Source file certificate.ml
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let debug = false
open Polynomial
module Mc = Micromega
module Ml2C = Mutils.CamlToCoq
module C2Ml = Mutils.CoqToCaml
open NumCompat
open Q.Notations
open Mutils
let use_simplex =
Goptions.declare_bool_option_and_ref ~depr:true ~key:["Simplex"] ~value:true
let dump_file =
Goptions.declare_stringopt_option_and_ref ~depr:false ~key:["Dump"; "Arith"]
type ('prf, 'model) res = Prf of 'prf | Model of 'model | Unknown
type zres = (Mc.zArithProof, int * Mc.z list) res
type qres = (Mc.q Mc.psatz, int * Mc.q list) res
type 'a number_spec =
{ bigint_to_number : Z.t -> 'a
; number_to_num : 'a -> Q.t
; zero : 'a
; unit : 'a
; mult : 'a -> 'a -> 'a
; eqb : 'a -> 'a -> bool }
let z_spec =
{ bigint_to_number = Ml2C.bigint
; number_to_num = (fun x -> Q.of_bigint (C2Ml.z_big_int x))
; zero = Mc.Z0
; unit = Mc.Zpos Mc.XH
; mult = Mc.Z.mul
; eqb = Mc.zeq_bool }
let q_spec =
{ bigint_to_number = (fun x -> {Mc.qnum = Ml2C.bigint x; Mc.qden = Mc.XH})
; number_to_num = C2Ml.q_to_num
; zero = {Mc.qnum = Mc.Z0; Mc.qden = Mc.XH}
; unit = {Mc.qnum = Mc.Zpos Mc.XH; Mc.qden = Mc.XH}
; mult = Mc.qmult
; eqb = Mc.qeq_bool }
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
let rec fixpoint f x =
let y' = f x in
if 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
let simplify_cone n_spec c = fixpoint (rec_simpl_cone n_spec) c
let constrain_variable v l =
let coeffs = List.fold_left (fun acc p -> Vect.get v p.coeffs :: acc) [] l in
{ coeffs =
Vect.from_list
(Q.of_bigint Z.zero :: Q.of_bigint Z.zero :: List.rev coeffs)
; op = Eq
; cst = Q.of_bigint Z.zero }
let constrain_constant l =
let coeffs = List.fold_left (fun acc p -> Q.neg p.cst :: acc) [] l in
{ coeffs =
Vect.from_list (Q.of_bigint Z.zero :: Q.of_bigint Z.one :: List.rev coeffs)
; op = Eq
; cst = Q.of_bigint Z.zero }
let positivity l =
let rec xpositivity i l =
match l with
| [] -> []
| c :: l -> (
match c.op with
| Eq -> xpositivity (i + 1) l
| _ ->
{ coeffs = Vect.update (i + 1) (fun _ -> Q.one) Vect.null
; op = Ge
; cst = Q.zero }
:: xpositivity (i + 1) l )
in
xpositivity 1 l
let cstr_of_poly (p, o) =
let c, l = Vect.decomp_cst p in
{coeffs = l; op = o; cst = Q.neg c}
let variables_of_cstr c = Vect.variables c.coeffs
let build_dual_linear_system l =
let variables =
List.fold_left
(fun acc p -> ISet.union acc (variables_of_cstr p))
ISet.empty l
in
let s0 =
ISet.fold (fun mn res -> constrain_variable mn l :: res) variables []
in
let c = constrain_constant l in
let strict =
{ coeffs =
Vect.from_list
( Q.of_bigint Z.zero :: Q.of_bigint Z.one
:: List.map
(fun c ->
if is_strict c then Q.of_bigint Z.one else Q.of_bigint Z.zero)
l )
; op = Ge
; cst = Q.of_bigint Z.one }
in
{ coeffs = Vect.from_list [Q.of_bigint Z.zero; Q.of_bigint Z.one]
; op = Ge
; cst = Q.of_bigint Z.zero }
:: ((strict :: positivity l) @ (c :: s0))
(** [direct_linear_prover l] does not handle strict inegalities *)
let fourier_linear_prover l =
let open Util in
match Mfourier.Fourier.find_point l with
| Inr prf ->
if debug then Printf.printf "AProof : %a\n" Mfourier.pp_proof prf;
