path: root/plugins/micromega/sos.ml

(* ========================================================================= *)
(* - This code originates from John Harrison's HOL LIGHT 2.30                *)
(*   (see file LICENSE.sos for license, copyright and disclaimer)            *)
(* - Laurent Théry (thery@sophia.inria.fr) has isolated the HOL              *)
(*   independent bits                                                        *)
(* - Frédéric Besson (fbesson@irisa.fr) is using it to feed  micromega       *)
(* ========================================================================= *)

(* ========================================================================= *)
(* Nonlinear universal reals procedure using SOS decomposition.              *)
(* ========================================================================= *)
open Num;;
open List;;
open Sos_types;;
open Sos_lib;;

(*
prioritize_real();;
*)

let debugging = ref false;;

exception Sanity;;

exception Unsolvable;;

(* ------------------------------------------------------------------------- *)
(* Turn a rational into a decimal string with d sig digits.                  *)
(* ------------------------------------------------------------------------- *)

let decimalize =
  let rec normalize y =
    if abs_num y </ Int 1 // Int 10 then normalize (Int 10 */ y) - 1
    else if abs_num y >=/ Int 1 then normalize (y // Int 10) + 1
    else 0 in
  fun d x ->
    if x =/ Int 0 then "0.0" else
    let y = abs_num x in
    let e = normalize y in
    let z = pow10(-e) */ y +/ Int 1 in
    let k = round_num(pow10 d */ z) in
    (if x </ Int 0 then "-0." else "0.") ^
    implode(tl(explode(string_of_num k))) ^
    (if e = 0 then "" else "e"^string_of_int e);;

(* ------------------------------------------------------------------------- *)
(* Iterations over numbers, and lists indexed by numbers.                    *)
(* ------------------------------------------------------------------------- *)

let rec itern k l f a =
  match l with
    [] -> a
  | h::t -> itern (k + 1) t f (f h k a);;

let rec iter (m,n) f a =
  if n < m then a
  else iter (m+1,n) f (f m a);;

(* ------------------------------------------------------------------------- *)
(* The main types.                                                           *)
(* ------------------------------------------------------------------------- *)

type vector = int*(int,num)func;;

type matrix = (int*int)*(int*int,num)func;;

type monomial = (vname,int)func;;

type poly = (monomial,num)func;;

(* ------------------------------------------------------------------------- *)
(* Assignment avoiding zeros.                                                *)
(* ------------------------------------------------------------------------- *)

let (|-->) x y a = if y =/ Int 0 then a else (x |-> y) a;;

(* ------------------------------------------------------------------------- *)
(* This can be generic.                                                      *)
(* ------------------------------------------------------------------------- *)

let element (d,v) i = tryapplyd v i (Int 0);;

let mapa f (d,v) =
  d,foldl (fun a i c -> (i |--> f(c)) a) undefined v;;

let is_zero (d,v) =
  match v with
    Empty -> true
  | _ -> false;;

(* ------------------------------------------------------------------------- *)
(* Vectors. Conventionally indexed 1..n.                                     *)
(* ------------------------------------------------------------------------- *)

let vector_0 n = (n,undefined:vector);;

let dim (v:vector) = fst v;;

let vector_const c n =
  if c =/ Int 0 then vector_0 n
  else (n,itlist (fun k -> k |-> c) (1--n) undefined :vector);;

let vector_1 = vector_const (Int 1);;

let vector_cmul c (v:vector) =
  let n = dim v in
  if c =/ Int 0 then vector_0 n
  else n,mapf (fun x -> c */ x) (snd v)

let vector_neg (v:vector) = (fst v,mapf minus_num (snd v) :vector);;

let vector_add (v1:vector) (v2:vector) =
  let m = dim v1 and n = dim v2 in
  if m <> n then failwith "vector_add: incompatible dimensions" else
  (n,combine (+/) (fun x -> x =/ Int 0) (snd v1) (snd v2) :vector);;

let vector_sub v1 v2 = vector_add v1 (vector_neg v2);;

let vector_dot (v1:vector) (v2:vector) =
  let m = dim v1 and n = dim v2 in
  if m <> n then failwith "vector_add: incompatible dimensions" else
  foldl (fun a i x -> x +/ a) (Int 0)
        (combine ( */ ) (fun x -> x =/ Int 0) (snd v1) (snd v2));;

let vector_of_list l =
  let n = length l in
  (n,itlist2 (|->) (1--n) l undefined :vector);;

(* ------------------------------------------------------------------------- *)
(* Matrices; again rows and columns indexed from 1.                          *)
(* ------------------------------------------------------------------------- *)

let matrix_0 (m,n) = ((m,n),undefined:matrix);;

let dimensions (m:matrix) = fst m;;

let matrix_const c (m,n as mn) =
  if m <> n then failwith "matrix_const: needs to be square"
  else if c =/ Int 0 then matrix_0 mn
  else (mn,itlist (fun k -> (k,k) |-> c) (1--n) undefined :matrix);;

let matrix_1 = matrix_const (Int 1);;

let matrix_cmul c (m:matrix) =
  let (i,j) = dimensions m in
  if c =/ Int 0 then matrix_0 (i,j)
  else (i,j),mapf (fun x -> c */ x) (snd m);;

let matrix_neg (m:matrix) = (dimensions m,mapf minus_num (snd m) :matrix);;

let matrix_add (m1:matrix) (m2:matrix) =
  let d1 = dimensions m1 and d2 = dimensions m2 in
  if d1 <> d2 then failwith "matrix_add: incompatible dimensions"
  else (d1,combine (+/) (fun x -> x =/ Int 0) (snd m1) (snd m2) :matrix);;

let matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);;

let row k (m:matrix) =
  let i,j = dimensions m in
  (j,
   foldl (fun a (i,j) c -> if i = k then (j |-> c) a else a) undefined (snd m)
   : vector);;

let column k (m:matrix) =
  let i,j = dimensions m in
  (i,
   foldl (fun a (i,j) c -> if j = k then (i |-> c) a else a) undefined (snd m)
   : vector);;

let transp (m:matrix) =
  let i,j = dimensions m in
  ((j,i),foldl (fun a (i,j) c -> ((j,i) |-> c) a) undefined (snd m) :matrix);;

let diagonal (v:vector) =
  let n = dim v in
  ((n,n),foldl (fun a i c -> ((i,i) |-> c) a) undefined (snd v) : matrix);;

let matrix_of_list l =
  let m = length l in
  if m = 0 then matrix_0 (0,0) else
  let n = length (hd l) in
  (m,n),itern 1 l (fun v i -> itern 1 v (fun c j -> (i,j) |-> c)) undefined;;

(* ------------------------------------------------------------------------- *)
(* Monomials.                                                                *)
(* ------------------------------------------------------------------------- *)

let monomial_eval assig (m:monomial) =
  foldl (fun a x k -> a */ power_num (apply assig x) (Int k))
        (Int 1) m;;

let monomial_1 = (undefined:monomial);;

let monomial_var x = (x |=> 1 :monomial);;

let (monomial_mul:monomial->monomial->monomial) =
  combine (+) (fun x -> false);;

let monomial_pow (m:monomial) k =
  if k = 0 then monomial_1
  else mapf (fun x -> k * x) m;;

let monomial_divides (m1:monomial) (m2:monomial) =
  foldl (fun a x k -> tryapplyd m2 x 0 >= k & a) true m1;;

let monomial_div (m1:monomial) (m2:monomial) =
  let m = combine (+) (fun x -> x = 0) m1 (mapf (fun x -> -x) m2) in
  if foldl (fun a x k -> k >= 0 & a) true m then m
  else failwith "monomial_div: non-divisible";;

let monomial_degree x (m:monomial) = tryapplyd m x 0;;

let monomial_lcm (m1:monomial) (m2:monomial) =
  (itlist (fun x -> x |-> max (monomial_degree x m1) (monomial_degree x m2))
          (union (dom m1) (dom m2)) undefined :monomial);;

let monomial_multidegree (m:monomial) = foldl (fun a x k -> k + a) 0 m;;

let monomial_variables m = dom m;;

(* ------------------------------------------------------------------------- *)
(* Polynomials.                                                              *)
(* ------------------------------------------------------------------------- *)

let eval assig (p:poly) =
  foldl (fun a m c -> a +/ c */ monomial_eval assig m) (Int 0) p;;

let poly_0 = (undefined:poly);;

let poly_isconst (p:poly) = foldl (fun a m c -> m = monomial_1 & a) true p;;

let poly_var x = ((monomial_var x) |=> Int 1 :poly);;

let poly_const c =
  if c =/ Int 0 then poly_0 else (monomial_1 |=> c);;

let poly_cmul c (p:poly) =
  if c =/ Int 0 then poly_0
  else mapf (fun x -> c */ x) p;;

let poly_neg (p:poly) = (mapf minus_num p :poly);;

let poly_add (p1:poly) (p2:poly) =
  (combine (+/) (fun x -> x =/ Int 0) p1 p2 :poly);;

let poly_sub p1 p2 = poly_add p1 (poly_neg p2);;

let poly_cmmul (c,m) (p:poly) =
  if c =/ Int 0 then poly_0
  else if m = monomial_1 then mapf (fun d -> c */ d) p
  else foldl (fun a m' d -> (monomial_mul m m' |-> c */ d) a) poly_0 p;;

let poly_mul (p1:poly) (p2:poly) =
  foldl (fun a m c -> poly_add (poly_cmmul (c,m) p2) a) poly_0 p1;;

let poly_div (p1:poly) (p2:poly) =
  if not(poly_isconst p2) then failwith "poly_div: non-constant" else
  let c = eval undefined p2 in
  if c =/ Int 0 then failwith "poly_div: division by zero"
  else poly_cmul (Int 1 // c) p1;;

let poly_square p = poly_mul p p;;

let rec poly_pow p k =
  if k = 0 then poly_const (Int 1)
  else if k = 1 then p
  else let q = poly_square(poly_pow p (k / 2)) in
       if k mod 2 = 1 then poly_mul p q else q;;

let poly_exp p1 p2 =
  if not(poly_isconst p2) then failwith "poly_exp: not a constant" else
  poly_pow p1 (Num.int_of_num (eval undefined p2));;

let degree x (p:poly) = foldl (fun a m c -> max (monomial_degree x m) a) 0 p;;

let multidegree (p:poly) =
  foldl (fun a m c -> max (monomial_multidegree m) a) 0 p;;

let poly_variables (p:poly) =
  foldr (fun m c -> union (monomial_variables m)) p [];;

