src/HOL/Library/Sum_of_Squares/sum_of_squares.ML
author haftmann
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parent 69064 5840724b1d71
permissions -rw-r--r--
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(*  Title:      HOL/Library/Sum_of_Squares/sum_of_squares.ML
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    Author:     Amine Chaieb, University of Cambridge
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    Author:     Philipp Meyer, TU Muenchen
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A tactic for proving nonlinear inequalities.
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*)
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signature SUM_OF_SQUARES =
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sig
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  datatype proof_method = Certificate of RealArith.pss_tree | Prover of string -> string
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  val sos_tac: (RealArith.pss_tree -> unit) -> proof_method -> Proof.context -> int -> tactic
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  val trace: bool Config.T
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  val debug: bool Config.T
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  val trace_message: Proof.context -> (unit -> string) -> unit
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  val debug_message: Proof.context -> (unit -> string) -> unit
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  exception Failure of string;
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end
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structure Sum_of_Squares: SUM_OF_SQUARES =
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struct
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val max = Integer.max;
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val denominator_rat = Rat.dest #> snd #> Rat.of_int;
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fun int_of_rat a =
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  (case Rat.dest a of
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    (i, 1) => i
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  | _ => error "int_of_rat: not an int");
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fun lcm_rat x y =
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  Rat.of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
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fun rat_pow r i =
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 let fun pow r i =
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   if i = 0 then @1 else
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   let val d = pow r (i div 2)
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   in d * d * (if i mod 2 = 0 then @1 else r)
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   end
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 in if i < 0 then pow (Rat.inv r) (~ i) else pow r i end;
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fun round_rat r =
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  let
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    val (a,b) = Rat.dest (abs r)
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    val d = a div b
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    val s = if r < @0 then ~ o Rat.of_int else Rat.of_int
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    val x2 = 2 * (a - (b * d))
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  in s (if x2 >= b then d + 1 else d) end
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val trace = Attrib.setup_config_bool \<^binding>\<open>sos_trace\<close> (K false);
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val debug = Attrib.setup_config_bool \<^binding>\<open>sos_debug\<close> (K false);
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fun trace_message ctxt msg =
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  if Config.get ctxt trace orelse Config.get ctxt debug then tracing (msg ()) else ();
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fun debug_message ctxt msg = if Config.get ctxt debug then tracing (msg ()) else ();
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exception Sanity;
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exception Unsolvable;
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exception Failure of string;
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datatype proof_method =
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    Certificate of RealArith.pss_tree
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  | Prover of (string -> string)
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(* Turn a rational into a decimal string with d sig digits.                  *)
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local
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fun normalize y =
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  if abs y < @1/10 then normalize (@10 * y) - 1
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  else if abs y >= @1 then normalize (y / @10) + 1
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  else 0
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in
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fun decimalize d x =
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  if x = @0 then "0.0"
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  else
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    let
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      val y = abs x
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      val e = normalize y
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      val z = rat_pow @10 (~ e) * y + @1
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      val k = int_of_rat (round_rat (rat_pow @10 d * z))
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    in
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      (if x < @0 then "-0." else "0.") ^
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      implode (tl (raw_explode(string_of_int k))) ^
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      (if e = 0 then "" else "e" ^ string_of_int e)
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    end
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end;
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(* Iterations over numbers, and lists indexed by numbers.                    *)
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fun itern k l f a =
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  (case l of
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    [] => a
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  | h::t => itern (k + 1) t f (f h k a));
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fun iter (m,n) f a =
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  if n < m then a
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  else iter (m + 1, n) f (f m a);
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(* The main types.                                                           *)
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type vector = int * Rat.rat FuncUtil.Intfunc.table;
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type matrix = (int * int) * Rat.rat FuncUtil.Intpairfunc.table;
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fun iszero (_, r) = r = @0;
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(* Vectors. Conventionally indexed 1..n.                                     *)
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fun vector_0 n = (n, FuncUtil.Intfunc.empty): vector;
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fun dim (v: vector) = fst v;
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fun vector_cmul c (v: vector) =
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  let val n = dim v in
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    if c = @0 then vector_0 n
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    else (n,FuncUtil.Intfunc.map (fn _ => fn x => c * x) (snd v))
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  end;
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fun vector_of_list l =
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  let val n = length l in
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    (n, fold_rev FuncUtil.Intfunc.update (1 upto n ~~ l) FuncUtil.Intfunc.empty): vector
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  end;
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(* Matrices; again rows and columns indexed from 1.                          *)
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fun dimensions (m: matrix) = fst m;
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fun row k (m: matrix) : vector =
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  let val (_, j) = dimensions m in
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    (j,
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      FuncUtil.Intpairfunc.fold (fn ((i, j), c) => fn a =>
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        if i = k then FuncUtil.Intfunc.update (j, c) a else a) (snd m) FuncUtil.Intfunc.empty)
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  end;
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(* Monomials.                                                                *)
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fun monomial_eval assig m =
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  FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => a * rat_pow (FuncUtil.Ctermfunc.apply assig x) k)
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    m @1;
