Source file polynomial.ml
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open NumCompat
open Q.Notations
open Mutils
module Mc = Micromega
let max_nb_cstr = ref max_int
type var = int
let debug = false
let ( <+> ) = ( +/ )
let ( <*> ) = ( */ )
module Monomial : sig
type t
val const : t
val is_const : t -> bool
val var : var -> t
val is_var : t -> bool
val get_var : t -> var option
val prod : t -> t -> t
val factor : t -> var -> t option
val compare : t -> t -> int
val pp : out_channel -> t -> unit
val fold : (var -> int -> 'a -> 'a) -> t -> 'a -> 'a
val sqrt : t -> t option
val variables : t -> ISet.t
val degree : t -> int
val subset : t -> t -> bool
val output : out_channel -> t -> unit
end =
struct
type t = int array
let const = [|0|]
let subset m1 m2 =
m1.(0) <= m2.(0) &&
let len1 = Array.length m1 in
let len2 = Array.length m2 in
let get_var m c v =
v+m.(c) , m.(c+1) in
let rec xsubset cur1 v1 e1 cur2 v2 e2 =
match Int.compare v1 v2 with
| 0 -> e1 <= e2 &&
(if cur1 + 2 = len1
then true
else if cur2 + 2 = len2
then false
else
let (v1,e1) = get_var m1 (cur1+2) v1 in
let (v2,e2) = get_var m2 (cur2+2) v2 in
xsubset (cur1+2) v1 e1 (cur2+2) v2 e2)
| -1 -> false
| _ -> if cur2 + 2 = len2
then false
else let (v2,e2) = get_var m2 (cur2+2) v2 in
xsubset cur1 v1 e1 (cur2+2) v2 e2
in
if len1 <= 1 then true
else if len2 <= 1 then false
else xsubset 1 m1.(1) m1.(2) 1 m2.(1) m2.(2)
let is_const (m : t) = match m with [|_|] -> true | _ -> false
let var x = [|1; x; 1|]
let is_var (m : t) = Int.equal m.(0) 1
let get_var (m : t) = match m with
| [|1; x; _|] -> Some x
| _ -> None
let prod (m1 : t) (m2 : t) =
let len1 = Array.length m1 in
let len2 = Array.length m2 in
let rec nvars accu cur1 cur2 i1 i2 =
if Int.equal i1 len1 && Int.equal i2 len2 then accu
else if Int.equal i1 len1 then accu + (len2 - i2)
else if Int.equal i2 len2 then accu + (len1 - i1)
else
let ncur1 = cur1 + m1.(i1) in
let ncur2 = cur2 + m2.(i2) in
if ncur1 < ncur2 then nvars (accu + 2) ncur1 cur2 (i1 + 2) i2
else if ncur1 > ncur2 then nvars (accu + 2) cur1 ncur2 i1 (i2 + 2)
else nvars (accu + 2) ncur1 ncur2 (i1 + 2) (i2 + 2)
in
let n = nvars 1 0 0 1 1 in
let m = Array.make n 0 in
let () = m.(0) <- m1.(0) + m2.(0) in
let rec set cur cur1 cur2 i i1 i2 =
if Int.equal i1 len1 && Int.equal i2 len2 then ()
else if Int.equal i1 len1 then
let ncur2 = cur2 + m2.(i2) in
let () = m.(i) <- ncur2 - cur in
let () = m.(i + 1) <- m2.(i2 + 1) in
set ncur2 cur1 ncur2 (i + 2) i1 (i2 + 2)
else if Int.equal i2 len2 then
let ncur1 = cur1 + m1.(i1) in
let () = m.(i) <- ncur1 - cur in
let () = m.(i + 1) <- m1.(i1 + 1) in
set ncur1 ncur1 cur2 (i + 2) (i1 + 2) i2
else
let ncur1 = cur1 + m1.(i1) in
let ncur2 = cur2 + m2.(i2) in
if ncur1 < ncur2 then
let () = m.(i) <- ncur1 - cur in
let () = m.(i + 1) <- m1.(i1 + 1) in
set ncur1 ncur1 cur2 (i + 2) (i1 + 2) i2
else if ncur1 > ncur2 then
let () = m.(i) <- ncur2 - cur in
let () = m.(i + 1) <- m2.(i2 + 1) in
set ncur2 cur1 ncur2 (i + 2) i1 (i2 + 2)
else
let () = m.(i) <- ncur1 - cur in
let () = m.(i + 1) <- m1.(i1 + 1) + m2.(i2 + 1) in
set ncur1 ncur1 ncur2 (i + 2) (i1 + 2) (i2 + 2)
in
let () = set 0 0 0 1 1 1 in
m
let factor (m : t) (x : var) =
let len = Array.length m in
let rec factor cur i =
if Int.equal i len then None
else
let ncur = cur + m.(i) in
let k = m.(i + 1) in
if ncur < x then factor ncur (i + 2)
else if x < ncur then None
else if Int.equal k 1 then
let ans = Array.make (len - 2) 0 in
let () = ans.(0) <- m.(0) - 1 in
let () = Array.blit m 1 ans 1 (i - 1) in
let () = Array.blit m (i + 2) ans i (len - i - 2) in
let () = if not (Int.equal len (i + 2)) then ans.(i) <- ans.(i) + m.(i) in
Some ans
else
let ans = Array.copy m in
let () = ans.(0) <- ans.(0) - 1 in
let () = ans.(i + 1) <- ans.(i + 1) - 1 in
Some ans
in
factor 0 1
let compare (m1 : t) (m2 : t) = CArray.compare Int.compare m1 m2
let sqrt (m : t) = match m with
| [|_|] -> Some const
| _ ->
