaller j i1 in
let impr_i2 = I.is_strict_smaller j i2 in
Debug.tighten_interval_modulo_eq_end p1 p2 impr_i1 impr_i2;
let env =
if impr_i1 then generic_add r1 j us1 is_mon_1 env else env
in
if impr_i2 then generic_add r2 j us2 is_mon_2 env else env
let rec loop_update_intervals are_eq env cpt =
let cpt = cpt + 1 in
let env = {env with improved_p=SP.empty; improved_x=SX.empty} in
let env = update_non_lin_monomes_intervals are_eq env in
let env = Sim_Wrap.solve env 1 in
let env = update_polynomes_intervals env in
let env = Sim_Wrap.solve env 1 in
if env.improved_p == SP.empty && env.improved_x == SX.empty || cpt > 10
then env
else loop_update_intervals are_eq env cpt
let assume ~query env uf la =
Oracle.incr_age ();
let env = count_splits env la in
let are_eq = Uf.are_equal uf ~added_terms:true in
let classes = Uf.cl_extract uf in
let rclass_of = Uf.rclass_of uf in
let env =
{env with
improved_p=SP.empty; improved_x=SX.empty; classes; new_uf = uf}
in
Debug.env env;
let nb_num = ref 0 in
let env, eqs, new_ineqs, to_remove =
List.fold_left
(fun ((env, eqs, new_ineqs, rm) as acc) (a, root, expl, orig) ->
let a = normal_form a in
Debug.assume a expl;
try
match a with
| L.Builtin(_, ((L.LE | L.LT) as n), [r1;r2]) ->
incr nb_num;
let p1 = poly_of r1 in
let p2 = poly_of r2 in
let ineq =
Oracle.create_ineq p1 p2 (n == L.LE) root expl in
begin
match ineq_status ineq with
| Bottom -> raise (Ex.Inconsistent (expl, env.classes))
| Trivial_eq | Trivial_ineq _ ->
{env with inequations=remove_ineq root env.inequations},
eqs, new_ineqs,
(match root with None -> rm | Some a -> a:: rm)
| Monome _
| Other ->
let env =
init_monomes_of_poly
are_eq env ineq.Oracle.ple0 SX.empty
Explanation.empty
in
let env =
update_ple0
are_eq env ineq.Oracle.ple0 (n == L.LE) expl
in
{env with inequations=add_ineq root ineq env.inequations},
eqs, true, rm
end
| L.Distinct (false, [r1; r2]) when is_num r1 && is_num r2 ->
incr nb_num;
let p = P.sub (poly_of r1) (poly_of r2) in
begin
match P.is_const p with
| Some c ->
if Q.is_zero c then
raise (Ex.Inconsistent (expl, env.classes))
else
let rm = match root with Some a -> a::rm | None -> rm in
env, eqs, new_ineqs, rm
| None ->
let env = init_monomes_of_poly are_eq env p SX.empty
Explanation.empty
in
let env, eqs = add_disequality are_eq env eqs p expl in
env, eqs, new_ineqs, rm
end
| L.Eq(r1, r2) when is_num r1 && is_num r2 ->
incr nb_num;
let p1 = poly_of r1 in
let p2 = poly_of r2 in
let p = P.sub p1 p2 in
let env = init_monomes_of_poly are_eq env p SX.empty
Explanation.empty
in
let env, eqs = add_equality are_eq env eqs p expl in
let env = tighten_eq_bounds env r1 r2 p1 p2 orig expl in
env, eqs, new_ineqs, rm
| _ -> acc
with I.NotConsistent expl ->
Debug.inconsistent_interval expl ;
raise (Ex.Inconsistent (expl, env.classes))
)
(env, [], false, []) la
in
try
let env = if query then env else Sim_Wrap.solve env 1 in
if !nb_num = 0 || query then env, {Sig_rel.assume=[]; remove = to_remove}
else
let env, eqs =
if new_ineqs && not (Options.no_fm ()) then
fm uf are_eq rclass_of env eqs
else env, eqs
in
let env = Sim_Wrap.solve env 1 in
let env = loop_update_intervals are_eq env 0 in
let env, eqs = equalities_from_intervals env eqs in
Debug.env env;
let eqs = remove_trivial_eqs eqs la in
Debug.implied_equalities eqs;
let to_assume =
List.rev_map (fun (sa, _, ex, orig) ->
(Sig_rel.LSem sa, ex, orig)) eqs
in
env, {Sig_rel.assume = to_assume; remove = to_remove}
with I.NotConsistent expl ->
Debug.inconsistent_interval expl ;
raise (Ex.Inconsistent (expl, env.classes))
let assume ~query env uf la =
let env, res = assume ~query env uf la in
let polys =
MP.fold
(fun p _ mp ->