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val monomial_1 = FuncUtil.Ctermfunc.empty;
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fun monomial_var x = FuncUtil.Ctermfunc.onefunc (x, 1);
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val monomial_mul =
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  FuncUtil.Ctermfunc.combine Integer.add (K false);
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fun monomial_multidegree m =
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  FuncUtil.Ctermfunc.fold (fn (_, k) => fn a => k + a) m 0;
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fun monomial_variables m = FuncUtil.Ctermfunc.dom m;
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(* Polynomials.                                                              *)
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fun eval assig p =
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  FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => a + c * monomial_eval assig m) p @0;
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val poly_0 = FuncUtil.Monomialfunc.empty;
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fun poly_isconst p =
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  FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a)
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    p true;
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fun poly_var x = FuncUtil.Monomialfunc.onefunc (monomial_var x, @1);
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fun poly_const c =
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  if c = @0 then poly_0 else FuncUtil.Monomialfunc.onefunc (monomial_1, c);
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fun poly_cmul c p =
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  if c = @0 then poly_0
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  else FuncUtil.Monomialfunc.map (fn _ => fn x => c * x) p;
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fun poly_neg p = FuncUtil.Monomialfunc.map (K ~) p;
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fun poly_add p1 p2 =
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  FuncUtil.Monomialfunc.combine (curry op +) (fn x => x = @0) p1 p2;
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fun poly_sub p1 p2 = poly_add p1 (poly_neg p2);
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fun poly_cmmul (c,m) p =
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  if c = @0 then poly_0
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  else
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    if FuncUtil.Ctermfunc.is_empty m
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    then FuncUtil.Monomialfunc.map (fn _ => fn d => c * d) p
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    else
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      FuncUtil.Monomialfunc.fold (fn (m', d) => fn a =>
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          (FuncUtil.Monomialfunc.update (monomial_mul m m', c * d) a)) p poly_0;
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fun poly_mul p1 p2 =
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  FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
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fun poly_square p = poly_mul p p;
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fun poly_pow p k =
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  if k = 0 then poly_const @1
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  else if k = 1 then p
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  else
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    let val q = poly_square(poly_pow p (k div 2))
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    in if k mod 2 = 1 then poly_mul p q else q end;
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fun multidegree p =
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  FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => max (monomial_multidegree m) a) p 0;
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fun poly_variables p =
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  sort Thm.fast_term_ord
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    (FuncUtil.Monomialfunc.fold_rev
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      (fn (m, _) => union (is_equal o Thm.fast_term_ord) (monomial_variables m)) p []);
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(* Conversion from HOL term.                                                 *)
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local
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  val neg_tm = \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close>
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  val add_tm = \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
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  val sub_tm = \<^cterm>\<open>(-) :: real \<Rightarrow> _\<close>
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  val mul_tm = \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
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  val inv_tm = \<^cterm>\<open>inverse :: real \<Rightarrow> _\<close>
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  val div_tm = \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
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  val pow_tm = \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close>
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  val zero_tm = \<^cterm>\<open>0:: real\<close>
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  val is_numeral = can (HOLogic.dest_number o Thm.term_of)
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  fun poly_of_term tm =
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    if tm aconvc zero_tm then poly_0
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    else
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      if RealArith.is_ratconst tm
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      then poly_const(RealArith.dest_ratconst tm)
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      else
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       (let
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          val (lop, r) = Thm.dest_comb tm
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        in
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          if lop aconvc neg_tm then poly_neg(poly_of_term r)
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          else if lop aconvc inv_tm then
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            let val p = poly_of_term r in
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              if poly_isconst p
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              then poly_const(Rat.inv (eval FuncUtil.Ctermfunc.empty p))
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              else error "poly_of_term: inverse of non-constant polyomial"
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            end
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          else
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           (let
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              val (opr,l) = Thm.dest_comb lop
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            in
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              if opr aconvc pow_tm andalso is_numeral r
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              then poly_pow (poly_of_term l) ((snd o HOLogic.dest_number o Thm.term_of) r)
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              else if opr aconvc add_tm
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              then poly_add (poly_of_term l) (poly_of_term r)
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              else if opr aconvc sub_tm
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              then poly_sub (poly_of_term l) (poly_of_term r)
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              else if opr aconvc mul_tm
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              then poly_mul (poly_of_term l) (poly_of_term r)
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              else if opr aconvc div_tm
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              then
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                let
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                  val p = poly_of_term l
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                  val q = poly_of_term r
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                in
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                  if poly_isconst q
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                  then poly_cmul (Rat.inv (eval FuncUtil.Ctermfunc.empty q)) p
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                  else error "poly_of_term: division by non-constant polynomial"
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                end
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              else poly_var tm
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            end handle CTERM ("dest_comb",_) => poly_var tm)
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        end handle CTERM ("dest_comb",_) => poly_var tm)
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in
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  val poly_of_term = fn tm =>