let m = Array.copy m in
let len = Array.length m in
let () = m.(0) <- m.(0) / 2 in
let rec set i =
if Int.equal i len then ()
else
let v = m.(i + 1) in
let () = if v mod 2 = 0 then m.(i + 1) <- v / 2 else raise_notrace Exit in
set (i + 2)
in
try let () = set 1 in Some m with Exit -> None
let degree (m : t) = m.(0)
let fold f (m : t) accu =
let len = Array.length m in
let rec fold accu cur i =
if Int.equal i len then accu
else
let cur = cur + m.(i) in
let accu = f cur m.(i + 1) accu in
fold accu cur (i + 2)
in
fold accu 0 1
let output o m = fold (fun v i () -> Printf.fprintf o "x%i^%i" v i) m ()
let variables (m : t) =
fold (fun x _ accu -> ISet.add x accu) m ISet.empty
let pp o m =
let pp_elt o (k, v) =
if v = 1 then Printf.fprintf o "x%i" k else Printf.fprintf o "x%i^%i" k v
in
let rec pp_list o l =
match l with
| [] -> ()
| [e] -> pp_elt o e
| e :: l -> Printf.fprintf o "%a*%a" pp_elt e pp_list l
in
pp_list o (List.rev @@ fold (fun x v accu -> (x, v) :: accu) m [])
end
module MonMap = struct
include Map.Make (Monomial)
let union f =
merge (fun x v1 v2 ->
match (v1, v2) with
| None, None -> None
| Some v, None | None, Some v -> Some v
| Some v1, Some v2 -> f x v1 v2)
end
let pp_mon o (m, i) =
if Monomial.is_const m then
if Q.zero =/ i then () else Printf.fprintf o "%s" (Q.to_string i)
else if Q.one =/ i then Monomial.pp o m
else if Q.minus_one =/ i then Printf.fprintf o "-%a" Monomial.pp m
else if Q.zero =/ i then ()
else Printf.fprintf o "%s*%a" (Q.to_string i) Monomial.pp m
module Poly :
sig
type t
val pp : out_channel -> t -> unit
val get : Monomial.t -> t -> Q.t
val variable : var -> t
val add : Monomial.t -> Q.t -> t -> t
val constant : Q.t -> t
val product : t -> t -> t
val addition : t -> t -> t
val uminus : t -> t
val fold : (Monomial.t -> Q.t -> 'a -> 'a) -> t -> 'a -> 'a
val factorise : var -> t -> t * t
end = struct
module P = Map.Make (Monomial)
open P
type t = Q.t P.t
let pp o p = P.iter (fun mn i -> Printf.fprintf o "%a + " pp_mon (mn, i)) p
let get : Monomial.t -> t -> Q.t =
fun mn p -> try find mn p with Not_found -> Q.zero
let variable : var -> t = fun x -> add (Monomial.var x) Q.one empty
let constant : Q.t -> t = fun c -> add Monomial.const c empty
let add : Monomial.t -> Q.t -> t -> t =
fun mn v p ->
if Q.sign v = 0 then p
else
let vl = get mn p <+> v in
if Q.sign vl = 0 then remove mn p else add mn vl p
(** Design choice: empty is not a polynomial
I do not remember why ....
**)
let mult : Monomial.t -> Q.t -> t -> t =
fun mn v p ->
if Q.sign v = 0 then constant Q.zero
else
fold
(fun mn' v' res -> P.add (Monomial.prod mn mn') (v <*> v') res)
p empty
let addition : t -> t -> t =
fun p1 p2 -> fold (fun mn v p -> add mn v p) p1 p2
let product : t -> t -> t =
fun p1 p2 -> fold (fun mn v res -> addition (mult mn v p2) res) p1 empty
let uminus : t -> t = fun p -> map (fun v -> Q.neg v) p
let fold = P.fold
let factorise x p =
P.fold
(fun m v (px, cx) ->
match Monomial.factor m x with
| None -> (px, add m v cx)
| Some mx ->
(add mx v px, cx))
p
(constant Q.zero, constant Q.zero)
end
type vector = Vect.t
type cstr = {coeffs : vector; op : op; cst : Q.t}
and op = Eq | Ge | Gt
exception Strict
let is_strict c = c.op = Gt
let eval_op = function Eq -> ( =/ ) | Ge -> ( >=/ ) | Gt -> ( >/ )
let string_of_op = function Eq -> "=" | Ge -> ">=" | Gt -> ">"
let compare_op o1 o2 =
match (o1, o2) with
| Eq, Eq -> 0
| Eq, _ -> -1
| _, Eq -> 1
| Ge, Ge -> 0
| Ge, _ -> -1
| _, Ge -> 1
| Gt, Gt -> 0
let output_cstr o {coeffs; op; cst} =
Printf.fprintf o "%a %s %s" Vect.pp coeffs (string_of_op op) (Q.to_string cst)
let opMult o1 o2 =
match (o1, o2) with Eq, _ | _, Eq -> Eq | Ge, _ | _, Ge -> Ge | Gt, Gt -> Gt
let opAdd o1 o2 =
match (o1, o2) with Eq, x | x, Eq -> x | Gt, x | x, Gt -> Gt | Ge, Ge -> Ge
module LinPoly = struct
(** A linear polynomial a0 + a1.x1 + ... + an.xn
By convention, the constant a0 is the coefficient of the variable 0.
