Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/sources/formale Sprachen/Isabelle/HOL/Library/Sum_of_Squares/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 38 kB

Quelle sum_of_squares.ML Sprache: SML

(*  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
  val debug: bool Config.T
  val trace_message: Proof.context -> (unit -> string) -> unit
  val debug_message: Proof.context -> (unit -> string) -> unit
  exception Failure of string;
end

structure Sum_of_Squares: SUM_OF_SQUARES =
struct

val max = Integer.max;

val denominator_rat = Rat.dest #> snd #> Rat.of_int;

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

fun lcm_rat x y =
  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 @1 else
   let val d = pow r (i div 2)
   in d * d * (if i mod 2 = 0 then @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.dest (abs r)
    val d = a div b
    val s = if r < @0 then ~ o Rat.of_int else Rat.of_int
    val x2 = 2 * (a - (b * d))
  in s (if x2 >= b then d + 1 else d) end

val trace = Attrib.setup_config_bool \<^binding>\<open>sos_trace\<close> (K false);
val debug = Attrib.setup_config_bool \<^binding>\<open>sos_debug\<close> (K false);

fun trace_message ctxt msg =
  if Config.get ctxt trace orelse Config.get ctxt debug then tracing (msg ()) else ();
fun debug_message ctxt msg = if Config.get ctxt debug then tracing (msg ()) else ();

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 y < @1/10 then normalize (@10 * y) - 1
  else if abs y >= @1 then normalize (y / @10) + 1
  else 0

in

fun decimalize d x =
  if x = @0 then "0.0"
  else
    let
      val y = abs x
      val e = normalize y
      val z = rat_pow @10 (~ e) * y + @1
      val k = int_of_rat (round_rat (rat_pow @10 d * z))
    in
      (if x < @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 = @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 = @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_rev 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) : vector =
  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)
  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 @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 @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, @1);

fun poly_const c =
  if c = @0 then poly_0 else FuncUtil.Monomialfunc.onefunc (monomial_1, c);

fun poly_cmul c p =
  if c = @0 then poly_0
  else FuncUtil.Monomialfunc.map (fn _ => fn x => c * x) p;

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

fun poly_add p1 p2 =
  FuncUtil.Monomialfunc.combine (curry op +) (fn x => x = @0) p1 p2;

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

fun poly_cmmul (c,m) p =
  if c = @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 @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 Thm.fast_term_ord
    (FuncUtil.Monomialfunc.fold_rev
      (fn (m, _) => union (is_equal o Thm.fast_term_ord) (monomial_variables m)) p []);

(* Conversion from HOL term.                                                 *)

local
  val neg_tm = \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close>
  val add_tm = \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
  val sub_tm = \<^cterm>\<open>(-) :: real \<Rightarrow> _\<close>
  val mul_tm = \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
  val inv_tm = \<^cterm>\<open>inverse :: real \<Rightarrow> _\<close>
  val div_tm = \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
  val pow_tm = \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close>
  val zero_tm = \<^cterm>\<open>0:: real\<close>
  val is_numeral = can (HOLogic.dest_number o Thm.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 Thm.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 (Thm.term_of tm) = \<^Type>\<open>real\<close>
    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 @0)) (1 upto n)
  in implode_space 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);

(* Parse back csdp output. *)

local

val decimal_digits = Scan.many1 Symbol.is_ascii_digit
val decimal_nat = decimal_digits >> (#1 o Library.read_int);
val decimal_int = decimal_nat >> Rat.of_int;

val decimal_sig =
  decimal_int -- Scan.option (Scan.$$ "." |-- decimal_digits) >>
  (fn (a, NONE) => a
    | (a, SOME bs) => a + Rat.of_int (#1 (Library.read_int bs)) / rat_pow @10 (length bs));

fun signed neg parse = $$ "-" |-- parse >> neg || $$ "+" |-- parse || parse;
val exponent = ($$ "e" || $$ "E") |-- signed ~ decimal_nat;

val decimal =
  signed ~ decimal_sig -- Scan.optional exponent 0
    >> (fn (a, b) => a * rat_pow @10 b);

val csdp_output =
  decimal -- Scan.repeat (Scan.$$ " " |-- Scan.option decimal) --| Scan.many Symbol.not_eof
    >> (fn (a, bs) => vector_of_list (a :: map_filter I bs));

in

fun parse_csdpoutput s =
  Symbol.scanner "Malformed CSDP output" csdp_output (raw_explode s);

end;

