(*  Title:      HOL/Library/Sum_of_Squares/sum_of_squares.ML

     Author:     Amine Chaieb, University of Cambridge

     Author:     Philipp Meyer, TU Muenchen

 A tactic for proving nonlinear inequalities.

 *)

 signature SUM_OF_SQUARES =

 sig

   datatype proof_method = Certificate of RealArith.pss_tree | Prover of string -> string

   val sos_tac: (RealArith.pss_tree -> unit) -> proof_method -> Proof.context -> int -> tactic

   val trace: bool Config.T

   exception Failure of string;

 end

 structure Sum_of_Squares: SUM_OF_SQUARES =

 struct

 val rat_0 = Rat.zero;

 val rat_1 = Rat.one;

 val rat_2 = Rat.two;

 val rat_10 = Rat.rat_of_int 10;

 val max = Integer.max;

 val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;

 fun int_of_rat a =

     case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";

 fun lcm_rat x y = Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));

 fun rat_pow r i =

  let fun pow r i =

    if i = 0 then rat_1 else

    let val d = pow r (i div 2)

    in d */ d */ (if i mod 2 = 0 then rat_1 else r)

    end

  in if i < 0 then pow (Rat.inv r) (~ i) else pow r i end;

 fun round_rat r =

  let val (a,b) = Rat.quotient_of_rat (Rat.abs r)

      val d = a div b

      val s = if r </ rat_0 then (Rat.neg o Rat.rat_of_int) else Rat.rat_of_int

      val x2 = 2 * (a - (b * d))

  in s (if x2 >= b then d + 1 else d) end

 val abs_rat = Rat.abs;

 val pow2 = rat_pow rat_2;

 val pow10 = rat_pow rat_10;

 val trace = Attrib.setup_config_bool @{binding sos_trace} (K false);

 exception Sanity;

 exception Unsolvable;

 exception Failure of string;

 datatype proof_method =

     Certificate of RealArith.pss_tree

   | Prover of (string -> string)

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

 local

 fun normalize y =

   if abs_rat y </ (rat_1 // rat_10) then normalize (rat_10 */ y) - 1

   else if abs_rat y >=/ rat_1 then normalize (y // rat_10) + 1

   else 0

  in

 fun decimalize d x =

   if x =/ rat_0 then "0.0" else

   let

    val y = Rat.abs x

    val e = normalize y

    val z = pow10(~ e) */ y +/ rat_1

    val k = int_of_rat (round_rat(pow10 d */ z))

   in (if x </ rat_0 then "-0." else "0.") ^

      implode(tl(raw_explode(string_of_int k))) ^

      (if e = 0 then "" else "e"^string_of_int e)

   end

 end;

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

 fun itern k l f a =

   case l of

     [] => a

   | h::t => itern (k + 1) t f (f h k a);

 fun 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* Rat.rat FuncUtil.Intfunc.table;

 type matrix = (int*int)*(Rat.rat FuncUtil.Intpairfunc.table);

 fun iszero (_,r) = r =/ rat_0;

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

 fun vector_0 n = (n,FuncUtil.Intfunc.empty):vector;

 fun dim (v:vector) = fst v;

 fun vector_cmul c (v:vector) =

  let val n = dim v

  in if c =/ rat_0 then vector_0 n

     else (n,FuncUtil.Intfunc.map (fn _ => fn x => c */ x) (snd v))

  end;

 fun vector_of_list l =

  let val n = length l

  in (n,fold_rev2 (curry FuncUtil.Intfunc.update) (1 upto n) l FuncUtil.Intfunc.empty) :vector

  end;

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

 fun dimensions (m:matrix) = fst m;

 fun row k (m:matrix) =

  let val (_,j) = dimensions m

  in (j,

    FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if i = k then FuncUtil.Intfunc.update (j,c) a else a) (snd m) FuncUtil.Intfunc.empty ) : vector

  end;

 (* Monomials.                                                                *)

 fun monomial_eval assig m =

   FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => a */ rat_pow (FuncUtil.Ctermfunc.apply assig x) k)

         m rat_1;

 val monomial_1 = FuncUtil.Ctermfunc.empty;

 fun monomial_var x = FuncUtil.Ctermfunc.onefunc (x, 1);

 val monomial_mul =

   FuncUtil.Ctermfunc.combine Integer.add (K false);

 fun monomial_multidegree m =

  FuncUtil.Ctermfunc.fold (fn (_, k) => fn a => k + a) m 0;;

 fun monomial_variables m = FuncUtil.Ctermfunc.dom m;;