(* ------------------------------------------------------------------------- *)
(* Order monomials for human presentation.                                   *)
(* ------------------------------------------------------------------------- *)

let humanorder_varpow (x1,k1) (x2,k2) = x1 < x2 or x1 = x2 & k1 > k2;;

let humanorder_monomial =
  let rec ord l1 l2 = match (l1,l2) with
    _,[] -> true
  | [],_ -> false
  | h1::t1,h2::t2 -> humanorder_varpow h1 h2 or h1 = h2 & ord t1 t2 in
  fun m1 m2 -> m1 = m2 or
               ord (sort humanorder_varpow (graph m1))
                   (sort humanorder_varpow (graph m2));;

(* ------------------------------------------------------------------------- *)
(* Conversions to strings.                                                   *)
(* ------------------------------------------------------------------------- *)

let string_of_vector min_size max_size (v:vector) =
  let n_raw = dim v in
  if n_raw = 0 then "[]" else
  let n = max min_size (min n_raw max_size) in
  let xs = map ((o) string_of_num (element v)) (1--n) in
  "[" ^ end_itlist (fun s t -> s ^ ", " ^ t) xs ^
  (if n_raw > max_size then ", ...]" else "]");;

let string_of_matrix max_size (m:matrix) =
  let i_raw,j_raw = dimensions m in
  let i = min max_size i_raw and j = min max_size j_raw in
  let rstr = map (fun k -> string_of_vector j j (row k m)) (1--i) in
  "["^end_itlist(fun s t -> s^";\n "^t) rstr ^
  (if j > max_size then "\n ...]" else "]");;

let string_of_vname (v:vname): string = (v: string);;

let rec string_of_term t =
  match t with
  Opp t1 -> "(- " ^ string_of_term t1 ^ ")"
| Add (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " + " ^ (string_of_term t2) ^ ")"
| Sub (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " - " ^ (string_of_term t2) ^ ")"
| Mul (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " * " ^ (string_of_term t2) ^ ")"
| Inv t1 -> "(/ " ^ string_of_term t1 ^ ")"
| Div (t1, t2) ->
   "(" ^ (string_of_term t1) ^ " / " ^ (string_of_term t2) ^ ")"
| Pow (t1, n1) ->
   "(" ^ (string_of_term t1) ^ " ^ " ^ (string_of_int n1) ^ ")"
| Zero -> "0"
| Var v -> "x" ^ (string_of_vname v)
| Const x -> string_of_num x;;


let string_of_varpow x k =
  if k = 1 then string_of_vname x else string_of_vname x^"^"^string_of_int k;;

let string_of_monomial m =
  if m = monomial_1 then "1" else
  let vps = List.fold_right (fun (x,k) a -> string_of_varpow x k :: a)
                            (sort humanorder_varpow (graph m)) [] in
  end_itlist (fun s t -> s^"*"^t) vps;;

let string_of_cmonomial (c,m) =
  if m = monomial_1 then string_of_num c
  else if c =/ Int 1 then string_of_monomial m
  else string_of_num c ^ "*" ^ string_of_monomial m;;

let string_of_poly (p:poly) =
  if p = poly_0 then "<<0>>" else
  let cms = sort (fun (m1,_) (m2,_) -> humanorder_monomial m1 m2) (graph p) in
  let s =
    List.fold_left (fun a (m,c) ->
             if c </ Int 0 then a ^ " - " ^ string_of_cmonomial(minus_num c,m)
             else a ^ " + " ^ string_of_cmonomial(c,m))
          "" cms in
  let s1 = String.sub s 0 3
  and s2 = String.sub s 3 (String.length s - 3) in
  "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>";;

(* ------------------------------------------------------------------------- *)
(* Printers.                                                                 *)
(* ------------------------------------------------------------------------- *)

let print_vector v = Format.print_string(string_of_vector 0 20 v);;

let print_matrix m = Format.print_string(string_of_matrix 20 m);;

let print_monomial m = Format.print_string(string_of_monomial m);;

let print_poly m = Format.print_string(string_of_poly m);;

(*
#install_printer print_vector;;
#install_printer print_matrix;;
#install_printer print_monomial;;
#install_printer print_poly;;
*)

(* ------------------------------------------------------------------------- *)
(* Conversion from  term.                                                 *)
(* ------------------------------------------------------------------------- *)

let rec poly_of_term t = match t with
  Zero -> poly_0
| Const n -> poly_const n
| Var x -> poly_var x
| Opp t1 -> poly_neg (poly_of_term t1)
| Inv t1 ->
      let p = poly_of_term t1 in
      if poly_isconst p then poly_const(Int 1 // eval undefined p)
      else failwith "poly_of_term: inverse of non-constant polyomial"
| Add (l, r) -> poly_add (poly_of_term l) (poly_of_term r)
| Sub (l, r) -> poly_sub (poly_of_term l) (poly_of_term r)
| Mul (l, r) -> poly_mul (poly_of_term l) (poly_of_term r)
| Div (l, r) ->
      let p = poly_of_term l and q = poly_of_term r in
      if poly_isconst q then poly_cmul (Int 1 // eval undefined q) p
      else failwith "poly_of_term: division by non-constant polynomial"
| Pow (t, n) ->
      poly_pow (poly_of_term t) n;;

(* ------------------------------------------------------------------------- *)
(* String of vector (just a list of space-separated numbers).                *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_vector (v:vector) =
  let n = dim v in
  let strs = map (o (decimalize 20)  (element v)) (1--n) in
  end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";;

(* ------------------------------------------------------------------------- *)
(* String for block diagonal matrix numbered k.                              *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

(* ------------------------------------------------------------------------- *)
(* String for a matrix numbered k, in SDPA sparse format.                    *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_matrix k (m:matrix) =
  let pfx = string_of_int k ^ " 1 " in
  let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a)
                 (snd m) [] in
  let mss = sort (increasing fst) ms in
  itlist (fun ((i,j),c) a ->
     pfx ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) mss "";;

(* ------------------------------------------------------------------------- *)
(* String in SDPA sparse format for standard SDP problem:                    *)
(*                                                                           *)
(*    X = v_1 * [M_1] + ... + v_m * [M_m] - [M_0] must be PSD                *)
(*    Minimize obj_1 * v_1 + ... obj_m * v_m                                 *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_problem comment obj mats =
  let m = length mats - 1
  and n,_ = dimensions (hd mats) in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  "1\n" ^
  string_of_int n ^ "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a)
          (1--length mats) mats "";;

(* ------------------------------------------------------------------------- *)
(* More parser basics.                                                       *)
(* ------------------------------------------------------------------------- *)

let word s =
   end_itlist (fun p1 p2 -> (p1 ++ p2) >> (fun (s,t) -> s^t))
              (map a (explode s));;
let token s =
  many (some isspace) ++ word s ++ many (some isspace)
  >> (fun ((_,t),_) -> t);;

let decimal =
  let numeral = some isnum in
  let decimalint = atleast 1 numeral >> ((o) Num.num_of_string  implode) in
  let decimalfrac = atleast 1 numeral
    >> (fun s -> Num.num_of_string(implode s) // pow10 (length s)) in
  let decimalsig =
    decimalint ++ possibly (a "." ++ decimalfrac >> snd)
    >> (function (h,[x]) -> h +/ x | (h,_) -> h) in
  let signed prs =
       a "-" ++ prs >> ((o) minus_num snd)
    || a "+" ++ prs >> snd
    || prs in
  let exponent = (a "e" || a "E") ++ signed decimalint >> snd in
    signed decimalsig ++ possibly exponent
    >> (function (h,[x]) -> h */ power_num (Int 10) x | (h,_) -> h);;

let mkparser p s =
  let x,rst = p(explode s) in
  if rst = [] then x else failwith "mkparser: unparsed input";;

let parse_decimal = mkparser decimal;;

(* ------------------------------------------------------------------------- *)
(* Parse back a vector.                                                      *)
(* ------------------------------------------------------------------------- *)

let parse_sdpaoutput,parse_csdpoutput =
  let vector =
    token "{" ++ listof decimal (token ",") "decimal" ++ token "}"
               >> (fun ((_,v),_) -> vector_of_list v) in
  let rec skipupto dscr prs inp =
      (dscr ++ prs >> snd
    || some (fun c -> true) ++ skipupto dscr prs >> snd) inp in
  let ignore inp = (),[] in
  let sdpaoutput =
    skipupto (word "xVec" ++ token "=")
             (vector ++ ignore >> fst) in
  let csdpoutput =
    (decimal ++ many(a " " ++ decimal >> snd) >> (fun (h,t) -> h::t)) ++
    (a " " ++ a "\n" ++ ignore) >> ((o) vector_of_list fst) in
  mkparser sdpaoutput,mkparser csdpoutput;;