*)
type t = Vect.t
module MonT = struct
module MonoMap = Map.Make (Monomial)
module IntMap = Map.Make (Int)
(** A hash table might be preferable but requires a hash function. *)
let (index_of_monomial : int MonoMap.t ref) = ref MonoMap.empty
let (monomial_of_index : Monomial.t IntMap.t ref) = ref IntMap.empty
let fresh = ref 0
let reserve vr =
if !fresh > vr then failwith (Printf.sprintf "Cannot reserve %i" vr)
else fresh := vr + 1
let safe_reserve vr = if !fresh > vr then () else fresh := vr + 1
let get_fresh () =
let vr = !fresh in
incr fresh; vr
let register m =
try MonoMap.find m !index_of_monomial
with Not_found ->
let res = !fresh in
index_of_monomial := MonoMap.add m res !index_of_monomial;
monomial_of_index := IntMap.add res m !monomial_of_index;
incr fresh;
res
let retrieve i = IntMap.find i !monomial_of_index
let clear () =
index_of_monomial := MonoMap.empty;
monomial_of_index := IntMap.empty;
fresh := 0;
ignore (register Monomial.const)
let _ = register Monomial.const
end
let var v = Vect.set (MonT.register (Monomial.var v)) Q.one Vect.null
let of_monomial m =
let v = MonT.register m in
Vect.set v Q.one Vect.null
let linpol_of_pol p =
Poly.fold
(fun mon num vct ->
let vr = MonT.register mon in
Vect.set vr num vct)
p Vect.null
let pol_of_linpol v =
Vect.fold
(fun p vr n -> Poly.add (MonT.retrieve vr) n p)
(Poly.constant Q.zero) v
let coq_poly_of_linpol cst p =
let pol_of_mon m =
Monomial.fold
(fun x v p ->
Mc.PEmul (Mc.PEpow (Mc.PEX (CamlToCoq.positive x), CamlToCoq.n v), p))
m
(Mc.PEc (cst Q.one))
in
Vect.fold
(fun acc x v ->
let mn = MonT.retrieve x in
Mc.PEadd (Mc.PEmul (Mc.PEc (cst v), pol_of_mon mn), acc))
(Mc.PEc (cst Q.zero))
p
let pp_var o vr =
try Monomial.pp o (MonT.retrieve vr)
with Not_found -> Printf.fprintf o "v%i" vr
let pp o p = Vect.pp_gen pp_var o p
let constant c = if Q.sign c = 0 then Vect.null else Vect.set 0 c Vect.null
let is_linear p =
Vect.for_all
(fun v _ ->
let mn = MonT.retrieve v in
Monomial.is_var mn || Monomial.is_const mn)
p
let is_variable p =
let (x, v), r = Vect.decomp_fst p in
if Vect.is_null r && v >/ Q.zero then Monomial.get_var (MonT.retrieve x)
else None
let factorise x p =
let px, cx = Poly.factorise x (pol_of_linpol p) in
(linpol_of_pol px, linpol_of_pol cx)
let is_linear_for x p =
let a, b = factorise x p in
Vect.is_constant a
let search_all_linear p l =
Vect.fold
(fun acc x v ->
if p v then
let x' = MonT.retrieve x in
match Monomial.get_var x' with
| None -> acc
| Some x -> if is_linear_for x l then x :: acc else acc
else acc)
[] l
let min_list (l : int list) =
match l with [] -> None | e :: l -> Some (List.fold_left min e l)
let search_linear p l = min_list (search_all_linear p l)
let product p1 p2 =
linpol_of_pol (Poly.product (pol_of_linpol p1) (pol_of_linpol p2))
let addition p1 p2 = Vect.add p1 p2
let of_vect v =
Vect.fold
(fun acc v vl -> addition (product (var v) (constant vl)) acc)
Vect.null v
let variables p =
Vect.fold
(fun acc v _ -> ISet.union (Monomial.variables (MonT.retrieve v)) acc)
ISet.empty p
let monomials p = Vect.fold (fun acc v _ -> ISet.add v acc) ISet.empty p
let pp_goal typ o l =
let vars =
List.fold_left
(fun acc p -> ISet.union acc (variables (fst p)))
ISet.empty l
in
let pp_vars o i =
ISet.iter (fun v -> Printf.fprintf o "(x%i : %s) " v typ) vars
in
Printf.fprintf o "forall %a\n" pp_vars vars;
List.iteri
(fun i (p, op) ->
Printf.fprintf o "(H%i : %a %s 0)\n" i pp p (string_of_op op))
l;
Printf.fprintf o ", False\n"
let collect_square p =
Vect.fold
(fun acc v _ ->
let m = MonT.retrieve v in
match Monomial.sqrt m with None -> acc | Some s -> MonMap.add s m acc)
MonMap.empty p
end
module ProofFormat = struct
type prf_rule =
| Annot of string * prf_rule
| Hyp of int
| Def of int
| Ref of int
| Cst of Q.t
| Zero
| Square of Vect.t
| MulC of Vect.t * prf_rule
| Gcd of Z.t * prf_rule
| MulPrf of prf_rule * prf_rule
| AddPrf of prf_rule * prf_rule
| CutPrf of prf_rule
| LetPrf of prf_rule * prf_rule
type proof =
| Done
| Step of int * prf_rule * proof
| Split of int * Vect.t * proof * proof
| Enum of int * prf_rule * Vect.t * prf_rule * proof list
| ExProof of int * int * int * var * var * var * proof
let rec output_prf_rule o = function
| Annot (s, p) -> Printf.fprintf o "(%a)@%s" output_prf_rule p s
| Hyp i -> Printf.fprintf o "Hyp %i" i
| Def i -> Printf.fprintf o "Def %i" i
| Ref i -> Printf.fprintf o "Ref %i" i
| LetPrf (p1, p2) ->
Printf.fprintf o "Let (%a) in %a" output_prf_rule p1 output_prf_rule p2
| Cst c -> Printf.fprintf o "Cst %s" (Q.to_string c)
| Zero -> Printf.fprintf o "Zero"
| Square s -> Printf.fprintf o "(%a)^2" Poly.pp (LinPoly.pol_of_linpol s)
| MulC (p, pr) ->
Printf.fprintf o "(%a) * (%a)" Poly.pp (LinPoly.pol_of_linpol p)
output_prf_rule pr
| MulPrf (p1, p2) ->
Printf.fprintf o "(%a) * (%a)" output_prf_rule p1 output_prf_rule p2
| AddPrf (p1, p2) ->
Printf.fprintf o "%a + %a" output_prf_rule p1 output_prf_rule p2
| CutPrf p -> Printf.fprintf o "[%a]" output_prf_rule p
| Gcd (c, p) -> Printf.fprintf o "(%a)/%s" output_prf_rule p (Z.to_string c)
let rec output_proof o = function
| Done -> Printf.fprintf o "."