let cert =
fst (List.hd (Mfourier.Proof.mk_proof l prf))
in
if debug then Printf.printf "CProof : %a" Vect.pp cert;
Some (Vect.normalise cert)
| Inl _ -> None
let direct_linear_prover l =
if use_simplex () then Simplex.find_unsat_certificate l
else fourier_linear_prover l
let find_point l =
let open Util in
if use_simplex () then Simplex.find_point l
else
match Mfourier.Fourier.find_point l with
| Inr _ -> None
| Inl cert -> Some cert
let optimise v l =
if use_simplex () then Simplex.optimise v l else Mfourier.Fourier.optimise v l
let output_cstr_sys o sys =
List.iter
(fun (c, wp) ->
Printf.fprintf o "%a by %a\n" output_cstr c ProofFormat.output_prf_rule wp)
sys
let output_sys o sys =
List.iter (fun s -> Printf.fprintf o "%a\n" WithProof.output s) sys
let tr_sys str f sys =
let sys' = f sys in
if debug then
Printf.fprintf stdout "[%s\n%a=>\n%a]\n" str output_sys sys output_sys sys';
sys'
let tr_cstr_sys str f sys =
let sys' = f sys in
if debug then
Printf.fprintf stdout "[%s\n%a=>\n%a]\n" str output_cstr_sys sys
output_cstr_sys sys';
sys'
let dual_raw_certificate l =
if debug then begin
Printf.printf "dual_raw_certificate\n";
List.iter (fun c -> Printf.fprintf stdout "%a\n" output_cstr c) l
end;
let sys = build_dual_linear_system l in
if debug then begin
Printf.printf "dual_system\n";
List.iter (fun c -> Printf.fprintf stdout "%a\n" output_cstr c) sys
end;
try
match find_point sys with
| None -> None
| Some cert -> (
match Vect.choose cert with
| None -> failwith "dual_raw_certificate: empty_certificate"
| Some _ ->
Some (Vect.normalise (Vect.decr_var 2 (Vect.set 1 Q.zero cert))) )
with x when CErrors.noncritical x ->
if debug then (
Printf.printf "dual raw certificate %s" (Printexc.to_string x);
flush stdout );
None
let simple_linear_prover l =
try direct_linear_prover l
with Strict ->
dual_raw_certificate l
let env_of_list l =
snd
(List.fold_left (fun (i, m) p -> (i + 1, IMap.add i p m)) (0, IMap.empty) l)
let linear_prover_cstr sys =
let sysi, prfi = List.split sys in
match simple_linear_prover sysi with
| None -> None
| Some cert -> Some (ProofFormat.proof_of_farkas (env_of_list prfi) cert)
let linear_prover_cstr =
if debug then ( fun sys ->
Printf.printf "<linear_prover";
flush stdout;
let res = linear_prover_cstr sys in
Printf.printf ">"; flush stdout; res )
else linear_prover_cstr
let compute_max_nb_cstr l d =
let len = List.length l in
max len (max d (len * d))
let develop_constraint z_spec (e, k) =
( dev_form z_spec e
, match k with
| Mc.NonStrict -> Ge
| Mc.Equal -> Eq
| Mc.Strict -> Gt
| _ -> assert false )
(** A single constraint can be unsat for the following reasons:
- 0 >= c for c a negative constant
- 0 = c for c a non-zero constant
- e = c when the coeffs of e are all integers and c is rational
*)
type checksat =
| Tauto
| Unsat of ProofFormat.prf_rule
| Cut of cstr * ProofFormat.prf_rule
| Normalise of cstr * ProofFormat.prf_rule
exception FoundProof of ProofFormat.prf_rule
(** [check_sat]
- detects constraints that are not satisfiable;
- normalises constraints and generate cuts.