(* ------------------------------------------------------------------------- *)
(* Also parse the SDPA output to test success (CSDP yields a return code).   *)
(* ------------------------------------------------------------------------- *)

let sdpa_run_succeeded =
   let rec skipupto dscr prs inp =
      (dscr ++ prs >> snd
    || some (fun c -> true) ++ skipupto dscr prs >> snd) inp in
  let prs = skipupto (word "phase.value" ++ token "=")
                     (possibly (a "p") ++ possibly (a "d") ++
                      (word "OPT" || word "FEAS")) in
  fun s -> try ignore (prs (explode s)); true with Noparse -> false;;

(* ------------------------------------------------------------------------- *)
(* The default parameters. Unfortunately this goes to a fixed file.          *)
(* ------------------------------------------------------------------------- *)

let sdpa_default_parameters =
"100     unsigned int maxIteration;\
\n1.0E-7  double 0.0 < epsilonStar;\
\n1.0E2   double 0.0 < lambdaStar;\
\n2.0     double 1.0 < omegaStar;\
\n-1.0E5  double lowerBound;\
\n1.0E5   double upperBound;\
\n0.1     double 0.0 <= betaStar <  1.0;\
\n0.2     double 0.0 <= betaBar  <  1.0, betaStar <= betaBar;\
\n0.9     double 0.0 < gammaStar  <  1.0;\
\n1.0E-7  double 0.0 < epsilonDash;\
\n";;

(* ------------------------------------------------------------------------- *)
(* These were suggested by Makoto Yamashita for problems where we are        *)
(* right at the edge of the semidefinite cone, as sometimes happens.         *)
(* ------------------------------------------------------------------------- *)

let sdpa_alt_parameters =
"1000    unsigned int maxIteration;\
\n1.0E-7  double 0.0 < epsilonStar;\
\n1.0E4   double 0.0 < lambdaStar;\
\n2.0     double 1.0 < omegaStar;\
\n-1.0E5  double lowerBound;\
\n1.0E5   double upperBound;\
\n0.1     double 0.0 <= betaStar <  1.0;\
\n0.2     double 0.0 <= betaBar  <  1.0, betaStar <= betaBar;\
\n0.9     double 0.0 < gammaStar  <  1.0;\
\n1.0E-7  double 0.0 < epsilonDash;\
\n";;

let sdpa_params = sdpa_alt_parameters;;

(* ------------------------------------------------------------------------- *)
(* CSDP parameters; so far I'm sticking with the defaults.                   *)
(* ------------------------------------------------------------------------- *)

let csdp_default_parameters =
"axtol=1.0e-8\
\natytol=1.0e-8\
\nobjtol=1.0e-8\
\npinftol=1.0e8\
\ndinftol=1.0e8\
\nmaxiter=100\
\nminstepfrac=0.9\
\nmaxstepfrac=0.97\
\nminstepp=1.0e-8\
\nminstepd=1.0e-8\
\nusexzgap=1\
\ntweakgap=0\
\naffine=0\
\nprintlevel=1\
\n";;

let csdp_params = csdp_default_parameters;;

(* ------------------------------------------------------------------------- *)
(* Now call CSDP on a problem and parse back the output.                     *)
(* ------------------------------------------------------------------------- *)

let run_csdp dbg obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file (sdpa_of_problem "" obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                        " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp obj mats =
  let rv,res = run_csdp (!debugging) obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
   else if rv = 3 then ()
    (* Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline() *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* Try some apparently sensible scaling first. Note that this is purely to   *)
(* get a cleaner translation to floating-point, and doesn't affect any of    *)
(* the results, in principle. In practice it seems a lot better when there   *)
(* are extreme numbers in the original problem.                              *)
(* ------------------------------------------------------------------------- *)

let scale_then =
  let common_denominator amat acc =
    foldl (fun a m c -> lcm_num (denominator c) a) acc amat
  and maximal_element amat acc =
    foldl (fun maxa m c -> max_num maxa (abs_num c)) acc amat in
  fun solver obj mats ->
    let cd1 = itlist common_denominator mats (Int 1)
    and cd2 = common_denominator (snd obj)  (Int 1) in
    let mats' = map (mapf (fun x -> cd1 */ x)) mats
    and obj' = vector_cmul cd2 obj in
    let max1 = itlist maximal_element mats' (Int 0)
    and max2 = maximal_element (snd obj') (Int 0) in
    let scal1 = pow2 (20-int_of_float(log(float_of_num max1) /. log 2.0))
    and scal2 = pow2 (20-int_of_float(log(float_of_num max2) /. log 2.0)) in
    let mats'' = map (mapf (fun x -> x */ scal1)) mats'
    and obj'' = vector_cmul scal2 obj' in
    solver obj'' mats'';;

(* ------------------------------------------------------------------------- *)
(* Round a vector to "nice" rationals.                                       *)
(* ------------------------------------------------------------------------- *)

let nice_rational n x = round_num (n */ x) // n;;

let nice_vector n = mapa (nice_rational n);;

(* ------------------------------------------------------------------------- *)
(* Reduce linear program to SDP (diagonal matrices) and test with CSDP. This *)
(* one tests A [-1;x1;..;xn] >= 0 (i.e. left column is negated constants).   *)
(* ------------------------------------------------------------------------- *)

let linear_program_basic a =
  let m,n = dimensions a in
  let mats =  map (fun j -> diagonal (column j a)) (1--n)
  and obj = vector_const (Int 1) m in
  let rv,res = run_csdp false obj mats in
  if rv = 1 or rv = 2 then false
  else if rv = 0 then true
  else failwith "linear_program: An error occurred in the SDP solver";;

(* ------------------------------------------------------------------------- *)
(* Alternative interface testing A x >= b for matrix A, vector b.            *)
(* ------------------------------------------------------------------------- *)

let linear_program a b =
  let m,n = dimensions a in
  if dim b <> m then failwith "linear_program: incompatible dimensions" else
  let mats = diagonal b :: map (fun j -> diagonal (column j a)) (1--n)
  and obj = vector_const (Int 1) m in
  let rv,res = run_csdp false obj mats in
  if rv = 1 or rv = 2 then false
  else if rv = 0 then true
  else failwith "linear_program: An error occurred in the SDP solver";;

(* ------------------------------------------------------------------------- *)
(* Test whether a point is in the convex hull of others. Rather than use     *)
(* computational geometry, express as linear inequalities and call CSDP.     *)
(* This is a bit lazy of me, but it's easy and not such a bottleneck so far. *)
(* ------------------------------------------------------------------------- *)

let in_convex_hull pts pt =
  let pts1 = (1::pt) :: map (fun x -> 1::x) pts in
  let pts2 = map (fun p -> map (fun x -> -x) p @ p) pts1 in
  let n = length pts + 1
  and v = 2 * (length pt + 1) in
  let m = v + n - 1 in
  let mat =
    (m,n),
    itern 1 pts2 (fun pts j -> itern 1 pts (fun x i -> (i,j) |-> Int x))
                 (iter (1,n) (fun i -> (v + i,i+1) |-> Int 1) undefined) in
  linear_program_basic mat;;

(* ------------------------------------------------------------------------- *)
(* Filter down a set of points to a minimal set with the same convex hull.   *)
(* ------------------------------------------------------------------------- *)

let minimal_convex_hull =
  let augment1 = function
    | [] -> assert false
    | (m::ms) -> if in_convex_hull ms m then ms else ms@[m] in
  let augment m ms = funpow 3 augment1 (m::ms) in
  fun mons ->
    let mons' = itlist augment (tl mons) [hd mons] in
    funpow (length mons') augment1 mons';;

(* ------------------------------------------------------------------------- *)
(* Stuff for "equations" (generic A->num functions).                         *)
(* ------------------------------------------------------------------------- *)

let equation_cmul c eq =
  if c =/ Int 0 then Empty else mapf (fun d -> c */ d) eq;;

let equation_add eq1 eq2 = combine (+/) (fun x -> x =/ Int 0) eq1 eq2;;

let equation_eval assig eq =
  let value v = apply assig v in
  foldl (fun a v c -> a +/ value(v) */ c) (Int 0) eq;;

(* ------------------------------------------------------------------------- *)
(* Eliminate among linear equations: return unconstrained variables and      *)
(* assignments for the others in terms of them. We give one pseudo-variable  *)
(* "one" that's used for a constant term.                                    *)
(* ------------------------------------------------------------------------- *)

let failstore = ref [];;

let eliminate_equations =
  let rec extract_first p l =
    match l with
      [] -> failwith "extract_first"
    | h::t -> if p(h) then h,t else
              let k,s = extract_first p t in
              k,h::s in
  let rec eliminate vars dun eqs =
    match vars with
      [] -> if forall is_undefined eqs then dun
            else (failstore := [vars,dun,eqs]; raise Unsolvable)
    | v::vs ->
            try let eq,oeqs = extract_first (fun e -> defined e v) eqs in
                let a = apply eq v in
                let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
                let elim e =
                  let b = tryapplyd e v (Int 0) in
                  if b =/ Int 0 then e else
                  equation_add e (equation_cmul (minus_num b // a) eq) in
                eliminate vs ((v |-> eq') (mapf elim dun)) (map elim oeqs)
            with Failure _ -> eliminate vs dun eqs in
  fun one vars eqs ->
    let assig = eliminate vars undefined eqs in
    let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
    setify vs,assig;;