 (* Polynomials.                                                              *)

 fun eval assig p =

   FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => a +/ c */ monomial_eval assig m) p rat_0;

 val poly_0 = FuncUtil.Monomialfunc.empty;

 fun poly_isconst p =

   FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a) p true;

 fun poly_var x = FuncUtil.Monomialfunc.onefunc (monomial_var x,rat_1);

 fun poly_const c =

   if c =/ rat_0 then poly_0 else FuncUtil.Monomialfunc.onefunc(monomial_1, c);

 fun poly_cmul c p =

   if c =/ rat_0 then poly_0

   else FuncUtil.Monomialfunc.map (fn _ => fn x => c */ x) p;

 fun poly_neg p = FuncUtil.Monomialfunc.map (K Rat.neg) p;;

 fun poly_add p1 p2 =

   FuncUtil.Monomialfunc.combine (curry op +/) (fn x => x =/ rat_0) p1 p2;

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

 fun poly_cmmul (c,m) p =

  if c =/ rat_0 then poly_0

  else if FuncUtil.Ctermfunc.is_empty m

       then FuncUtil.Monomialfunc.map (fn _ => fn d => c */ d) p

       else FuncUtil.Monomialfunc.fold (fn (m', d) => fn a => (FuncUtil.Monomialfunc.update (monomial_mul m m', c */ d) a)) p poly_0;

 fun poly_mul p1 p2 =

   FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;

 fun poly_square p = poly_mul p p;

 fun poly_pow p k =

  if k = 0 then poly_const rat_1

  else if k = 1 then p

  else let val q = poly_square(poly_pow p (k div 2)) in

       if k mod 2 = 1 then poly_mul p q else q end;

 fun multidegree p =

   FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => max (monomial_multidegree m) a) p 0;

 fun poly_variables p =

   sort FuncUtil.cterm_ord (FuncUtil.Monomialfunc.fold_rev (fn (m, _) => union (is_equal o FuncUtil.cterm_ord) (monomial_variables m)) p []);;

 (* Conversion from HOL term.                                                 *)

 local

  val neg_tm = @{cterm "uminus :: real => _"}

  val add_tm = @{cterm "op + :: real => _"}

  val sub_tm = @{cterm "op - :: real => _"}

  val mul_tm = @{cterm "op * :: real => _"}

  val inv_tm = @{cterm "inverse :: real => _"}

  val div_tm = @{cterm "op / :: real => _"}

  val pow_tm = @{cterm "op ^ :: real => _"}

  val zero_tm = @{cterm "0:: real"}

  val is_numeral = can (HOLogic.dest_number o term_of)

  fun poly_of_term tm =

   if tm aconvc zero_tm then poly_0

   else if RealArith.is_ratconst tm

        then poly_const(RealArith.dest_ratconst tm)

   else

   (let val (lop,r) = Thm.dest_comb tm

    in if lop aconvc neg_tm then poly_neg(poly_of_term r)

       else if lop aconvc inv_tm then

        let val p = poly_of_term r

        in if poly_isconst p

           then poly_const(Rat.inv (eval FuncUtil.Ctermfunc.empty p))

           else error "poly_of_term: inverse of non-constant polyomial"

        end

    else (let val (opr,l) = Thm.dest_comb lop

          in

           if opr aconvc pow_tm andalso is_numeral r

           then poly_pow (poly_of_term l) ((snd o HOLogic.dest_number o term_of) r)

           else if opr aconvc add_tm

            then poly_add (poly_of_term l) (poly_of_term r)

           else if opr aconvc sub_tm

            then poly_sub (poly_of_term l) (poly_of_term r)

           else if opr aconvc mul_tm

            then poly_mul (poly_of_term l) (poly_of_term r)

           else if opr aconvc div_tm

            then let

                   val p = poly_of_term l

                   val q = poly_of_term r

                 in if poly_isconst q then poly_cmul (Rat.inv (eval FuncUtil.Ctermfunc.empty q)) p

                    else error "poly_of_term: division by non-constant polynomial"

                 end

           else poly_var tm

          end

          handle CTERM ("dest_comb",_) => poly_var tm)

    end

    handle CTERM ("dest_comb",_) => poly_var tm)

 in

 val poly_of_term = fn tm =>

  if type_of (term_of tm) = @{typ real} then poly_of_term tm

  else error "poly_of_term: term does not have real type"

 end;

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

 fun sdpa_of_vector (v:vector) =

  let

   val n = dim v

   val strs = map (decimalize 20 o (fn i => FuncUtil.Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)

  in space_implode " " strs ^ "\n"

  end;

 fun triple_int_ord ((a,b,c),(a',b',c')) =

  prod_ord int_ord (prod_ord int_ord int_ord)

     ((a,(b,c)),(a',(b',c')));

 structure Inttriplefunc = FuncFun(type key = int*int*int val ord = triple_int_ord);

 fun index_char str chr pos =

   if pos >= String.size str then ~1

   else if String.sub(str,pos) = chr then pos

   else index_char str chr (pos + 1);

 fun rat_of_quotient (a,b) = if b = 0 then rat_0 else Rat.rat_of_quotient (a,b);

 fun rat_of_string s =

  let val n = index_char s #"/" 0 in

   if n = ~1 then s |> Int.fromString |> the |> Rat.rat_of_int

   else

    let val SOME numer = Int.fromString(String.substring(s,0,n))

        val SOME den = Int.fromString (String.substring(s,n+1,String.size s - n - 1))

    in rat_of_quotient(numer, den)

    end

  end;

 fun isnum x = member (op =) ["0","1","2","3","4","5","6","7","8","9"] x;

