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 { Goptions.get = dump_file } =
Goptions.declare_stringopt_option_and_ref
~key:["Dump"; "Arith"]
~value:None
()
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 make_cstr_system sys =
let map wp =
let ((p, o), prf) = WithProof.repr wp in
(cstr_of_poly (p, o), prf)
in
List.map map sys
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))
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 Simplex.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 Simplex.find_unsat_certificate 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 (WithProof.polynomial pc) || LinPoly.is_linear_for v (WithProof.polynomial pc)
let is_linear_substitution sys wp =
let (p, o), _ = WithProof.repr wp in
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 elim_redundant sys =
let module VectMap = Map.Make (Vect) in
let elim_eq sys =
List.fold_left
(fun acc wp ->
let (_, o), _ = WithProof.repr wp in
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 wp ->
let (v, o), _ = WithProof.repr wp in
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
let bound_monomials (sys : WithProof.t list) =
let (all_bounds,_) = extract_all BoundWithProof.make sys in
let mon = List.mapi (fun i b ->
let v = (BoundWithProof.bound b).Vect.Bound.var in
let m = LinPoly.MonT.retrieve v in
(i,(v,m,b))) all_bounds in
let vars = List.fold_left
(fun acc wp -> ISet.union (LinPoly.monomials (WithProof.polynomial wp)) acc)
ISet.empty sys in
let rec build_constraints l =
match l with
|[] -> Linsolve.empty
| (i,(_,m',_)) ::l ->
let c = build_constraints l in
let cm = Monomial.fold (fun x d acc -> Linsolve.make_mon x i d acc) m' Linsolve.empty in
Linsolve.merge c cm
in
let eqn = build_constraints mon in
let set_constants_for m e =
Monomial.fold (fun x d acc -> Linsolve.set_constant x d e :: acc) m [] in
let rec exp_bound b j =
if j = 1 then Some b
else
let b1 = exp_bound b (j/2) in
match b1 with
| None -> None
| Some b1 ->
match BoundWithProof.mul_bound b1 b1 with
| None -> None
| Some b1_b1 ->
if j mod 2 = 0
then Some b1_b1
else BoundWithProof.mul_bound b b1_b1 in
let rec bound_using_sol sol =
match sol with
| [] -> None
| [x,j] -> let (_,_,b) = List.assoc x mon in
exp_bound b j
| (x,j)::sol'-> let (_,_,b) = List.assoc x mon in
match exp_bound b j with
| None -> None
| Some b -> match bound_using_sol sol' with
|None -> None
| Some b' -> BoundWithProof.mul_bound b b' in
let bound_one_monomial x =
let m = LinPoly.MonT.retrieve x in
if Monomial.degree m <= 1
then []
else
let eqn = set_constants_for m eqn in
if debug then Printf.printf "Equations : %a\n" Linsolve.output_equations eqn ; flush stdout;
let sol = Linsolve.solve_and_enum eqn in
if debug then Printf.printf "Solutions %i \n %a\n" (List.length sol) Linsolve.output_solutions sol;
let l = elim_redundant (CList.map_filter (fun s -> Option.map BoundWithProof.proof (bound_using_sol s)) sol) in
if debug then Printf.printf "New bounds %a" output_sys l; l
in
ISet.fold (fun m acc -> List.rev_append (bound_one_monomial m) acc) vars []
let bound_monomials sys= tr_sys "bound_monomials" bound_monomials sys
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
let sys = List.mapi (fun i (p, o) -> WithProof.mkhyp (LinPoly.linpol_of_pol p) o i) sys in
ProofFormat.Env.make (List.length sys), sys
let square_of_var i =
let x = LinPoly.var i in
WithProof.square (LinPoly.product x x) 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 wp -> LinPoly.is_linear @@ WithProof.polynomial wp) sys in
if is_linear then sys
else
let collect_square =
List.fold_left
(fun acc wp ->
MonMap.union (fun k e1 e2 -> Some e1) acc (LinPoly.collect_square @@ WithProof.polynomial wp))
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
(WithProof.square m s) :: acc)
collect_square sys
in
let collect_vars =
List.fold_left
(fun acc p -> ISet.union acc (LinPoly.variables (WithProof.polynomial 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 env, 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 = make_cstr_system sys 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 env, sys = develop_constraints prfdepth q_spec sys in
let sys = make_cstr_system sys in
match linear_prover_cstr sys with
| None -> Unknown
| Some cert ->
Prf
(ProofFormat.cmpl_prf_rule Mc.normQ CamlToCoq.q
env
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
(** Proof generating pivoting over variable v.