(* ------------------------------------------------------------------------- *)
(* Eliminate all variables, in an essentially arbitrary order.               *)
(* ------------------------------------------------------------------------- *)

let eliminate_all_equations one =
  let choose_variable eq =
    let (v,_) = choose eq in
    if v = one then
      let eq' = undefine v eq in
      if is_undefined eq' then failwith "choose_variable" else
      let (w,_) = choose eq' in w
    else v in
  let rec eliminate dun eqs =
    match eqs with
      [] -> dun
    | eq::oeqs ->
        if is_undefined eq then eliminate dun oeqs else
        let v = choose_variable eq in
        let a = apply eq v in
        let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
        let elim e =
          let b = tryapplyd e v (Int 0) in
          if b =/ Int 0 then e else
          equation_add e (equation_cmul (minus_num b // a) eq) in
        eliminate ((v |-> eq') (mapf elim dun)) (map elim oeqs) in
  fun eqs ->
    let assig = eliminate undefined eqs in
    let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
    setify vs,assig;;

(* ------------------------------------------------------------------------- *)
(* Solve equations by assigning arbitrary numbers.                           *)
(* ------------------------------------------------------------------------- *)

let solve_equations one eqs =
  let vars,assigs = eliminate_all_equations one eqs in
  let vfn = itlist (fun v -> (v |-> Int 0)) vars (one |=> Int(-1)) in
  let ass =
    combine (+/) (fun c -> false) (mapf (equation_eval vfn) assigs) vfn in
  if forall (fun e -> equation_eval ass e =/ Int 0) eqs
  then undefine one ass else raise Sanity;;

(* ------------------------------------------------------------------------- *)
(* Hence produce the "relevant" monomials: those whose squares lie in the    *)
(* Newton polytope of the monomials in the input. (This is enough according  *)
(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal,       *)
(* vol 45, pp. 363--374, 1978.                                               *)
(*                                                                           *)
(* These are ordered in sort of decreasing degree. In particular the         *)
(* constant monomial is last; this gives an order in diagonalization of the  *)
(* quadratic form that will tend to display constants.                       *)
(* ------------------------------------------------------------------------- *)

let newton_polytope pol =
  let vars = poly_variables pol in
  let mons = map (fun m -> map (fun x -> monomial_degree x m) vars) (dom pol)
  and ds = map (fun x -> (degree x pol + 1) / 2) vars in
  let all = itlist (fun n -> allpairs (fun h t -> h::t) (0--n)) ds [[]]
  and mons' = minimal_convex_hull mons in
  let all' =
    filter (fun m -> in_convex_hull mons' (map (fun x -> 2 * x) m)) all in
  map (fun m -> itlist2 (fun v i a -> if i = 0 then a else (v |-> i) a)
                        vars m monomial_1) (rev all');;

(* ------------------------------------------------------------------------- *)
(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form.  *)
(* ------------------------------------------------------------------------- *)

let diag m =
  let nn = dimensions m in
  let n = fst nn in
  if snd nn <> n then failwith "diagonalize: non-square matrix" else
  let rec diagonalize i m =
    if is_zero m then [] else
    let a11 = element m (i,i) in
    if a11 </ Int 0 then failwith "diagonalize: not PSD"
    else if a11 =/ Int 0 then
      if is_zero(row i m) then diagonalize (i + 1) m
      else failwith "diagonalize: not PSD"
    else
      let v = row i m in
      let v' = mapa (fun a1k -> a1k // a11) v in
      let m' =
      (n,n),
      iter (i+1,n) (fun j ->
          iter (i+1,n) (fun k ->
              ((j,k) |--> (element m (j,k) -/ element v j */ element v' k))))
          undefined in
      (a11,v')::diagonalize (i + 1) m' in
  diagonalize 1 m;;

(* ------------------------------------------------------------------------- *)
(* Adjust a diagonalization to collect rationals at the start.               *)
(* ------------------------------------------------------------------------- *)

let deration d =
  if d = [] then Int 0,d else
  let adj(c,l) =
    let a = foldl (fun a i c -> lcm_num a (denominator c)) (Int 1) (snd l) //
            foldl (fun a i c -> gcd_num a (numerator c)) (Int 0) (snd l) in
    (c // (a */ a)),mapa (fun x -> a */ x) l in
  let d' = map adj d in
  let a = itlist ((o) lcm_num ( (o) denominator fst)) d' (Int 1) //
          itlist ((o) gcd_num ( (o) numerator fst)) d' (Int 0)  in
  (Int 1 // a),map (fun (c,l) -> (a */ c,l)) d';;

(* ------------------------------------------------------------------------- *)
(* Enumeration of monomials with given multidegree bound.                    *)
(* ------------------------------------------------------------------------- *)

let rec enumerate_monomials d vars =
  if d < 0 then []
  else if d = 0 then [undefined]
  else if vars = [] then [monomial_1] else
  let alts =
    map (fun k -> let oths = enumerate_monomials (d - k) (tl vars) in
                  map (fun ks -> if k = 0 then ks else (hd vars |-> k) ks) oths)
        (0--d) in
  end_itlist (@) alts;;

(* ------------------------------------------------------------------------- *)
(* Enumerate products of distinct input polys with degree <= d.              *)
(* We ignore any constant input polynomials.                                 *)
(* Give the output polynomial and a record of how it was derived.            *)
(* ------------------------------------------------------------------------- *)

let rec enumerate_products d pols =
  if d = 0 then [poly_const num_1,Rational_lt num_1] else if d < 0 then [] else
  match pols with
    [] -> [poly_const num_1,Rational_lt num_1]
  | (p,b)::ps -> let e = multidegree p in
                 if e = 0 then enumerate_products d ps else
                 enumerate_products d ps @
                 map (fun (q,c) -> poly_mul p q,Product(b,c))
                     (enumerate_products (d - e) ps);;

(* ------------------------------------------------------------------------- *)
(* Multiply equation-parametrized poly by regular poly and add accumulator.  *)
(* ------------------------------------------------------------------------- *)

let epoly_pmul p q acc =
  foldl (fun a m1 c ->
           foldl (fun b m2 e ->
                        let m =  monomial_mul m1 m2 in
                        let es = tryapplyd b m undefined in
                        (m |-> equation_add (equation_cmul c e) es) b)
                 a q) acc p;;

(* ------------------------------------------------------------------------- *)
(* Usual operations on equation-parametrized poly.                           *)
(* ------------------------------------------------------------------------- *)

let epoly_cmul c l =
  if c =/ Int 0 then undefined else mapf (equation_cmul c) l;;

let epoly_neg = epoly_cmul (Int(-1));;

let epoly_add = combine equation_add is_undefined;;

let epoly_sub p q = epoly_add p (epoly_neg q);;

(* ------------------------------------------------------------------------- *)
(* Convert regular polynomial. Note that we treat (0,0,0) as -1.             *)
(* ------------------------------------------------------------------------- *)

let epoly_of_poly p =
  foldl (fun a m c -> (m |-> ((0,0,0) |=> minus_num c)) a) undefined p;;

(* ------------------------------------------------------------------------- *)
(* String for block diagonal matrix numbered k.                              *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

(* ------------------------------------------------------------------------- *)
(* SDPA for problem using block diagonal (i.e. multiple SDPs)                *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_blockproblem comment nblocks blocksizes obj mats =
  let m = length mats - 1 in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  string_of_int nblocks ^ "\n" ^
  (end_itlist (fun s t -> s^" "^t) (map string_of_int blocksizes)) ^
  "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_blockdiagonal (k - 1) m ^ a)
          (1--length mats) mats "";;

(* ------------------------------------------------------------------------- *)
(* Hence run CSDP on a problem in block diagonal form.                       *)
(* ------------------------------------------------------------------------- *)

let run_csdp dbg nblocks blocksizes obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file
   (sdpa_of_blockproblem "" nblocks blocksizes obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                        " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp nblocks blocksizes obj mats =
  let rv,res = run_csdp (!debugging) nblocks blocksizes obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
   else if rv = 3 then ()
    (*Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline() *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* 3D versions of matrix operations to consider blocks separately.           *)
(* ------------------------------------------------------------------------- *)

let bmatrix_add = combine (+/) (fun x -> x =/ Int 0);;

let bmatrix_cmul c bm =
  if c =/ Int 0 then undefined
  else mapf (fun x -> c */ x) bm;;

let bmatrix_neg = bmatrix_cmul (Int(-1));;

let bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;

(* ------------------------------------------------------------------------- *)
(* Smash a block matrix into components.                                     *)
(* ------------------------------------------------------------------------- *)

let blocks blocksizes bm =
  map (fun (bs,b0) ->
        let m = foldl
          (fun a (b,i,j) c -> if b = b0 then ((i,j) |-> c) a else a)
          undefined bm in
        (((bs,bs),m):matrix))
      (zip blocksizes (1--length blocksizes));;