| Step (i, p, pf) ->
Printf.fprintf o "%i:= %a\n ; %a" i output_prf_rule p output_proof pf
| Split (i, v, p1, p2) ->
Printf.fprintf o "%i:=%a ; { %a } { %a }" i Vect.pp v output_proof p1
output_proof p2
| Enum (i, p1, v, p2, pl) ->
Printf.fprintf o "%i{%a<=%a<=%a}%a" i output_prf_rule p1 Vect.pp v
output_prf_rule p2 (pp_list ";" output_proof) pl
| ExProof (i, j, k, x, z, t, pr) ->
Printf.fprintf o "%i := %i = %i - %i ; %i := %i >= 0 ; %i := %i >= 0 ; %a"
i x z t j z k t output_proof pr
module OrdPrfRule = struct
type t = prf_rule
let id_of_constr = function
| Annot _ -> 0
| Hyp _ -> 1
| Def _ -> 2
| Ref _ -> 3
| Cst _ -> 4
| Zero -> 5
| Square _ -> 6
| MulC _ -> 7
| Gcd _ -> 8
| MulPrf _ -> 9
| AddPrf _ -> 10
| CutPrf _ -> 11
| LetPrf _ -> 12
let cmp_pair c1 c2 (x1, x2) (y1, y2) =
match c1 x1 y1 with 0 -> c2 x2 y2 | i -> i
let rec compare p1 p2 =
match (p1, p2) with
| Annot (s1, p1), Annot (s2, p2) ->
if s1 = s2 then compare p1 p2 else String.compare s1 s2
| Hyp i, Hyp j -> Int.compare i j
| Def i, Def j -> Int.compare i j
| Ref i, Ref j -> Int.compare i j
| Cst n, Cst m -> Q.compare n m
| Zero, Zero -> 0
| Square v1, Square v2 -> Vect.compare v1 v2
| MulC (v1, p1), MulC (v2, p2) ->
cmp_pair Vect.compare compare (v1, p1) (v2, p2)
| Gcd (b1, p1), Gcd (b2, p2) ->
cmp_pair Z.compare compare (b1, p1) (b2, p2)
| MulPrf (p1, q1), MulPrf (p2, q2) ->
cmp_pair compare compare (p1, q1) (p2, q2)
| AddPrf (p1, q1), AddPrf (p2, q2) ->
cmp_pair compare compare (p1, q1) (p2, q2)
| CutPrf p, CutPrf p' -> compare p p'
| LetPrf(p1,q1) , LetPrf(p2,q2) ->
cmp_pair compare compare (p1, q1) (p2, q2)
| _, _ -> Int.compare (id_of_constr p1) (id_of_constr p2)
end
module PrfRuleMap = Map.Make (OrdPrfRule)
let rec pr_size = function
| Annot (_, p) -> pr_size p
| Zero | Square _ -> Q.zero
| Hyp _ -> Q.one
| Def _ -> Q.one
| Ref _ -> Q.one
| Cst n -> n
| Gcd (i, p) -> pr_size p // Q.of_bigint i
| MulPrf (p1, p2) | AddPrf (p1, p2) | LetPrf (p1, p2) ->
pr_size p1 +/ pr_size p2
| CutPrf p -> pr_size p
| MulC (v, p) -> pr_size p
let rec pr_unit = function
| Annot (_, p) -> pr_unit p
| Zero | Square _ -> true
| Hyp _ -> true
| Def _ -> true
| Cst n -> true
| _ -> false
let rec pr_rule_max_hyp = function
| Annot (_, p) -> pr_rule_max_hyp p
| Hyp i -> i
| Def i -> -1
| Ref i -> -1
| Cst _ | Zero | Square _ -> -1
| MulC (_, p) | CutPrf p | Gcd (_, p) -> pr_rule_max_hyp p
| MulPrf (p1, p2) | AddPrf (p1, p2) | LetPrf (p1, p2) ->
max (pr_rule_max_hyp p1) (pr_rule_max_hyp p2)
let rec pr_rule_max_def = function
| Annot (_, p) -> pr_rule_max_hyp p
| Hyp i -> -1
| Def i -> i
| Ref _ -> -1
| Cst _ | Zero | Square _ -> -1
| MulC (_, p) | CutPrf p | Gcd (_, p) -> pr_rule_max_def p
| MulPrf (p1, p2) | AddPrf (p1, p2) | LetPrf (p1, p2) ->
max (pr_rule_max_def p1) (pr_rule_max_def p2)
let rec proof_max_def = function
| Done -> -1
| Step (i, pr, prf) -> max i (max (pr_rule_max_def pr) (proof_max_def prf))
| Split (i, _, p1, p2) -> max i (max (proof_max_def p1) (proof_max_def p2))
| Enum (i, p1, _, p2, l) ->
let m = max (pr_rule_max_def p1) (pr_rule_max_def p2) in
List.fold_left (fun i prf -> max i (proof_max_def prf)) (max i m) l
| ExProof (i, j, k, _, _, _, prf) ->
max (max (max i j) k) (proof_max_def prf)
(** [pr_rule_def_cut id pr] gives an explicit [id] to cut rules.