 (* More parser basics.                                                       *)

  val numeral = Scan.one isnum

  val decimalint = Scan.repeat1 numeral >> (rat_of_string o implode)

  val decimalfrac = Scan.repeat1 numeral

     >> (fn s => rat_of_string(implode s) // pow10 (length s))

  val decimalsig =

     decimalint -- Scan.option (Scan.$$ "." |-- decimalfrac)

     >> (fn (h,NONE) => h | (h,SOME x) => h +/ x)

  fun signed prs =

        $$ "-" |-- prs >> Rat.neg

     || $$ "+" |-- prs

     || prs;

 fun emptyin def xs = if null xs then (def,xs) else Scan.fail xs

  val exponent = ($$ "e" || $$ "E") |-- signed decimalint;

  val decimal = signed decimalsig -- (emptyin rat_0|| exponent)

     >> (fn (h, x) => h */ pow10 (int_of_rat x));

  fun mkparser p s =

   let val (x,rst) = p (raw_explode s)

   in if null rst then x

      else error "mkparser: unparsed input"

   end;;

 (* Parse back csdp output.                                                      *)

  fun ignore _ = ((),[])

  fun csdpoutput inp =

    ((decimal -- Scan.repeat (Scan.$$ " " |-- Scan.option decimal) >>

     (fn (h,to) => map_filter I ((SOME h)::to))) --| ignore >> vector_of_list) inp

  val parse_csdpoutput = mkparser csdpoutput

 (* 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.                              *)

   (* Version for (int*int*int) keys *)

 local

   fun max_rat x y = if x </ y then y else x

   fun common_denominator fld amat acc =

       fld (fn (_,c) => fn a => lcm_rat (denominator_rat c) a) amat acc

   fun maximal_element fld amat acc =

     fld (fn (_,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc

 fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x

                      in Real.fromInt a / Real.fromInt b end;

 fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)

 in

 fun tri_scale_then solver (obj:vector)  mats =

  let

   val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats (rat_1)

   val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj)  (rat_1)

   val mats' = map (Inttriplefunc.map (fn _ => fn x => cd1 */ x)) mats

   val obj' = vector_cmul cd2 obj

   val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' (rat_0)

   val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') (rat_0)

   val scal1 = pow2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))

   val scal2 = pow2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))

   val mats'' = map (Inttriplefunc.map (fn _ => fn x => x */ scal1)) mats'

   val obj'' = vector_cmul scal2 obj'

  in solver obj'' mats''

   end

 end;

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

 fun nice_rational n x = round_rat (n */ x) // n;;

 fun nice_vector n ((d,v) : vector) =

  (d, FuncUtil.Intfunc.fold (fn (i,c) => fn a =>

    let val y = nice_rational n c

    in if c =/ rat_0 then a

       else FuncUtil.Intfunc.update (i,y) a end) v FuncUtil.Intfunc.empty):vector

 fun dest_ord f x = is_equal (f x);

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

 fun tri_equation_cmul c eq =

   if c =/ rat_0 then Inttriplefunc.empty else Inttriplefunc.map (fn _ => fn d => c */ d) eq;

 fun tri_equation_add eq1 eq2 = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0) eq1 eq2;

 fun tri_equation_eval assig eq =

  let fun value v = Inttriplefunc.apply assig v

  in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0

  end;

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

 fun tri_eliminate_all_equations one =

  let

   fun choose_variable eq =

    let val (v,_) = Inttriplefunc.choose eq

    in if is_equal (triple_int_ord(v,one)) then

       let val eq' = Inttriplefunc.delete_safe v eq

       in if Inttriplefunc.is_empty eq' then error "choose_variable"

          else fst (Inttriplefunc.choose eq')

       end

     else v

    end

   fun eliminate dun eqs = case eqs of

     [] => dun

   | eq::oeqs =>

     if Inttriplefunc.is_empty eq then eliminate dun oeqs else

     let val v = choose_variable eq

         val a = Inttriplefunc.apply eq v

         val eq' = tri_equation_cmul ((Rat.rat_of_int ~1) // a)

                    (Inttriplefunc.delete_safe v eq)

         fun elim e =

          let val b = Inttriplefunc.tryapplyd e v rat_0

          in if b =/ rat_0 then e

             else tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)

          end

     in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map (K elim) dun))

                  (map elim oeqs)

     end

 in fn eqs =>

  let

   val assig = eliminate Inttriplefunc.empty eqs

   val vs = Inttriplefunc.fold (fn (_, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []

  in (distinct (dest_ord triple_int_ord) vs,assig)

  end

 end;