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    if type_of (Thm.term_of tm) = \<^typ>\<open>real\<close>
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    then poly_of_term tm
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    else error "poly_of_term: term does not have real type"
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end;
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(* String of vector (just a list of space-separated numbers). *)
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fun sdpa_of_vector (v: vector) =
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  let
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    val n = dim v
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    val strs =
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      map (decimalize 20 o (fn i => FuncUtil.Intfunc.tryapplyd (snd v) i @0)) (1 upto n)
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  in space_implode " " strs ^ "\n" end;
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fun triple_int_ord ((a, b, c), (a', b', c')) =
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  prod_ord int_ord (prod_ord int_ord int_ord) ((a, (b, c)), (a', (b', c')));
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structure Inttriplefunc = FuncFun(type key = int * int * int val ord = triple_int_ord);
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(* Parse back csdp output. *)
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local
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val decimal_digits = Scan.many1 Symbol.is_ascii_digit
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val decimal_nat = decimal_digits >> (#1 o Library.read_int);
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val decimal_int = decimal_nat >> Rat.of_int;
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val decimal_sig =
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  decimal_int -- Scan.option (Scan.$$ "." |-- decimal_digits) >>
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  (fn (a, NONE) => a
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    | (a, SOME bs) => a + Rat.of_int (#1 (Library.read_int bs)) / rat_pow @10 (length bs));
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fun signed neg parse = $$ "-" |-- parse >> neg || $$ "+" |-- parse || parse;
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val exponent = ($$ "e" || $$ "E") |-- signed ~ decimal_nat;
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val decimal =
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  signed ~ decimal_sig -- Scan.optional exponent 0
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    >> (fn (a, b) => a * rat_pow @10 b);
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val csdp_output =
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  decimal -- Scan.repeat (Scan.$$ " " |-- Scan.option decimal) --| Scan.many Symbol.not_eof
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    >> (fn (a, bs) => vector_of_list (a :: map_filter I bs));
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in
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fun parse_csdpoutput s =
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  Symbol.scanner "Malformed CSDP output" csdp_output (raw_explode s);
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end;
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(* Try some apparently sensible scaling first. Note that this is purely to   *)
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(* get a cleaner translation to floating-point, and doesn't affect any of    *)
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(* the results, in principle. In practice it seems a lot better when there   *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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(* are extreme numbers in the original problem.                              *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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(* Version for (int*int*int) keys *)
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local
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  fun max_rat x y = if x < y then y else x
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  fun common_denominator fld amat acc =
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    fld (fn (_,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
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  fun maximal_element fld amat acc =
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    fld (fn (_,c) => fn maxa => max_rat maxa (abs c)) amat acc
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   337
  fun float_of_rat x =
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    let val (a,b) = Rat.dest x
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   339
    in Real.fromInt a / Real.fromInt b end;
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   340
  fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)
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2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   341
in
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   342
55508
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fun tri_scale_then solver (obj:vector) mats =
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  let
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    val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats @1
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    val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) @1
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    val mats' = map (Inttriplefunc.map (fn _ => fn x => cd1 * x)) mats
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    val obj' = vector_cmul cd2 obj
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   349
    val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' @0
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   350
    val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') @0
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   351
    val scal1 = rat_pow @2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))
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   352
    val scal2 = rat_pow @2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))
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   353
    val mats'' = map (Inttriplefunc.map (fn _ => fn x => x * scal1)) mats'
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   354
    val obj'' = vector_cmul scal2 obj'
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   355
  in solver obj'' mats'' end
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2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   356
end;
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   357
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   358
(* Round a vector to "nice" rationals.                                       *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   359
63198
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   360
fun nice_rational n x = round_rat (n * x) / n;
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   361
fun nice_vector n ((d,v) : vector) =
55508
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   362
  (d, FuncUtil.Intfunc.fold (fn (i,c) => fn a =>
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   363
      let val y = nice_rational n c in
63205
97b721666890 prefer rat numberals;
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diff changeset
   364
        if c = @0 then a
55508
90c42b130652 tuned whitespace;
wenzelm
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diff changeset
   365
        else FuncUtil.Intfunc.update (i,y) a
90c42b130652 tuned whitespace;
wenzelm
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   366
      end) v FuncUtil.Intfunc.empty): vector
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   367
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   368
fun dest_ord f x = is_equal (f x);
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   369
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   370
(* Stuff for "equations" ((int*int*int)->num functions).                         *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   371
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   372
fun tri_equation_cmul c eq =
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   373
  if c = @0 then Inttriplefunc.empty
63198
c583ca33076a ad-hoc overloading for standard operations on type Rat.rat;
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   374
  else Inttriplefunc.map (fn _ => fn d => c * d) eq;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
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   375
55508
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   376
fun tri_equation_add eq1 eq2 =
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   377
  Inttriplefunc.combine (curry op +) (fn x => x = @0) eq1 eq2;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
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diff changeset
   378
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
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   379
fun tri_equation_eval assig eq =
55508
90c42b130652 tuned whitespace;
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   380
  let
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wenzelm
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diff changeset
   381
    fun value v = Inttriplefunc.apply assig v
63205
97b721666890 prefer rat numberals;
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   382
  in Inttriplefunc.fold (fn (v, c) => fn a => a + value v * c) eq @0 end;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