if Uf.is_normalized uf (alien_of p) then mp else MP.remove p mp)
env.polynomes env.polynomes
in
{env with polynomes = polys}, res
let query env uf a_ex =
try
ignore(assume ~query:true env uf [a_ex]);
None
with Ex.Inconsistent (expl, classes) -> Some (expl, classes)
let assume env uf la =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Arith Timers.F_assume;
let res =assume ~query:false env uf la in
Timers.exec_timer_pause Timers.M_Arith Timers.F_assume;
res
with e ->
Timers.exec_timer_pause Timers.M_Arith Timers.F_assume;
raise e
else assume ~query:false env uf la
let query env uf la =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Arith Timers.F_query;
let res = query env uf la in
Timers.exec_timer_pause Timers.M_Arith Timers.F_query;
res
with e ->
Timers.exec_timer_pause Timers.M_Arith Timers.F_query;
raise e
else query env uf la
let case_split_polynomes env =
let o = MP.fold
(fun p i o ->
match I.finite_size i with
| Some s when Q.compare s Q.one > 0 ->
begin
match o with
| Some (s', _, _) when Q.compare s' s < 0 -> o
| _ ->
let n, _, is_large = I.borne_inf i in
assert (is_large);
Some (s, p, n)
end
| _ -> o
) env.polynomes None in
match o with
| Some (s, p, n) ->
let r1 = alien_of p in
let r2 = alien_of (P.create [] n (P.type_info p)) in
Debug.case_split r1 r2;
[LR.mkv_eq r1 r2, true, Th_util.CS (Th_util.Th_arith, s)], s
| None ->
Debug.no_case_split "polynomes";
[], Q.zero
let case_split_monomes env =
let o = MX.fold
(fun x (i,_) o ->
match I.finite_size i with
| Some s when Q.compare s Q.one > 0 ->
begin
match o with
| Some (s', _, _) when Q.compare s' s < 0 -> o
| _ ->
let n, _, is_large = I.borne_inf i in
assert (is_large);
Some (s, x, n)
end
| _ -> o
) env.monomes None in
match o with
| Some (s,x,n) ->
let ty = X.type_info x in
let r1 = x in
let r2 = alien_of (P.create [] n ty) in
Debug.case_split r1 r2;
[LR.mkv_eq r1 r2, true, Th_util.CS (Th_util.Th_arith, s)], s
| None ->
Debug.no_case_split "monomes";
[], Q.zero
let check_size for_model env res =
if for_model then res
else
match res with
| [] -> res
| [_, _, Th_util.CS (Th_util.Th_arith, s)] ->
if Numbers.Q.compare (Q.mult s env.size_splits) (max_split ()) <= 0 ||
Numbers.Q.sign (max_split ()) < 0 then res
else []
| _ -> assert false
let default_case_split env uf ~for_model =
Options.tool_req 4 "TR-Arith-CaseSplit";
match check_size for_model env (Sim_Wrap.case_split env uf) with
[] ->
begin
let cs1, sz1 = case_split_polynomes env in
let cs2, sz2 = case_split_monomes env in
match check_size for_model env cs1, check_size for_model env cs2
with
| [], cs | cs, [] -> cs
| cs1, cs2 -> if Q.compare sz1 sz2 < 0 then cs1 else cs2
end
| res -> res
let add =
let are_eq t1 t2 =
if E.equal t1 t2 then Some (Explanation.empty, []) else None
in
fun env new_uf r _ ->
try
let env = {env with new_uf} in
if is_num r then
init_monomes_of_poly are_eq env
(poly_of r) SX.empty Explanation.empty
else env
with I.NotConsistent expl ->
Debug.inconsistent_interval expl ;
raise (Ex.Inconsistent (expl, env.classes))
let print_model fmt env rs = match rs with
| [] -> ()
| _ ->
fprintf fmt "Relation:";
List.iter (fun (t, r) ->
let p = poly_of r in
let ty = P.type_info p in
if ty == Ty.Tint || ty == Ty.Treal then
let p', c, d = P.normal_form_pos p in
let pu' =
try MP.n_find p' env.polynomes
with Not_found -> I.undefined ty
in
let pm' =
try intervals_from_monomes ~monomes_inited:false env p'
with Not_found -> I.undefined ty
in
let u' = I.intersect pu' pm' in
if I.is_point u' == None && I.is_undefined u' then
let u =
I.scale d
(I.add u'
(I.point c ty Explanation.empty)) in
fprintf fmt "\n %a ∈ %a" E.print t I.pretty_print u
) rs;
fprintf fmt "\n@."