This is because the Coq proof format only accept they as a proof-step *)
let pr_rule_def_cut m id p =
let rec pr_rule_def_cut m id = function
| Annot (_, p) -> pr_rule_def_cut m id p
| MulC (p, prf) ->
let bds, m, id', prf' = pr_rule_def_cut m id prf in
(bds, m, id', MulC (p, prf'))
| MulPrf (p1, p2) ->
let bds1, m, id, p1 = pr_rule_def_cut m id p1 in
let bds2, m, id, p2 = pr_rule_def_cut m id p2 in
(bds2 @ bds1, m, id, MulPrf (p1, p2))
| AddPrf (p1, p2) ->
let bds1, m, id, p1 = pr_rule_def_cut m id p1 in
let bds2, m, id, p2 = pr_rule_def_cut m id p2 in
(bds2 @ bds1, m, id, AddPrf (p1, p2))
| LetPrf (p1, p2) ->
let bds1, m, id, p1 = pr_rule_def_cut m id p1 in
let bds2, m, id, p2 = pr_rule_def_cut m id p2 in
(bds2 @ bds1, m, id, LetPrf (p1, p2))
| CutPrf p | Gcd (_, p) -> (
let bds, m, id, p = pr_rule_def_cut m id p in
try
let id' = PrfRuleMap.find p m in
(bds, m, id, Def id')
with Not_found ->
let m = PrfRuleMap.add p id m in
((id, p) :: bds, m, id + 1, Def id) )
| (Square _ | Cst _ | Def _ | Hyp _ | Ref _ | Zero) as x -> ([], m, id, x)
in
pr_rule_def_cut m id p
let pr_rule_def_cut m id = function
| CutPrf p ->
let bds, m, ids, p' = pr_rule_def_cut m id p in
(bds, m, ids, CutPrf p')
| p -> pr_rule_def_cut m id p
let rec implicit_cut p = match p with CutPrf p -> implicit_cut p | _ -> p
let rec pr_rule_collect_defs pr =
match pr with
| Annot (_, pr) -> pr_rule_collect_defs pr
| Def i -> ISet.add i ISet.empty
| Hyp i -> ISet.empty
| Ref i -> ISet.empty
| Cst _ | Zero | Square _ -> ISet.empty
| MulC (_, pr) | Gcd (_, pr) | CutPrf pr -> pr_rule_collect_defs pr
| MulPrf (p1, p2) | AddPrf (p1, p2) | LetPrf (p1, p2) ->
ISet.union (pr_rule_collect_defs p1) (pr_rule_collect_defs p2)
let add_proof x y =
match (x, y) with Zero, p | p, Zero -> p | _ -> AddPrf (x, y)
let rec mul_cst_proof c p =
match p with
| Annot (s, p) -> Annot (s, mul_cst_proof c p)
| MulC (v, p') -> MulC (Vect.mul c v, p')
| _ -> (
match Q.sign c with
| 0 -> Zero
| -1 ->
MulC (LinPoly.constant c, p)
| 1 -> if Q.one =/ c then p else MulPrf (Cst c, p)
| _ -> assert false )
let sMulC v p =
let c, v' = Vect.decomp_cst v in
if Vect.is_null v' then mul_cst_proof c p else MulC (v, p)
let mul_proof p1 p2 =
match (p1, p2) with
| Zero, _ | _, Zero -> Zero
| Cst c, p | p, Cst c -> mul_cst_proof c p
| _, _ -> MulPrf (p1, p2)
let prf_rule_of_map m =
PrfRuleMap.fold (fun k v acc -> add_proof (sMulC v k) acc) m Zero
let rec dev_prf_rule p =
match p with
| Annot (s, p) -> dev_prf_rule p
| Hyp _ | Def _ | Ref _ | Cst _ | Zero | Square _ ->
PrfRuleMap.singleton p (LinPoly.constant Q.one)
| MulC (v, p) ->
PrfRuleMap.map (fun v1 -> LinPoly.product v v1) (dev_prf_rule p)
| AddPrf (p1, p2) ->
PrfRuleMap.merge
(fun k o1 o2 ->
match (o1, o2) with
| None, None -> None
| None, Some v | Some v, None -> Some v
| Some v1, Some v2 -> Some (LinPoly.addition v1 v2))
(dev_prf_rule p1) (dev_prf_rule p2)
| MulPrf (p1, p2) -> (
let p1' = dev_prf_rule p1 in
let p2' = dev_prf_rule p2 in
let p1'' = prf_rule_of_map p1' in
let p2'' = prf_rule_of_map p2' in
match p1'' with
| Cst c -> PrfRuleMap.map (fun v1 -> Vect.mul c v1) p2'
| _ -> PrfRuleMap.singleton (MulPrf (p1'', p2'')) (LinPoly.constant Q.one)
)
| Gcd (c, p) ->
PrfRuleMap.singleton
(Gcd (c, prf_rule_of_map (dev_prf_rule p)))
(LinPoly.constant Q.one)
| CutPrf p ->
PrfRuleMap.singleton
(CutPrf (prf_rule_of_map (dev_prf_rule p)))
(LinPoly.constant Q.one)
| LetPrf (p1, p2) ->
let p1' = dev_prf_rule p1 in
let p2' = dev_prf_rule p2 in
let p1'' = prf_rule_of_map p1' in
let p2'' = prf_rule_of_map p2' in
PrfRuleMap.singleton (LetPrf (p1'', p2'')) (LinPoly.constant Q.one)
let simplify_prf_rule p = prf_rule_of_map (dev_prf_rule p)
(** [simplify_proof p] removes proof steps that are never re-used. *)
let rec simplify_proof p =
match p with
| Done -> (Done, ISet.empty)
| Step (i, pr, Done) -> (p, ISet.add i (pr_rule_collect_defs pr))
| Step (i, pr, prf) ->
let prf', hyps = simplify_proof prf in
if not (ISet.mem i hyps) then (prf', hyps)
else
( Step (i, pr, prf')
, ISet.add i (ISet.union (pr_rule_collect_defs pr) hyps) )
| Split (i, v, p1, p2) ->
let p1, h1 = simplify_proof p1 in
let p2, h2 = simplify_proof p2 in