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   383
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   384
(* Eliminate all variables, in an essentially arbitrary order.               *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   385
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   386
fun tri_eliminate_all_equations one =
55508
90c42b130652 tuned whitespace;
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   387
  let
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wenzelm
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diff changeset
   388
    fun choose_variable eq =
90c42b130652 tuned whitespace;
wenzelm
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diff changeset
   389
      let val (v,_) = Inttriplefunc.choose eq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   390
      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   391
        if is_equal (triple_int_ord(v,one)) then
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   392
          let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   393
            val eq' = Inttriplefunc.delete_safe v eq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   394
          in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   395
            if Inttriplefunc.is_empty eq' then error "choose_variable"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   396
            else fst (Inttriplefunc.choose eq')
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   397
          end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   398
        else v
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   399
      end
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   400
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   401
    fun eliminate dun eqs =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   402
      (case eqs of
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   403
        [] => dun
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   404
      | eq :: oeqs =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   405
          if Inttriplefunc.is_empty eq then eliminate dun oeqs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   406
          else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   407
            let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   408
              val v = choose_variable eq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   409
              val a = Inttriplefunc.apply eq v
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   410
              val eq' =
63201
f151704c08e4 tuned signature;
wenzelm
parents: 63198
diff changeset
   411
                tri_equation_cmul ((Rat.of_int ~1) / a) (Inttriplefunc.delete_safe v eq)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   412
              fun elim e =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   413
                let val b = Inttriplefunc.tryapplyd e v @0 in
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   414
                  if b = @0 then e
63211
0bec0d1d9998 more adhoc overloading;
wenzelm
parents: 63205
diff changeset
   415
                  else tri_equation_add e (tri_equation_cmul (~ b / a) eq)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   416
                end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   417
            in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   418
              eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map (K elim) dun))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   419
                (map elim oeqs)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   420
            end)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   421
  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   422
    fn eqs =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   423
      let
90c42b130652 tuned whitespace;
wenzelm
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diff changeset
   424
        val assig = eliminate Inttriplefunc.empty eqs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   425
        val vs = Inttriplefunc.fold (fn (_, f) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   426
          remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   427
      in (distinct (dest_ord triple_int_ord) vs,assig) end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   428
  end;
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   429
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   430
(* Multiply equation-parametrized poly by regular poly and add accumulator.  *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   431
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   432
fun tri_epoly_pmul p q acc =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   433
  FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   434
    FuncUtil.Monomialfunc.fold (fn (m2, e) => fn b =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   435
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   436
        val m =  monomial_mul m1 m2
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   437
        val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   438
      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   439
        FuncUtil.Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   440
      end) q a) p acc;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   441
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   442
(* Hence produce the "relevant" monomials: those whose squares lie in the    *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   443
(* Newton polytope of the monomials in the input. (This is enough according  *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   444
(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal,       *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   445
(* vol 45, pp. 363--374, 1978.                                               *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   446
(*                                                                           *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   447
(* These are ordered in sort of decreasing degree. In particular the         *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   448
(* constant monomial is last; this gives an order in diagonalization of the  *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   449
(* quadratic form that will tend to display constants.                       *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   450
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   451
(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form.  *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   452
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   453
local
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   454
  fun diagonalize n i m =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   455
    if FuncUtil.Intpairfunc.is_empty (snd m) then []
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   456
    else
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   457
      let
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   458
        val a11 = FuncUtil.Intpairfunc.tryapplyd (snd m) (i,i) @0
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   459
      in
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   460
        if a11 < @0 then raise Failure "diagonalize: not PSD"
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   461
        else if a11 = @0 then
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   462
          if FuncUtil.Intfunc.is_empty (snd (row i m))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   463
          then diagonalize n (i + 1) m
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   464
          else raise Failure "diagonalize: not PSD ___ "
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   465
        else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   466
          let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   467
            val v = row i m
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   468
            val v' =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   469
              (fst v, FuncUtil.Intfunc.fold (fn (i, c) => fn a =>
63198
c583ca33076a ad-hoc overloading for standard operations on type Rat.rat;
wenzelm
parents: 60867
diff changeset
   470
                let val y = c / a11
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   471
                in if y = @0 then a else FuncUtil.Intfunc.update (i,y) a
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   472
                end) (snd v) FuncUtil.Intfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   473
            fun upt0 x y a =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   474
              if y = @0 then a
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   475
              else FuncUtil.Intpairfunc.update (x,y) a
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   476
            val m' =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   477
              ((n, n),
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   478
                iter (i + 1, n) (fn j =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   479
                  iter (i + 1, n) (fn k =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   480
                    (upt0 (j, k)
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   481
                      (FuncUtil.Intpairfunc.tryapplyd (snd m) (j, k) @0 -
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   482