(* ------------------------------------------------------------------------- *)
(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
(* ------------------------------------------------------------------------- *)

let real_positivnullstellensatz_general linf d eqs leqs pol =
  let vars = itlist ((o) union poly_variables) (pol::eqs @ map fst leqs) [] in
  let monoid =
    if linf then
      (poly_const num_1,Rational_lt num_1)::
      (filter (fun (p,c) -> multidegree p <= d) leqs)
    else enumerate_products d leqs in
  let nblocks = length monoid in
  let mk_idmultiplier k p =
    let e = d - multidegree p in
    let mons = enumerate_monomials e vars in
    let nons = zip mons (1--length mons) in
    mons,
    itlist (fun (m,n) -> (m |-> ((-k,-n,n) |=> Int 1))) nons undefined in
  let mk_sqmultiplier k (p,c) =
    let e = (d - multidegree p) / 2 in
    let mons = enumerate_monomials e vars in
    let nons = zip mons (1--length mons) in
    mons,
    itlist (fun (m1,n1) ->
      itlist (fun (m2,n2) a ->
          let m = monomial_mul m1 m2 in
          if n1 > n2 then a else
          let c = if n1 = n2 then Int 1 else Int 2 in
          let e = tryapplyd a m undefined in
          (m |-> equation_add ((k,n1,n2) |=> c) e) a)
         nons)
       nons undefined in
  let sqmonlist,sqs = unzip(map2 mk_sqmultiplier (1--length monoid) monoid)
  and idmonlist,ids =  unzip(map2 mk_idmultiplier (1--length eqs) eqs) in
  let blocksizes = map length sqmonlist in
  let bigsum =
    itlist2 (fun p q a -> epoly_pmul p q a) eqs ids
            (itlist2 (fun (p,c) s a -> epoly_pmul p s a) monoid sqs
                     (epoly_of_poly(poly_neg pol))) in
  let eqns = foldl (fun a m e -> e::a) [] bigsum in
  let pvs,assig = eliminate_all_equations (0,0,0) eqns in
  let qvars = (0,0,0)::pvs in
  let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
  let mk_matrix v =
    foldl (fun m (b,i,j) ass -> if b < 0 then m else
                                let c = tryapplyd ass v (Int 0) in
                                if c =/ Int 0 then m else
                                ((b,j,i) |-> c) (((b,i,j) |-> c) m))
          undefined allassig in
  let diagents = foldl
    (fun a (b,i,j) e -> if b > 0 & i = j then equation_add e a else a)
    undefined allassig in
  let mats = map mk_matrix qvars
  and obj = length pvs,
            itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
                        undefined in
  let raw_vec = if pvs = [] then vector_0 0
                else scale_then (csdp nblocks blocksizes) obj mats in
  let find_rounding d =
   (if !debugging then
     (Format.print_string("Trying rounding with limit "^string_of_num d);
      Format.print_newline())
    else ());
    let vec = nice_vector d raw_vec in
    let blockmat = iter (1,dim vec)
     (fun i a -> bmatrix_add (bmatrix_cmul (element vec i) (el i mats)) a)
     (bmatrix_neg (el 0 mats)) in
    let allmats = blocks blocksizes blockmat in
    vec,map diag allmats in
  let vec,ratdias =
    if pvs = [] then find_rounding num_1
    else tryfind find_rounding (map Num.num_of_int (1--31) @
                                map pow2 (5--66)) in
  let newassigs =
    itlist (fun k -> el (k - 1) pvs |-> element vec k)
           (1--dim vec) ((0,0,0) |=> Int(-1)) in
  let finalassigs =
    foldl (fun a v e -> (v |-> equation_eval newassigs e) a) newassigs
          allassig in
  let poly_of_epoly p =
    foldl (fun a v e -> (v |--> equation_eval finalassigs e) a)
          undefined p in
  let mk_sos mons =
    let mk_sq (c,m) =
        c,itlist (fun k a -> (el (k - 1) mons |--> element m k) a)
                 (1--length mons) undefined in
    map mk_sq in
  let sqs = map2 mk_sos sqmonlist ratdias
  and cfs = map poly_of_epoly ids in
  let msq = filter (fun (a,b) -> b <> []) (map2 (fun a b -> a,b) monoid sqs) in
  let eval_sq sqs = itlist
   (fun (c,q) -> poly_add (poly_cmul c (poly_mul q q))) sqs poly_0 in
  let sanity =
    itlist (fun ((p,c),s) -> poly_add (poly_mul p (eval_sq s))) msq
           (itlist2 (fun p q -> poly_add (poly_mul p q)) cfs eqs
                    (poly_neg pol)) in
  if not(is_undefined sanity) then raise Sanity else
  cfs,map (fun (a,b) -> snd a,b) msq;;

(* ------------------------------------------------------------------------- *)
(* Iterative deepening.                                                      *)
(* ------------------------------------------------------------------------- *)

let rec deepen f n =
  try print_string "Searching with depth limit ";
      print_int n; print_newline(); f n
  with Failure _ -> deepen f (n + 1);;

(* ------------------------------------------------------------------------- *)
(* The ordering so we can create canonical HOL polynomials.                  *)
(* ------------------------------------------------------------------------- *)

let dest_monomial mon = sort (increasing fst) (graph mon);;

let monomial_order =
  let rec lexorder l1 l2 =
    match (l1,l2) with
      [],[] -> true
    | vps,[] -> false
    | [],vps -> true
    | ((x1,n1)::vs1),((x2,n2)::vs2) ->
          if x1 < x2 then true
          else if x2 < x1 then false
          else if n1 < n2 then false
          else if n2 < n1 then true
          else lexorder vs1 vs2 in
  fun m1 m2 ->
    if m2 = monomial_1 then true else if m1 = monomial_1 then false else
    let mon1 = dest_monomial m1 and mon2 = dest_monomial m2 in
    let deg1 = itlist ((o) (+) snd) mon1 0
    and deg2 = itlist ((o) (+) snd) mon2 0 in
    if deg1 < deg2 then false else if deg1 > deg2 then true
    else lexorder mon1 mon2;;

let dest_poly p =
  map (fun (m,c) -> c,dest_monomial m)
      (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p));;

(* ------------------------------------------------------------------------- *)
(* Map back polynomials and their composites to HOL.                         *)
(* ------------------------------------------------------------------------- *)

let term_of_varpow =
  fun x k ->
    if k = 1 then Var x else Pow (Var x, k);;

let term_of_monomial =
  fun m -> if m = monomial_1 then Const num_1 else
           let m' = dest_monomial m in
           let vps = itlist (fun (x,k) a -> term_of_varpow x k :: a) m' [] in
           end_itlist (fun s t -> Mul (s,t)) vps;;

let term_of_cmonomial =
  fun (m,c) ->
    if m = monomial_1 then Const c
    else if c =/ num_1 then term_of_monomial m
    else Mul (Const c,term_of_monomial m);;

let term_of_poly =
  fun p ->
    if p = poly_0 then Zero else
    let cms = map term_of_cmonomial
     (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p)) in
    end_itlist (fun t1 t2 -> Add (t1,t2)) cms;;

let term_of_sqterm (c,p) =
  Product(Rational_lt c,Square(term_of_poly p));;

let term_of_sos (pr,sqs) =
  if sqs = [] then pr
  else Product(pr,end_itlist (fun a b -> Sum(a,b)) (map term_of_sqterm sqs));;