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

 fun tri_epoly_pmul p q acc =

  FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>

   FuncUtil.Monomialfunc.fold (fn (m2,e) => fn b =>

    let val m =  monomial_mul m1 m2

        val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty

    in FuncUtil.Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b

    end) q a) p acc ;

 (* 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.                       *)

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

 local

 fun diagonalize n i m =

  if FuncUtil.Intpairfunc.is_empty (snd m) then []

  else

   let val a11 = FuncUtil.Intpairfunc.tryapplyd (snd m) (i,i) rat_0

   in if a11 </ rat_0 then raise Failure "diagonalize: not PSD"

     else if a11 =/ rat_0 then

           if FuncUtil.Intfunc.is_empty (snd (row i m)) then diagonalize n (i + 1) m

           else raise Failure "diagonalize: not PSD ___ "

     else

      let

       val v = row i m

       val v' = (fst v, FuncUtil.Intfunc.fold (fn (i, c) => fn a =>

        let val y = c // a11

        in if y = rat_0 then a else FuncUtil.Intfunc.update (i,y) a

        end)  (snd v) FuncUtil.Intfunc.empty)

       fun upt0 x y a = if y = rat_0 then a else FuncUtil.Intpairfunc.update (x,y) a

       val m' =

       ((n,n),

       iter (i+1,n) (fn j =>

           iter (i+1,n) (fn k =>

               (upt0 (j,k) (FuncUtil.Intpairfunc.tryapplyd (snd m) (j,k) rat_0 -/ FuncUtil.Intfunc.tryapplyd (snd v) j rat_0 */ FuncUtil.Intfunc.tryapplyd (snd v') k rat_0))))

           FuncUtil.Intpairfunc.empty)

      in (a11,v')::diagonalize n (i + 1) m'

      end

   end

 in

 fun diag m =

  let

    val nn = dimensions m

    val n = fst nn

  in if snd nn <> n then error "diagonalize: non-square matrix"

     else diagonalize n 1 m

  end

 end;

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

 fun enumerate_monomials d vars =

  if d < 0 then []

  else if d = 0 then [FuncUtil.Ctermfunc.empty]

  else if null vars then [monomial_1] else

  let val alts =

   map_range (fn k => let val oths = enumerate_monomials (d - k) (tl vars)

                in map (fn ks => if k = 0 then ks else FuncUtil.Ctermfunc.update (hd vars, k) ks) oths end) (d + 1)

  in flat alts

  end;

 (* 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.            *)

 fun enumerate_products d pols =

 if d = 0 then [(poly_const rat_1,RealArith.Rational_lt rat_1)]

 else if d < 0 then [] else

 case pols of

    [] => [(poly_const rat_1,RealArith.Rational_lt rat_1)]

  | (p,b)::ps =>

     let val e = multidegree p

     in if e = 0 then enumerate_products d ps else

        enumerate_products d ps @

        map (fn (q,c) => (poly_mul p q,RealArith.Product(b,c)))

          (enumerate_products (d - e) ps)

     end

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

 fun epoly_of_poly p =

   FuncUtil.Monomialfunc.fold (fn (m,c) => fn a => FuncUtil.Monomialfunc.update (m, Inttriplefunc.onefunc ((0,0,0), Rat.neg c)) a) p FuncUtil.Monomialfunc.empty;

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

 fun sdpa_of_blockdiagonal k m =

  let

   val pfx = string_of_int k ^" "

   val ents =

     Inttriplefunc.fold

       (fn ((b,i,j),c) => fn a => if i > j then a else ((b,i,j),c)::a)

       m []

   val entss = sort (triple_int_ord o pairself fst) ents

  in fold_rev (fn ((b,i,j),c) => fn a =>

      pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^

      " " ^ decimalize 20 c ^ "\n" ^ a) entss ""

  end;

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

 fun sdpa_of_blockproblem nblocks blocksizes obj mats =

  let val m = length mats - 1

  in

   string_of_int m ^ "\n" ^

   string_of_int nblocks ^ "\n" ^

   (space_implode " " (map string_of_int blocksizes)) ^

   "\n" ^

   sdpa_of_vector obj ^

   fold_rev2 (fn k => fn m => fn a => sdpa_of_blockdiagonal (k - 1) m ^ a)

     (1 upto length mats) mats ""

  end;

 (* Run prover on a problem in block diagonal form.                       *)

 fun run_blockproblem prover nblocks blocksizes obj mats=

   parse_csdpoutput (prover (sdpa_of_blockproblem nblocks blocksizes obj mats))

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

 val bmatrix_add = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0);

 fun bmatrix_cmul c bm =

   if c =/ rat_0 then Inttriplefunc.empty

   else Inttriplefunc.map (fn _ => fn x => c */ x) bm;

 val bmatrix_neg = bmatrix_cmul (Rat.rat_of_int ~1);