*)
let check_int_sat (cstr, prf) =
let {coeffs; op; cst} = cstr in
match Vect.choose coeffs with
| None -> if eval_op op Q.zero cst then Tauto else Unsat prf
| _ -> (
let gcdi = Vect.gcd coeffs in
let gcd = Q.of_bigint gcdi in
if gcd =/ Q.one then Normalise (cstr, prf)
else if Int.equal (Q.sign (Q.mod_ cst gcd)) 0 then begin
assert (Q.sign gcd >= 1);
let cstr = {coeffs = Vect.div gcd coeffs; op; cst = cst // gcd} in
Normalise (cstr, ProofFormat.Gcd (gcdi, prf))
end
else
match op with
| Eq -> Unsat (ProofFormat.CutPrf prf)
| Ge ->
let cstr =
{coeffs = Vect.div gcd coeffs; op; cst = Q.ceiling (cst // gcd)}
in
Cut (cstr, ProofFormat.CutPrf prf)
| Gt -> failwith "check_sat : Unexpected operator" )
let apply_and_normalise check f psys =
List.fold_left
(fun acc pc' ->
match f pc' with
| None -> pc' :: acc
| Some pc' -> (
match check pc' with
| Tauto -> acc
| Unsat prf -> raise (FoundProof prf)
| Cut (c, p) -> (c, p) :: acc
| Normalise (c, p) -> (c, p) :: acc ))
[] psys
let is_linear_for v pc =
LinPoly.is_linear (fst (fst pc)) || LinPoly.is_linear_for v (fst (fst pc))
let is_linear_substitution sys ((p, o), prf) =
let pred v = v =/ Q.one || v =/ Q.minus_one in
match o with
| Eq -> (
match
List.filter
(fun v -> List.for_all (is_linear_for v) sys)
(LinPoly.search_all_linear pred p)
with
| [] -> None
| v :: _ -> Some v )
| _ -> None
let elim_simple_linear_equality sys0 =
let elim sys =
let oeq, sys' = extract (is_linear_substitution sys) sys in
match oeq with
| None -> None
| Some (v, pc) -> simplify (WithProof.linear_pivot sys0 pc v) sys'
in
iterate_until_stable elim sys0
let subst sys = tr_sys "subst" WithProof.subst sys
(** [saturate_linear_equality sys] generate new constraints
obtained by eliminating linear equalities by pivoting.
For integers, the obtained constraints are sound but not complete.
*)
let saturate_by_linear_equalities sys0 = WithProof.saturate_subst false sys0
let saturate_by_linear_equalities sys =
tr_sys "saturate_by_linear_equalities" saturate_by_linear_equalities sys
let bound_monomials (sys : WithProof.t list) =
let (bounds,_) =
extract_all
(fun p ->
match BoundWithProof.make p with
| None -> None
| Some b ->
let Vect.Bound.{cst; var; coeff} = BoundWithProof.bound b in
Some (Monomial.degree (LinPoly.MonT.retrieve var), b))
sys
in
let deg =
List.fold_left (fun acc ((p, o), _) -> max acc (LinPoly.degree p)) 0 sys
in
let vars =
List.fold_left
(fun acc ((p, o), _) -> ISet.union (LinPoly.monomials p) acc)
ISet.empty sys
in
let module SetWP = Set.Make (struct
type t = int * BoundWithProof.t
let compare (_, x) (_, y) = BoundWithProof.compare x y
end) in
let bounds =
saturate_bin
(module SetWP : Set.S with type elt = int * BoundWithProof.t)
(fun (i1, w1) (i2, w2) ->
if i1 + i2 > deg then None
else
match BoundWithProof.mul_bound w1 w2 with
| None -> None
| Some b -> Some (i1 + i2, b))
bounds
in
let has_mon (_, b) =
let Vect.Bound.{cst; var; coeff} = BoundWithProof.bound b in
if ISet.mem var vars then Some (BoundWithProof.proof b) else None
in
CList.map_filter has_mon bounds
let bound_monomials = tr_sys "bound_monomials" bound_monomials
let develop_constraints prfdepth n_spec sys =
LinPoly.MonT.clear ();
max_nb_cstr := compute_max_nb_cstr sys prfdepth;
let sys = List.map (develop_constraint n_spec) sys in
List.mapi
(fun i (p, o) -> ((LinPoly.linpol_of_pol p, o), ProofFormat.Hyp i))
sys
let square_of_var i =
let x = LinPoly.var i in
((LinPoly.product x x, Ge), ProofFormat.Square x)
(** [nlinear_preprocess sys] augments the system [sys] by performing some limited non-linear reasoning.