(* ------------------------------------------------------------------------- *)
(* Interface to HOL.                                                         *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_NONLINEAR_PROVER translator (eqs,les,lts) =
  let eq0 = map (poly_of_term o lhand o concl) eqs
  and le0 = map (poly_of_term o lhand o concl) les
  and lt0 = map (poly_of_term o lhand o concl) lts in
  let eqp0 = map (fun (t,i) -> t,Axiom_eq i) (zip eq0 (0--(length eq0 - 1)))
  and lep0 = map (fun (t,i) -> t,Axiom_le i) (zip le0 (0--(length le0 - 1)))
  and ltp0 = map (fun (t,i) -> t,Axiom_lt i) (zip lt0 (0--(length lt0 - 1))) in
  let keq,eq = partition (fun (p,_) -> multidegree p = 0) eqp0
  and klep,lep = partition (fun (p,_) -> multidegree p = 0) lep0
  and kltp,ltp = partition (fun (p,_) -> multidegree p = 0) ltp0 in
  let trivial_axiom (p,ax) =
    match ax with
      Axiom_eq n when eval undefined p <>/ num_0 -> el n eqs
    | Axiom_le n when eval undefined p </ num_0 -> el n les
    | Axiom_lt n when eval undefined p <=/ num_0 -> el n lts
    | _ -> failwith "not a trivial axiom" in
  try let th = tryfind trivial_axiom (keq @ klep @ kltp) in
      CONV_RULE (LAND_CONV REAL_POLY_CONV THENC REAL_RAT_RED_CONV) th
  with Failure _ ->
  let pol = itlist poly_mul (map fst ltp) (poly_const num_1) in
  let leq = lep @ ltp in
  let tryall d =
    let e = multidegree pol in
    let k = if e = 0 then 0 else d / e in
    let eq' = map fst eq in
    tryfind (fun i -> d,i,real_positivnullstellensatz_general false d eq' leq
                          (poly_neg(poly_pow pol i)))
            (0--k) in
  let d,i,(cert_ideal,cert_cone) = deepen tryall 0 in
  let proofs_ideal =
    map2 (fun q (p,ax) -> Eqmul(term_of_poly q,ax)) cert_ideal eq
  and proofs_cone = map term_of_sos cert_cone
  and proof_ne =
    if ltp = [] then Rational_lt num_1 else
    let p = end_itlist (fun s t -> Product(s,t)) (map snd ltp) in
    funpow i (fun q -> Product(p,q)) (Rational_lt num_1) in
  let proof = end_itlist (fun s t -> Sum(s,t))
                         (proof_ne :: proofs_ideal @ proofs_cone) in
  print_string("Translating proof certificate to HOL");
  print_newline();
  translator (eqs,les,lts) proof;;
*)
(* ------------------------------------------------------------------------- *)
(* A wrapper that tries to substitute away variables first.                  *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_NONLINEAR_SUBST_PROVER =
  let zero = `&0:real`
  and mul_tm = `( * ):real->real->real`
  and shuffle1 =
    CONV_RULE(REWR_CONV(REAL_ARITH `a + x = (y:real) <=> x = y - a`))
  and shuffle2 =
    CONV_RULE(REWR_CONV(REAL_ARITH `x + a = (y:real) <=> x = y - a`)) in
  let rec substitutable_monomial fvs tm =
    match tm with
      Var(_,Tyapp("real",[])) when not (mem tm fvs) -> Int 1,tm
    | Comb(Comb(Const("real_mul",_),c),(Var(_,_) as t))
         when is_ratconst c & not (mem t fvs)
          -> rat_of_term c,t
    | Comb(Comb(Const("real_add",_),s),t) ->
         (try substitutable_monomial (union (frees t) fvs) s
          with Failure _ -> substitutable_monomial (union (frees s) fvs) t)
    | _ -> failwith "substitutable_monomial"
  and isolate_variable v th =
    match lhs(concl th) with
      x when x = v -> th
    | Comb(Comb(Const("real_add",_),(Var(_,Tyapp("real",[])) as x)),t)
        when x = v -> shuffle2 th
    | Comb(Comb(Const("real_add",_),s),t) ->
        isolate_variable v(shuffle1 th) in
  let make_substitution th =
    let (c,v) = substitutable_monomial [] (lhs(concl th)) in
    let th1 = AP_TERM (mk_comb(mul_tm,term_of_rat(Int 1 // c))) th in
    let th2 = CONV_RULE(BINOP_CONV REAL_POLY_MUL_CONV) th1 in
    CONV_RULE (RAND_CONV REAL_POLY_CONV) (isolate_variable v th2) in
  fun translator ->
    let rec substfirst(eqs,les,lts) =
      try let eth = tryfind make_substitution eqs in
          let modify =
            CONV_RULE(LAND_CONV(SUBS_CONV[eth] THENC REAL_POLY_CONV)) in
          substfirst(filter (fun t -> lhand(concl t) <> zero) (map modify eqs),
                     map modify les,map modify lts)
      with Failure _ -> REAL_NONLINEAR_PROVER translator (eqs,les,lts) in
    substfirst;;
*)
(* ------------------------------------------------------------------------- *)
(* Overall function.                                                         *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_SOS =
  let init = GEN_REWRITE_CONV ONCE_DEPTH_CONV [DECIMAL]
  and pure = GEN_REAL_ARITH REAL_NONLINEAR_SUBST_PROVER in
  fun tm -> let th = init tm in EQ_MP (SYM th) (pure(rand(concl th)));;
*)
(* ------------------------------------------------------------------------- *)
(* Add hacks for division.                                                   *)
(* ------------------------------------------------------------------------- *)
(*
let REAL_SOSFIELD =
  let inv_tm = `inv:real->real` in
  let prenex_conv =
    TOP_DEPTH_CONV BETA_CONV THENC
    PURE_REWRITE_CONV[FORALL_SIMP; EXISTS_SIMP; real_div;
                      REAL_INV_INV; REAL_INV_MUL; GSYM REAL_POW_INV] THENC
    NNFC_CONV THENC DEPTH_BINOP_CONV `(/\)` CONDS_CELIM_CONV THENC
    PRENEX_CONV
  and setup_conv = NNF_CONV THENC WEAK_CNF_CONV THENC CONJ_CANON_CONV
  and core_rule t =
    try REAL_ARITH t
    with Failure _ -> try REAL_RING t
    with Failure _ -> REAL_SOS t
  and is_inv =
    let is_div = is_binop `(/):real->real->real` in
    fun tm -> (is_div tm or (is_comb tm & rator tm = inv_tm)) &
              not(is_ratconst(rand tm)) in
  let BASIC_REAL_FIELD tm =
    let is_freeinv t = is_inv t & free_in t tm in
    let itms = setify(map rand (find_terms is_freeinv tm)) in
    let hyps = map (fun t -> SPEC t REAL_MUL_RINV) itms in
    let tm' = itlist (fun th t -> mk_imp(concl th,t)) hyps tm in
    let itms' = map (curry mk_comb inv_tm) itms in
    let gvs = map (genvar o type_of) itms' in
    let tm'' = subst (zip gvs itms') tm' in
    let th1 = setup_conv tm'' in
    let cjs = conjuncts(rand(concl th1)) in
    let ths = map core_rule cjs in
    let th2 = EQ_MP (SYM th1) (end_itlist CONJ ths) in
    rev_itlist (C MP) hyps (INST (zip itms' gvs) th2) in
  fun tm ->
    let th0 = prenex_conv tm in
    let tm0 = rand(concl th0) in
    let avs,bod = strip_forall tm0 in
    let th1 = setup_conv bod in
    let ths = map BASIC_REAL_FIELD (conjuncts(rand(concl th1))) in
    EQ_MP (SYM th0) (GENL avs (EQ_MP (SYM th1) (end_itlist CONJ ths)));;
*)
(* ------------------------------------------------------------------------- *)
(* Integer version.                                                          *)
(* ------------------------------------------------------------------------- *)
(*
let INT_SOS =
  let atom_CONV =
    let pth = prove
      (`(~(x <= y) <=> y + &1 <= x:int) /\
        (~(x < y) <=> y <= x) /\
        (~(x = y) <=> x + &1 <= y \/ y + &1 <= x) /\
        (x < y <=> x + &1 <= y)`,
       REWRITE_TAC[INT_NOT_LE; INT_NOT_LT; INT_NOT_EQ; INT_LT_DISCRETE]) in
    GEN_REWRITE_CONV I [pth]
  and bub_CONV = GEN_REWRITE_CONV TOP_SWEEP_CONV
   [int_eq; int_le; int_lt; int_ge; int_gt;
    int_of_num_th; int_neg_th; int_add_th; int_mul_th;
    int_sub_th; int_pow_th; int_abs_th; int_max_th; int_min_th] in
  let base_CONV = TRY_CONV atom_CONV THENC bub_CONV in
  let NNF_NORM_CONV = GEN_NNF_CONV false
   (base_CONV,fun t -> base_CONV t,base_CONV(mk_neg t)) in
  let init_CONV =
    GEN_REWRITE_CONV DEPTH_CONV [FORALL_SIMP; EXISTS_SIMP] THENC
    GEN_REWRITE_CONV DEPTH_CONV [INT_GT; INT_GE] THENC
    CONDS_ELIM_CONV THENC NNF_NORM_CONV in
  let p_tm = `p:bool`
  and not_tm = `(~)` in
  let pth = TAUT(mk_eq(mk_neg(mk_neg p_tm),p_tm)) in
  fun tm ->
    let th0 = INST [tm,p_tm] pth
    and th1 = NNF_NORM_CONV(mk_neg tm) in
    let th2 = REAL_SOS(mk_neg(rand(concl th1))) in
    EQ_MP th0 (EQ_MP (AP_TERM not_tm (SYM th1)) th2);;
*)
(* ------------------------------------------------------------------------- *)
(* Natural number version.                                                   *)
(* ------------------------------------------------------------------------- *)
(*
let SOS_RULE tm =
  let avs = frees tm in
  let tm' = list_mk_forall(avs,tm) in
  let th1 = NUM_TO_INT_CONV tm' in
  let th2 = INT_SOS (rand(concl th1)) in
  SPECL avs (EQ_MP (SYM th1) th2);;
*)
(* ------------------------------------------------------------------------- *)
(* Now pure SOS stuff.                                                       *)
(* ------------------------------------------------------------------------- *)

(*prioritize_real();;*)

(* ------------------------------------------------------------------------- *)
(* Some combinatorial helper functions.                                      *)
(* ------------------------------------------------------------------------- *)

let rec allpermutations l =
  if l = [] then [[]] else
  itlist (fun h acc -> map (fun t -> h::t)
                (allpermutations (subtract l [h])) @ acc) l [];;

let allvarorders l =
  map (fun vlis x -> index x vlis) (allpermutations l);;

let changevariables_monomial zoln (m:monomial) =
  foldl (fun a x k -> (assoc x zoln |-> k) a) monomial_1 m;;

let changevariables zoln pol =
  foldl (fun a m c -> (changevariables_monomial zoln m |-> c) a)
        poly_0 pol;;

(* ------------------------------------------------------------------------- *)
(* Return to original non-block matrices.                                    *)
(* ------------------------------------------------------------------------- *)

let sdpa_of_vector (v:vector) =
  let n = dim v in
  let strs = map (o (decimalize 20) (element v)) (1--n) in
  end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";;

let sdpa_of_blockdiagonal k m =
  let pfx = string_of_int k ^" " in
  let ents =
    foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
  let entss = sort (increasing fst) ents in
  itlist (fun ((b,i,j),c) a ->
     pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;

let sdpa_of_matrix k (m:matrix) =
  let pfx = string_of_int k ^ " 1 " in
  let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a)
                 (snd m) [] in
  let mss = sort (increasing fst) ms in
  itlist (fun ((i,j),c) a ->
     pfx ^ string_of_int i ^ " " ^ string_of_int j ^
     " " ^ decimalize 20 c ^ "\n" ^ a) mss "";;

let sdpa_of_problem comment obj mats =
  let m = length mats - 1
  and n,_ = dimensions (hd mats) in
  "\"" ^ comment ^ "\"\n" ^
  string_of_int m ^ "\n" ^
  "1\n" ^
  string_of_int n ^ "\n" ^
  sdpa_of_vector obj ^
  itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a)
          (1--length mats) mats "";;

let run_csdp dbg obj mats =
  let input_file = Filename.temp_file "sos" ".dat-s" in
  let output_file =
    String.sub input_file 0 (String.length input_file - 6) ^ ".out"
  and params_file = Filename.concat (!temp_path) "param.csdp" in
  file_of_string input_file (sdpa_of_problem "" obj mats);
  file_of_string params_file csdp_params;
  let rv = Sys.command("cd "^(!temp_path)^"; csdp "^input_file ^
                       " " ^ output_file ^
                       (if dbg then "" else "> /dev/null")) in
  let op = string_of_file output_file in
  let res = parse_csdpoutput op in
  ((if dbg then ()
    else (Sys.remove input_file; Sys.remove output_file));
    rv,res);;

let csdp obj mats =
  let rv,res = run_csdp (!debugging) obj mats in
  (if rv = 1 or rv = 2 then failwith "csdp: Problem is infeasible"
    else if rv = 3 then ()
(*    (Format.print_string "csdp warning: Reduced accuracy";
     Format.print_newline()) *)
   else if rv <> 0 then failwith("csdp: error "^string_of_int rv)
   else ());
  res;;