 (* Smash a block matrix into components.                                     *)

 fun blocks blocksizes bm =

  map (fn (bs,b0) =>

       let val m = Inttriplefunc.fold

           (fn ((b,i,j),c) => fn a => if b = b0 then FuncUtil.Intpairfunc.update ((i,j),c) a else a) bm FuncUtil.Intpairfunc.empty

           val _ = FuncUtil.Intpairfunc.fold (fn ((i,j),_) => fn a => max a (max i j)) m 0

       in (((bs,bs),m):matrix) end)

  (blocksizes ~~ (1 upto length blocksizes));;

 (* FIXME : Get rid of this !!!*)

 local

   fun tryfind_with msg _ [] = raise Failure msg

     | tryfind_with _ f (x::xs) = (f x handle Failure s => tryfind_with s f xs);

 in

   fun tryfind f = tryfind_with "tryfind" f

 end

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

 fun real_positivnullstellensatz_general ctxt prover linf d eqs leqs pol =

 let

  val vars = fold_rev (union (op aconvc) o poly_variables)

    (pol :: eqs @ map fst leqs) []

  val monoid = if linf then

       (poly_const rat_1,RealArith.Rational_lt rat_1)::

       (filter (fn (p,_) => multidegree p <= d) leqs)

     else enumerate_products d leqs

  val nblocks = length monoid

  fun mk_idmultiplier k p =

   let

    val e = d - multidegree p

    val mons = enumerate_monomials e vars

    val nons = mons ~~ (1 upto length mons)

   in (mons,

       fold_rev (fn (m,n) => FuncUtil.Monomialfunc.update(m,Inttriplefunc.onefunc((~k,~n,n),rat_1))) nons FuncUtil.Monomialfunc.empty)

   end

  fun mk_sqmultiplier k (p,_) =

   let

    val e = (d - multidegree p) div 2

    val mons = enumerate_monomials e vars

    val nons = mons ~~ (1 upto length mons)

   in (mons,

       fold_rev (fn (m1,n1) =>

        fold_rev (fn (m2,n2) => fn  a =>

         let val m = monomial_mul m1 m2

         in if n1 > n2 then a else

           let val c = if n1 = n2 then rat_1 else rat_2

               val e = FuncUtil.Monomialfunc.tryapplyd a m Inttriplefunc.empty

           in FuncUtil.Monomialfunc.update(m, tri_equation_add (Inttriplefunc.onefunc((k,n1,n2), c)) e) a

           end

         end)  nons)

        nons FuncUtil.Monomialfunc.empty)

   end

   val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)

   val (_(*idmonlist*),ids) =  split_list(map2 mk_idmultiplier (1 upto length eqs) eqs)

   val blocksizes = map length sqmonlist

   val bigsum =

     fold_rev2 (fn p => fn q => fn a => tri_epoly_pmul p q a) eqs ids

             (fold_rev2 (fn (p,_) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs

                      (epoly_of_poly(poly_neg pol)))

   val eqns = FuncUtil.Monomialfunc.fold (fn (_,e) => fn a => e::a) bigsum []

   val (pvs,assig) = tri_eliminate_all_equations (0,0,0) eqns

   val qvars = (0,0,0)::pvs

   val allassig = fold_rev (fn v => Inttriplefunc.update(v,(Inttriplefunc.onefunc(v,rat_1)))) pvs assig

   fun mk_matrix v =

     Inttriplefunc.fold (fn ((b,i,j), ass) => fn m =>

         if b < 0 then m else

          let val c = Inttriplefunc.tryapplyd ass v rat_0

          in if c = rat_0 then m else

             Inttriplefunc.update ((b,j,i), c) (Inttriplefunc.update ((b,i,j), c) m)

          end)

           allassig Inttriplefunc.empty

   val diagents = Inttriplefunc.fold

     (fn ((b,i,j), e) => fn a => if b > 0 andalso i = j then tri_equation_add e a else a)

     allassig Inttriplefunc.empty

   val mats = map mk_matrix qvars

   val obj = (length pvs,

             itern 1 pvs (fn v => fn i => FuncUtil.Intfunc.updatep iszero (i,Inttriplefunc.tryapplyd diagents v rat_0))

                         FuncUtil.Intfunc.empty)

   val raw_vec = if null pvs then vector_0 0

                 else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats

   fun int_element (_,v) i = FuncUtil.Intfunc.tryapplyd v i rat_0

   fun find_rounding d =

    let

     val _ =

       if Config.get ctxt trace

       then writeln ("Trying rounding with limit "^Rat.string_of_rat d ^ "\n")

       else ()

     val vec = nice_vector d raw_vec

     val blockmat = iter (1,dim vec)

      (fn i => fn a => bmatrix_add (bmatrix_cmul (int_element vec i) (nth mats i)) a)

