diff options
Diffstat (limited to 'plugins/micromega/certificate.ml')
| -rw-r--r-- | plugins/micromega/certificate.ml | 1565 |
1 files changed, 812 insertions, 753 deletions
diff --git a/plugins/micromega/certificate.ml b/plugins/micromega/certificate.ml index a5fceb56..459c72f9 100644 --- a/plugins/micromega/certificate.ml +++ b/plugins/micromega/certificate.ml @@ -63,82 +63,82 @@ let r_spec = z_spec let dev_form n_spec p = let rec dev_form p = match p with - | Mc.PEc z -> Poly.constant (n_spec.number_to_num z) - | Mc.PEX v -> Poly.variable (C2Ml.positive v) - | Mc.PEmul(p1,p2) -> - let p1 = dev_form p1 in - let p2 = dev_form p2 in - Poly.product p1 p2 - | Mc.PEadd(p1,p2) -> Poly.addition (dev_form p1) (dev_form p2) - | Mc.PEopp p -> Poly.uminus (dev_form p) - | Mc.PEsub(p1,p2) -> Poly.addition (dev_form p1) (Poly.uminus (dev_form p2)) - | Mc.PEpow(p,n) -> - let p = dev_form p in - let n = C2Ml.n n in - let rec pow n = - if Int.equal n 0 - then Poly.constant (n_spec.number_to_num n_spec.unit) - else Poly.product p (pow (n-1)) in - pow n in - dev_form p + | Mc.PEc z -> Poly.constant (n_spec.number_to_num z) + | Mc.PEX v -> Poly.variable (C2Ml.positive v) + | Mc.PEmul(p1,p2) -> + let p1 = dev_form p1 in + let p2 = dev_form p2 in + Poly.product p1 p2 + | Mc.PEadd(p1,p2) -> Poly.addition (dev_form p1) (dev_form p2) + | Mc.PEopp p -> Poly.uminus (dev_form p) + | Mc.PEsub(p1,p2) -> Poly.addition (dev_form p1) (Poly.uminus (dev_form p2)) + | Mc.PEpow(p,n) -> + let p = dev_form p in + let n = C2Ml.n n in + let rec pow n = + if Int.equal n 0 + then Poly.constant (n_spec.number_to_num n_spec.unit) + else Poly.product p (pow (n-1)) in + pow n in + dev_form p let monomial_to_polynomial mn = Monomial.fold (fun v i acc -> - let v = Ml2C.positive v in - let mn = if Int.equal i 1 then Mc.PEX v else Mc.PEpow (Mc.PEX v ,Ml2C.n i) in - if Pervasives.(=) acc (Mc.PEc (Mc.Zpos Mc.XH)) (** FIXME *) - then mn - else Mc.PEmul(mn,acc)) - mn - (Mc.PEc (Mc.Zpos Mc.XH)) + let v = Ml2C.positive v in + let mn = if Int.equal i 1 then Mc.PEX v else Mc.PEpow (Mc.PEX v ,Ml2C.n i) in + if Pervasives.(=) acc (Mc.PEc (Mc.Zpos Mc.XH)) (** FIXME *) + then mn + else Mc.PEmul(mn,acc)) + mn + (Mc.PEc (Mc.Zpos Mc.XH)) let list_to_polynomial vars l = assert (List.for_all (fun x -> ceiling_num x =/ x) l); let var x = monomial_to_polynomial (List.nth vars x) in - + let rec xtopoly p i = function | [] -> p | c::l -> if c =/ (Int 0) then xtopoly p (i+1) l - else let c = Mc.PEc (Ml2C.bigint (numerator c)) in - let mn = - if Pervasives.(=) c (Mc.PEc (Mc.Zpos Mc.XH)) - then var i - else Mc.PEmul (c,var i) in - let p' = if Pervasives.(=) p (Mc.PEc Mc.Z0) then mn else - Mc.PEadd (mn, p) in - xtopoly p' (i+1) l in - - xtopoly (Mc.PEc Mc.Z0) 0 l + else let c = Mc.PEc (Ml2C.bigint (numerator c)) in + let mn = + if Pervasives.(=) c (Mc.PEc (Mc.Zpos Mc.XH)) + then var i + else Mc.PEmul (c,var i) in + let p' = if Pervasives.(=) p (Mc.PEc Mc.Z0) then mn else + Mc.PEadd (mn, p) in + xtopoly p' (i+1) l in + + xtopoly (Mc.PEc Mc.Z0) 0 l let rec fixpoint f x = let y' = f x in - if Pervasives.(=) y' x then y' - else fixpoint f y' + if Pervasives.(=) y' x then y' + else fixpoint f y' let rec_simpl_cone n_spec e = let simpl_cone = Mc.simpl_cone n_spec.zero n_spec.unit n_spec.mult n_spec.eqb in let rec rec_simpl_cone = function - | Mc.PsatzMulE(t1, t2) -> - simpl_cone (Mc.PsatzMulE (rec_simpl_cone t1, rec_simpl_cone t2)) - | Mc.PsatzAdd(t1,t2) -> - simpl_cone (Mc.PsatzAdd (rec_simpl_cone t1, rec_simpl_cone t2)) - | x -> simpl_cone x in - rec_simpl_cone e - - + | Mc.PsatzMulE(t1, t2) -> + simpl_cone (Mc.PsatzMulE (rec_simpl_cone t1, rec_simpl_cone t2)) + | Mc.PsatzAdd(t1,t2) -> + simpl_cone (Mc.PsatzAdd (rec_simpl_cone t1, rec_simpl_cone t2)) + | x -> simpl_cone x in + rec_simpl_cone e + + let simplify_cone n_spec c = fixpoint (rec_simpl_cone n_spec) c type cone_prod = - Const of cone - | Ideal of cone *cone - | Mult of cone * cone - | Other of cone + Const of cone +| Ideal of cone *cone +| Mult of cone * cone +| Other of cone and cone = Mc.zWitness @@ -147,32 +147,32 @@ let factorise_linear_cone c = let rec cone_list c l = match c with - | Mc.PsatzAdd (x,r) -> cone_list r (x::l) - | _ -> c :: l in - + | Mc.PsatzAdd (x,r) -> cone_list r (x::l) + | _ -> c :: l in + let factorise c1 c2 = match c1 , c2 with - | Mc.PsatzMulC(x,y) , Mc.PsatzMulC(x',y') -> - if Pervasives.(=) x x' then Some (Mc.PsatzMulC(x, Mc.PsatzAdd(y,y'))) else None - | Mc.PsatzMulE(x,y) , Mc.PsatzMulE(x',y') -> - if Pervasives.(=) x x' then Some (Mc.PsatzMulE(x, Mc.PsatzAdd(y,y'))) else None - | _ -> None in - + | Mc.PsatzMulC(x,y) , Mc.PsatzMulC(x',y') -> + if Pervasives.(=) x x' then Some (Mc.PsatzMulC(x, Mc.PsatzAdd(y,y'))) else None + | Mc.PsatzMulE(x,y) , Mc.PsatzMulE(x',y') -> + if Pervasives.(=) x x' then Some (Mc.PsatzMulE(x, Mc.PsatzAdd(y,y'))) else None + | _ -> None in + let rec rebuild_cone l pending = match l with - | [] -> (match pending with - | None -> Mc.PsatzZ - | Some p -> p - ) - | e::l -> - (match pending with - | None -> rebuild_cone l (Some e) - | Some p -> (match factorise p e with - | None -> Mc.PsatzAdd(p, rebuild_cone l (Some e)) - | Some f -> rebuild_cone l (Some f) ) - ) in + | [] -> (match pending with + | None -> Mc.PsatzZ + | Some p -> p + ) + | e::l -> + (match pending with + | None -> rebuild_cone l (Some e) + | Some p -> (match factorise p e with + | None -> Mc.PsatzAdd(p, rebuild_cone l (Some e)) + | Some f -> rebuild_cone l (Some f) ) + ) in - (rebuild_cone (List.sort Pervasives.compare (cone_list c [])) None) + (rebuild_cone (List.sort Pervasives.compare (cone_list c [])) None) @@ -199,28 +199,28 @@ open Mfourier let constrain_monomial mn l = let coeffs = List.fold_left (fun acc p -> (Poly.get mn p)::acc) [] l in - if Pervasives.(=) mn Monomial.const - then - { coeffs = Vect.from_list ((Big_int unit_big_int):: (List.rev coeffs)) ; - op = Eq ; - cst = Big_int zero_big_int } - else - { coeffs = Vect.from_list ((Big_int zero_big_int):: (List.rev coeffs)) ; - op = Eq ; - cst = Big_int zero_big_int } + if Pervasives.(=) mn Monomial.const + then + { coeffs = Vect.from_list ((Big_int unit_big_int):: (List.rev coeffs)) ; + op = Eq ; + cst = Big_int zero_big_int } + else + { coeffs = Vect.from_list ((Big_int zero_big_int):: (List.rev coeffs)) ; + op = Eq ; + cst = Big_int zero_big_int } - + let positivity l = let rec xpositivity i l = match l with - | [] -> [] - | (_,Mc.Equal)::l -> xpositivity (i+1) l - | (_,_)::l -> - {coeffs = Vect.update (i+1) (fun _ -> Int 1) Vect.null ; - op = Ge ; - cst = Int 0 } :: (xpositivity (i+1) l) + | [] -> [] + | (_,Mc.Equal)::l -> xpositivity (i+1) l + | (_,_)::l -> + {coeffs = Vect.update (i+1) (fun _ -> Int 1) Vect.null ; + op = Ge ; + cst = Int 0 } :: (xpositivity (i+1) l) in - xpositivity 0 l + xpositivity 0 l let string_of_op = function @@ -241,23 +241,23 @@ let build_linear_system l = let monomials = List.fold_left (fun acc p -> - Poly.fold (fun m _ acc -> MonSet.add m acc) p acc) - (MonSet.singleton Monomial.const) l' + Poly.fold (fun m _ acc -> MonSet.add m acc) p acc) + (MonSet.singleton Monomial.const) l' in (* For each monomial, compute a constraint *) let s0 = MonSet.fold (fun mn res -> (constrain_monomial mn l')::res) monomials [] in - (* I need at least something strictly positive *) + (* I need at least something strictly positive *) let strict = { coeffs = Vect.from_list ((Big_int unit_big_int):: (List.map (fun (x,y) -> - match y