For instance, it asserts that the x² ≥0 but also that if c₁ ≥ 0 ∈ sys and c₂ ≥ 0 ∈ sys then c₁ × c₂ ≥ 0.
The resulting system is linearised.
*)
let nlinear_preprocess (sys : WithProof.t list) =
let is_linear = List.for_all (fun ((p, _), _) -> LinPoly.is_linear p) sys in
if is_linear then sys
else
let collect_square =
List.fold_left
(fun acc ((p, _), _) ->
MonMap.union (fun k e1 e2 -> Some e1) acc (LinPoly.collect_square p))
MonMap.empty sys
in
let sys =
MonMap.fold
(fun s m acc ->
let s = LinPoly.of_monomial s in
let m = LinPoly.of_monomial m in
((m, Ge), ProofFormat.Square s) :: acc)
collect_square sys
in
let collect_vars =
List.fold_left
(fun acc p -> ISet.union acc (LinPoly.variables (fst (fst p))))
ISet.empty sys
in
let sys =
ISet.fold (fun i acc -> square_of_var i :: acc) collect_vars sys
in
let sys = sys @ all_pairs WithProof.product sys in
List.map (WithProof.annot "P") sys
let nlinear_preprocess = tr_sys "nlinear_preprocess" nlinear_preprocess
let nlinear_prover prfdepth sys =
let sys = develop_constraints prfdepth q_spec sys in
let sys1 = elim_simple_linear_equality sys in
let sys2 = saturate_by_linear_equalities sys1 in
let sys = nlinear_preprocess sys1 @ sys2 in
let sys = List.map (fun ((p, o), prf) -> (cstr_of_poly (p, o), prf)) sys in
let id =
List.fold_left
(fun acc (_, r) -> max acc (ProofFormat.pr_rule_max_hyp r))
0 sys
in
let env = List.map (fun i -> ProofFormat.Hyp i) (CList.interval 0 id) in
match linear_prover_cstr sys with
| None -> Unknown
| Some cert -> Prf (ProofFormat.cmpl_prf_rule Mc.normQ CamlToCoq.q env cert)
let linear_prover_with_cert prfdepth sys =
let sys = develop_constraints prfdepth q_spec sys in
let sys = List.map (fun (c, p) -> (cstr_of_poly c, p)) sys in
match linear_prover_cstr sys with
| None -> Unknown
| Some cert ->
Prf
(ProofFormat.cmpl_prf_rule Mc.normQ CamlToCoq.q
(List.mapi (fun i e -> ProofFormat.Hyp i) sys)
cert)
open Sos_types
let rec scale_term t =
match t with
| Zero -> (Z.one, Zero)
| Const n -> (Q.den n, Const (Q.of_bigint (Q.num n)))
| Var n -> (Z.one, Var n)
| 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 = Z.gcd s1 s2 in
let s1' = Z.div s1 g in
let s2' = Z.div s2 g in
let e = Z.mul g (Z.mul s1' s2') in
if Int.equal (Z.compare e Z.one) 0 then (Z.one, Add (y1, y2))
else
( e
, Add
(Mul (Const (Q.of_bigint s2'), y1), Mul (Const (Q.of_bigint s1'), y2))
)
| Sub _ -> failwith "scale term: not implemented"
| Mul (y, z) ->
let s1, y1 = scale_term y and s2, y2 = scale_term z in
(Z.mul s1 s2, Mul (y1, y2))
| Pow (t, n) ->
let s, t = scale_term t in
(Z.power_int s n, Pow (t, n))
let scale_term t =
let s, t' = scale_term t in
(s, t')
let rec scale_certificate pos =
match pos with
| Axiom_eq i -> (Z.one, Axiom_eq i)
| Axiom_le i -> (Z.one, Axiom_le i)
| Axiom_lt i -> (Z.one, Axiom_lt i)
| Monoid l -> (Z.one, Monoid l)
| Rational_eq n -> (Q.den n, Rational_eq (Q.of_bigint (Q.num n)))