Assumes [a] is the non-zero coefficient for [v] in [c1]. *)
let pivot v (a, c1, p1) (c2, p2) =
let {coeffs = v1; op = op1; cst = n1} = c1
and {coeffs = v2; op = op2; cst = n2} = c2 in
let () = assert (op1 == Eq) in
let xpivot cv1 cv2 =
( { coeffs = Vect.add (Vect.mul cv1 v1) (Vect.mul cv2 v2)
; op = opAdd Eq op2
; cst = (n1 */ cv1) +/ (n2 */ cv2) }
, ProofFormat.add_proof
(ProofFormat.mul_cst_proof cv1 p1)
(ProofFormat.mul_cst_proof cv2 p2) )
in
let b = Vect.get v v2 in
if 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
let cv1 = Q.neg (b */ Q.of_int (Q.sign a)) and cv2 = Q.abs a in
Some (xpivot cv1 cv2)
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
(** [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) =
let () = assert (c1.op == Eq) in
if 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 psys =
let rec rl l =
match l with
| [] -> (None, rl)
| e :: l ->
match (fst e).op with
| Eq ->
begin match extract (extract_coprime e) l with
| None, _ -> xextract2 (e :: rl) l
| Some (r, e'), l' -> (Some (r, e, e'), List.rev_append rl l')
end
| Gt | Ge -> xextract2 (e :: rl) l
in
xextract2 [] psys
let pivot_sys v (cstr, prf) psys =
let a = Vect.get v cstr.coeffs in
if a =/ Q.zero then List.rev psys
else apply_and_normalise check_int_sat (pivot v (a, cstr, prf)) 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, (cstr, prf)) ->
let () = assert (cstr.op == Eq) in
Some (pivot_sys v (cstr, prf) 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
open ProofFormat
let xlia env sys =
let sys = make_cstr_system sys in
match reduction_equations sys with
| sys ->
let sys = List.map WithProof.of_cstr sys in
begin match Simplex.integer_solver sys with
| None -> Unknown
| Some prf -> Prf (compile_proof env prf)
end
| exception FoundProof prf ->
Prf (compile_proof env (Step (0, prf, Done)))
let gen_bench (tac, prover) prfdepth sys =
let res = prover 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 (fun wp -> fst @@ WithProof.repr wp) 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
(** [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 try_pivot qx wp wp' =
match WithProof.simple_pivot qx wp wp' with
| None -> None
| Some wp2 ->
match WithProof.cutting_plane wp2 with
| Some wp2 -> Some wp2
| None -> None
let fourier_small (sys : WithProof.t list) =
let module WPset = Set.Make(WithProof) in
let gen_pivot acc qx wp l =
let fold acc wp' =
match try_pivot qx wp wp' with
| None -> acc
| Some wp2 -> WPset.add wp2 acc
in
let acc = WPset.fold (fun wp acc -> fold acc wp) acc acc in
List.fold_left (fun acc (_,wp') -> fold acc wp') acc l
in
let rec all_pivots acc l =
match l with
| [] -> acc
| ((_, qx), wp) :: l' -> all_pivots (gen_pivot acc qx wp l') l'
in
let sys = WithProof.sort sys in
let res = all_pivots WPset.empty sys in
WPset.elements res
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 (prfdepth : int) sys =
let env, 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 wp ->
let ((p, op), _) = WithProof.repr wp in
Printf.fprintf stdout "(assert (%s %a))\n" (string_of_op op) Vect.pp_smt
p)
sys
end;
let sys = pre_process sys in
xlia env sys
let nlia prfdepth sys =
let env, sys = develop_constraints prfdepth z_spec sys in
let is_linear = List.for_all (fun wp -> LinPoly.is_linear @@ WithProof.polynomial wp) 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 env (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
xlia env sys3
let lia prfdepth sys = gen_bench ("lia", lia) prfdepth sys
let nlia prfdepth sys = gen_bench ("nia", nlia) prfdepth sys