                        FuncUtil.Intfunc.tryapplyd (snd v) j @0 *
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   483
                        FuncUtil.Intfunc.tryapplyd (snd v') k @0))))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   484
                    FuncUtil.Intpairfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   485
          in (a11, v') :: diagonalize n (i + 1) m' end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   486
      end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   487
in
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   488
  fun diag m =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   489
    let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   490
      val nn = dimensions m
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   491
      val n = fst nn
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   492
    in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   493
      if snd nn <> n then error "diagonalize: non-square matrix"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   494
      else diagonalize n 1 m
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   495
    end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   496
end;
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   497
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   498
(* Enumeration of monomials with given multidegree bound.                    *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   499
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   500
fun enumerate_monomials d vars =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   501
  if d < 0 then []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   502
  else if d = 0 then [FuncUtil.Ctermfunc.empty]
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   503
  else if null vars then [monomial_1]
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   504
  else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   505
    let val alts =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   506
      map_range (fn k =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   507
        let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   508
          val oths = enumerate_monomials (d - k) (tl vars)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   509
        in map (fn ks => if k = 0 then ks else FuncUtil.Ctermfunc.update (hd vars, k) ks) oths end)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   510
        (d + 1)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   511
  in flat alts end;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   512
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   513
(* Enumerate products of distinct input polys with degree <= d.              *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   514
(* We ignore any constant input polynomials.                                 *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   515
(* Give the output polynomial and a record of how it was derived.            *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   516
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   517
fun enumerate_products d pols =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   518
  if d = 0 then [(poly_const @1, RealArith.Rational_lt @1)]
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   519
  else if d < 0 then []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   520
  else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   521
    (case pols of
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   522
      [] => [(poly_const @1, RealArith.Rational_lt @1)]
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   523
    | (p, b) :: ps =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   524
        let val e = multidegree p in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   525
          if e = 0 then enumerate_products d ps
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   526
          else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   527
            enumerate_products d ps @
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   528
            map (fn (q, c) => (poly_mul p q, RealArith.Product (b, c)))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   529
              (enumerate_products (d - e) ps)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   530
        end)
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   531
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   532
(* Convert regular polynomial. Note that we treat (0,0,0) as -1.             *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   533
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   534
fun epoly_of_poly p =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   535
  FuncUtil.Monomialfunc.fold (fn (m, c) => fn a =>
63211
0bec0d1d9998 more adhoc overloading;
wenzelm
parents: 63205
diff changeset
   536
      FuncUtil.Monomialfunc.update (m, Inttriplefunc.onefunc ((0, 0, 0), ~ c)) a)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   537
    p FuncUtil.Monomialfunc.empty;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   538
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   539
(* String for block diagonal matrix numbered k.                              *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   540
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   541
fun sdpa_of_blockdiagonal k m =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   542
  let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   543
    val pfx = string_of_int k ^" "
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   544
    val ents =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   545
      Inttriplefunc.fold
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   546
        (fn ((b, i, j), c) => fn a => if i > j then a else ((b, i, j), c) :: a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   547
        m []
59058
a78612c67ec0 renamed "pairself" to "apply2", in accordance to @{apply 2};
wenzelm
parents: 58635
diff changeset
   548
    val entss = sort (triple_int_ord o apply2 fst) ents
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   549
  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   550
    fold_rev (fn ((b,i,j),c) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   551
      pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   552
      " " ^ decimalize 20 c ^ "\n" ^ a) entss ""
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   553
  end;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   554
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   555
(* SDPA for problem using block diagonal (i.e. multiple SDPs)                *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   556
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   557
fun sdpa_of_blockproblem nblocks blocksizes obj mats =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   558
  let val m = length mats - 1
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   559
  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   560
    string_of_int m ^ "\n" ^
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   561
    string_of_int nblocks ^ "\n" ^
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   562
    (space_implode " " (map string_of_int blocksizes)) ^
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   563
    "\n" ^
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   564
    sdpa_of_vector obj ^
58635
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   565
    fold_rev (fn (k, m) => fn a => sdpa_of_blockdiagonal (k - 1) m ^ a)
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   566
      (1 upto length mats ~~ mats) ""
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   567
  end;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   568
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   569
(* Run prover on a problem in block diagonal form.                       *)
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   570
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   571
fun run_blockproblem prover nblocks blocksizes obj mats =
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   572
  parse_csdpoutput (prover (sdpa_of_blockproblem nblocks blocksizes obj mats))
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   573
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   574
(* 3D versions of matrix operations to consider blocks separately.           *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   575
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   576
val bmatrix_add = Inttriplefunc.combine (curry op +) (fn x => x = @0);
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   577
fun bmatrix_cmul c bm =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   578
  if c = @0 then Inttriplefunc.empty
63198
c583ca33076a ad-hoc overloading for standard operations on type Rat.rat;
wenzelm
parents: 60867
diff changeset
   579
  else Inttriplefunc.map (fn _ => fn x => c * x) bm;
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   580
63201
f151704c08e4 tuned signature;
wenzelm
parents: 63198
diff changeset
   581
val bmatrix_neg = bmatrix_cmul (Rat.of_int ~1);
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   582
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   583
(* Smash a block matrix into components.                                     *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   584
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   585
fun blocks blocksizes bm =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   586