let new_terms _ = SE.empty
let case_split_union_of_intervals =
let aux acc uf i z =
if Uf.is_normalized uf z then
match I.bounds_of i with
| [] -> assert false
| [_] -> ()
| (_,(v, ex))::_ -> acc := Some (z, v, ex); raise Exit
in fun env uf ->
let cs = ref None in
try
MP.iter (fun p i -> aux cs uf i (alien_of p)) env.polynomes;
MX.iter (fun x (i,_) -> aux cs uf i x) env.monomes;
[]
with Exit ->
match !cs with
| None -> assert false
| Some(_,None, _) -> assert false
| Some(r1,Some (n, eps), _) ->
let ty = X.type_info r1 in
let r2 = alien_of (P.create [] n ty) in
let pred =
if Q.is_zero eps then L.LE else (assert (Q.is_m_one eps); L.LT)
in
[LR.mkv_builtin true pred [r1; r2], true,
Th_util.CS (Th_util.Th_arith, Q.one)]
let int_constraints_from_map_intervals =
let aux p xp i uf acc =
if Uf.is_normalized uf xp && I.is_point i == None
&& P.type_info p == Ty.Tint
then (p, I.bounds_of i) :: acc
else acc
in
fun env uf ->
let acc =
MP.fold (fun p i acc -> aux p (alien_of p) i uf acc) env.polynomes []
in
MX.fold (fun x (i,_) acc -> aux (poly_of x) x i uf acc) env.monomes acc
let fm_simplex_unbounded_integers_encoding env uf =
let simplex = Sim.Core.empty ~is_int:true ~check_invs:true ~debug:0 in
let int_ctx = int_constraints_from_map_intervals env uf in
List.fold_left
(fun simplex (p, uints) ->
match uints with
| [] ->
fprintf fmt "Intervals already empty !!!!@.";
assert false
| _::_::_ ->
fprintf fmt "case-split over unions of intervals is needed !!!!@.";
assert false
| [(mn, ex_mn), (mx, ex_mx)] ->
let l, c = P.to_list p in
let l = List.rev_map (fun (c, x) -> x, c) (List.rev l) in
assert (Q.sign c = 0);
let cst0 =
List.fold_left (fun z (_, c) -> Q.add z (Q.abs c))Q.zero l
in
let cst = Q.div cst0 (Q.from_int 2) in
assert (mn == None || mx == None);
let mn =
match mn with
| None -> None
| Some (q, q') -> Some (Q.add q cst, q')
in
let mx =
match mx with
| None -> None
| Some (q, q') -> Some (Q.sub q cst, q')
in
match l with
| [] -> assert false
| [x, c] ->
assert (Q.is_one c);
Sim.Assert.var simplex x mn ex_mn mx ex_mx |> fst
| _ ->
let xp = alien_of p in
let sim_p =
match Sim.Core.poly_of_slake simplex xp with
| Some res -> res
| None -> Sim.Core.P.from_list l
in
Sim.Assert.poly simplex sim_p xp mn ex_mn mx ex_mx |> fst
) simplex int_ctx
let round_to_integers list =
List.rev_map (fun (x, q1) ->
let f = Q.floor q1 in
let c = Q.ceiling q1 in
x, if Q.compare (Q.sub q1 f) (Q.sub c q1) > 0 then f else c