with Mc.Strict -> - Big_int unit_big_int - | _ -> Big_int zero_big_int) l)); + match y with Mc.Strict -> + Big_int unit_big_int + | _ -> Big_int zero_big_int) l)); op = Ge ; cst = Big_int unit_big_int } in (* Add the positivity constraint *) - {coeffs = Vect.from_list ([Big_int unit_big_int]) ; - op = Ge ; - cst = Big_int zero_big_int}::(strict::(positivity l)@s0) + {coeffs = Vect.from_list ([Big_int unit_big_int]) ; + op = Ge ; + cst = Big_int zero_big_int}::(strict::(positivity l)@s0) let big_int_to_z = Ml2C.bigint @@ -266,32 +266,32 @@ let big_int_to_z = Ml2C.bigint -- at a lower layer, certificates are using nums... *) let make_certificate n_spec (cert,li) = let bint_to_cst = n_spec.bigint_to_number in - match cert with - | [] -> failwith "empty_certificate" - | e::cert' -> -(* let cst = match compare_big_int e zero_big_int with - | 0 -> Mc.PsatzZ - | 1 -> Mc.PsatzC (bint_to_cst e) - | _ -> failwith "positivity error" - in *) - let rec scalar_product cert l = - match cert with - | [] -> Mc.PsatzZ - | c::cert -> - match l with - | [] -> failwith "make_certificate(1)" - | i::l -> - let r = scalar_product cert l in - match compare_big_int c zero_big_int with - | -1 -> Mc.PsatzAdd ( - Mc.PsatzMulC (Mc.Pc ( bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)), - r) - | 0 -> r - | _ -> Mc.PsatzAdd ( - Mc.PsatzMulE (Mc.PsatzC (bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)), - r) in - (factorise_linear_cone - (simplify_cone n_spec (scalar_product cert' li))) + match cert with + | [] -> failwith "empty_certificate" + | e::cert' -> + (* let cst = match compare_big_int e zero_big_int with + | 0 -> Mc.PsatzZ + | 1 -> Mc.PsatzC (bint_to_cst e) + | _ -> failwith "positivity error" + in *) + let rec scalar_product cert l = + match cert with + | [] -> Mc.PsatzZ + | c::cert -> + match l with + | [] -> failwith "make_certificate(1)" + | i::l -> + let r = scalar_product cert l in + match compare_big_int c zero_big_int with + | -1 -> Mc.PsatzAdd ( + Mc.PsatzMulC (Mc.Pc ( bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)), + r) + | 0 -> r + | _ -> Mc.PsatzAdd ( + Mc.PsatzMulE (Mc.PsatzC (bint_to_cst c), Mc.PsatzIn (Ml2C.nat i)), + r) in + (factorise_linear_cone + (simplify_cone n_spec (scalar_product cert' li))) exception Found of Monomial.t @@ -301,92 +301,155 @@ exception Strict module MonMap = Map.Make(Monomial) let primal l = - let vr = ref 0 in - - let vect_of_poly map p = - Poly.fold (fun mn vl (map,vect) -> - if Pervasives.(=) mn Monomial.const - then (map,vect) - else - let (mn,m) = try (MonMap.find mn map,map) with Not_found -> let res = (!vr, MonMap.add mn !vr map) in incr vr ; res in - (m,if Int.equal (sign_num vl) 0 then vect else (mn,vl)::vect)) p (map,[]) in - - let op_op = function Mc.NonStrict -> Ge |Mc.Equal -> Eq | _ -> raise Strict in + let vr = ref 0 in + + let vect_of_poly map p = + Poly.fold (fun mn vl (map,vect) -> + if Pervasives.(=) mn Monomial.const + then (map,vect) + else + let (mn,m) = try (MonMap.find mn map,map) with Not_found -> let res = (!vr, MonMap.add mn !vr map) in incr vr ; res in + (m,if Int.equal (sign_num vl) 0 then vect else (mn,vl)::vect)) p (map,[]) in + + let op_op = function Mc.NonStrict -> Ge |Mc.Equal -> Eq | _ -> raise Strict in - let cmp x y = Int.compare (fst x) (fst y) in + let cmp x y = Int.compare (fst x) (fst y) in - snd (List.fold_right (fun (p,op) (map,l) -> - let (mp,vect) = vect_of_poly map p in - let cstr = {coeffs = List.sort cmp vect; op = op_op op ; cst = minus_num (Poly.get Monomial.const p)} in + snd (List.fold_right (fun (p,op) (map,l) -> + let (mp,vect) = vect_of_poly map p in + let cstr = {coeffs = List.sort cmp vect; op = op_op op ; cst = minus_num (Poly.get Monomial.const p)} in - (mp,cstr::l)) l (MonMap.empty,[])) + (mp,cstr::l)) l (MonMap.empty,[])) let dual_raw_certificate (l: (Poly.t * Mc.op1) list) = -(* List.iter (fun (p,op) -> Printf.fprintf stdout "%a %s 0\n" Poly.pp p (string_of_op op) ) l ; *) - + (* List.iter (fun (p,op) -> Printf.fprintf stdout "%a %s 0\n" Poly.pp p (string_of_op op) ) l ; *) + let sys = build_linear_system l in - try - match Fourier.find_point sys with - | Inr _ -> None - | Inl cert -> Some (rats_to_ints (Vect.to_list cert)) - (* should not use rats_to_ints *) - with x when Errors.noncritical x -> - if debug - then (Printf.printf "raw certificate %s" (Printexc.to_string x); - flush stdout) ; - None + try + match Fourier.find_point sys with + | Inr _ -> None + | Inl cert -> Some (rats_to_ints (Vect.to_list cert)) + (* should not use rats_to_ints *) + with x when CErrors.noncritical x -> + if debug + then (Printf.printf "raw certificate %s" (Printexc.to_string x); + flush stdout) ; + None let raw_certificate l = - try - let p = primal l in - match Fourier.find_point p with - | Inr prf -> - if debug then Printf.printf "AProof : %a\n" pp_proof prf ; - let cert = List.map (fun (x,n) -> x+1,n) (fst (List.hd (Proof.mk_proof p prf))) in - if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; - Some (rats_to_ints (Vect.to_list cert)) - | Inl _ -> None - with Strict -> + try + let p = primal l in + match Fourier.find_point p with + | Inr prf -> + if debug then Printf.printf "AProof : %a\n" pp_proof prf ; + let cert = List.map (fun (x,n) -> x+1,n) (fst (List.hd (Proof.mk_proof p prf))) in + if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; + Some (rats_to_ints (Vect.to_list cert)) + | Inl _ -> None + with Strict -> (* Fourier elimination should handle > *) - dual_raw_certificate l + dual_raw_certificate l let simple_linear_prover l = let (lc,li) = List.split l in - match raw_certificate lc with - | None -> None (* No certificate *) - | Some cert -> Some (cert,li) - + match raw_certificate lc with + | None -> None (* No certificate *) + | Some cert -> Some (cert,li) + let linear_prover n_spec l = - let build_system n_spec l = - let li = List.combine l (interval 0 (List.length l -1)) in - let (l1,l') = List.partition - (fun (x,_) -> if Pervasives.(=) (snd x) Mc.NonEqual then true else false) li in - List.map - (fun ((x,y),i) -> match y with - Mc.NonEqual -> failwith "cannot happen" - | y -> ((dev_form n_spec x, y),i)) l' in - let l' = build_system n_spec l in - simple_linear_prover (*n_spec*) l' + let build_system n_spec l = + let li = List.combine l (interval 0 (List.length l -1)) in + let (l1,l') = List.partition + (fun (x,_) -> if Pervasives.