      (bmatrix_neg (nth mats 0))

     val allmats = blocks blocksizes blockmat

    in (vec,map diag allmats)

    end

   val (vec,ratdias) =

     if null pvs then find_rounding rat_1

     else tryfind find_rounding (map Rat.rat_of_int (1 upto 31) @

                                 map pow2 (5 upto 66))

   val newassigs =

     fold_rev (fn k => Inttriplefunc.update (nth pvs (k - 1), int_element vec k))

            (1 upto dim vec) (Inttriplefunc.onefunc ((0,0,0), Rat.rat_of_int ~1))

   val finalassigs =

     Inttriplefunc.fold (fn (v,e) => fn a => Inttriplefunc.update(v, tri_equation_eval newassigs e) a) allassig newassigs

   fun poly_of_epoly p =

     FuncUtil.Monomialfunc.fold (fn (v,e) => fn a => FuncUtil.Monomialfunc.updatep iszero (v,tri_equation_eval finalassigs e) a)

           p FuncUtil.Monomialfunc.empty

   fun  mk_sos mons =

    let fun mk_sq (c,m) =

     (c,fold_rev (fn k=> fn a => FuncUtil.Monomialfunc.updatep iszero (nth mons (k - 1), int_element m k) a)

                  (1 upto length mons) FuncUtil.Monomialfunc.empty)

    in map mk_sq

    end

   val sqs = map2 mk_sos sqmonlist ratdias

   val cfs = map poly_of_epoly ids

   val msq = filter (fn (_,b) => not (null b)) (map2 pair monoid sqs)

   fun eval_sq sqs = fold_rev (fn (c,q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0

   val sanity =

     fold_rev (fn ((p,_),s) => poly_add (poly_mul p (eval_sq s))) msq

            (fold_rev2 (fn p => fn q => poly_add (poly_mul p q)) cfs eqs

                     (poly_neg pol))

 in if not(FuncUtil.Monomialfunc.is_empty sanity) then raise Sanity else

   (cfs,map (fn (a,b) => (snd a,b)) msq)

  end

 (* Iterative deepening.                                                      *)

 fun deepen f n =

   (writeln ("Searching with depth limit " ^ string_of_int n);

     (f n handle Failure s => (writeln ("failed with message: " ^ s); deepen f (n + 1))));

 (* Map back polynomials and their composites to a positivstellensatz.        *)

 fun cterm_of_sqterm (c,p) = RealArith.Product(RealArith.Rational_lt c,RealArith.Square p);

 fun cterm_of_sos (pr,sqs) = if null sqs then pr

   else RealArith.Product(pr,foldr1 RealArith.Sum (map cterm_of_sqterm sqs));

 (* Interface to HOL.                                                         *)

 local

   open Conv

   val concl = Thm.dest_arg o cprop_of

   fun simple_cterm_ord t u = Term_Ord.fast_term_ord (term_of t, term_of u) = LESS

 in

   (* FIXME: Replace tryfind by get_first !! *)

 fun real_nonlinear_prover proof_method ctxt =

  let

   val {add = _, mul = _, neg = _, pow = _,

        sub = _, main = real_poly_conv} =

       Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt

       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))

      simple_cterm_ord

   fun mainf cert_choice translator (eqs,les,lts) =

   let

    val eq0 = map (poly_of_term o Thm.dest_arg1 o concl) eqs

    val le0 = map (poly_of_term o Thm.dest_arg o concl) les

    val lt0 = map (poly_of_term o Thm.dest_arg o concl) lts

    val eqp0 = map_index (fn (i, t) => (t,RealArith.Axiom_eq i)) eq0

    val lep0 = map_index (fn (i, t) => (t,RealArith.Axiom_le i)) le0

    val ltp0 = map_index (fn (i, t) => (t,RealArith.Axiom_lt i)) lt0

    val (keq,eq) = List.partition (fn (p,_) => multidegree p = 0) eqp0

    val (klep,lep) = List.partition (fn (p,_) => multidegree p = 0) lep0

    val (kltp,ltp) = List.partition (fn (p,_) => multidegree p = 0) ltp0

    fun trivial_axiom (p,ax) =

     case ax of

        RealArith.Axiom_eq n => if eval FuncUtil.Ctermfunc.empty p <>/ Rat.zero then nth eqs n

                      else raise Failure "trivial_axiom: Not a trivial axiom"

      | RealArith.Axiom_le n => if eval FuncUtil.Ctermfunc.empty p </ Rat.zero then nth les n

                      else raise Failure "trivial_axiom: Not a trivial axiom"

      | RealArith.Axiom_lt n => if eval FuncUtil.Ctermfunc.empty p <=/ Rat.zero then nth lts n

                      else raise Failure "trivial_axiom: Not a trivial axiom"

      | _ => error "trivial_axiom: Not a trivial axiom"