  map (fn (bs, b0) =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   587
    let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   588
      val m =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   589
        Inttriplefunc.fold
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   590
          (fn ((b, i, j), c) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   591
            if b = b0 then FuncUtil.Intpairfunc.update ((i, j), c) a else a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   592
        bm FuncUtil.Intpairfunc.empty
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   593
      val _ = FuncUtil.Intpairfunc.fold (fn ((i, j), _) => fn a => max a (max i j)) m 0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   594
    in (((bs, bs), m): matrix) end)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   595
  (blocksizes ~~ (1 upto length blocksizes));
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   596
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   597
(* FIXME : Get rid of this !!!*)
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   598
local
44453
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   599
  fun tryfind_with msg _ [] = raise Failure msg
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   600
    | tryfind_with _ f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   601
in
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   602
  fun tryfind f = tryfind_with "tryfind" f
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   603
end
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   604
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   605
(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   606
38805
b09d8603f865 Sum_Of_Squares: proper configuration options;
wenzelm
parents: 38795
diff changeset
   607
fun real_positivnullstellensatz_general ctxt prover linf d eqs leqs pol =
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   608
  let
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   609
    val vars =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   610
      fold_rev (union (op aconvc) o poly_variables)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   611
        (pol :: eqs @ map fst leqs) []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   612
    val monoid =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   613
      if linf then
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   614
        (poly_const @1, RealArith.Rational_lt @1)::
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   615
        (filter (fn (p,_) => multidegree p <= d) leqs)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   616
      else enumerate_products d leqs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   617
    val nblocks = length monoid
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   618
    fun mk_idmultiplier k p =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   619
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   620
        val e = d - multidegree p
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   621
        val mons = enumerate_monomials e vars
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   622
        val nons = mons ~~ (1 upto length mons)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   623
      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   624
        (mons,
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   625
          fold_rev (fn (m, n) =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   626
            FuncUtil.Monomialfunc.update (m, Inttriplefunc.onefunc ((~k, ~n, n), @1)))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   627
          nons FuncUtil.Monomialfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   628
      end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   629
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   630
    fun mk_sqmultiplier k (p,_) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   631
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   632
        val e = (d - multidegree p) div 2
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   633
        val mons = enumerate_monomials e vars
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   634
        val nons = mons ~~ (1 upto length mons)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   635
      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   636
        (mons,
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   637
          fold_rev (fn (m1, n1) =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   638
            fold_rev (fn (m2, n2) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   639
              let val m = monomial_mul m1 m2 in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   640
                if n1 > n2 then a
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   641
                else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   642
                  let
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   643
                    val c = if n1 = n2 then @1 else @2
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   644
                    val e = FuncUtil.Monomialfunc.tryapplyd a m Inttriplefunc.empty
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   645
                  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   646
                    FuncUtil.Monomialfunc.update
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   647
                      (m, tri_equation_add (Inttriplefunc.onefunc ((k, n1, n2), c)) e) a
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   648
                  end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   649
              end) nons) nons FuncUtil.Monomialfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   650
      end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   651
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   652
    val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   653
    val (_(*idmonlist*),ids) =  split_list (map2 mk_idmultiplier (1 upto length eqs) eqs)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   654
    val blocksizes = map length sqmonlist
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   655
    val bigsum =
58635
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   656
      fold_rev (fn (p, q) => fn a => tri_epoly_pmul p q a) (eqs ~~ ids)
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   657
        (fold_rev (fn ((p, _), s) => fn a => tri_epoly_pmul p s a) (monoid ~~ sqs)
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   658
          (epoly_of_poly (poly_neg pol)))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   659
    val eqns = FuncUtil.Monomialfunc.fold (fn (_, e) => fn a => e :: a) bigsum []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   660
    val (pvs, assig) = tri_eliminate_all_equations (0, 0, 0) eqns
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   661
    val qvars = (0, 0, 0) :: pvs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   662
    val allassig =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   663
      fold_rev (fn v => Inttriplefunc.update (v, (Inttriplefunc.onefunc (v, @1)))) pvs assig
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   664
    fun mk_matrix v =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   665
      Inttriplefunc.fold (fn ((b, i, j), ass) => fn m =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   666
          if b < 0 then m
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   667
          else
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   668
            let val c = Inttriplefunc.tryapplyd ass v @0 in
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   669
              if c = @0 then m
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   670
              else Inttriplefunc.update ((b, j, i), c) (Inttriplefunc.update ((b, i, j), c) m)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   671
            end)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   672
        allassig Inttriplefunc.empty
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   673
    val diagents =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   674
      Inttriplefunc.fold
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   675
        (fn ((b, i, j), e) => fn a => if b > 0 andalso i = j then tri_equation_add e a else a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   676
        allassig Inttriplefunc.empty
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   677
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   678
    val mats = map mk_matrix qvars
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   679
    val obj =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   680
      (length pvs,
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   681
        itern 1 pvs (fn v => fn i =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   682
          FuncUtil.Intfunc.updatep iszero (i,Inttriplefunc.tryapplyd diagents v @0))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   683
          FuncUtil.Intfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   684
    val raw_vec =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   685
      if null pvs then vector_0 0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   686
      else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   687
    fun int_element (_, v) i = FuncUtil.Intfunc.tryapplyd v i @0
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   688
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   689
    fun find_rounding d =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   690