if not (ISet.mem i h1) then (p1, h1)
else if not (ISet.mem i h2) then (p2, h2)
else (Split (i, v, p1, p2), ISet.add i (ISet.union h1 h2))
| Enum (i, p1, v, p2, pl) ->
let pl, hl = List.split (List.map simplify_proof pl) in
let hyps = List.fold_left ISet.union ISet.empty hl in
( Enum (i, p1, v, p2, pl)
, ISet.add i
(ISet.union
(ISet.union (pr_rule_collect_defs p1) (pr_rule_collect_defs p2))
hyps) )
| ExProof (i, j, k, x, z, t, prf) ->
let prf', hyps = simplify_proof prf in
if
(not (ISet.mem i hyps))
&& (not (ISet.mem j hyps))
&& not (ISet.mem k hyps)
then (prf', hyps)
else
( ExProof (i, j, k, x, z, t, prf')
, ISet.add i (ISet.add j (ISet.add k hyps)) )
let rec normalise_proof id prf =
match prf with
| Done -> (id, Done)
| Step (i, Gcd (c, p), Done) -> normalise_proof id (Step (i, p, Done))
| Step (i, p, prf) ->
let bds, m, id, p' =
pr_rule_def_cut PrfRuleMap.empty id (simplify_prf_rule p)
in
let id, prf = normalise_proof id prf in
let prf =
List.fold_left
(fun acc (i, p) -> Step (i, CutPrf p, acc))
(Step (i, p', prf))
bds
in
(id, prf)
| Split (i, v, p1, p2) ->
let id, p1 = normalise_proof id p1 in
let id, p2 = normalise_proof id p2 in
(id, Split (i, v, p1, p2))
| ExProof (i, j, k, x, z, t, prf) ->
let id, prf = normalise_proof id prf in
(id, ExProof (i, j, k, x, z, t, prf))
| Enum (i, p1, v, p2, pl) ->
let bds1, m, id, p1' =
pr_rule_def_cut PrfRuleMap.empty id (implicit_cut p1)
in
let bds2, m, id, p2' = pr_rule_def_cut m id (implicit_cut p2) in
let ids, prfs = List.split (List.map (normalise_proof id) pl) in
( List.fold_left max 0 ids
, List.fold_left
(fun acc (i, p) -> Step (i, CutPrf p, acc))
(Enum (i, p1', v, p2', prfs))
(bds2 @ bds1) )
let normalise_proof id prf =
let prf = fst (simplify_proof prf) in
let res = normalise_proof id prf in
if debug then
Printf.printf "normalise_proof %a -> %a" output_proof prf output_proof
(snd res);
res
let proof_of_farkas env vect =
Vect.fold
(fun prf x n -> add_proof (mul_cst_proof n (IMap.find x env)) prf)
Zero vect
module Env :
sig
type t
val make : int -> t
val id_of_def : int -> t -> int
val id_of_hyp : int -> t -> int
val push_ref : t -> t
val push_def : int -> t -> t
end =
struct
type t =
{
lref : int;
ndefs : int;
ldefs : int Int.Map.t;
nhyps : int;
}
let push_ref { nhyps; ndefs; lref; ldefs } =
{ nhyps; ndefs; lref = lref + 1; ldefs }
let push_def i { nhyps; ndefs; lref; ldefs } =
let () = if lref <> 0 then failwith "Cannot push def" in
{ nhyps; ndefs = ndefs + 1; lref; ldefs = Int.Map.add i ndefs ldefs }
let make n = { nhyps = n; ndefs = 0; lref = 0; ldefs = Int.Map.empty }
let id_of_def def { nhyps; ndefs; lref; ldefs } =
try
let pos = Int.Map.find def ldefs in
lref + (ndefs - pos - 1)
with Not_found -> failwith "Cannot find def"
let id_of_hyp h { nhyps; ndefs; lref; ldefs } =
if 0 <= h && h < nhyps then lref + ndefs + h
else failwith "Cannot find hyp"
end
let cmpl_prf_rule norm (cst : Q.t -> 'a) env prf =
let rec cmpl env = function
| Annot (s, p) -> cmpl env p
| Ref i -> Mc.PsatzIn (CamlToCoq.nat i)
| Hyp h -> Mc.PsatzIn (CamlToCoq.nat (Env.id_of_hyp h env))
| Def d -> Mc.PsatzIn (CamlToCoq.nat (Env.id_of_def d env))
| Cst i -> Mc.PsatzC (cst i)
| Zero -> Mc.PsatzZ
| MulPrf (p1, p2) -> Mc.PsatzMulE (cmpl env p1, cmpl env p2)
| AddPrf (p1, p2) -> Mc.PsatzAdd (cmpl env p1, cmpl env p2)
| LetPrf (p1, p2) -> Mc.PsatzLet (cmpl env p1, cmpl (Env.push_ref env) p2)
| MulC (lp, p) ->
let lp = norm (LinPoly.coq_poly_of_linpol cst lp) in
Mc.PsatzMulC (lp, cmpl env p)
| Square lp -> Mc.PsatzSquare (norm (LinPoly.coq_poly_of_linpol cst lp))
| _ -> failwith "Cuts should already be compiled"
in
cmpl env prf
let cmpl_prf_rule_z env r =
cmpl_prf_rule Mc.normZ (fun x -> CamlToCoq.bigint (Q.num x)) env r
let cmpl_pol_z lp =
try
let cst x = CamlToCoq.bigint (Q.num x) in
Mc.normZ (LinPoly.coq_poly_of_linpol cst lp)
with x ->
Printf.printf "cmpl_pol_z %s %a\n" (Printexc.to_string x) LinPoly.pp lp;
raise x
let rec cmpl_proof env prf =
match prf with
| Done -> Mc.DoneProof
| Step (i, p, prf) -> (
match p with
| CutPrf p' ->
Mc.CutProof (cmpl_prf_rule_z env p', cmpl_proof (Env.push_def i env) prf)