) (List.rev list)
let model_from_simplex sim is_int env uf =
match Sim.Result.get None sim with
| Sim.Core.Unknown | Sim.Core.Unbounded _ | Sim.Core.Max _ -> assert false
| Sim.Core.Unsat ex ->
raise (Ex.Inconsistent(Lazy.force ex, env.classes))
| Sim.Core.Sat sol ->
let {Sim.Core.main_vars; slake_vars; int_sol} = Lazy.force sol in
let main_vars, _slake_vars =
if int_sol || not is_int then main_vars, slake_vars
else round_to_integers main_vars, round_to_integers slake_vars
in
let fct = if is_int then E.int else E.real in
List.fold_left
(fun acc (v, q) ->
assert (not is_int || Q.is_int q);
if SX.mem v env.known_eqs || not (Uf.is_normalized uf v) then
acc
else
let t = fct (Q.to_string q) in
let r, _ = X.make t in
if debug_interpretation() then
fprintf fmt "[%s simplex] %a = %a@."
(if is_int then "integer" else "rational") X.print v X.print r;
(v, r, Explanation.empty) :: acc
)[] (List.rev main_vars)
let model_from_unbounded_domains =
let mk_cs acc (x, v, _ex) =
((LR.view (LR.mk_eq x v)), true,
Th_util.CS (Th_util.Th_arith, Q.from_int 2)) :: acc
in
fun env uf ->
assert (env.int_sim.Sim.Core.status == Sim.Core.SAT);
assert (env.rat_sim.Sim.Core.status == Sim.Core.SAT);
let rat_sim = env.rat_sim in
let int_sim =
let sim = fm_simplex_unbounded_integers_encoding env uf in
Sim.Solve.solve sim
in
let l1 = model_from_simplex rat_sim false env uf in
let l2 = model_from_simplex int_sim true env uf in
List.fold_left mk_cs (List.fold_left mk_cs [] l1) l2
let case_split env uf ~for_model =
let res = default_case_split env uf ~for_model in
match res with
| [] ->
if not for_model then []
else
begin
match case_split_union_of_intervals env uf with
| [] -> model_from_unbounded_domains env uf
| l -> l
end
| _ -> res
let best_interval_of optimized env p =
match P.is_const p with
| Some c -> env, I.point c (P.type_info p) Explanation.empty
| None ->
let i =
try let i, _, _ = generic_find (alien_of p) env in i
with Not_found -> I.undefined (P.type_info p)
in
if SP.mem p !optimized then env, i
else
try
let j = Sim_Wrap.infer_best_bounds env p in
optimized := SP.add p !optimized;
let k = I.intersect i j in
if not (I.is_strict_smaller k i) then env, i
else
let env = MP.n_add p k i env in
Sim_Wrap.solve env 1, k
with I.NotConsistent expl ->
if true then begin
[@ocaml.ppwarning "TODO: find an example triggering this case!"]
fprintf fmt "TODO: should check that this is correct !!!!@."