(=) (snd x) Mc.NonEqual then true else false) li in + List.map + (fun ((x,y),i) -> match y with + Mc.NonEqual -> failwith "cannot happen" + | y -> ((dev_form n_spec x, y),i)) l' in + let l' = build_system n_spec l in + simple_linear_prover (*n_spec*) l' let linear_prover n_spec l = try linear_prover n_spec l - with x when Errors.noncritical x -> - (print_string (Printexc.to_string x); None) + with x when CErrors.noncritical x -> + (print_string (Printexc.to_string x); None) + +let compute_max_nb_cstr l d = + let len = List.length l in + max len (max d (len * d)) + +let linear_prover_with_cert prfdepth spec l = + max_nb_cstr := compute_max_nb_cstr l prfdepth ; + match linear_prover spec l with + | None -> None + | Some cert -> Some (make_certificate spec cert) + +let nlinear_prover prfdepth (sys: (Mc.q Mc.pExpr * Mc.op1) list) = + LinPoly.MonT.clear (); + max_nb_cstr := compute_max_nb_cstr sys prfdepth ; + (* Assign a proof to the initial hypotheses *) + let sys = mapi (fun c i -> (c,Mc.PsatzIn (Ml2C.nat i))) sys in + + + (* Add all the product of hypotheses *) + let prod = all_pairs (fun ((c,o),p) ((c',o'),p') -> + ((Mc.PEmul(c,c') , Mc.opMult o o') , Mc.PsatzMulE(p,p'))) sys in + + (* Only filter those have a meaning *) + let prod = List.fold_left (fun l ((c,o),p) -> + match o with + | None -> l + | Some o -> ((c,o),p) :: l) [] prod in + + let sys = sys @ prod in + + let square = + (* Collect the squares and state that they are positive *) + let pols = List.map (fun ((p,_),_) -> dev_form q_spec p) sys in + let square = + List.fold_left (fun acc p -> + Poly.fold + (fun m _ acc -> + match Monomial.sqrt m with + | None -> acc + | Some s -> MonMap.add s m acc) p acc) MonMap.empty pols in -let linear_prover_with_cert spec l = - match linear_prover spec l with - | None -> None - | Some cert -> Some (make_certificate spec cert) + let pol_of_mon m = + Monomial.fold (fun x v p -> Mc.PEmul(Mc.PEpow(Mc.PEX(Ml2C.positive x),Ml2C.n v),p)) m (Mc.PEc q_spec.unit) in + + let norm0 = + Mc.norm q_spec.zero q_spec.unit Mc.qplus Mc.qmult Mc.qminus Mc.qopp Mc.qeq_bool in + + + MonMap.fold (fun s m acc -> ((pol_of_mon m , Mc.NonStrict), Mc.PsatzSquare(norm0 (pol_of_mon s)))::acc) square [] in + + let sys = sys @ square in + (* Call the linear prover without the proofs *) + let sys_no_prf = List.map fst sys in + + match linear_prover q_spec sys_no_prf with + | None -> None + | Some cert -> + let cert = make_certificate q_spec cert in + let rec map_psatz = function + | Mc.PsatzIn n -> snd (List.nth sys (C2Ml.nat n)) + | Mc.PsatzSquare c -> Mc.PsatzSquare c + | Mc.PsatzMulC(c,p) -> Mc.PsatzMulC(c, map_psatz p) + | Mc.PsatzMulE(p1,p2) -> Mc.PsatzMulE(map_psatz p1,map_psatz p2) + | Mc.PsatzAdd(p1,p2) -> Mc.PsatzAdd(map_psatz p1,map_psatz p2) + | Mc.PsatzC c -> Mc.PsatzC c + | Mc.PsatzZ -> Mc.PsatzZ in + Some (map_psatz cert) + let make_linear_system l = @@ -395,11 +458,11 @@ let make_linear_system l = (Poly.constant (Int 0)) l' in let monomials = Poly.fold (fun mn _ l -> if Pervasives.(=) mn Monomial.const then l else mn::l) monomials [] in - (List.map (fun (c,op) -> - {coeffs = Vect.from_list (List.map (fun mn -> (Poly.get mn c)) monomials) ; - op = op ; - cst = minus_num ( (Poly.get Monomial.const c))}) l - ,monomials) + (List.map (fun (c,op) -> + {coeffs = Vect.from_list (List.map (fun mn -> (Poly.get mn c)) monomials) ; + op = op ; + cst = minus_num ( (Poly.get Monomial.const c))}) l + ,monomials) let pplus x y = Mc.PEadd(x,y) @@ -413,7 +476,7 @@ let rec mem p x l = let rec remove_assoc p x l = match l with [] -> [] | e::l -> if p x (fst e) then - remove_assoc p x l else e::(remove_assoc p x l) + remove_assoc p x l else e::(remove_assoc p x l) let eq x y = Int.equal (Vect.compare x y) 0 @@ -424,39 +487,39 @@ let remove e l = List.fold_left (fun l x -> if eq x e then l else x::l) [] l only searching for naive cutting planes *) let develop_constraint z_spec (e,k) = - match k with - | Mc.NonStrict -> (dev_form z_spec e , Ge) - | Mc.Equal -> (dev_form z_spec e , Eq) - | _ -> assert false + match k with + | Mc.NonStrict -> (dev_form z_spec e , Ge) + | Mc.Equal -> (dev_form z_spec e , Eq) + | _ -> assert false let op_of_op_compat = function - | Ge -> Mc.NonStrict - | Eq -> Mc.Equal + | Ge -> Mc.NonStrict + | Eq -> Mc.Equal let integer_vector coeffs = - let vars , coeffs = List.split coeffs in - List.combine vars (List.map (fun x -> Big_int x) (rats_to_ints coeffs)) + let vars , coeffs = List.split coeffs in + List.combine vars (List.map (fun x -> Big_int x) (rats_to_ints coeffs)) let integer_cstr {coeffs = coeffs ; op = op ; cst = cst } = - let vars , coeffs = List.split coeffs in - match rats_to_ints (cst::coeffs) with - | cst :: coeffs -> - { - coeffs = List.combine vars (List.map (fun x -> Big_int x) coeffs) ; - op = op ; cst = Big_int cst} - | _ -> assert false - + let vars , coeffs = List.split coeffs in + match rats_to_ints (cst::coeffs) with + | cst :: coeffs -> + { + coeffs = List.combine vars (List.map (fun x -> Big_int x) coeffs) ; + op = op ; cst = Big_int cst} + | _ -> assert false + let pexpr_of_cstr_compat var cstr = - let {coeffs = coeffs ; op = op ; cst = cst } = integer_cstr cstr in - try - let expr = list_to_polynomial var (Vect.to_list coeffs) in - let d = Ml2C.bigint (denominator cst) in - let n = Ml2C.bigint (numerator cst) in - (pplus (pmult (pconst d) expr) (popp (pconst n)), op_of_op_compat op) - with Failure _ -> failwith "pexpr_of_cstr_compat" + let {coeffs = coeffs ; op = op ; cst = cst } = integer_cstr cstr in + try + let expr = list_to_polynomial var (Vect.to_list coeffs) in + let d = Ml2C.bigint (denominator cst) in + let n = Ml2C.bigint (numerator cst) in + (pplus (pmult (pconst d) expr) (popp (pconst n)), op_of_op_compat op) + with Failure _ -> failwith "pexpr_of_cstr_compat" @@ -465,41 +528,41 @@ open Sos_types let rec scale_term t = match t with - | Zero -> unit_big_int , Zero - | Const n -> (denominator n) , Const (Big_int (numerator n)) - | Var n -> unit_big_int , Var n - | Inv _ -> failwith "scale_term : not implemented" - | Opp t -> let s, t = scale_term t in s, Opp t - | Add(t1,t2) -> let s1,y1 = scale_term t1 and s2,y2 = scale_term t2 in - let g = gcd_big_int s1 s2 in - let s1' = div_big_int s1 g in - let s2' = div_big_int s2 g in - let e = mult_big_int g (mult_big_int s1' s2') in - if Int.equal (compare_big_int e unit_big_int) 0 - then (unit_big_int, Add (y1,y2)) - else e, Add (Mul(Const (Big_int s2'), y1), - Mul (Const (Big_int s1'), y2)) - | Sub _ -> failwith "scale term: not implemented" - | Mul(y,z) -> let s1,y1 = scale_term y and s2,y2 = scale_term z in - mult_big_int s1 s2 , Mul (y1, y2) - | Pow(t,n) -> let s,t = scale_term t in - power_big_int_positive_int s n , Pow(t,n) - | _ -> failwith "scale_term : not implemented" + | Zero -> unit_big_int , Zero + | Const n -> (denominator n) , Const (Big_int (numerator n)) + | Var n -> unit_big_int , Var n + | Inv _ -> failwith "scale_term : not implemented" + | Opp t -> let s, t = scale_term t in s, Opp t + | Add(t1,t2) -> let s1,y1 = scale_term t1 and s2,y2 = scale_term t2 in + let g = gcd_big_int s1 s2 in + let s1' = div_big_int s1 g in + let s2' = div_big_int s2 g in + let e = mult_big_int g (mult_big_int s1' s2') in + if Int.equal (compare_big_int e unit_big_int) 0 + then (unit_big_int, Add (y1,y2)) + else e, Add (Mul(Const (Big_int s2'), y1), + Mul (Const (Big_int s1'), y2)) + | Sub _ -> failwith "scale term: not implemented" + | Mul(y,z) -> let s1,y1 = scale_term y and s2,y2 = scale_term z in + mult_big_int s1 s2 , Mul (y1, y2) + | Pow(t,n) -> let s,t = scale_term t in + power_big_int_positive_int s n , Pow(t,n) + | _ -> failwith "scale_term : not implemented" let scale_term t = let (s,t') = scale_term t in - s,t' + s,t' let get_index_of_ith_match f i l = let rec get j res l = match l with - | [] -> failwith "bad index" - | e::l -> if f e - then - (if Int.equal j i then res else get (j+1) (res+1) l ) - else get j (res+1) l in - get 0 0 l + | [] -> failwith "bad index" + | e::l -> if f e + then + (if Int.equal j i then res else get (j+1) (res+1) l ) + else get j (res+1) l in + get 0 0 l let rec scale_certificate pos = match pos with @@ -511,97 +574,97 @@ let rec scale_certificate pos = match pos with | Rational_le n -> (denominator n) , Rational_le (Big_int (numerator n)) | Rational_lt n -> (denominator n) , Rational_lt (Big_int (numerator n)) | Square t -> let s,t' = scale_term t in - mult_big_int s s , Square t' + mult_big_int s s , Square t' | Eqmul (t, y) -> let s1,y1 = scale_term t and s2,y2 = scale_certificate y in - mult_big_int s1 s2 , Eqmul (y1,y2) + mult_big_int s1 s2 , Eqmul (y1,y2) | Sum (y, z) -> let s1,y1 = scale_certificate y - and s2,y2 = scale_certificate z in - let g = gcd_big_int s1 s2 in - let s1' = div_big_int s1 g in - let s2' = div_big_int s2 g in - mult_big_int g (mult_big_int s1' s2'), - Sum (Product(Rational_le (Big_int s2'), y1), - Product (Rational_le (Big_int s1'), y2)) + and s2,y2 = scale_certificate z in + let g = gcd_big_int s1 s2 in + let s1' = div_big_int s1 g in + let s2' = div_big_int s2 g in + mult_big_int g (mult_big_int s1' s2'), + Sum (Product(Rational_le (Big_int s2'), y1), + Product (Rational_le (Big_int s1'), y2)) | Product (y, z) -> - let s1,y1 = scale_certificate y and s2,y2 = scale_certificate z in - mult_big_int s1 s2 , Product (y1,y2) + let s1,y1 = scale_certificate y and s2,y2 = scale_certificate z in + mult_big_int s1 s2 , Product (y1,y2) open Micromega - let rec term_to_q_expr = function - | Const n -> PEc (Ml2C.q n) - | Zero -> PEc ( Ml2C.q (Int 0)) - | Var s -> PEX (Ml2C.index - (int_of_string (String.sub s 1 (String.length s - 1)))) - | Mul(p1,p2) -> PEmul(term_to_q_expr p1, term_to_q_expr p2) - | Add(p1,p2) -> PEadd(term_to_q_expr p1, term_to_q_expr p2) - | Opp p -> PEopp (term_to_q_expr p) - | Pow(t,n) -> PEpow (term_to_q_expr t,Ml2C.n n) - | Sub(t1,t2) -> PEsub (term_to_q_expr t1, term_to_q_expr t2) - | _ -> failwith "term_to_q_expr: not implemented" - - let term_to_q_pol e = Mc.norm_aux (Ml2C.q (Int 0)) (Ml2C.q (Int 1)) Mc.qplus Mc.qmult Mc.qminus Mc.qopp Mc.qeq_bool (term_to_q_expr e) - - - let rec product l = - match l with - | [] -> Mc.PsatzZ - | [i] -> Mc.PsatzIn (Ml2C.nat i) - | i ::l -> Mc.PsatzMulE(Mc.PsatzIn (Ml2C.nat i), product l) +let rec term_to_q_expr = function + | Const n -> PEc (Ml2C.q n) + | Zero -> PEc ( Ml2C.q (Int 0)) + | Var s -> PEX (Ml2C.index + (int_of_string (String.sub s 1 (String.length s - 1)))) + | Mul(p1,p2) -> PEmul(term_to_q_expr p1, term_to_q_expr p2) + | Add(p1,p2) -> PEadd(term_to_q_expr p1, term_to_q_expr p2) + | Opp p -> PEopp (term_to_q_expr p) + | Pow(t,n) -> PEpow (term_to_q_expr t,Ml2C.n n) + | Sub(t1,t2) -> PEsub (term_to_q_expr t1, term_to_q_expr t2) + | _ -> failwith "term_to_q_expr: not implemented" + +let term_to_q_pol e = Mc.norm_aux (Ml2C.q (Int 0)) (Ml2C.q (Int 1)) Mc.qplus Mc.qmult Mc.qminus Mc.qopp Mc.qeq_bool (term_to_q_expr e) + + +let rec product l = + match l with + | [] -> Mc.PsatzZ + | [i] -> Mc.PsatzIn (Ml2C.nat i) + | i ::l -> Mc.PsatzMulE(Mc.PsatzIn (Ml2C.nat i), product l) let q_cert_of_pos pos = let rec _cert_of_pos = function - Axiom_eq i -> Mc.PsatzIn (Ml2C.nat i) + Axiom_eq i -> Mc.PsatzIn (Ml2C.nat i) | Axiom_le i -> Mc.PsatzIn (Ml2C.nat i) | Axiom_lt i -> Mc.PsatzIn (Ml2C.nat i) | Monoid l -> product l | Rational_eq n | Rational_le n | Rational_lt n -> - if Int.equal (compare_num n (Int 0)) 0 then Mc.PsatzZ else - Mc.PsatzC (Ml2C.q n) + if Int.equal (compare_num n (Int 0)) 0 then Mc.PsatzZ else + Mc.PsatzC (Ml2C.q n) | Square t -> Mc.PsatzSquare (term_to_q_pol t) | Eqmul (t, y) -> Mc.PsatzMulC(term_to_q_pol t, _cert_of_pos y) | Sum (y, z) -> Mc.PsatzAdd (_cert_of_pos y, _cert_of_pos z) | Product (y, z) -> Mc.PsatzMulE (_cert_of_pos y, _cert_of_pos z) in - simplify_cone q_spec (_cert_of_pos pos) + simplify_cone q_spec (_cert_of_pos pos) - let rec term_to_z_expr = function - | Const n -> PEc (Ml2C.bigint (big_int_of_num n)) - | Zero -> PEc ( Z0) - | Var s -> PEX (Ml2C.index - (int_of_string (String.sub s 1 (String.length s - 1)))) - | Mul(p1,p2) -> PEmul(term_to_z_expr p1, term_to_z_expr p2) - | Add(p1,p2) -> PEadd(term_to_z_expr p1, term_to_z_expr p2) - | Opp p -> PEopp (term_to_z_expr p) - | Pow(t,n) -> PEpow (term_to_z_expr t,Ml2C.n n) - | Sub(t1,t2) -> PEsub (term_to_z_expr t1, term_to_z_expr t2) - | _ -> failwith "term_to_z_expr: not implemented" +let rec term_to_z_expr = function + | Const n -> PEc (Ml2C.bigint (big_int_of_num n)) + | Zero -> PEc ( Z0) + | Var s -> PEX (Ml2C.index + (int_of_string (String.sub s 1 (String.length s - 1)))) + | Mul(p1,p2) -> PEmul(term_to_z_expr p1, term_to_z_expr p2) + | Add(p1,p2) -> PEadd(term_to_z_expr p1, term_to_z_expr p2) + | Opp p -> PEopp (term_to_z_expr p) + | Pow(t,n) -> PEpow (term_to_z_expr t,Ml2C.n n) + | Sub(t1,t2) -> PEsub (term_to_z_expr t1, term_to_z_expr t2) + | _ -> failwith "term_to_z_expr: not implemented" - let term_to_z_pol e = Mc.norm_aux (Ml2C.z 0) (Ml2C.z 1) Mc.Z.add Mc.Z.mul Mc.Z.sub Mc.Z.opp Mc.zeq_bool (term_to_z_expr e) +let term_to_z_pol e = Mc.norm_aux (Ml2C.z 0) (Ml2C.z 1) Mc.Z.add Mc.Z.mul Mc.Z.sub Mc.Z.opp Mc.zeq_bool (term_to_z_expr e) let z_cert_of_pos pos = let s,pos = (scale_certificate pos) in let rec _cert_of_pos = function - Axiom_eq i -> Mc.PsatzIn (Ml2C.nat i) + Axiom_eq i -> Mc.PsatzIn (Ml2C.nat i) | Axiom_le i -> Mc.PsatzIn (Ml2C.nat i) | Axiom_lt i -> Mc.PsatzIn (Ml2C.nat i) | Monoid l -> product l | Rational_eq n | Rational_le n | Rational_lt n -> - if Int.equal (compare_num n (Int 0)) 0 then Mc.PsatzZ else - Mc.PsatzC (Ml2C.bigint (big_int_of_num n)) + if Int.equal (compare_num n (Int 0)) 0 then Mc.PsatzZ else + Mc.PsatzC (Ml2C.bigint (big_int_of_num n)) | Square t -> Mc.PsatzSquare (term_to_z_pol t) | Eqmul (t, y) -> - let is_unit = - match t with - | Const n -> n =/ Int 1 - | _ -> false in - if is_unit - then _cert_of_pos y - else Mc.PsatzMulC(term_to_z_pol t, _cert_of_pos y) + let is_unit = + match t with + | Const n -> n =/ Int 1 + | _ -> false in + if is_unit + then _cert_of_pos y + else Mc.PsatzMulC(term_to_z_pol t, _cert_of_pos y) | Sum (y, z) -> Mc.PsatzAdd (_cert_of_pos y, _cert_of_pos z) | Product (y, z) -> Mc.PsatzMulE (_cert_of_pos y, _cert_of_pos z) in - simplify_cone z_spec (_cert_of_pos pos) + simplify_cone z_spec (_cert_of_pos pos) (** All constraints (initial or derived) have an index and have a justification i.e., proof. Given a constraint, all the coefficients are always integers. @@ -612,116 +675,109 @@ open Num open Big_int open Polynomial -(*module Mc = Micromega*) -(*module Ml2C = Mutils.CamlToCoq -module C2Ml = Mutils.CoqToCaml -*) -let debug = false - - module Env = struct - type t = int list + type t = int list - let id_of_hyp hyp l = - let rec xid_of_hyp i l = - match l with - | [] -> failwith "id_of_hyp" - | hyp'::l -> if Pervasives.(=) hyp hyp' then i else xid_of_hyp (i+1) l in - xid_of_hyp 0 l + let id_of_hyp hyp l = + let rec xid_of_hyp i l = + match l with + | [] -> failwith "id_of_hyp" + | hyp'::l -> if Pervasives.