    in

   (let val th = tryfind trivial_axiom (keq @ klep @ kltp)

    in

     (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Numeral_Simprocs.field_comp_conv) th, RealArith.Trivial)

    end)

    handle Failure _ =>

      (let val proof =

        (case proof_method of Certificate certs =>

          (* choose certificate *)

          let

            fun chose_cert [] (RealArith.Cert c) = c

              | chose_cert (RealArith.Left::s) (RealArith.Branch (l, _)) = chose_cert s l

              | chose_cert (RealArith.Right::s) (RealArith.Branch (_, r)) = chose_cert s r

              | chose_cert _ _ = error "certificate tree in invalid form"

          in

            chose_cert cert_choice certs

          end

        | Prover prover =>

          (* call prover *)

          let

           val pol = fold_rev poly_mul (map fst ltp) (poly_const Rat.one)

           val leq = lep @ ltp

           fun tryall d =

            let val e = multidegree pol

                val k = if e = 0 then 0 else d div e

                val eq' = map fst eq

            in tryfind (fn i => (d,i,real_positivnullstellensatz_general ctxt prover false d eq' leq

                                  (poly_neg(poly_pow pol i))))

                    (0 upto k)

            end

          val (_,i,(cert_ideal,cert_cone)) = deepen tryall 0

          val proofs_ideal =

            map2 (fn q => fn (_,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq

          val proofs_cone = map cterm_of_sos cert_cone

          val proof_ne = if null ltp then RealArith.Rational_lt Rat.one else

            let val p = foldr1 RealArith.Product (map snd ltp)

            in  funpow i (fn q => RealArith.Product(p,q)) (RealArith.Rational_lt Rat.one)

            end

          in

            foldr1 RealArith.Sum (proof_ne :: proofs_ideal @ proofs_cone)

          end)

      in

         (translator (eqs,les,lts) proof, RealArith.Cert proof)

      end)

    end

  in mainf end

 end

 fun C f x y = f y x;

   (* FIXME : This is very bad!!!*)

 fun subst_conv eqs t =

  let

   val t' = fold (Thm.lambda o Thm.lhs_of) eqs t

  in Conv.fconv_rule (Thm.beta_conversion true) (fold (C Thm.combination) eqs (Thm.reflexive t'))

  end

 (* A wrapper that tries to substitute away variables first.                  *)

 local

  open Conv

   fun simple_cterm_ord t u = Term_Ord.fast_term_ord (term_of t, term_of u) = LESS

  val concl = Thm.dest_arg o cprop_of

  val shuffle1 =

    fconv_rule (rewr_conv @{lemma "(a + x == y) == (x == y - (a::real))" by (atomize (full)) (simp add: field_simps) })

  val shuffle2 =

     fconv_rule (rewr_conv @{lemma "(x + a == y) ==  (x == y - (a::real))" by (atomize (full)) (simp add: field_simps)})

  fun substitutable_monomial fvs tm = case term_of tm of

     Free(_,@{typ real}) => if not (member (op aconvc) fvs tm) then (Rat.one,tm)

                            else raise Failure "substitutable_monomial"

   | @{term "op * :: real => _"}$_$(Free _) =>

      if RealArith.is_ratconst (Thm.dest_arg1 tm) andalso not (member (op aconvc) fvs (Thm.dest_arg tm))

          then (RealArith.dest_ratconst (Thm.dest_arg1 tm),Thm.dest_arg tm) else raise Failure "substitutable_monomial"

   | @{term "op + :: real => _"}$_$_ =>

        (substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg tm) fvs) (Thm.dest_arg1 tm)

         handle Failure _ => substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg1 tm) fvs) (Thm.dest_arg tm))

   | _ => raise Failure "substitutable_monomial"

   fun isolate_variable v th =

    let val w = Thm.dest_arg1 (cprop_of th)

    in if v aconvc w then th

       else case term_of w of

            @{term "op + :: real => _"}$_$_ =>

               if Thm.dest_arg1 w aconvc v then shuffle2 th

               else isolate_variable v (shuffle1 th)

           | _ => error "isolate variable : This should not happen?"

    end

 in

 fun real_nonlinear_subst_prover prover ctxt =

  let

   val {add = _, mul = real_poly_mul_conv, neg = _,

        pow = _, sub = _, main = real_poly_conv} =

       Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt

       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))

      simple_cterm_ord

   fun make_substitution th =

    let

     val (c,v) = substitutable_monomial [] (Thm.dest_arg1(concl th))

     val th1 = Drule.arg_cong_rule (Thm.apply @{cterm "op * :: real => _"} (RealArith.cterm_of_rat (Rat.inv c))) (mk_meta_eq th)

     val th2 = fconv_rule (binop_conv real_poly_mul_conv) th1

    in fconv_rule (arg_conv real_poly_conv) (isolate_variable v th2)