end;
raise (Ex.Inconsistent (expl, env.classes))
let mk_const_term ty s =
match ty with
| Ty.Tint -> E.int (Q.to_string s)
| Ty.Treal -> E.real (Q.to_string s)
| _ -> assert false
let integrate_mapsTo_bindings sbs maps_to =
try
let sbs =
List.fold_left
(fun ((sbt, sty) as sbs) (x, tx) ->
let x = Sy.Var x in
assert (not (Symbols.Map.mem x sbt));
let t = E.apply_subst sbs tx in
let mk, _ = X.make t in
match P.is_const (poly_of mk) with
| None ->
if debug_fpa() >= 2 then begin
fprintf fmt "bad semantic trigger %a |-> %a"
Sy.print x E.print tx;
fprintf fmt " left-hand side is not a constant!@.";
end;
raise Exit
| Some c ->
let tc = mk_const_term (E.type_info t) c in
Symbols.Map.add x tc sbt, sty
)sbs maps_to
in
Some sbs
with Exit -> None
let extend_with_domain_substitution =
let eps = Q.div_2exp Q.one 1076 in
let aux idoms sbt =
Var.Map.fold
(fun v_hs (lv, uv, ty) sbt ->
let s = Hstring.view (Var.view v_hs).Var.hs in
match s.[0] with
| '?' -> sbt
| _ ->
let lb_var = Sy.var v_hs in
let lb_val = match lv, uv with
| None, None -> raise Exit
| Some (q1, false), Some (q2, false) when Q.equal q1 q2 ->
mk_const_term ty q1
| Some (q1,_), Some (q2,_) ->
fprintf fmt "%a <= %a <= %a@."
Q.print q1 Sy.print lb_var Q.print q2;
Format.eprintf "Which value should we choose ?@.";
assert (Q.compare q2 q1 >= 0);
assert false
| Some (q, is_strict), None ->
mk_const_term ty (if is_strict then Q.add q eps else q)
| None, Some (q, is_strict) ->
mk_const_term ty (if is_strict then Q.sub q eps else q)
in
Sy.Map.add lb_var lb_val sbt
) idoms sbt
in
fun (sbt, sbty) idoms ->
try Some (aux idoms sbt, sbty)
with Exit ->
if debug_fpa() >=2 then
fprintf fmt "[IC] extend_with_domain_substitution failed !@.";
None
let terms_linear_dep { linear_dep ; _ } lt =
match lt with
| [] | [_] -> true
| e::l ->
try
let st = ME.find e linear_dep in
List.for_all (fun t -> SE.mem t st) l
with Not_found -> false
exception Sem_match_fails of t
let domain_matching _lem_name tr sbt env uf optimized =
try
let idoms, maps_to, env, _uf =
List.fold_left
(fun (idoms, maps_to, env, uf) s ->
match s with
| E.MapsTo (x, t) ->
idoms, (x, t) :: maps_to, env, uf
| E.Interval (t, lb, ub) ->
let tt = E.apply_subst sbt t in
assert (E.is_ground tt);
let uf, _ = Uf.add uf tt in
let rr, _ex = Uf.find uf tt in
let p = poly_of rr in
let p', c', d = P.normal_form_pos p in
let env, i' = best_interval_of optimized env p' in
let ic = I.point c' (P.type_info p') Explanation.empty in
let i = I.scale d (I.add i' ic) in
begin match I.match_interval lb ub i idoms with
| None -> raise (Sem_match_fails env)
| Some idoms -> idoms, maps_to, env, uf
end
| E.NotTheoryConst t ->
let tt = E.apply_subst sbt t in
let uf, _ = Uf.add uf tt in
if X.leaves (fst (Uf.find uf tt)) == [] ||
X.leaves (fst (X.make tt)) == [] then
raise (Sem_match_fails env);
idoms, maps_to, env, uf
| E.IsTheoryConst t ->
let tt = E.apply_subst sbt t in
let uf, _ = Uf.add uf tt in
let r, _ = X.make tt in
if X.leaves r != [] then raise (Sem_match_fails env);
idoms, maps_to, env, uf
| E.LinearDependency (x, y) ->
let x = E.apply_subst sbt x in
let y = E.apply_subst sbt y in
if not (terms_linear_dep env [x;y]) then
raise (Sem_match_fails env);
let uf, _ = Uf.add uf x in
let uf, _ = Uf.add uf y in
idoms, maps_to, env, uf
)(Var.Map.empty, [], env, uf) tr.E.semantic
in
env, Some (idoms, maps_to)
with Sem_match_fails env -> env, None
let semantic_matching lem_name tr sbt env uf optimized =
match domain_matching lem_name tr sbt env uf optimized with