(* ------------------------------------------------------------------------- *)
(* Sum-of-squares function with some lowbrow symmetry reductions.            *)
(* ------------------------------------------------------------------------- *)

let sumofsquares_general_symmetry tool pol =
  let vars = poly_variables pol
  and lpps = newton_polytope pol in
  let n = length lpps in
  let sym_eqs =
    let invariants = filter
     (fun vars' ->
        is_undefined(poly_sub pol (changevariables (zip vars vars') pol)))
     (allpermutations vars) in
    let lpns = zip lpps (1--length lpps) in
    let lppcs =
      filter (fun (m,(n1,n2)) -> n1 <= n2)
             (allpairs
               (fun (m1,n1) (m2,n2) -> (m1,m2),(n1,n2)) lpns lpns) in
    let clppcs = end_itlist (@)
       (map (fun ((m1,m2),(n1,n2)) ->
                map (fun vars' ->
                        (changevariables_monomial (zip vars vars') m1,
                         changevariables_monomial (zip vars vars') m2),(n1,n2))
                    invariants)
            lppcs) in
    let clppcs_dom = setify(map fst clppcs) in
    let clppcs_cls = map (fun d -> filter (fun (e,_) -> e = d) clppcs)
                         clppcs_dom in
    let eqvcls = map (o setify (map snd)) clppcs_cls in
    let mk_eq cls acc =
      match cls with
        [] -> raise Sanity
      | [h] -> acc
      | h::t -> map (fun k -> (k |-> Int(-1)) (h |=> Int 1)) t @ acc in
    itlist mk_eq eqvcls [] in
  let eqs = foldl (fun a x y -> y::a) []
   (itern 1 lpps (fun m1 n1 ->
        itern 1 lpps (fun m2 n2 f ->
                let m = monomial_mul m1 m2 in
                if n1 > n2 then f else
                let c = if n1 = n2 then Int 1 else Int 2 in
                (m |-> ((n1,n2) |-> c) (tryapplyd f m undefined)) f))
       (foldl (fun a m c -> (m |-> ((0,0)|=>c)) a)
              undefined pol)) @
    sym_eqs in
  let pvs,assig = eliminate_all_equations (0,0) eqs in
  let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
  let qvars = (0,0)::pvs in
  let diagents =
    end_itlist equation_add (map (fun i -> apply allassig (i,i)) (1--n)) in
  let mk_matrix v =
   ((n,n),
    foldl (fun m (i,j) ass -> let c = tryapplyd ass v (Int 0) in
                              if c =/ Int 0 then m else
                              ((j,i) |-> c) (((i,j) |-> c) m))
          undefined allassig :matrix) in
  let mats = map mk_matrix qvars
  and obj = length pvs,
            itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
                undefined in
  let raw_vec = if pvs = [] then vector_0 0 else tool obj mats in
  let find_rounding d =
   (if !debugging then
     (Format.print_string("Trying rounding with limit "^string_of_num d);
      Format.print_newline())
    else ());
    let vec = nice_vector d raw_vec in
    let mat = iter (1,dim vec)
     (fun i a -> matrix_add (matrix_cmul (element vec i) (el i mats)) a)
     (matrix_neg (el 0 mats)) in
    deration(diag mat) in
  let rat,dia =
    if pvs = [] then
       let mat = matrix_neg (el 0 mats) in
       deration(diag mat)
    else
       tryfind find_rounding (map Num.num_of_int (1--31) @
                              map pow2 (5--66)) in
  let poly_of_lin(d,v) =
    d,foldl(fun a i c -> (el (i - 1) lpps |-> c) a) undefined (snd v) in
  let lins = map poly_of_lin dia in
  let sqs = map (fun (d,l) -> poly_mul (poly_const d) (poly_pow l 2)) lins in
  let sos = poly_cmul rat (end_itlist poly_add sqs) in
  if is_undefined(poly_sub sos pol) then rat,lins else raise Sanity;;

let sumofsquares = sumofsquares_general_symmetry csdp;;

(* ------------------------------------------------------------------------- *)
(* Pure HOL SOS conversion.                                                  *)
(* ------------------------------------------------------------------------- *)
(*
let SOS_CONV =
  let mk_square =
    let pow_tm = `(pow)` and two_tm = `2` in
    fun tm -> mk_comb(mk_comb(pow_tm,tm),two_tm)
  and mk_prod = mk_binop `( * )`
  and mk_sum = mk_binop `(+)` in
  fun tm ->
    let k,sos = sumofsquares(poly_of_term tm) in
    let mk_sqtm(c,p) =
      mk_prod (term_of_rat(k */ c)) (mk_square(term_of_poly p)) in
    let tm' = end_itlist mk_sum (map mk_sqtm sos) in
    let th = REAL_POLY_CONV tm and th' = REAL_POLY_CONV tm' in
    TRANS th (SYM th');;
*)
(* ------------------------------------------------------------------------- *)
(* Attempt to prove &0 <= x by direct SOS decomposition.                     *)
(* ------------------------------------------------------------------------- *)
(*
let PURE_SOS_TAC =
  let tac =
    MATCH_ACCEPT_TAC(REWRITE_RULE[GSYM REAL_POW_2] REAL_LE_SQUARE) ORELSE
    MATCH_ACCEPT_TAC REAL_LE_SQUARE ORELSE
    (MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC) ORELSE
    (MATCH_MP_TAC REAL_LE_MUL THEN CONJ_TAC) ORELSE
    CONV_TAC(RAND_CONV REAL_RAT_REDUCE_CONV THENC REAL_RAT_LE_CONV) in
  REPEAT GEN_TAC THEN REWRITE_TAC[real_ge] THEN
  GEN_REWRITE_TAC I [GSYM REAL_SUB_LE] THEN
  CONV_TAC(RAND_CONV SOS_CONV) THEN
  REPEAT tac THEN NO_TAC;;

let PURE_SOS tm = prove(tm,PURE_SOS_TAC);;
*)
(* ------------------------------------------------------------------------- *)
(* Examples.                                                                 *)
(* ------------------------------------------------------------------------- *)

(*****

time REAL_SOS
  `a1 >= &0 /\ a2 >= &0 /\
   (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + &2) /\
   (a1 * b1 + a2 * b2 = &0)
   ==> a1 * a2 - b1 * b2 >= &0`;;

time REAL_SOS `&3 * x + &7 * a < &4 /\ &3 < &2 * x ==> a < &0`;;

time REAL_SOS
  `b pow 2 < &4 * a * c ==> ~(a * x pow 2 + b * x + c = &0)`;;

time REAL_SOS
  `(a * x pow 2 + b * x + c = &0) ==> b pow 2 >= &4 * a * c`;;

time REAL_SOS
 `&0 <= x /\ x <= &1 /\ &0 <= y /\ y <= &1
  ==> x pow 2 + y pow 2 < &1 \/
      (x - &1) pow 2 + y pow 2 < &1 \/
      x pow 2 + (y - &1) pow 2 < &1 \/
      (x - &1) pow 2 + (y - &1) pow 2 < &1`;;

time REAL_SOS
 `&0 <= b /\ &0 <= c /\ &0 <= x /\ &0 <= y /\
  (x pow 2 = c) /\ (y pow 2 = a pow 2 * c + b)
  ==> a * c <= y * x`;;

time REAL_SOS
 `&0 <= x /\ &0 <= y /\ &0 <= z /\ x + y + z <= &3
  ==> x * y + x * z + y * z >= &3 * x * y * z`;;

time REAL_SOS
 `(x pow 2 + y pow 2 + z pow 2 = &1) ==> (x + y + z) pow 2 <= &3`;;

time REAL_SOS
 `(w pow 2 + x pow 2 + y pow 2 + z pow 2 = &1)
  ==> (w + x + y + z) pow 2 <= &4`;;

time REAL_SOS
 `x >= &1 /\ y >= &1 ==> x * y >= x + y - &1`;;

time REAL_SOS
 `x > &1 /\ y > &1 ==> x * y > x + y - &1`;;

time REAL_SOS
 `abs(x) <= &1
  ==> abs(&64 * x pow 7 - &112 * x pow 5 + &56 * x pow 3 - &7 * x) <= &1`;;

time REAL_SOS
 `abs(x - z) <= e /\ abs(y - z) <= e /\ &0 <= u /\ &0 <= v /\ (u + v = &1)
  ==> abs((u * x + v * y) - z) <= e`;;

(* ------------------------------------------------------------------------- *)
(* One component of denominator in dodecahedral example.                     *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
  `&2 <= x /\ x <= &125841 / &50000 /\
   &2 <= y /\ y <= &125841 / &50000 /\
   &2 <= z /\ z <= &125841 / &50000
   ==> &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= &0`;;