| Rational_le n -> (Q.den n, Rational_le (Q.of_bigint (Q.num n)))
| Rational_lt n -> (Q.den n, Rational_lt (Q.of_bigint (Q.num n)))
| Square t ->
let s, t' = scale_term t in
(Z.mul s s, Square t')
| Eqmul (t, y) ->
let s1, y1 = scale_term t and s2, y2 = scale_certificate y in
(Z.mul s1 s2, Eqmul (y1, y2))
| Sum (y, z) ->
let s1, y1 = scale_certificate y and s2, y2 = scale_certificate z in
let g = Z.gcd s1 s2 in
let s1' = Z.div s1 g in
let s2' = Z.div s2 g in
( Z.mul g (Z.mul s1' s2')
, Sum
( Product (Rational_le (Q.of_bigint s2'), y1)
, Product (Rational_le (Q.of_bigint s1'), y2) ) )
| Product (y, z) ->
let s1, y1 = scale_certificate y and s2, y2 = scale_certificate z in
(Z.mul s1 s2, Product (y1, y2))
module Z_ = Z
open Micromega
let rec term_to_q_expr = function
| Const n -> PEc (Ml2C.q n)
| Zero -> PEc (Ml2C.q Q.zero)
| 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)
let term_to_q_pol e =
Mc.norm_aux (Ml2C.q Q.zero) (Ml2C.q Q.one) 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_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 (Q.compare n Q.zero) 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)
let rec term_to_z_expr = function
| Const n -> PEc (Ml2C.bigint (Q.to_bigint 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)
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_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 (Q.compare n Q.zero) 0 then Mc.PsatzZ
else Mc.PsatzC (Ml2C.bigint (Q.to_bigint n))
| Square t -> Mc.PsatzSquare (term_to_z_pol t)
| Eqmul (t, y) ->
let is_unit = match t with Const n -> n =/ Q.one | _ -> 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)
(** All constraints (initial or derived) have an index and have a justification i.e., proof.
Given a constraint, all the coefficients are always integers.
*)
open Mutils
open Polynomial
type prf_sys = (cstr * ProofFormat.prf_rule) list
(** 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 xpivot cv1 cv2 =
( { coeffs = Vect.add (Vect.mul cv1 v1) (Vect.mul cv2 v2)
; op = opAdd op1 op2
; cst = (n1 */ cv1) +/ (n2 */ cv2) }
, ProofFormat.add_proof
(ProofFormat.mul_cst_proof cv1 p1)
(ProofFormat.mul_cst_proof cv2 p2) )
in
let a, b = (Vect.get v v1, Vect.get v v2) in
if a =/ Q.zero || b =/ Q.zero then None
else if Int.equal (Q.sign a * Q.sign b) (-1) then
let cv1 = Q.abs b and cv2 = Q.abs a in
Some (xpivot cv1 cv2)
else if op1 == Eq then
let cv1 = Q.neg (b */ Q.of_int (Q.sign a)) and cv2 = Q.abs a in
Some (xpivot cv1 cv2)
else if op2 == Eq then
let cv1 = Q.abs b and cv2 = Q.neg (a */ Q.of_int (Q.sign b)) in
Some (xpivot cv1 cv2)
else None
let pivot v c1 c2 =
let res = pivot v c1 c2 in
( match res with
| None -> ()
| Some (c, _) ->
if Vect.get v c.coeffs =/ Q.zero then ()
else Printf.printf "pivot error %a\n" output_cstr c );
res
let simpl_sys sys =
List.fold_left
(fun acc (c, p) ->
match check_int_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.