| _ -> Mc.RatProof (cmpl_prf_rule_z env p, cmpl_proof (Env.push_def i env) prf)
)
| Split (i, v, p1, p2) ->
Mc.SplitProof
( cmpl_pol_z v
, cmpl_proof (Env.push_def i env) p1
, cmpl_proof (Env.push_def i env) p2 )
| Enum (i, p1, _, p2, l) ->
Mc.EnumProof
( cmpl_prf_rule_z env p1
, cmpl_prf_rule_z env p2
, List.map (cmpl_proof (Env.push_def i env)) l )
| ExProof (i, j, k, x, _, _, prf) ->
Mc.ExProof
(CamlToCoq.positive x, cmpl_proof (Env.push_def i (Env.push_def j (Env.push_def k env))) prf)
let compile_proof env prf =
let id = 1 + proof_max_def prf in
let _, prf = normalise_proof id prf in
cmpl_proof env prf
end
module WithProof = struct
type t = (LinPoly.t * op) * ProofFormat.prf_rule
let compare : t -> t -> int =
fun ((lp1, o1), _) ((lp2, o2), _) ->
let c = Vect.compare lp1 lp2 in
if c = 0 then compare_op o1 o2 else c
let annot s (p, prf) = (p, ProofFormat.Annot (s, prf))
let output o ((lp, op), prf) =
Printf.fprintf o "%a %s 0 by %a\n" LinPoly.pp lp (string_of_op op)
ProofFormat.output_prf_rule prf
let output_sys o l = List.iter (Printf.fprintf o "%a\n" output) l
exception InvalidProof
let zero = ((Vect.null, Eq), ProofFormat.Zero)
let const n = ((LinPoly.constant n, Ge), ProofFormat.Cst n)
let of_cstr (c, prf) = ((Vect.set 0 (Q.neg c.cst) c.coeffs, c.op), prf)
let product : t -> t -> t =
fun ((p1, o1), prf1) ((p2, o2), prf2) ->
((LinPoly.product p1 p2, opMult o1 o2), ProofFormat.mul_proof prf1 prf2)
let addition : t -> t -> t =
fun ((p1, o1), prf1) ((p2, o2), prf2) ->
((Vect.add p1 p2, opAdd o1 o2), ProofFormat.add_proof prf1 prf2)
let neg : t -> t =
fun ((p1, o1), prf1) ->
match o1 with
| Eq ->
((Vect.mul Q.minus_one p1, o1), ProofFormat.mul_cst_proof Q.minus_one prf1)
| _ -> failwith "neg: invalid proof"
let mult p ((p1, o1), prf1) =
match o1 with
| Eq -> ((LinPoly.product p p1, o1), ProofFormat.sMulC p prf1)
| Gt | Ge ->
let n, r = Vect.decomp_cst p in
if Vect.is_null r && n >/ Q.zero then
((LinPoly.product p p1, o1), ProofFormat.mul_cst_proof n prf1)
else (
if debug then
Printf.printf "mult_error %a [*] %a\n" LinPoly.pp p output
((p1, o1), prf1);
raise InvalidProof )
let cutting_plane ((p, o), prf) =
let c, p' = Vect.decomp_cst p in
let g = Vect.gcd p' in
if Z.equal Z.one g || c =/ Q.zero || not (Z.equal (Q.den c) Z.one) then None
else
let c1 = c // Q.of_bigint g in
let c1' = Q.floor c1 in
if c1 =/ c1' then None
else
match o with
| Eq ->
Some ((Vect.set 0 Q.minus_one Vect.null, Eq), ProofFormat.CutPrf prf)
| Gt -> failwith "cutting_plane ignore strict constraints"
| Ge ->
Some
( (Vect.set 0 c1' (Vect.div (Q.of_bigint g) p), o)
, ProofFormat.CutPrf prf )
let construct_sign p =
let c, p' = Vect.decomp_cst p in
if Vect.is_null p' then
Some
( match Q.sign c with
| 0 -> (true, Eq, ProofFormat.Zero)
| 1 -> (true, Gt, ProofFormat.Cst c)
| _ -> (false, Gt, ProofFormat.Cst (Q.neg c)) )
else None
let get_sign l p =
match construct_sign p with
| None -> (
try
let (p', o), prf =
List.find (fun ((p', o), prf) -> Vect.equal p p') l
in
Some (true, o, prf)
with Not_found -> (
let p = Vect.uminus p in
try
let (p', o), prf =
List.find (fun ((p', o), prf) -> Vect.equal p p') l
in
Some (false, o, prf)
with Not_found -> None ) )
| Some s -> Some s
let mult_sign : bool -> t -> t =
fun b ((p, o), prf) -> if b then ((p, o), prf) else ((Vect.uminus p, o), prf)
let rec linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2) =
let a1, b1 = LinPoly.factorise x lp1 in
let a2, b2 = LinPoly.factorise x lp2 in
if Vect.is_null a2 then
Some ((lp2, op2), prf2)
else
match (op1, op2) with
| Eq, (Ge | Gt) -> (
match get_sign sys a1 with
| None -> None
| Some (b, o, prf) ->
let sa1 = mult_sign b ((a1, o), prf) in
let sa2 = if b then Vect.uminus a2 else a2 in
let (lp2, op2), prf2 =
addition
(product sa1 ((lp2, op2), prf2))
(mult sa2 ((lp1, op1), prf1))
in
linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2) )
| Eq, Eq ->
let (lp2, op2), prf2 =
addition
(mult a1 ((lp2, op2), prf2))
(mult (Vect.uminus a2) ((lp1, op1), prf1))
in
linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2)
| (Ge | Gt), (Ge | Gt) -> (
match (get_sign sys a1, get_sign sys a2) with
| Some (b1, o1, p1), Some (b2, o2, p2) ->
if b1 <> b2 then