(* ------------------------------------------------------------------------- *)
(* Over a larger but simpler interval.                                       *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
 `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4
  ==> &0 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;;

(* ------------------------------------------------------------------------- *)
(* We can do 12. I think 12 is a sharp bound; see PP's certificate.          *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
 `&2 <= x /\ x <= &4 /\ &2 <= y /\ y <= &4 /\ &2 <= z /\ z <= &4
  ==> &12 <= &2 * (x * z + x * y + y * z) - (x * x + y * y + z * z)`;;

(* ------------------------------------------------------------------------- *)
(* Gloptipoly example.                                                       *)
(* ------------------------------------------------------------------------- *)

(*** This works but normalization takes minutes

time REAL_SOS
 `(x - y - &2 * x pow 4 = &0) /\ &0 <= x /\ x <= &2 /\ &0 <= y /\ y <= &3
  ==> y pow 2 - &7 * y - &12 * x + &17 >= &0`;;

 ***)

(* ------------------------------------------------------------------------- *)
(* Inequality from sci.math (see "Leon-Sotelo, por favor").                  *)
(* ------------------------------------------------------------------------- *)

time REAL_SOS
  `&0 <= x /\ &0 <= y /\ (x * y = &1)
   ==> x + y <= x pow 2 + y pow 2`;;

time REAL_SOS
  `&0 <= x /\ &0 <= y /\ (x * y = &1)
   ==> x * y * (x + y) <= x pow 2 + y pow 2`;;

time REAL_SOS
  `&0 <= x /\ &0 <= y ==> x * y * (x + y) pow 2 <= (x pow 2 + y pow 2) pow 2`;;

(* ------------------------------------------------------------------------- *)
(* Some examples over integers and natural numbers.                          *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `!m n. 2 * m + n = (n + m) + m`;;
time SOS_RULE `!n. ~(n = 0) ==> (0 MOD n = 0)`;;
time SOS_RULE  `!m n. m < n ==> (m DIV n = 0)`;;
time SOS_RULE `!n:num. n <= n * n`;;
time SOS_RULE  `!m n. n * (m DIV n) <= m`;;
time SOS_RULE `!n. ~(n = 0) ==> (0 DIV n = 0)`;;
time SOS_RULE `!m n p. ~(p = 0) /\ m <= n ==> m DIV p <= n DIV p`;;
time SOS_RULE `!a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b)`;;

(* ------------------------------------------------------------------------- *)
(* This is particularly gratifying --- cf hideous manual proof in arith.ml   *)
(* ------------------------------------------------------------------------- *)

(*** This doesn't now seem to work as well as it did; what changed?

time SOS_RULE
 `!a b c d. ~(b = 0) /\ b * c < (a + 1) * d ==> c DIV d <= a DIV b`;;

 ***)

(* ------------------------------------------------------------------------- *)
(* Key lemma for injectivity of Cantor-type pairing functions.               *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE
 `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1)
                ==> (x1 + y1 = x2 + y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2. ((x1 + y1) EXP 2 + x1 + 1 = (x2 + y2) EXP 2 + x2 + 1) /\
                (x1 + y1 = x2 + y2)
                ==> (x1 = x2) /\ (y1 = y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2.
      (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 =
       ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2)
       ==> (x1 + y1 = x2 + y2)`;;

time SOS_RULE
 `!x1 y1 x2 y2.
      (((x1 + y1) EXP 2 + 3 * x1 + y1) DIV 2 =
       ((x2 + y2) EXP 2 + 3 * x2 + y2) DIV 2) /\
      (x1 + y1 = x2 + y2)
      ==> (x1 = x2) /\ (y1 = y2)`;;

(* ------------------------------------------------------------------------- *)
(* Reciprocal multiplication (actually just ARITH_RULE does these).          *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `x <= 127 ==> ((86 * x) DIV 256 = x DIV 3)`;;

time SOS_RULE `x < 2 EXP 16 ==> ((104858 * x) DIV (2 EXP 20) = x DIV 10)`;;

(* ------------------------------------------------------------------------- *)
(* This is more impressive since it's really nonlinear. See REMAINDER_DECODE *)
(* ------------------------------------------------------------------------- *)

time SOS_RULE `0 < m /\ m < n  ==> ((m * ((n * x) DIV m + 1)) DIV n = x)`;;

(* ------------------------------------------------------------------------- *)
(* Some conversion examples.                                                 *)
(* ------------------------------------------------------------------------- *)

time SOS_CONV
 `&2 * x pow 4 + &2 * x pow 3 * y - x pow 2 * y pow 2 + &5 * y pow 4`;;

time SOS_CONV
 `x pow 4 - (&2 * y * z + &1) * x pow 2 +
  (y pow 2 * z pow 2 + &2 * y * z + &2)`;;

time SOS_CONV `&4 * x pow 4 +
          &4 * x pow 3 * y - &7 * x pow 2 * y pow 2 - &2 * x * y pow 3 +
          &10 * y pow 4`;;

time SOS_CONV `&4 * x pow 4 * y pow 6 + x pow 2 - x * y pow 2 + y pow 2`;;

time SOS_CONV
  `&4096 * (x pow 4 + x pow 2 + z pow 6 - &3 * x pow 2 * z pow 2) + &729`;;

time SOS_CONV
 `&120 * x pow 2 - &63 * x pow 4 + &10 * x pow 6 +
  &30 * x * y - &120 * y pow 2 + &120 * y pow 4 + &31`;;

time SOS_CONV
 `&9 * x pow 2 * y pow 4 + &9 * x pow 2 * z pow 4 + &36 * x pow 2 * y pow 3 +
  &36 * x pow 2 * y pow 2 - &48 * x * y * z pow 2 + &4 * y pow 4 +
  &4 * z pow 4 - &16 * y pow 3 + &16 * y pow 2`;;

time SOS_CONV
 `(x pow 2 + y pow 2 + z pow 2) *
  (x pow 4 * y pow 2 + x pow 2 * y pow 4 +
   z pow 6 - &3 * x pow 2 * y pow 2 *   z pow 2)`;;

time SOS_CONV
 `x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3`;;

(*** I think this will work, but normalization is slow

time SOS_CONV
 `&100 * (x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z) + &212`;;

 ***)

time SOS_CONV
 `&100 * ((&2 * x - &2) pow 2 + (x pow 3 - &8 * x - &2) pow 2) - &588`;;

time SOS_CONV
 `x pow 2 * (&120 - &63 * x pow 2 + &10 * x pow 4) + &30 * x * y +
    &30 * y pow 2 * (&4 * y pow 2 - &4) + &31`;;

(* ------------------------------------------------------------------------- *)
(* Example of basic rule.                                                    *)
(* ------------------------------------------------------------------------- *)

time PURE_SOS
  `!x. x pow 4 + y pow 4 + z pow 4 - &4 * x * y * z + x + y + z + &3
       >= &1 / &7`;;

time PURE_SOS
 `&0 <= &98 * x pow 12 +
        -- &980 * x pow 10 +
        &3038 * x pow 8 +
        -- &2968 * x pow 6 +
        &1022 * x pow 4 +
        -- &84 * x pow 2 +
        &2`;;

time PURE_SOS
 `!x. &0 <= &2 * x pow 14 +
            -- &84 * x pow 12 +
            &1022 * x pow 10 +
            -- &2968 * x pow 8 +
            &3038 * x pow 6 +
            -- &980 * x pow 4 +
            &98 * x pow 2`;;

(* ------------------------------------------------------------------------- *)
(* From Zeng et al, JSC vol 37 (2004), p83-99.                               *)
(* All of them work nicely with pure SOS_CONV, except (maybe) the one noted. *)
(* ------------------------------------------------------------------------- *)

PURE_SOS
  `x pow 6 + y pow 6 + z pow 6 - &3 * x pow 2 * y pow 2 * z pow 2 >= &0`;;

PURE_SOS `x pow 4 + y pow 4 + z pow 4 + &1 - &4*x*y*z >= &0`;;

PURE_SOS `x pow 4 + &2*x pow 2*z + x pow 2 - &2*x*y*z + &2*y pow 2*z pow 2 +
&2*y*z pow 2 + &2*z pow 2 - &2*x + &2* y*z + &1 >= &0`;;

(**** This is harder. Interestingly, this fails the pure SOS test, it seems.
      Yet only on rounding(!?) Poor Newton polytope optimization or something?
      But REAL_SOS does finally converge on the second run at level 12!

REAL_SOS
`x pow 4*y pow 4 - &2*x pow 5*y pow 3*z pow 2 + x pow 6*y pow 2*z pow 4 + &2*x
pow 2*y pow 3*z - &4* x pow 3*y pow 2*z pow 3 + &2*x pow 4*y*z pow 5 + z pow
2*y pow 2 - &2*z pow 4*y*x + z pow 6*x pow 2 >= &0`;;

 ****)

PURE_SOS
`x pow 4 + &4*x pow 2*y pow 2 + &2*x*y*z pow 2 + &2*x*y*w pow 2 + y pow 4 + z
pow 4 + w pow 4 + &2*z pow 2*w pow 2 + &2*x pow 2*w + &2*y pow 2*w + &2*x*y +
&3*w pow 2 + &2*z pow 2 + &1 >= &0`;;

PURE_SOS
`w pow 6 + &2*z pow 2*w pow 3 + x pow 4 + y pow 4 + z pow 4 + &2*x pow 2*w +
&2*x pow 2*z + &3*x pow 2 + w pow 2 + &2*z*w + z pow 2 + &2*z + &2*w + &1 >=
&0`;;

*****)