[ext_gcd a b = (x,y,g)] iff [ax+by=g]
Source: http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
*)
let rec ext_gcd a b =
if Int.equal (Z_.sign b) 0 then (Z_.one, Z_.zero)
else
let q, r = Z_.quomod a b in
let s, t = ext_gcd b r in
(t, Z_.sub s (Z_.mul q t))
let (c1, p1) (c2, p2) =
if c1.op == Eq && c2.op == Eq then
Vect.exists2
(fun n1 n2 ->
Int.equal (Z_.compare (Z_.gcd (Q.num n1) (Q.num n2)) Z_.one) 0)
c1.coeffs c2.coeffs
else None
let pred l =
let rec rl l =
match l with
| [] -> (None, rl)
| 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 psys = extract2 extract_coprime psys
let pivot_sys v pc psys = apply_and_normalise check_int_sat (pivot v pc) psys
let reduce_coprime psys =
let oeq, sys = extract_coprime_equation psys in
match oeq with
| None -> None
| Some ((v, n1, n2), (c1, p1), (c2, p2)) ->
let l1, l2 = ext_gcd (Q.num n1) (Q.num n2) in
let l1' = Q.of_bigint l1 and l2' = Q.of_bigint 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 =
ProofFormat.add_proof
(ProofFormat.mul_cst_proof l1' p1)
(ProofFormat.mul_cst_proof 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
Vect.find
(fun v n -> if n =/ Q.one || n =/ Q.minus_one then Some v else None)
cstr.coeffs
else None
in
let oeq, sys = extract is_unary_equation psys in
match oeq with
| None -> None
| Some (v, pc) -> Some (pivot_sys v pc sys)
let reduce_var_change psys =
let rec rel_prime vect =
match Vect.choose vect with
| None -> None
| Some (x, v, vect) -> (
let v = Q.num v in
match
Vect.find
(fun x' v' ->
let v' = Q.num v' in
if Z_.equal (Z_.gcd v v') Z_.one then Some (x', v') else None)
vect
with
| Some (x', v') -> Some ((x, v), (x', v'))
| None -> 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 = (Q.of_bigint l1, Q.of_bigint l2) in
let pivot_eq (c', p') =
let {coeffs; op; cst} = c' in
let vx = Vect.get x coeffs in
let vx' = Vect.get x' coeffs in
let m = Q.neg ((vx */ l1) +/ (vx' */ l2)) in
Some
( { coeffs = Vect.add (Vect.mul m c.coeffs) coeffs
; op
; cst = (m */ c.cst) +/ cst }
, ProofFormat.add_proof (ProofFormat.mul_cst_proof m p) p' )
in
Some (apply_and_normalise check_int_sat pivot_eq sys)
let reduction_equations psys =
iterate_until_stable
(app_funs
[reduce_unary; reduce_coprime; reduce_var_change ])
psys
let reduction_equations = tr_cstr_sys "reduction_equations" reduction_equations
(** [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 <=/ Q.one
in
let select_best (x1, i1) (x2, i2) =
if Itv.smaller_itv i1 i2 then (x1, i1) else (x2, i2)
in
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 optimise vect sys with
| None -> acc
| Some i ->
if debug then
Printf.printf "Found a new bound %a in %a" Vect.pp vect Itv.pp i;
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
in
match smallest_interval with
| Some (lb, e, ub) -> (
let lbn, lbd = (Z_.sub (Q.num lb) Z_.one, Q.den lb) in
let ubn, ubd = (Z_.add Z_.one (Q.num ub), Q.den ub) in
match
( direct_linear_prover
( { coeffs = Vect.mul (Q.of_bigint ubd) e
; op = Ge
; cst = Q.of_bigint ubn }
:: sys )
,
direct_linear_prover
( { coeffs = Vect.mul (Q.neg (Q.of_bigint lbd)) e
; op = Ge
; cst = Q.neg (Q.of_bigint lbn) }
:: sys ) )
with
| Some cub, Some clb ->
Some (List.tl (Vect.to_list clb), (lb, e, ub), List.tl (Vect.to_list cub))
| _ -> failwith "Interval without proof" )
| None -> None
let check_sys sys =
List.for_all
(fun (c, p) -> Vect.for_all (fun _ n -> Q.sign n <> 0) c.coeffs)
sys
open ProofFormat
let xlia (can_enum : bool) reduction_equations sys =
let rec enum_proof (id : int) (sys : prf_sys) =
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 -> Unknown
| Some (prf1, (lb, e, ub), prf2) -> (
if debug then