(* 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 c)) amat acc
  fun float_of_rat x =
    let val (a,b) = Rat.dest 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 @1
    val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) @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' @0
    val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') @0
    val scal1 = rat_pow @2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))
    val scal2 = rat_pow @2 (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 = @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 = @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 = @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 @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.of_int ~1) / a) (Inttriplefunc.delete_safe v eq)
              fun elim e =
                let val b = Inttriplefunc.tryapplyd e v @0 in
                  if b = @0 then e
                  else tri_equation_add e (tri_equation_cmul (~ 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) @0
      in
        if a11 < @0 then raise Failure "diagonalize: not PSD"
        else if a11 = @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 = @0 then a else FuncUtil.Intfunc.update (i,y) a
                end) (snd v) FuncUtil.Intfunc.empty)
            fun upt0 x y a =
              if y = @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) @0 -
                        FuncUtil.Intfunc.tryapplyd (snd v) j @0 *
                        FuncUtil.Intfunc.tryapplyd (snd v') k @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 @1, RealArith.Rational_lt @1)]
  else if d < 0 then []
  else
    (case pols of
      [] => [(poly_const @1, RealArith.Rational_lt @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), ~ 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 apply2 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" ^
    (implode_space (map string_of_int blocksizes)) ^
    "\n" ^
    sdpa_of_vector obj ^
    fold_rev (fn (k, 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 = @0);
fun bmatrix_cmul c bm =
  if c = @0 then Inttriplefunc.empty
  else Inttriplefunc.map (fn _ => fn x => c * x) bm;

val bmatrix_neg = bmatrix_cmul (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 @1, RealArith.Rational_lt @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), @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 @1 else @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_rev (fn (p, q) => fn a => tri_epoly_pmul p q a) (eqs ~~ ids)
        (fold_rev (fn ((p, _), 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, @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 @0 in
              if c = @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 @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 @0

    fun find_rounding d =
      let
        val _ =
          debug_message ctxt (fn () => "Trying rounding with limit "^Rat.string_of_rat d ^ "\n")
        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 @1
      else tryfind find_rounding (map Rat.of_int (1 upto 31) @ map (rat_pow @2) (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.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_rev (fn (p, 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 ctxt f n =
  (trace_message ctxt (fn () => "Searching with depth limit " ^ string_of_int n);
    (f n handle Failure s =>
      (trace_message ctxt (fn () => "failed with message: " ^ s); deepen ctxt 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 Thm.cprop_of
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>\<open>(0::real) + 1\<close>))
        Thm.term_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 <> @0 then nth eqs n
              else raise Failure "trivial_axiom: Not a trivial axiom"
          | RealArith.Axiom_le n =>
              if eval FuncUtil.Ctermfunc.empty p < @0 then nth les n
              else raise Failure "trivial_axiom: Not a trivial axiom"
          | RealArith.Axiom_lt n =>
              if eval FuncUtil.Ctermfunc.empty p <= @0 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 @1)
                    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 ctxt 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 @1
                      else
                        let val p = foldr1 RealArith.Product (map snd ltp) in
                          funpow i (fn q => RealArith.Product (p, q))
                            (RealArith.Rational_lt @1)
                        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

(* 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 (fn a => fn b => Thm.combination b a) eqs (Thm.reflexive t'))
  end

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

local
  open Conv
  val concl = Thm.dest_arg o Thm.cprop_of
  val shuffle1 =
    fconv_rule (rewr_conv @{lemma "(a + x \ y) \ (x \ y - a)" for a x y :: real
      by (atomize (full)) (simp add: field_simps)})
  val shuffle2 =
    fconv_rule (rewr_conv @{lemma "(x + a \ y) \ (x \ y - a)" for a x y :: real
      by (atomize (full)) (simp add: field_simps)})
  fun substitutable_monomial fvs tm =
    (case Thm.term_of tm of
      Free (_, \<^Type>\<open>real\<close>) =>
        if not (Cterms.defined fvs tm) then (@1, tm)
        else raise Failure "substitutable_monomial"
    | \<^Const_>\<open>times \<^typ>\<open>real\<close> for _ \<open>Free _\<close>\<close> =>
        if RealArith.is_ratconst (Thm.dest_arg1 tm) andalso
          not (Cterms.defined fvs (Thm.dest_arg tm))
        then (RealArith.dest_ratconst (Thm.dest_arg1 tm), Thm.dest_arg tm)
        else raise Failure "substitutable_monomial"
    | \<^Const_>\<open>plus \<^Type>\<open>real\<close> for _ _\<close> =>
         (substitutable_monomial (Drule.add_frees_cterm (Thm.dest_arg tm) fvs)
           (Thm.dest_arg1 tm)
           handle Failure _ =>
            substitutable_monomial (Drule.add_frees_cterm (Thm.dest_arg1 tm) fvs)
              (Thm.dest_arg tm))
    | _ => raise Failure "substitutable_monomial")