| env, None -> env, None
| env, Some(idom, mapsTo) ->
begin match extend_with_domain_substitution sbt idom with
| None -> env, None
| Some sbs -> env, integrate_mapsTo_bindings sbs mapsTo
end
let record_this_instance f accepted lorig =
if Options.profiling() then
match E.form_view lorig with
| E.Lemma { E.name; loc; _ } ->
Profiling.new_instance_of name f loc accepted
| E.Unit _ | E.Clause _ | E.Literal _ | E.Skolem _
| E.Let _ | E.Iff _ | E.Xor _ | E.Not_a_form -> assert false
let profile_produced_terms menv lorig nf s trs =
if Options.profiling() then
let st0 =
List.fold_left (fun st t -> E.sub_terms st (E.apply_subst s t))
SE.empty trs
in
let name, loc, _ = match E.form_view lorig with
| E.Lemma { E.name; main; loc; _ } -> name, loc, main
| E.Unit _ | E.Clause _ | E.Literal _ | E.Skolem _
| E.Let _ | E.Iff _ | E.Xor _ | E.Not_a_form -> assert false
in
let st1 = E.max_ground_terms_rec_of_form nf in
let diff = SE.diff st1 st0 in
let info, _ = EM.terms_info menv in
let _new = SE.filter (fun t -> not (ME.mem t info)) diff in
Profiling.register_produced_terms name loc st0 st1 diff _new
let new_facts_for_axiom
~do_syntactic_matching menv uf selector optimized substs accu =
List.fold_left
(fun acc ({trigger_formula=f; trigger_age=age; trigger_dep=dep;
trigger_orig=orig; trigger = tr}, subst_list) ->
List.fold_left
(fun (env, acc)
{sbs = sbs;
sty = sty;
gen = g;
goal = b;
s_term_orig = torig;
s_lem_orig = lorig} ->
let lem_name = E.name_of_lemma orig in
let s = sbs, sty in
if debug_fpa () >= 2 then begin
fprintf fmt "[IC] try to extend synt sbt %a of ax %a@."
(Symbols.Map.print E.print) sbs E.print orig;
end;
match tr.E.guard with
| Some _ -> assert false
| None when tr.E.semantic == [] && not do_syntactic_matching ->
env, acc
| None when not (terms_linear_dep env torig) ->
if debug_fpa () >= 2 then
fprintf fmt "semantic matching failed(1)@.";
env, acc
| None ->
match semantic_matching lem_name tr s env uf optimized with
| env, None ->
if debug_fpa () >= 2 then
fprintf fmt "semantic matching failed(2)@.";
env, acc
| env, Some sbs ->
if debug_fpa () >= 2 then
fprintf fmt "semantic matching succeeded:@.%a@."
(Symbols.Map.print E.print) (fst sbs);
let nf = E.apply_subst sbs f in
let accepted = selector nf orig in
record_this_instance nf accepted lorig;
if accepted then begin
let hyp =
List.map (fun f -> E.apply_subst sbs f) tr.E.hyp
in
let p =
{ E.ff = nf;
origin_name = E.name_of_lemma lorig;
trigger_depth = tr.E.t_depth;
gdist = -1;
hdist = -1;
nb_reductions = 0;
age = 1+(max g age);
mf = true;
gf = b;
lem = Some lorig;
from_terms = torig;
theory_elim = false;
}
in
profile_produced_terms menv lorig nf s tr.E.content;
let dep =
if not (Options.unsat_core() || Options.profiling())
then
dep
else Ex.union dep (Ex.singleton (Ex.Dep lorig))
in
env, (hyp, p, dep) :: acc
end
else
env, acc
) acc subst_list
) accu substs
let syntactic_matching menv env uf _selector =
let mconf =
{Util.nb_triggers = nb_triggers ();
no_ematching = no_Ematching();
triggers_var = triggers_var ();
use_cs = false;
backward = Util.Normal;
greedy = greedy ();
}
in
let synt_match =
ME.fold
(fun f (_th_ax, dep) accu ->
let forms = ME.singleton f (0 , dep) in
let menv = EM.add_triggers mconf menv forms in
let res = EM.query mconf menv uf in
if debug_fpa () >= 2 then begin
let cpt = ref 0 in
List.iter (fun (_, l) -> List.iter (fun _ -> incr cpt) l) res;
fprintf fmt "syntactic matching of Ax %s: got %d substs@."