Printf.printf "Found interval: %a in [%s;%s] -> " Vect.pp e
(Q.to_string lb) (Q.to_string ub);
match start_enum id e (Q.ceiling lb) (Q.floor ub) sys with
| Prf prfl ->
Prf
(ProofFormat.Enum
( id
, ProofFormat.proof_of_farkas (env_of_list prf)
(Vect.from_list prf1)
, e
, ProofFormat.proof_of_farkas (env_of_list prf)
(Vect.from_list prf2)
, prfl ))
| _ -> Unknown )
and start_enum id e clb cub sys =
if clb >/ cub then Prf []
else
let eq = {coeffs = e; op = Eq; cst = clb} in
match aux_lia (id + 1) ((eq, ProofFormat.Def id) :: sys) with
| Unknown | Model _ -> Unknown
| Prf prf -> (
match start_enum id e (clb +/ Q.one) cub sys with
| Prf l -> Prf (prf :: l)
| _ -> Unknown )
and aux_lia (id : int) (sys : prf_sys) =
assert (check_sys sys);
if debug then Printf.printf "xlia: %a \n" output_cstr_sys sys;
try
let sys = reduction_equations sys in
if debug then Printf.printf "after reduction: %a \n" output_cstr_sys sys;
match linear_prover_cstr sys with
| Some prf -> Prf (Step (id, prf, Done))
| None -> if can_enum then enum_proof id sys else Unknown
with FoundProof prf ->
Prf (Step (id, prf, Done))
in
let id =
1
+ List.fold_left
(fun acc (_, r) -> max acc (ProofFormat.pr_rule_max_hyp r))
0 sys
in
let orpf =
try
let sys = simpl_sys sys in
aux_lia id sys
with FoundProof pr -> Prf (Step (id, pr, Done))
in
match orpf with
| Unknown | Model _ -> Unknown
| Prf prf ->
let env = CList.interval 0 (id - 1) in
if debug then begin
Printf.fprintf stdout "direct proof %a\n" output_proof prf;
flush stdout
end;
let prf = compile_proof env prf in
Prf prf
let xlia_simplex env red sys =
let compile_prf sys prf =
let id =
1
+ List.fold_left
(fun acc (_, r) -> max acc (ProofFormat.pr_rule_max_hyp r))
0 sys
in
let env = CList.interval 0 (id - 1) in
Prf (compile_proof env prf)
in
try
let sys = red sys in
match Simplex.integer_solver sys with
| None -> Unknown
| Some prf -> compile_prf sys prf
with FoundProof prf -> compile_prf sys (Step (0, prf, Done))
let xlia env0 en red sys =
if use_simplex () then xlia_simplex env0 red sys else xlia en red sys
let gen_bench (tac, prover) can_enum prfdepth sys =
let res = prover can_enum prfdepth sys in
( match dump_file () with
| None -> ()
| Some file ->
let o = open_out (Filename.temp_file ~temp_dir:(Sys.getcwd ()) file ".v") in
let sys = develop_constraints prfdepth z_spec sys in
Printf.fprintf o "Require Import ZArith Lia. Open Scope Z_scope.\n";
Printf.fprintf o "Goal %a.\n" (LinPoly.pp_goal "Z") (List.map fst sys);
begin
match res with
| Unknown | Model _ ->
Printf.fprintf o "Proof.\n intros. Fail %s.\nAbort.\n" tac
| Prf res -> Printf.fprintf o "Proof.\n intros. %s.\nQed.\n" tac
end;
flush o; close_out o );
res
let normalise sys =
List.fold_left
(fun acc s ->
match WithProof.cutting_plane s with
| None -> s :: acc
| Some s' -> s' :: acc)
[] sys
let normalise = tr_sys "normalise" normalise
let elim_redundant sys =
let module VectMap = Map.Make (Vect) in
let elim_eq sys =
List.fold_left
(fun acc (((v, o), prf) as wp) ->
match o with
| Gt -> assert false
| Ge -> wp :: acc
| Eq -> wp :: WithProof.neg wp :: acc)
[] sys
in
let of_list l =
List.fold_left
(fun m (((v, o), prf) as wp) ->
let q, v' = Vect.decomp_cst v in
try
let q', wp' = VectMap.find v' m in
match Q.compare q q' with
| 0 -> if o = Eq then VectMap.add v' (q, wp) m else m
| 1 -> m
| _ -> VectMap.add v' (q, wp) m
with Not_found -> VectMap.add v' (q, wp) m)
VectMap.empty l
in
let to_list m = VectMap.fold (fun _ (_, wp) sys -> wp :: sys) m [] in
to_list (of_list (elim_eq sys))
let elim_redundant sys = tr_sys "elim_redundant" elim_redundant sys
(** [fourier_small] performs some variable elimination and keeps the cutting planes.