(=) hyp hyp' then i else xid_of_hyp (i+1) l in + xid_of_hyp 0 l end let coq_poly_of_linpol (p,c) = - let pol_of_mon m = - Monomial.fold (fun x v p -> Mc.PEmul(Mc.PEpow(Mc.PEX(Ml2C.positive x),Ml2C.n v),p)) m (Mc.PEc (Mc.Zpos Mc.XH)) in + let pol_of_mon m = + Monomial.fold (fun x v p -> Mc.PEmul(Mc.PEpow(Mc.PEX(Ml2C.positive x),Ml2C.n v),p)) m (Mc.PEc (Mc.Zpos Mc.XH)) in - List.fold_left (fun acc (x,v) -> - let mn = LinPoly.MonT.retrieve x in - Mc.PEadd(Mc.PEmul(Mc.PEc (Ml2C.bigint (numerator v)), pol_of_mon mn),acc)) (Mc.PEc (Ml2C.bigint (numerator c))) p - + List.fold_left (fun acc (x,v) -> + let mn = LinPoly.MonT.retrieve x in + Mc.PEadd(Mc.PEmul(Mc.PEc (Ml2C.bigint (numerator v)), pol_of_mon mn),acc)) (Mc.PEc (Ml2C.bigint (numerator c))) p + let rec cmpl_prf_rule env = function - | Hyp i | Def i -> Mc.PsatzIn (Ml2C.nat (Env.id_of_hyp i env)) - | Cst i -> Mc.PsatzC (Ml2C.bigint i) - | Zero -> Mc.PsatzZ - | MulPrf(p1,p2) -> Mc.PsatzMulE(cmpl_prf_rule env p1, cmpl_prf_rule env p2) - | AddPrf(p1,p2) -> Mc.PsatzAdd(cmpl_prf_rule env p1 , cmpl_prf_rule env p2) - | MulC(lp,p) -> let lp = Mc.norm0 (coq_poly_of_linpol lp) in - Mc.PsatzMulC(lp,cmpl_prf_rule env p) - | Square lp -> Mc.PsatzSquare (Mc.norm0 (coq_poly_of_linpol lp)) - | _ -> failwith "Cuts should already be compiled" - + | Hyp i | Def i -> Mc.PsatzIn (Ml2C.nat (Env.id_of_hyp i env)) + | Cst i -> Mc.PsatzC (Ml2C.bigint i) + | Zero -> Mc.PsatzZ + | MulPrf(p1,p2) -> Mc.PsatzMulE(cmpl_prf_rule env p1, cmpl_prf_rule env p2) + | AddPrf(p1,p2) -> Mc.PsatzAdd(cmpl_prf_rule env p1 , cmpl_prf_rule env p2) + | MulC(lp,p) -> let lp = Mc.norm0 (coq_poly_of_linpol lp) in + Mc.PsatzMulC(lp,cmpl_prf_rule env p) + | Square lp -> Mc.PsatzSquare (Mc.norm0 (coq_poly_of_linpol lp)) + | _ -> failwith "Cuts should already be compiled" + let rec cmpl_proof env = function - | Done -> Mc.DoneProof - | Step(i,p,prf) -> - begin - match p with - | CutPrf p' -> - Mc.CutProof(cmpl_prf_rule env p', cmpl_proof (i::env) prf) - | _ -> Mc.RatProof(cmpl_prf_rule env p,cmpl_proof (i::env) prf) - end - | Enum(i,p1,_,p2,l) -> - Mc.EnumProof(cmpl_prf_rule env p1,cmpl_prf_rule env p2,List.map (cmpl_proof (i::env)) l) + | Done -> Mc.DoneProof + | Step(i,p,prf) -> + begin + match p with + | CutPrf p' -> + Mc.CutProof(cmpl_prf_rule env p', cmpl_proof (i::env) prf) + | _ -> Mc.RatProof(cmpl_prf_rule env p,cmpl_proof (i::env) prf) + end + | Enum(i,p1,_,p2,l) -> + Mc.EnumProof(cmpl_prf_rule env p1,cmpl_prf_rule env p2,List.map (cmpl_proof (i::env)) l) let compile_proof env prf = - let id = 1 + proof_max_id prf in - let _,prf = normalise_proof id prf in - if debug then Printf.fprintf stdout "compiled proof %a\n" output_proof prf; - cmpl_proof env prf + let id = 1 + proof_max_id prf in + let _,prf = normalise_proof id prf in + if debug then Printf.fprintf stdout "compiled proof %a\n" output_proof prf; + cmpl_proof env prf type prf_sys = (cstr_compat * prf_rule) list let xlinear_prover sys = - match Fourier.find_point sys with - | Inr prf -> - if debug then Printf.printf "AProof : %a\n" pp_proof prf ; - let cert = (*List.map (fun (x,n) -> x+1,n)*) (fst (List.hd (Proof.mk_proof sys prf))) in - if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; - Some (rats_to_ints (Vect.to_list cert)) - | Inl _ -> None + match Fourier.find_point sys with + | Inr prf -> + if debug then Printf.printf "AProof : %a\n" pp_proof prf ; + let cert = (*List.map (fun (x,n) -> x+1,n)*) (fst (List.hd (Proof.mk_proof sys prf))) in + if debug then Printf.printf "CProof : %a" Vect.pp_vect cert ; + Some (rats_to_ints (Vect.to_list cert)) + | Inl _ -> None let output_num o n = output_string o (string_of_num n) let output_bigint o n = output_string o (string_of_big_int n) let proof_of_farkas prf cert = -(* Printf.printf "\nproof_of_farkas %a , %a \n" (pp_list output_prf_rule) prf (pp_list output_bigint) cert ; *) - let rec mk_farkas acc prf cert = - match prf, cert with - | _ , [] -> acc - | [] , _ -> failwith "proof_of_farkas : not enough hyps" - | p::prf,c::cert -> - mk_farkas (add_proof (mul_proof c p) acc) prf cert in - let res = mk_farkas Zero prf cert in + (* Printf.printf "\nproof_of_farkas %a , %a \n" (pp_list output_prf_rule) prf (pp_list output_bigint) cert ; *) + let rec mk_farkas acc prf cert = + match prf, cert with + | _ , [] -> acc + | [] , _ -> failwith "proof_of_farkas : not enough hyps" + | p::prf,c::cert -> + mk_farkas (add_proof (mul_proof c p) acc) prf cert in + let res = mk_farkas Zero prf cert in (*Printf.printf "==> %a" output_prf_rule res ; *) - res + res let linear_prover sys = - let (sysi,prfi) = List.split sys in - match xlinear_prover sysi with - | None -> None - | Some cert -> Some (proof_of_farkas prfi cert) + let (sysi,prfi) = List.split sys in + match xlinear_prover sysi with + | None -> None + | Some cert -> Some (proof_of_farkas prfi cert) let linear_prover = - if debug - then - fun sys -> - Printf.printf "<linear_prover"; flush stdout ; - let res = linear_prover sys in - Printf.printf ">"; flush stdout ; - res - else linear_prover + if debug + then + fun sys -> + Printf.printf "<linear_prover"; flush stdout ; + let res = linear_prover sys in + Printf.printf ">"; flush stdout ; + res + else linear_prover @@ -733,11 +789,11 @@ let linear_prover = *) type checksat = - | Tauto (* Tautology *) - | Unsat of prf_rule (* Unsatisfiable *) - | Cut of cstr_compat * prf_rule (* Cutting plane *) - | Normalise of cstr_compat * prf_rule (* coefficients are relatively prime *) - +| Tauto (* Tautology *) +| Unsat of prf_rule (* Unsatisfiable *) +| Cut of cstr_compat * prf_rule (* Cutting plane *) +| Normalise of cstr_compat * prf_rule (* coefficients are relatively prime *) + (** [check_sat] - detects constraints that are not satisfiable; @@ -745,83 +801,83 @@ type checksat = *) let check_sat (cstr,prf) = - let {coeffs=coeffs ; op=op ; cst=cst} = cstr in - match coeffs with - | [] -> - if eval_op op (Int 0) cst then Tauto else Unsat prf - | _ -> - let gcdi = (gcd_list (List.map snd coeffs)) in - let gcd = Big_int gcdi in - if eq_num gcd (Int 1) - then Normalise(cstr,prf) - else - if Int.equal (sign_num (mod_num cst gcd)) 0 - then (* We can really normalise *) - begin - assert (sign_num gcd >=1 ) ; - let cstr = { - coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; - op = op ; cst = cst // gcd - } in - Normalise(cstr,Gcd(gcdi,prf)) - (* Normalise(cstr,CutPrf prf)*) - end - else - match op with - | Eq -> Unsat (CutPrf prf) - | Ge -> - let cstr = { - coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; - op = op ; cst = ceiling_num (cst // gcd) - } in Cut(cstr,CutPrf prf) + let {coeffs=coeffs ; op=op ; cst=cst} = cstr in + match coeffs with + | [] -> + if eval_op op (Int 0) cst then Tauto else Unsat prf + | _ -> + let gcdi = (gcd_list (List.map snd coeffs)) in + let gcd = Big_int gcdi in + if eq_num gcd (Int 1) + then Normalise(cstr,prf) + else + if Int.equal (sign_num (mod_num cst gcd)) 0 + then (* We can really normalise *) + begin + assert (sign_num gcd >=1 ) ; + let cstr = { + coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; + op = op ; cst = cst // gcd + } in + Normalise(cstr,Gcd(gcdi,prf)) + (* Normalise(cstr,CutPrf prf)*) + end + else + match op with + | Eq -> Unsat (CutPrf prf) + | Ge -> + let cstr = { + coeffs = List.map (fun (x,v) -> (x, v // gcd)) coeffs; + op = op ; cst = ceiling_num (cst // gcd) + } in Cut(cstr,CutPrf prf) (** Proof generating pivoting over variable v *) let pivot v (c1,p1) (c2,p2) = - let {coeffs = v1 ; op = op1 ; cst = n1} = c1 - and {coeffs = v2 ; op = op2 ; cst = n2} = c2 in + let {coeffs = v1 ; op = op1 ; cst = n1} = c1 + and {coeffs = v2 ; op = op2 ; cst = n2} = c2 in (* Could factorise gcd... *) - let xpivot cv1 cv2 = - ( - {coeffs = Vect.add (Vect.mul cv1 v1) (Vect.mul cv2 v2) ; - op = Proof.add_op op1 op2 ; - cst = n1 */ cv1 +/ n2 */ cv2 }, + let xpivot cv1 cv2 = + ( + {coeffs = Vect.add (Vect.mul cv1 v1) (Vect.mul cv2 v2) ; + op = Proof.add_op op1 op2 ; + cst = n1 */ cv1 +/ n2 */ cv2 }, - AddPrf(mul_proof (numerator cv1) p1,mul_proof (numerator cv2) p2)) in + AddPrf(mul_proof (numerator cv1) p1,mul_proof (numerator cv2) p2)) in + + match Vect.get v v1 , Vect.get v v2 with + | None , _ | _ , None -> None + | Some a , Some b -> + if Int.equal ((sign_num a) * (sign_num b)) (-1) + then + let cv1 = abs_num b + and cv2 = abs_num a in + Some (xpivot cv1 cv2) + else + if op1 == Eq + then + let cv1 = minus_num (b */ (Int (sign_num a))) + and cv2 = abs_num a in + Some (xpivot cv1 cv2) + else if op2 == Eq + then + let cv1 = abs_num b + and cv2 = minus_num (a */ (Int (sign_num b))) in + Some (xpivot cv1 cv2) + else None (* op2 could be Eq ... this might happen *) - match Vect.get v v1 , Vect.get v v2 with - | None , _ | _ , None -> None - | Some a , Some b -> - if Int.equal ((sign_num a) * (sign_num b)) (-1) - then - let cv1 = abs_num b - and cv2 = abs_num a in - Some (xpivot cv1 cv2) - else - if op1 == Eq - then - let cv1 = minus_num (b */ (Int (sign_num a))) - and cv2 = abs_num a in - Some (xpivot cv1 cv2) - else if op2 == Eq - then - let cv1 = abs_num b - and cv2 = minus_num (a */ (Int (sign_num b))) in - Some (xpivot cv1 cv2) - else None (* op2 could be Eq ... this might happen *) - exception FoundProof of prf_rule let simpl_sys sys = - List.fold_left (fun acc (c,p) -> - match check_sat (c,p) with - | Tauto -> acc - | Unsat prf -> raise (FoundProof prf) - | Cut(c,p) -> (c,p)::acc - | Normalise (c,p) -> (c,p)::acc) [] sys + List.fold_left (fun acc (c,p) -> + match check_sat (c,p) with + | Tauto -> acc + | Unsat prf -> raise (FoundProof prf) + | Cut(c,p) -> (c,p)::acc + | Normalise (c,p) -> (c,p)::acc) [] sys (** [ext_gcd a b] is the extended Euclid algorithm. @@ -829,77 +885,77 @@ let simpl_sys sys = Source: http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm *) let rec ext_gcd a b = - if Int.equal (sign_big_int b) 0 - then (unit_big_int,zero_big_int) - else - let (q,r) = quomod_big_int a b in - let (s,t) = ext_gcd b r in - (t, sub_big_int s (mult_big_int q t)) + if Int.equal (sign_big_int b) 0 + then (unit_big_int,zero_big_int) + else + let (q,r) = quomod_big_int a b in + let (s,t) = ext_gcd b r in + (t, sub_big_int s (mult_big_int q t)) let pp_ext_gcd a b = - let a' = big_int_of_int a in - let b' = big_int_of_int b in - - let (x,y) = ext_gcd a' b' in - Printf.fprintf stdout "%s * %s + %s * %s = %s\n" - (string_of_big_int x) (string_of_big_int a') - (string_of_big_int y) (string_of_big_int b') - (string_of_big_int (add_big_int (mult_big_int x a') (mult_big_int y b'))) + let a' = big_int_of_int a in + let b' = big_int_of_int b in + + let (x,y) = ext_gcd a' b' in + Printf.fprintf stdout "%s * %s + %s * %s = %s\n" + (string_of_big_int x) (string_of_big_int a') + (string_of_big_int y) (string_of_big_int b') + (string_of_big_int (add_big_int (mult_big_int x a') (mult_big_int y b'))) exception Result of (int * (proof * cstr_compat)) let split_equations psys = - List.partition (fun (c,p) -> c.op == Eq) + List.partition (fun (c,p) -> c.op == Eq) let extract_coprime (c1,p1) (c2,p2) = - let rec exist2 vect1 vect2 = - match vect1 , vect2 with - | _ , [] | [], _ -> None - | (v1,n1)::vect1' , (v2, n2) :: vect2' -> - if Pervasives.(=) v1 v2 - then - if Int.equal (compare_big_int (gcd_big_int (numerator n1) (numerator n2)) unit_big_int) 0 - then Some (v1,n1,n2) - else - exist2 vect1' vect2' - else - if v1 < v2 - then exist2 vect1' vect2 - else exist2 vect1 vect2' in - - if c1.op == Eq && c2.op == Eq - then exist2 c1.coeffs c2.coeffs - else None + let rec exist2 vect1 vect2 = + match vect1 , vect2 with + | _ , [] | [], _ -> None + | (v1,n1)::vect1' , (v2, n2) :: vect2' -> + if Pervasives.(=) v1 v2 + then + if Int.equal (compare_big_int (gcd_big_int (numerator n1) (numerator n2)) unit_big_int) 0 + then Some (v1,n1,n2) + else + exist2 vect1' vect2' + else + if v1 < v2 + then exist2 vect1' vect2 + else exist2 vect1 vect2' in + + if c1.op == Eq && c2.op == Eq + then exist2 c1.coeffs c2.coeffs + else None let extract2 pred l = - let rec xextract2 rl l = - match l with - | [] -> (None,rl) (* Did not find *) - | e::l -> - match extract (pred e) l with - | None,_ -> xextract2 (e::rl) l - | Some (r,e'),l' -> Some (r,e,e'), List.rev_append rl l' in - - xextract2 [] l + let rec xextract2 rl l = + match l with + | [] -> (None,rl) (* Did not find *) + | e::l -> + match extract (pred e) l with + | None,_ -> xextract2 (e::rl) l + | Some (r,e'),l' -> Some (r,e,e'), List.rev_append rl l' in + + xextract2 [] l let extract_coprime_equation psys = - extract2 extract_coprime psys + extract2 extract_coprime psys let apply_and_normalise f psys = - List.fold_left (fun acc pc' -> - match f pc' with - | None -> pc'::acc - | Some pc' -> - match check_sat pc' with - | Tauto -> acc - | Unsat prf -> raise (FoundProof prf) - | Cut(c,p) -> (c,p)::acc - | Normalise (c,p) -> (c,p)::acc - ) [] psys + List.fold_left (fun acc pc' -> + match f pc' with + | None -> pc'::acc + | Some pc' -> + match check_sat pc' with + | Tauto -> acc + | Unsat prf -> raise (FoundProof prf) + | Cut(c,p) -> (c,p)::acc + | Normalise (c,p) -> (c,p)::acc + ) [] psys @@ -908,314 +964,317 @@ let pivot_sys v pc psys = apply_and_normalise (pivot v pc) psys let reduce_coprime psys = - let oeq,sys = extract_coprime_equation psys in - match oeq with - | None -> None (* Nothing to do *) - | Some((v,n1,n2),(c1,p1),(c2,p2) ) -> - let (l1,l2) = ext_gcd (numerator n1) (numerator n2) in - let l1' = Big_int l1 and l2' = Big_int l2 in - let cstr = - {coeffs = Vect.add (Vect.mul l1' c1.coeffs) (Vect.mul l2' c2.coeffs); - op = Eq ; - cst = (l1' */ c1.cst) +/ (l2' */ c2.cst) - } in - let prf = add_proof (mul_proof (numerator l1') p1) (mul_proof (numerator l2') p2) in - - Some (pivot_sys v (cstr,prf) ((c1,p1)::sys)) + let oeq,sys = extract_coprime_equation psys in + match oeq with + | None -> None (* Nothing to do *) + | Some((v,n1,n2),(c1,p1),(c2,p2) ) -> + let (l1,l2) = ext_gcd (numerator n1) (numerator n2) in + let l1' = Big_int l1 and l2' = Big_int l2 in + let cstr = + {coeffs = Vect.add (Vect.mul l1' c1.coeffs) (Vect.mul l2' c2.coeffs); + op = Eq ; + cst = (l1' */ c1.cst) +/ (l2' */ c2.cst) + } in + let prf = add_proof (mul_proof (numerator l1') p1) (mul_proof (numerator l2') p2) in + + Some (pivot_sys v (cstr,prf) ((c1,p1)::sys)) (** If there is an equation [eq] of the form 1.x + e = c, do a pivot over x with equation [eq] *) let reduce_unary psys = - let is_unary_equation (cstr,prf) = - if cstr.op == Eq - then - try - Some (fst (List.find (fun (_,n) -> n =/ (Int 1) || n=/ (Int (-1))) cstr.coeffs)) - with Not_found -> None - else None in - - let (oeq,sys) = extract is_unary_equation psys in - match oeq with - | None -> None (* Nothing to do *) - | Some(v,pc) -> - Some(pivot_sys v pc sys) + let is_unary_equation (cstr,prf) = + if cstr.op == Eq + then + try + Some (fst (List.find (fun (_,n) -> n =/ (Int 1) || n=/ (Int (-1))) cstr.coeffs)) + with Not_found -> None + else None in + + let (oeq,sys) = extract is_unary_equation psys in + match oeq with + | None -> None (* Nothing to do *) + | Some(v,pc) -> + Some(pivot_sys v pc sys) let reduce_non_lin_unary psys = - let is_unary_equation (cstr,prf) = - if cstr.op == Eq - then - try - let x = fst (List.find (fun (x,n) -> (n =/ (Int 1) || n=/ (Int (-1))) && Monomial.is_var (LinPoly.MonT.retrieve x) ) cstr.coeffs) in - let x' = LinPoly.MonT.retrieve x in - if List.for_all (fun (y,_) -> Pervasives.(=) y x || Int.equal (snd (Monomial.div (LinPoly.MonT.retrieve y) x')) 0) cstr.coeffs - then Some x - else None - with Not_found -> None - else None in - - - let (oeq,sys) = extract is_unary_equation psys in - match oeq with - | None -> None (* Nothing to do *) - | Some(v,pc) -> - Some(apply_and_normalise (LinPoly.pivot_eq v pc) sys) + let is_unary_equation (cstr,prf) = + if cstr.op == Eq + then + try + let x = fst (List.find (fun (x,n) -> (n =/ (Int 1) || n=/ (Int (-1))) && Monomial.is_var (LinPoly.MonT.retrieve x) ) cstr.coeffs) in + let x' = LinPoly.MonT.retrieve x in + if List.for_all (fun (y,_) -> Pervasives.