let (lp2, op2), prf2 =
addition
(product (mult_sign b1 ((a1, o1), p1)) ((lp2, op2), prf2))
(product (mult_sign b2 ((a2, o2), p2)) ((lp1, op1), prf1))
in
linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2)
else None
| _ -> None )
| (Ge | Gt), Eq -> failwith "pivot: equality as second argument"
let linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2) =
match linear_pivot sys ((lp1, op1), prf1) x ((lp2, op2), prf2) with
| None -> None
| Some (c, p) -> Some (c, ProofFormat.simplify_prf_rule p)
let is_substitution strict ((p, o), prf) =
let pred v = if strict then v =/ Q.one || v =/ Q.minus_one else true in
match o with Eq -> LinPoly.search_linear pred p | _ -> None
let sort (sys : t list) =
let size ((p, o), prf) =
let _, p' = Vect.decomp_cst p in
let (x, q), p' = Vect.decomp_fst p' in
Vect.fold
(fun (l, (q, x)) x' q' ->
let q' = Q.abs q' in
(l + 1, if q </ q then (q, x) else (q', x')))
(1, (Q.abs q, x))
p
in
let cmp ((l1, (q1, _)), ((_, o), _)) ((l2, (q2, _)), ((_, o'), _)) =
if l1 < l2 then -1 else if l1 = l2 then Q.compare q1 q2 else 1
in
List.sort cmp (List.rev_map (fun wp -> (size wp, wp)) sys)
let iterate_pivot p sys0 =
let elim sys =
let oeq, sys' = extract p sys in
match oeq with
| None -> None
| Some (v, pc) -> simplify (linear_pivot sys0 pc v) sys'
in
iterate_until_stable elim (List.map snd (sort sys0))
let subst_constant is_int sys =
let is_integer q = Q.(q =/ floor q) in
let is_constant ((c, o), p) =
match o with
| Ge | Gt -> None
| Eq -> (
Vect.Bound.(
match of_vect c with
| None -> None
| Some b ->
if (not is_int) || is_integer (b.cst // b.coeff) then
Monomial.get_var (LinPoly.MonT.retrieve b.var)
else None) )
in
iterate_pivot is_constant sys
let subst sys0 = iterate_pivot (is_substitution true) sys0
let saturate_subst b sys0 =
let select = is_substitution b in
let gen (v, pc) ((c, op), prf) =
if ISet.mem v (LinPoly.variables c) then
linear_pivot sys0 pc v ((c, op), prf)
else None
in
saturate select gen sys0
let simple_pivot (q1, x) ((v1, o1), prf1) ((v2, o2), prf2) =
let q2 = Vect.get x v2 in
if q2 =/ Q.zero then None
else
let cv1, cv2 =
if Q.sign q1 <> Q.sign q2 then (Q.abs q2, Q.abs q1)
else
match (o1, o2) with
| Eq, _ -> (q2, Q.abs q1)
| _, Eq -> (Q.abs q2, q2)
| _, _ -> (Q.zero, Q.zero)
in
if cv2 =/ Q.zero then None
else
Some
( (Vect.mul_add cv1 v1 cv2 v2, opAdd o1 o2)
, ProofFormat.add_proof
(ProofFormat.mul_cst_proof cv1 prf1)
(ProofFormat.mul_cst_proof cv2 prf2) )
end
module BoundWithProof =
struct
type t = Vect.Bound.t * op * ProofFormat.prf_rule
let make ((p, o), prf) = match Vect.Bound.of_vect p with
| None -> None
| Some b -> Some (b, o, prf)
let padd (o1, prf1) (o2, prf2) = (opAdd o1 o2, ProofFormat.add_proof prf1 prf2)
let pmul (o1, prf1) (o2, prf2) = (opMult o1 o2, ProofFormat.mul_proof prf1 prf2)
let plet (o1,p1) (o2,p2) f =
match ProofFormat.pr_unit p1 , ProofFormat.pr_unit p2 with
| true , true -> f (o1,p1) (o2,p2)
| false , false ->
let (o,prf) = f (o1,ProofFormat.Ref 1) (o2,ProofFormat.Ref 0) in
(o, ProofFormat.LetPrf(p1,ProofFormat.LetPrf(p2,prf)))
| true , false ->
let (o,prf) = f (o1,p1) (o2,ProofFormat.Ref 0) in
(o, ProofFormat.LetPrf(p2,prf))
| false , true ->
let (o,prf) = f (o1,ProofFormat.Ref 0) (o2,p2) in
(o,ProofFormat.LetPrf(p1,prf))
let pext c (o, prf) =
if c =/ Q.zero then (Eq, ProofFormat.Zero)
else (o, ProofFormat.mul_cst_proof c prf)
let mul_bound (b1, o1, prf1) (b2, o2, prf2) =
let open Vect.Bound in
match (b1, b2) with
| {cst = c1; var = v1; coeff = c1'},
{cst = c2; var = v2; coeff = c2'} ->
let good_coeff b o =
match o with
| Eq -> Some (Q.neg b)
| _ -> if b <=/ Q.zero then Some (Q.neg b) else None
in
match (good_coeff c1 o2, good_coeff c2 o1) with
| None, _ | _, None -> None
| Some c1, Some c2 ->
let w1 = (o1, prf1) in
let w2 = (o2, prf2) in
let (o, prf) = plet w1 w2 (fun w1 w2 -> padd (padd (pmul w1 w2) (pext c1 w2)) (pext c2 w1)) in
let b = {
cst = Q.neg (c1 */ c2);
var = LinPoly.MonT.register (Monomial.prod (LinPoly.MonT.retrieve v1) (LinPoly.MonT.retrieve v2));
coeff = c1' */ c2';
} in
Some (b, o, prf)
let bound (b, _, _) = b
let proof (b, o, w) =
let p = Vect.Bound.to_vect b in
((p, o), w)
end