To decide which elimination to perform, the constraints are sorted according to
1 - the number of variables
2 - the value of the smallest coefficient
Given the smallest constraint, we eliminate the variable with the smallest coefficient.
The rational is that a constraint with a single variable provides some bound information.
When there are several variables, we hope to eliminate all the variables.
A necessary condition is to take the variable with the smallest coefficient *)
let fourier_small (sys : WithProof.t list) =
let gen_pivot acc (q, x) wp l =
List.fold_left
(fun acc (s, wp') ->
match WithProof.simple_pivot (q, x) wp wp' with
| None -> acc
| Some wp2 -> (
match WithProof.cutting_plane wp2 with
| Some wp2 -> (s, wp2) :: acc
| _ -> acc ))
acc l
in
let rec all_pivots acc l =
match l with
| [] -> acc
| ((_, qx), wp) :: l' -> all_pivots (gen_pivot acc qx wp (acc @ l')) l'
in
List.rev_map snd (all_pivots [] (WithProof.sort sys))
let fourier_small = tr_sys "fourier_small" fourier_small
(** [propagate_bounds sys] generate new constraints by exploiting bounds.
A bound is a constraint of the form c + a.x >= 0
*)
let rev_concat l =
let rec conc acc l =
match l with [] -> acc | l1 :: lr -> conc (List.rev_append l1 acc) lr
in
conc [] l
let pre_process sys =
let sys = normalise sys in
let bnd1 = bound_monomials sys in
let sys1 = normalise (subst (List.rev_append sys bnd1)) in
let pbnd1 = fourier_small sys1 in
let sys2 = elim_redundant (List.rev_append pbnd1 sys1) in
let bnd2 = bound_monomials sys2 in
let sys =
rev_concat [bnd2; saturate_by_linear_equalities sys2; sys2]
in
sys
let lia (can_enum : bool) (prfdepth : int) sys =
let sys = develop_constraints prfdepth z_spec sys in
if debug then begin
Printf.fprintf stdout "Input problem\n";
List.iter (fun s -> Printf.fprintf stdout "%a\n" WithProof.output s) sys;
Printf.fprintf stdout "Input problem\n";
let string_of_op = function Eq -> "=" | Ge -> ">=" | Gt -> ">" in
List.iter
(fun ((p, op), _) ->
Printf.fprintf stdout "(assert (%s %a))\n" (string_of_op op) Vect.pp_smt
p)
sys
end;
let sys = pre_process sys in
let sys' = List.map (fun ((p, o), prf) -> (cstr_of_poly (p, o), prf)) sys in
xlia (List.map fst sys) can_enum reduction_equations sys'
let make_cstr_system sys =
List.map (fun ((p, o), prf) -> (cstr_of_poly (p, o), prf)) sys
let nlia enum prfdepth sys =
let sys = develop_constraints prfdepth z_spec sys in
let is_linear = List.for_all (fun ((p, _), _) -> LinPoly.is_linear p) sys in
if debug then begin
Printf.fprintf stdout "Input problem\n";
List.iter (fun s -> Printf.fprintf stdout "%a\n" WithProof.output s) sys
end;
if is_linear then
xlia (List.map fst sys) enum reduction_equations
(make_cstr_system (pre_process sys))
else
let sys1 =
normalise
(elim_simple_linear_equality (WithProof.subst_constant true sys))
in
let bnd1 = bound_monomials sys1 in
let sys2 = saturate_by_linear_equalities sys1 in
let sys3 = nlinear_preprocess (rev_concat [bnd1; sys1; sys2]) in
let sys4 = make_cstr_system sys3 in
xlia (List.map fst sys) enum reduction_equations sys4
let lia can_enum prfdepth sys = gen_bench ("lia", lia) can_enum prfdepth sys
let nlia enum prfdepth sys = gen_bench ("nia", nlia) enum prfdepth sys