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   691
        val _ =
58631
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   692
          debug_message ctxt (fn () => "Trying rounding with limit "^Rat.string_of_rat d ^ "\n")
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   693
        val vec = nice_vector d raw_vec
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   694
        val blockmat =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   695
          iter (1, dim vec)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   696
            (fn i => fn a => bmatrix_add (bmatrix_cmul (int_element vec i) (nth mats i)) a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   697
            (bmatrix_neg (nth mats 0))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   698
        val allmats = blocks blocksizes blockmat
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   699
      in (vec, map diag allmats) end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   700
    val (vec, ratdias) =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   701
      if null pvs then find_rounding @1
63211
0bec0d1d9998 more adhoc overloading;
wenzelm
parents: 63205
diff changeset
   702
      else tryfind find_rounding (map Rat.of_int (1 upto 31) @ map (rat_pow @2) (5 upto 66))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   703
    val newassigs =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   704
      fold_rev (fn k => Inttriplefunc.update (nth pvs (k - 1), int_element vec k))
63201
f151704c08e4 tuned signature;
wenzelm
parents: 63198
diff changeset
   705
        (1 upto dim vec) (Inttriplefunc.onefunc ((0, 0, 0), Rat.of_int ~1))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   706
    val finalassigs =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   707
      Inttriplefunc.fold (fn (v, e) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   708
        Inttriplefunc.update (v, tri_equation_eval newassigs e) a) allassig newassigs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   709
    fun poly_of_epoly p =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   710
      FuncUtil.Monomialfunc.fold (fn (v, e) => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   711
          FuncUtil.Monomialfunc.updatep iszero (v, tri_equation_eval finalassigs e) a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   712
        p FuncUtil.Monomialfunc.empty
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   713
    fun mk_sos mons =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   714
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   715
        fun mk_sq (c, m) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   716
          (c, fold_rev (fn k => fn a =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   717
              FuncUtil.Monomialfunc.updatep iszero (nth mons (k - 1), int_element m k) a)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   718
            (1 upto length mons) FuncUtil.Monomialfunc.empty)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   719
      in map mk_sq end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   720
    val sqs = map2 mk_sos sqmonlist ratdias
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   721
    val cfs = map poly_of_epoly ids
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   722
    val msq = filter (fn (_, b) => not (null b)) (map2 pair monoid sqs)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   723
    fun eval_sq sqs = fold_rev (fn (c, q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   724
    val sanity =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   725
      fold_rev (fn ((p, _), s) => poly_add (poly_mul p (eval_sq s))) msq
58635
010b610eb55d eliminated some exotic combinators;
wenzelm
parents: 58631
diff changeset
   726
        (fold_rev (fn (p, q) => poly_add (poly_mul p q)) (cfs ~~ eqs) (poly_neg pol))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   727
  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   728
    if not(FuncUtil.Monomialfunc.is_empty sanity) then raise Sanity
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   729
    else (cfs, map (fn (a, b) => (snd a, b)) msq)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   730
  end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   731
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   732
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   733
(* Iterative deepening.                                                      *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   734
58631
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   735
fun deepen ctxt f n =
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   736
  (trace_message ctxt (fn () => "Searching with depth limit " ^ string_of_int n);
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   737
    (f n handle Failure s =>
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   738
      (trace_message ctxt (fn () => "failed with message: " ^ s); deepen ctxt f (n + 1))));
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   739
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   740
32645
1cc5b24f5a01 sos method generates and uses proof certificates
Philipp Meyer
parents: 32332
diff changeset
   741
(* Map back polynomials and their composites to a positivstellensatz.        *)
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   742
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   743
fun cterm_of_sqterm (c, p) = RealArith.Product (RealArith.Rational_lt c, RealArith.Square p);
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   744
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   745
fun cterm_of_sos (pr,sqs) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   746
  if null sqs then pr
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   747
  else RealArith.Product (pr, foldr1 RealArith.Sum (map cterm_of_sqterm sqs));
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   748
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   749
(* Interface to HOL.                                                         *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   750
local
32828
ad76967c703d removed opening of structures
Philipp Meyer
parents: 32740
diff changeset
   751
  open Conv
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   752
  val concl = Thm.dest_arg o Thm.cprop_of
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   753
in
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   754
(* FIXME: Replace tryfind by get_first !! *)
32645
1cc5b24f5a01 sos method generates and uses proof certificates
Philipp Meyer
parents: 32332
diff changeset
   755
fun real_nonlinear_prover proof_method ctxt =
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   756
  let
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   757
    val {add = _, mul = _, neg = _, pow = _, sub = _, main = real_poly_conv} =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   758
      Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   759
        (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
67562
2427d3e72b6e clarified signature;
wenzelm
parents: 67560
diff changeset
   760
        Thm.term_ord
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   761
    fun mainf cert_choice translator (eqs, les, lts) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   762
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   763
        val eq0 = map (poly_of_term o Thm.dest_arg1 o concl) eqs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   764
        val le0 = map (poly_of_term o Thm.dest_arg o concl) les
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   765
        val lt0 = map (poly_of_term o Thm.dest_arg o concl) lts
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   766
        val eqp0 = map_index (fn (i, t) => (t,RealArith.Axiom_eq i)) eq0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   767
        val lep0 = map_index (fn (i, t) => (t,RealArith.Axiom_le i)) le0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   768
        val ltp0 = map_index (fn (i, t) => (t,RealArith.Axiom_lt i)) lt0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   769
        val (keq,eq) = List.partition (fn (p, _) => multidegree p = 0) eqp0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   770
        val (klep,lep) = List.partition (fn (p, _) => multidegree p = 0) lep0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   771
        val (kltp,ltp) = List.partition (fn (p, _) => multidegree p = 0) ltp0
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   772
        fun trivial_axiom (p, ax) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   773
          (case ax of
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   774
            RealArith.Axiom_eq n =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   775
              if eval FuncUtil.Ctermfunc.empty p <> @0 then nth eqs n
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   776
              else raise Failure "trivial_axiom: Not a trivial axiom"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   777
          | RealArith.Axiom_le n =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   778
              if eval FuncUtil.Ctermfunc.empty p < @0 then nth les n
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   779
              else raise Failure "trivial_axiom: Not a trivial axiom"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   780
          | RealArith.Axiom_lt n =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   781
              if eval FuncUtil.Ctermfunc.empty p <= @0 then nth lts n