(=) y x || Int.equal (snd (Monomial.div (LinPoly.MonT.retrieve y) x')) 0) cstr.coeffs + then Some x + else None + with Not_found -> None + else None in + + + let (oeq,sys) = extract is_unary_equation psys in + match oeq with + | None -> None (* Nothing to do *) + | Some(v,pc) -> + Some(apply_and_normalise (LinPoly.pivot_eq v pc) sys) let reduce_var_change psys = - let rec rel_prime vect = - match vect with - | [] -> None - | (x,v)::vect -> - let v = numerator v in - try - let (x',v') = List.find (fun (_,v') -> - let v' = numerator v' in - eq_big_int (gcd_big_int v v') unit_big_int) vect in - Some ((x,v),(x',numerator v')) - with Not_found -> rel_prime vect in - - let rel_prime (cstr,prf) = if cstr.op == Eq then rel_prime cstr.coeffs else None in - - let (oeq,sys) = extract rel_prime psys in - - match oeq with - | None -> None - | Some(((x,v),(x',v')),(c,p)) -> - let (l1,l2) = ext_gcd v v' in - let l1,l2 = Big_int l1 , Big_int l2 in + let rec rel_prime vect = + match vect with + | [] -> None + | (x,v)::vect -> + let v = numerator v in + try + let (x',v') = List.find (fun (_,v') -> + let v' = numerator v' in + eq_big_int (gcd_big_int v v') unit_big_int) vect in + Some ((x,v),(x',numerator v')) + with Not_found -> rel_prime vect in + + let rel_prime (cstr,prf) = if cstr.op == Eq then rel_prime cstr.coeffs else None in - let get v vect = - match Vect.get v vect with - | None -> Int 0 - | Some n -> n in + let (oeq,sys) = extract rel_prime psys in + + match oeq with + | None -> None + | Some(((x,v),(x',v')),(c,p)) -> + let (l1,l2) = ext_gcd v v' in + let l1,l2 = Big_int l1 , Big_int l2 in - let pivot_eq (c',p') = - let {coeffs = coeffs ; op = op ; cst = cst} = c' in - let vx = get x coeffs in - let vx' = get x' coeffs in - let m = minus_num (vx */ l1 +/ vx' */ l2) in - Some ({coeffs = - Vect.add (Vect.mul m c.coeffs) coeffs ; op = op ; cst = m */ c.cst +/ cst} , - AddPrf(MulC(([], m),p),p')) in + let get v vect = + match Vect.get v vect with + | None -> Int 0 + | Some n -> n in - Some (apply_and_normalise pivot_eq sys) + let pivot_eq (c',p') = + let {coeffs = coeffs ; op = op ; cst = cst} = c' in + let vx = get x coeffs in + let vx' = get x' coeffs in + let m = minus_num (vx */ l1 +/ vx' */ l2) in + Some ({coeffs = + Vect.add (Vect.mul m c.coeffs) coeffs ; op = op ; cst = m */ c.cst +/ cst} , + AddPrf(MulC(([], m),p),p')) in + Some (apply_and_normalise pivot_eq sys) - let reduce_pivot psys = - let is_equation (cstr,prf) = - if cstr.op == Eq - then - try - Some (fst (List.hd cstr.coeffs)) - with Not_found -> None - else None in - let (oeq,sys) = extract is_equation psys in - match oeq with - | None -> None (* Nothing to do *) - | Some(v,pc) -> - if debug then - Printf.printf "Bad news : loss of completeness %a=%s" Vect.pp_vect (fst pc).coeffs (string_of_num (fst pc).cst); - Some(pivot_sys v pc sys) +let reduce_pivot psys = + let is_equation (cstr,prf) = + if cstr.op == Eq + then + try + Some (fst (List.hd cstr.coeffs)) + with Not_found -> None + else None in + let (oeq,sys) = extract is_equation psys in + match oeq with + | None -> None (* Nothing to do *) + | Some(v,pc) -> + if debug then + Printf.printf "Bad news : loss of completeness %a=%s" Vect.pp_vect (fst pc).coeffs (string_of_num (fst pc).cst); + Some(pivot_sys v pc sys) - let iterate_until_stable f x = - let rec iter x = - match f x with - | None -> x - | Some x' -> iter x' in - iter x - let rec app_funs l x = - match l with - | [] -> None - | f::fl -> - match f x with - | None -> app_funs fl x - | Some x' -> Some x' +let iterate_until_stable f x = + let rec iter x = + match f x with + | None -> x + | Some x' -> iter x' in + iter x - let reduction_equations psys = - iterate_until_stable (app_funs - [reduce_unary ; reduce_coprime ; - reduce_var_change (*; reduce_pivot*)]) psys +let rec app_funs l x = + match l with + | [] -> None + | f::fl -> + match f x with + | None -> app_funs fl x + | Some x' -> Some x' - let reduction_non_lin_equations psys = - iterate_until_stable (app_funs - [reduce_non_lin_unary (*; reduce_coprime ; - reduce_var_change ; reduce_pivot *)]) psys +let reduction_equations psys = + iterate_until_stable (app_funs + [reduce_unary ; reduce_coprime ; + reduce_var_change (*; reduce_pivot*)]) psys + +let reduction_non_lin_equations psys = + iterate_until_stable (app_funs + [reduce_non_lin_unary (*; reduce_coprime ; + reduce_var_change ; reduce_pivot *)]) psys (** [get_bound sys] returns upon success an interval (lb,e,ub) with proofs *) - let get_bound sys = - let is_small (v,i) = - match Itv.range i with - | None -> false - | Some i -> i <=/ (Int 1) in - - let select_best (x1,i1) (x2,i2) = - if Itv.smaller_itv i1 i2 - then (x1,i1) else (x2,i2) in +let get_bound sys = + let is_small (v,i) = + match Itv.range i with + | None -> false + | Some i -> i <=/ (Int 1) in + + let select_best (x1,i1) (x2,i2) = + if Itv.smaller_itv i1 i2 + then (x1,i1) else (x2,i2) in (* For lia, there are no equations => these precautions are not needed *) (* For nlia, there are equations => do not enumerate over equations! *) - let all_planes sys = - let (eq,ineq) = List.partition (fun c -> c.op == Eq) sys in - match eq with - | [] -> List.rev_map (fun c -> c.coeffs) ineq - | _ -> - List.fold_left (fun acc c -> - if List.exists (fun c' -> Vect.equal c.coeffs c'.coeffs) eq - then acc else c.coeffs ::acc) [] ineq in - - let smallest_interval = - List.fold_left - (fun acc vect -> - if is_small acc - then acc - else - match Fourier.optimise vect sys with - | None -> acc - | Some i -> - if debug then Printf.printf "Found a new bound %a" Vect.pp_vect vect ; - select_best (vect,i) acc) (Vect.null, (None,None)) (all_planes sys) in - let smallest_interval = - match smallest_interval - with - | (x,(Some i, Some j)) -> Some(i,x,j) - | x -> None (* This should not be possible *) - in - match smallest_interval with - | Some (lb,e,ub) -> - let (lbn,lbd) = (sub_big_int (numerator lb) unit_big_int, denominator lb) in - let (ubn,ubd) = (add_big_int unit_big_int (numerator ub) , denominator ub) in - (match + let all_planes sys = + let (eq,ineq) = List.partition (fun c -> c.op == Eq) sys in + match eq with + | [] -> List.rev_map (fun c -> c.coeffs) ineq + | _ -> + List.fold_left (fun acc c -> + if List.exists (fun c' -> Vect.equal c.coeffs c'.coeffs) eq + then acc else c.coeffs ::acc) [] ineq in + + let smallest_interval = + List.fold_left + (fun acc vect -> + if is_small acc + then acc + else + match Fourier.optimise vect sys with + | None -> acc + | Some i -> + if debug then Printf.printf "Found a new bound %a" Vect.pp_vect vect ; + select_best (vect,i) acc) (Vect.null, (None,None)) (all_planes sys) in + let smallest_interval = + match smallest_interval + with + | (x,(Some i, Some j)) -> Some(i,x,j) + | x -> None (* This should not be possible *) + in + match smallest_interval with + | Some (lb,e,ub) -> + let (lbn,lbd) = (sub_big_int (numerator lb) unit_big_int, denominator lb) in + let (ubn,ubd) = (add_big_int unit_big_int (numerator ub) , denominator ub) in + (match (* x <= ub -> x > ub *) - xlinear_prover ({coeffs = Vect.mul (Big_int ubd) e ; op = Ge ; cst = Big_int ubn} :: sys), + xlinear_prover ({coeffs = Vect.mul (Big_int ubd) e ; op = Ge ; cst = Big_int ubn} :: sys), (* lb <= x -> lb > x *) - xlinear_prover - ({coeffs = Vect.mul (minus_num (Big_int lbd)) e ; op = Ge ; cst = minus_num (Big_int lbn)} :: sys) - with - | Some cub , Some clb -> Some(List.tl clb,(lb,e,ub), List.tl cub) - | _ -> failwith "Interval without proof" - ) - | None -> None - - - let check_sys sys = - List.for_all (fun (c,p) -> List.for_all (fun (_,n) -> sign_num n <> 0) c.coeffs) sys - - - let xlia reduction_equations sys = - - let rec enum_proof (id:int) (sys:prf_sys) : proof option = - if debug then (Printf.printf "enum_proof\n" ; flush stdout) ; - assert (check_sys sys) ; - - let nsys,prf = List.split sys in - match get_bound nsys with - | None -> None (* Is the systeme really unbounded ? *) - | Some(prf1,(lb,e,ub),prf2) -> - if debug then Printf.printf "Found interval: %a in [%s;%s] -> " Vect.pp_vect e (string_of_num lb) (string_of_num ub) ; - (match start_enum id e (ceiling_num lb) (floor_num ub) sys - with - | Some prfl -> - Some(Enum(id,proof_of_farkas prf prf1,e, proof_of_farkas prf prf2,prfl)) - | None -> None - ) - - and start_enum id e clb cub sys = - if clb >/ cub - then Some [] - else - let eq = {coeffs = e ; op = Eq ; cst = clb} in - match aux_lia (id+1) ((eq, Def id) :: sys) with - | None -> None - | Some prf -> - match start_enum id e (clb +/ (Int 1)) cub sys with - | None -> None - | Some l -> Some (prf::l) - - and aux_lia (id:int) (sys:prf_sys) : proof option = - assert (check_sys sys) ; - if debug then Printf.printf "xlia: %a \n" (pp_list (fun o (c,_) -> output_cstr o c)) sys ; - try - let sys = reduction_equations sys in - if debug then + xlinear_prover + ({coeffs = Vect.mul (minus_num (Big_int lbd)) e ; op = Ge ; cst = minus_num (Big_int lbn)} :: sys) + with + | Some cub , Some clb -> Some(List.tl clb,(lb,e,ub), List.tl cub) + | _ -> failwith "Interval without proof" + ) + | None -> None + + +let check_sys sys = + List.for_all (fun (c,p) -> List.for_all (fun (_,n) -> sign_num n <> 0) c.coeffs) sys + + +let xlia (can_enum:bool) reduction_equations sys = + + + let rec enum_proof (id:int) (sys:prf_sys) : proof option = + if debug then (Printf.printf "enum_proof\n" ; flush stdout) ; + assert (check_sys sys) ; + + let nsys,prf = List.split sys in + match get_bound nsys with + | None -> None (* Is the systeme really unbounded ? *) + | Some(prf1,(lb,e,ub),prf2) -> + if debug then Printf.printf "Found interval: %a in [%s;%s] -> " Vect.pp_vect e (string_of_num lb) (string_of_num ub) ; + (match start_enum id e (ceiling_num lb) (floor_num ub) sys + with + | Some prfl -> + Some(Enum(id,proof_of_farkas prf prf1,e, proof_of_farkas prf prf2,prfl)) + | None -> None + ) + + and start_enum id e clb cub sys = + if clb >/ cub + then Some [] + else + let eq = {coeffs = e ; op = Eq ; cst = clb} in + match aux_lia (id+1) ((eq, Def id) :: sys) with + | None -> None + | Some prf -> + match start_enum id e (clb +/ (Int 1)) cub sys with + | None -> None + | Some l -> Some (prf::l) + + and aux_lia (id:int) (sys:prf_sys) : proof option = + assert (check_sys sys) ; + if debug then Printf.printf "xlia: %a \n" (pp_list (fun o (c,_) -> output_cstr o c)) sys ; + try + let sys = reduction_equations sys in + if debug then Printf.printf "after reduction: %a \n" (pp_list (fun o (c,_) -> output_cstr o c)) sys ; - match linear_prover sys with - | Some prf -> Some (Step(id,prf,Done)) - | None -> enum_proof id sys - with FoundProof prf -> + match linear_prover sys with + | Some prf -> Some (Step(id,prf,Done)) + | None -> if can_enum then enum_proof id sys else None + with FoundProof prf -> (* [reduction_equations] can find a proof *) - Some(Step(id,prf,Done)) in + Some(Step(id,prf,Done)) in (* let sys' = List.map (fun (p,o) -> Mc.norm0 p , o) sys in*) - let id = List.length sys in - let orpf = - try - let sys = simpl_sys sys in - aux_lia id sys - with FoundProof pr -> Some(Step(id,pr,Done)) in - match orpf with - | None -> None - | Some prf -> + let id = List.length sys in + let orpf = + try + let sys = simpl_sys sys in + aux_lia id sys + with FoundProof pr -> Some(Step(id,pr,Done)) in + match orpf with + | None -> None + | Some prf -> (*Printf.printf "direct proof %a\n" output_proof prf ; *) - let env = mapi (fun _ i -> i) sys in - let prf = compile_proof env prf in + let env = mapi (fun _ i -> i) sys in + let prf = compile_proof env prf in (*try if Mc.zChecker sys' prf then Some prf else raise Certificate.BadCertificate with Failure s -> (Printf.printf "%s" s ; Some prf) *) Some prf - - - let cstr_compat_of_poly (p,o) = - let (v,c) = LinPoly.linpol_of_pol p in - {coeffs = v ; op = o ; cst = minus_num c } - - - let lia sys = - LinPoly.MonT.clear (); - let sys = List.map (develop_constraint z_spec) sys in - let (sys:cstr_compat list) = List.map cstr_compat_of_poly sys in - let sys = mapi (fun c i -> (c,Hyp i)) sys in - xlia reduction_equations sys - - - let nlia sys = - LinPoly.MonT.clear (); - let sys = List.map (develop_constraint z_spec) sys in - let sys = mapi (fun c i -> (c,Hyp i)) sys in - - let is_linear = List.for_all (fun ((p,_),_) -> Poly.is_linear p) sys in - - let collect_square = - List.fold_left (fun acc ((p,_),_) -> Poly.fold - (fun m _ acc -> - match Monomial.sqrt m with - | None -> acc - | Some s -> MonMap.add s m acc) p acc) MonMap.empty sys in - let sys = MonMap.fold (fun s m acc -> - let s = LinPoly.linpol_of_pol (Poly.add s (Int 1) (Poly.constant (Int 0))) in - let m = Poly.add m (Int 1) (Poly.constant (Int 0)) in - ((m, Ge), (Square s))::acc) collect_square sys in - -(* List.iter (fun ((p,_),_) -> Printf.printf "square %a\n" Poly.pp p) gen_square*) - - let sys = - if is_linear then sys - else sys @ (all_sym_pairs (fun ((c,o),p) ((c',o'),p') -> - ((Poly.product c c',opMult o o'), MulPrf(p,p'))) sys) in + - let sys = List.map (fun (c,p) -> cstr_compat_of_poly c,p) sys in - assert (check_sys sys) ; - xlia (if is_linear then reduction_equations else reduction_non_lin_equations) sys +let cstr_compat_of_poly (p,o) = + let (v,c) = LinPoly.linpol_of_pol p in + {coeffs = v ; op = o ; cst = minus_num c } + + +let lia (can_enum:bool) (prfdepth:int) sys = + LinPoly.MonT.clear (); + max_nb_cstr := compute_max_nb_cstr sys prfdepth ; + let sys = List.map (develop_constraint z_spec) sys in + let (sys:cstr_compat list) = List.map cstr_compat_of_poly sys in + let sys = mapi (fun c i -> (c,Hyp i)) sys in + xlia can_enum reduction_equations sys + + +let nlia enum prfdepth sys = + LinPoly.MonT.clear (); + max_nb_cstr := compute_max_nb_cstr sys prfdepth; + let sys = List.map (develop_constraint z_spec) sys in + let sys = mapi (fun c i -> (c,Hyp i)) sys in + + let is_linear = List.for_all (fun ((p,_),_) -> Poly.is_linear p) sys in + + let collect_square = + List.fold_left (fun acc ((p,_),_) -> Poly.fold + (fun m _ acc -> + match Monomial.sqrt m with + | None -> acc + | Some s -> MonMap.add s m acc) p acc) MonMap.empty sys in + let sys = MonMap.fold (fun s m acc -> + let s = LinPoly.linpol_of_pol (Poly.add s (Int 1) (Poly.constant (Int 0))) in + let m = Poly.add m (Int 1) (Poly.constant (Int 0)) in + ((m, Ge), (Square s))::acc) collect_square sys in + + (* List.iter (fun ((p,_),_) -> Printf.printf "square %a\n" Poly.pp p) gen_square*) + + let sys = + if is_linear then sys + else sys @ (all_sym_pairs (fun ((c,o),p) ((c',o'),p') -> + ((Poly.product c c',opMult o o'), MulPrf(p,p'))) sys) in + + let sys = List.map (fun (c,p) -> cstr_compat_of_poly c,p) sys in + assert (check_sys sys) ; + xlia enum (if is_linear then reduction_equations else reduction_non_lin_equations) sys |