  fun isolate_variable v th =
    let
      val w = Thm.dest_arg1 (Thm.cprop_of th)
    in
      if v aconvc w then th
      else
        (case Thm.term_of w of
          \<^Const_>\<open>plus \<^Type>\<open>real\<close> for _ _\<close> =>
            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>\<open>(0::real) + 1\<close>))
        Thm.term_ord

    fun make_substitution th =
      let
        val (c,v) = substitutable_monomial Cterms.empty (Thm.dest_arg1(concl th))
        val th1 =
          Drule.arg_cong_rule
            (Thm.apply \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> (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>\<open>(\<le>) :: real \<Rightarrow> _\<close> orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
        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 Thm.cprop_of) t aconvc \<^cterm>\<open>0::real\<close>)
                  (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 =
  [\<^Const>\<open>Pure.imp\<close>, \<^Const>\<open>Trueprop\<close>,
   \<^Const>\<open>False\<close>, \<^Const>\<open>implies\<close>, \<^Const>\<open>conj\<close>, \<^Const>\<open>disj\<close>,
   \<^Const>\<open>Not\<close>, \<^Const>\<open>HOL.eq \<^Type>\<open>bool\<close>\<close>,
   \<^Const>\<open>All \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>Ex \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>HOL.eq \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>less \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>less_eq \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>plus \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>minus \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>times \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>uminus \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>divide \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>inverse \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>power \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>abs \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>min \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>max \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>zero_class.zero \<^Type>\<open>real\<close>\<close>, \<^Const>\<open>one_class.one \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>numeral \<^Type>\<open>nat\<close>\<close>,
   \<^Const>\<open>numeral \<^Type>\<open>real\<close>\<close>,
   \<^Const>\<open>Num.Bit0\<close>, \<^Const>\<open>Num.Bit1\<close>, \<^Const>\<open>Num.One\<close>];

fun check_sos kcts ct =
  let
    val t = Thm.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)) => T <> \<^Type>\<open>real\<close>) fs
      then error "SOS: not sos. Variables with type not real"
      else ()
    val vs = Term.add_vars t []
    val _ =
      if exists (fn ((_,T)) => T <> \<^Type>\<open>real\<close>) 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 = ctxt, ...} =>
  let
    val _ = check_sos known_sos_constants concl
    val (th, certificates) = real_sos prover ctxt (Thm.dest_arg concl)
    val _ = print_cert certificates
  in resolve_tac ctxt [th] 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 Thm.term_of)
in
  fun get_denom b ct =
    (case Thm.term_of ct of
      \<^Const_>\<open>divide \<^Type>\<open>real\<close> for _ _\<close> =>
        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)
    | \<^Const_>\<open>less \<^Type>\<open>real\<close> for _ _\<close> =>
        lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
    | \<^Const_>\<open>less_eq \<^Type>\<open>real\<close> for _ _\<close> =>
        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;

val sos_ss = @{context}
  |> fold Simplifier.add_simp @{thms field_simps}
  |> Simplifier.del_simp @{thm mult_nonneg_nonneg}
  |> Simplifier.simpset_of;

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 |> Simplifier.put_simpset sos_ss;
        val th =
          if ord then
            \<^instantiate>\<open>d and P = \<open>Thm.dest_arg P\<close> in
              lemma \<open>(d = 0 \<longrightarrow> P) \<and> (d > 0 \<longrightarrow> P) \<and> (d < 0 \<longrightarrow> P) \<Longrightarrow> P\<close> for d :: real by auto\<close>
          else
            \<^instantiate>\<open>d and P = \<open>Thm.dest_arg P\<close> in
              lemma \<open>(d = 0 \<longrightarrow> P) \<and> (d \<noteq> 0 \<longrightarrow> P) \<Longrightarrow> P\<close> for d :: real by blast\<close>
      in resolve_tac ctxt [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 ctxt THEN'
  elim_denom_tac ctxt THEN'
  core_sos_tac print_cert prover ctxt;

end;

quality99%

¤ Dauer der Verarbeitung: 0.30 Sekunden ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.