src/HOL/Library/Sum_of_Squares/sum_of_squares.ML

eliminiated neg_numeral in favour of - (numeral _)

(*  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. *)
(* FIXME improper use of parser combinators ahead *)

 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.                                                      *)
(* FIXME improper use of parser combinators ahead *)

 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 ctxt))
      then_conv Numeral_Simprocs.field_comp_conv ctxt) 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 ctxt)) th1
   in fconv_rule (arg_conv (real_poly_conv ctxt)) (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 ctxt))))
       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 "numeral :: num => nat"},
   @{term "numeral :: num => real"},
   @{term "Num.Bit0"}, @{term "Num.Bit1"}, @{term "Num.One"}];

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 simp_ctxt =
        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 simp_ctxt 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;