(E.name_of_lemma f) !cpt
end;
res:: accu
)env.th_axioms []
in
{env with syntactic_matching = synt_match}
let instantiate ~do_syntactic_matching match_terms env uf selector =
if debug_fpa () >= 2 then fprintf fmt "entering IC.instantiate@.";
let optimized = ref (SP.empty) in
let t_infos, t_subterms = match_terms in
let menv = EM.make ~max_t_depth:100 t_infos t_subterms [] in
let env =
if not do_syntactic_matching then env
else syntactic_matching menv env uf selector
in
let env, insts =
List.fold_left
(fun accu substs ->
new_facts_for_axiom
~do_syntactic_matching menv uf selector optimized substs accu
)(env, []) env.syntactic_matching
in
if debug_fpa () >= 2 then
fprintf fmt "IC.instantiate: %d insts generated@." (List.length insts);
env, insts
let separate_semantic_triggers =
let not_theory_const = Hstring.make "not_theory_constant" in
let is_theory_const = Hstring.make "is_theory_constant" in
let linear_dep = Hstring.make "linear_dependency" in
fun th_form ->
let { E.user_trs; _ } as q =
match E.form_view th_form with
| E.Lemma q -> q
| E.Unit _ | E.Clause _ | E.Literal _ | E.Skolem _
| E.Let _ | E.Iff _ | E.Xor _ | E.Not_a_form -> assert false
in
let r_triggers =
List.rev_map
(fun tr ->
assert (tr.E.semantic == []);
let syn, sem =
List.fold_left
(fun (syn, sem) t ->
match E.term_view t with
| E.Not_a_term _ -> assert false
| E.Term { E.f = Symbols.In (lb, ub); xs = [x]; _ } ->
syn, (E.Interval (x, lb, ub)) :: sem
| E.Term { E.f = Symbols.MapsTo x; xs = [t]; _ } ->
syn, (E.MapsTo (x, t)) :: sem
| E.Term { E.f = Sy.Name (hs,_); xs = [x]; _ }
when Hstring.equal hs not_theory_const ->
syn, (E.NotTheoryConst x) :: sem
| E.Term { E.f = Sy.Name (hs,_); xs = [x]; _ }
when Hstring.equal hs is_theory_const ->
syn, (E.IsTheoryConst x) :: sem
| E.Term { E.f = Sy.Name (hs,_); xs = [x;y]; _ }
when Hstring.equal hs linear_dep ->
syn, (E.LinearDependency(x,y)) :: sem
| _ -> t::syn, sem
)([], []) (List.rev tr.E.content)
in
{tr with E.content = syn; semantic = sem}
)user_trs
in
E.mk_forall
q.E.name q.E.loc q.E.binders (List.rev r_triggers) q.E.main
(E.id th_form) ~toplevel:true ~decl_kind:E.Dtheory
let assume_th_elt t th_elt dep =
let { Expr.axiom_kind; ax_form; th_name; extends; _ } = th_elt in
let kd_str =
if axiom_kind == Util.Propagator then "Th propagator" else "Th CS"
in
match extends with
| Util.NIA | Util.NRA | Util.FPA ->
let th_form = separate_semantic_triggers ax_form in
let th_elt = {th_elt with Expr.ax_form} in
if debug_fpa () >= 2 then
fprintf fmt "[IC][Theory %s][%s] %a@."
th_name kd_str E.print th_form;
assert (not (ME.mem th_form t.th_axioms));
{t with th_axioms = ME.add th_form (th_elt, dep) t.th_axioms}
| _ -> t
let instantiate ~do_syntactic_matching env uf selector =
if Options.timers() then
try
Timers.exec_timer_start Timers.M_Arith Timers.F_instantiate;
let res = instantiate ~do_syntactic_matching env uf selector in
Timers.exec_timer_pause Timers.M_Arith Timers.F_instantiate;
res
with e ->
Timers.exec_timer_pause Timers.M_Arith Timers.F_instantiate;
raise e
else instantiate ~do_syntactic_matching env uf selector