    end

    fun oprconv cv ct =

     let val g = Thm.dest_fun2 ct

     in if g aconvc @{cterm "op <= :: real => _"}

          orelse g aconvc @{cterm "op < :: real => _"}

        then arg_conv cv ct else arg1_conv cv ct

     end

   fun mainf cert_choice translator =

    let

     fun substfirst(eqs,les,lts) =

       ((let

            val eth = tryfind make_substitution eqs

            val modify = fconv_rule (arg_conv (oprconv(subst_conv [eth] then_conv real_poly_conv)))

        in  substfirst

              (filter_out (fn t => (Thm.dest_arg1 o Thm.dest_arg o cprop_of) t

                                    aconvc @{cterm "0::real"}) (map modify eqs),

                                    map modify les,map modify lts)

        end)

        handle Failure  _ => real_nonlinear_prover prover ctxt cert_choice translator (rev eqs, rev les, rev lts))

     in substfirst

    end

  in mainf

  end

 (* Overall function. *)

 fun real_sos prover ctxt =

   RealArith.gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt)

 end;

 val known_sos_constants =

   [@{term "op ==>"}, @{term "Trueprop"},

    @{term HOL.implies}, @{term HOL.conj}, @{term HOL.disj},

    @{term "Not"}, @{term "op = :: bool => _"},

    @{term "All :: (real => _) => _"}, @{term "Ex :: (real => _) => _"},

    @{term "op = :: real => _"}, @{term "op < :: real => _"},

    @{term "op <= :: real => _"},

    @{term "op + :: real => _"}, @{term "op - :: real => _"},

    @{term "op * :: real => _"}, @{term "uminus :: real => _"},

    @{term "op / :: real => _"}, @{term "inverse :: real => _"},

    @{term "op ^ :: real => _"}, @{term "abs :: real => _"},

    @{term "min :: real => _"}, @{term "max :: real => _"},

    @{term "0::real"}, @{term "1::real"}, @{term "number_of :: int => real"},

    @{term "number_of :: int => nat"},

    @{term "Int.Bit0"}, @{term "Int.Bit1"},

    @{term "Int.Pls"}, @{term "Int.Min"}];

 fun check_sos kcts ct =

  let

   val t = term_of ct

   val _ = if not (null (Term.add_tfrees t [])

                   andalso null (Term.add_tvars t []))

           then error "SOS: not sos. Additional type varables" else ()

   val fs = Term.add_frees t []

   val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs

           then error "SOS: not sos. Variables with type not real" else ()

   val vs = Term.add_vars t []

   val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) vs

           then error "SOS: not sos. Variables with type not real" else ()

   val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])

   val _ = if  null ukcs then ()

               else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))

 in () end

 fun core_sos_tac print_cert prover = SUBPROOF (fn {concl, context, ...} =>

   let

     val _ = check_sos known_sos_constants concl

     val (ths, certificates) = real_sos prover context (Thm.dest_arg concl)

     val _ = print_cert certificates

   in rtac ths 1 end)

 fun default_SOME _ NONE v = SOME v

   | default_SOME _ (SOME v) _ = SOME v;

 fun lift_SOME f NONE a = f a

   | lift_SOME _ (SOME a) _ = SOME a;

 local

  val is_numeral = can (HOLogic.dest_number o term_of)

 in

 fun get_denom b ct = case term_of ct of

   @{term "op / :: real => _"} $ _ $ _ =>

      if is_numeral (Thm.dest_arg ct) then get_denom b (Thm.dest_arg1 ct)

      else default_SOME (get_denom b) (get_denom b (Thm.dest_arg ct))   (Thm.dest_arg ct, b)

  | @{term "op < :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)

  | @{term "op <= :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)

  | _ $ _ => lift_SOME (get_denom b) (get_denom b (Thm.dest_fun ct)) (Thm.dest_arg ct)

  | _ => NONE

 end;

 fun elim_one_denom_tac ctxt =

 CSUBGOAL (fn (P,i) =>

  case get_denom false P of

    NONE => no_tac

  | SOME (d,ord) =>

      let

       val ss = simpset_of ctxt addsimps @{thms field_simps}

                addsimps [@{thm nonzero_power_divide}, @{thm power_divide}]

       val th = instantiate' [] [SOME d, SOME (Thm.dest_arg P)]

          (if ord then @{lemma "(d=0 --> P) & (d>0 --> P) & (d<(0::real) --> P) ==> P" by auto}

           else @{lemma "(d=0 --> P) & (d ~= (0::real) --> P) ==> P" by blast})

      in rtac th i THEN Simplifier.asm_full_simp_tac ss i end);

 fun elim_denom_tac ctxt i = REPEAT (elim_one_denom_tac ctxt i);

 fun sos_tac print_cert prover ctxt =

   Object_Logic.full_atomize_tac THEN'

   elim_denom_tac ctxt THEN'

   core_sos_tac print_cert prover ctxt;

 end;