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   782
              else raise Failure "trivial_axiom: Not a trivial axiom"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   783
          | _ => error "trivial_axiom: Not a trivial axiom")
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   784
      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   785
        let val th = tryfind trivial_axiom (keq @ klep @ kltp) in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   786
          (fconv_rule (arg_conv (arg1_conv (real_poly_conv ctxt))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   787
            then_conv Numeral_Simprocs.field_comp_conv ctxt) th,
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   788
            RealArith.Trivial)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   789
        end handle Failure _ =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   790
          let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   791
            val proof =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   792
              (case proof_method of
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   793
                Certificate certs =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   794
                  (* choose certificate *)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   795
                  let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   796
                    fun chose_cert [] (RealArith.Cert c) = c
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   797
                      | chose_cert (RealArith.Left::s) (RealArith.Branch (l, _)) = chose_cert s l
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   798
                      | chose_cert (RealArith.Right::s) (RealArith.Branch (_, r)) = chose_cert s r
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   799
                      | chose_cert _ _ = error "certificate tree in invalid form"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   800
                  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   801
                    chose_cert cert_choice certs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   802
                  end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   803
              | Prover prover =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   804
                  (* call prover *)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   805
                  let
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   806
                    val pol = fold_rev poly_mul (map fst ltp) (poly_const @1)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   807
                    val leq = lep @ ltp
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   808
                    fun tryall d =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   809
                      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   810
                        val e = multidegree pol
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   811
                        val k = if e = 0 then 0 else d div e
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   812
                        val eq' = map fst eq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   813
                      in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   814
                        tryfind (fn i =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   815
                            (d, i, real_positivnullstellensatz_general ctxt prover false d eq' leq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   816
                              (poly_neg(poly_pow pol i))))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   817
                          (0 upto k)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   818
                      end
58631
41333b45bff9 clarified messages depending on options;
wenzelm
parents: 57889
diff changeset
   819
                    val (_,i,(cert_ideal,cert_cone)) = deepen ctxt tryall 0
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   820
                    val proofs_ideal =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   821
                      map2 (fn q => fn (_,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   822
                    val proofs_cone = map cterm_of_sos cert_cone
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   823
                    val proof_ne =
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   824
                      if null ltp then RealArith.Rational_lt @1
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   825
                      else
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   826
                        let val p = foldr1 RealArith.Product (map snd ltp) in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   827
                          funpow i (fn q => RealArith.Product (p, q))
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   828
                            (RealArith.Rational_lt @1)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   829
                        end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   830
                  in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   831
                    foldr1 RealArith.Sum (proof_ne :: proofs_ideal @ proofs_cone)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   832
                  end)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   833
          in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   834
            (translator (eqs,les,lts) proof, RealArith.Cert proof)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   835
          end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   836
      end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   837
  in mainf end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   838
end
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   839
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   840
(* FIXME : This is very bad!!!*)
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   841
fun subst_conv eqs t =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   842
  let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   843
    val t' = fold (Thm.lambda o Thm.lhs_of) eqs t
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   844
  in
60329
e85ed7a36b2f eliminated odd C combinator -- Isabelle/ML usually has canonical argument order;
wenzelm
parents: 59582
diff changeset
   845
    Conv.fconv_rule (Thm.beta_conversion true)
e85ed7a36b2f eliminated odd C combinator -- Isabelle/ML usually has canonical argument order;
wenzelm
parents: 59582
diff changeset
   846
      (fold (fn a => fn b => Thm.combination b a) eqs (Thm.reflexive t'))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   847
  end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   848
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   849
(* A wrapper that tries to substitute away variables first.                  *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   850
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   851
local
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   852
  open Conv
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   853
  val concl = Thm.dest_arg o Thm.cprop_of
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   854
  val shuffle1 =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   855
    fconv_rule (rewr_conv @{lemma "(a + x == y) == (x == y - (a::real))"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   856
      by (atomize (full)) (simp add: field_simps)})
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   857
  val shuffle2 =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   858
    fconv_rule (rewr_conv @{lemma "(x + a == y) ==  (x == y - (a::real))"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   859
      by (atomize (full)) (simp add: field_simps)})
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   860
  fun substitutable_monomial fvs tm =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   861
    (case Thm.term_of tm of
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   862
      Free (_, \<^typ>\<open>real\<close>) =>
63205
97b721666890 prefer rat numberals;
wenzelm
parents: 63201
diff changeset
   863
        if not (member (op aconvc) fvs tm) then (@1, tm)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   864
        else raise Failure "substitutable_monomial"
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 67562
diff changeset
   865
    | \<^term>\<open>(*) :: real \<Rightarrow> _\<close> $ _ $ (Free _) =>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   866
        if RealArith.is_ratconst (Thm.dest_arg1 tm) andalso
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   867
          not (member (op aconvc) fvs (Thm.dest_arg tm))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   868
        then (RealArith.dest_ratconst (Thm.dest_arg1 tm), Thm.dest_arg tm)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   869
        else raise Failure "substitutable_monomial"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   870
    | \<^term>\<open>(+) :: real \<Rightarrow> _\<close>$_$_ =>
60818
5e11a6e2b044 more direct access to atomic cterms;
wenzelm
parents: 60801
diff changeset
   871
         (substitutable_monomial (Drule.cterm_add_frees (Thm.dest_arg tm) fvs) (Thm.dest_arg1 tm)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   872
           handle Failure _ =>
60818
5e11a6e2b044 more direct access to atomic cterms;
wenzelm
parents: 60801
diff changeset
   873
            substitutable_monomial (Drule.cterm_add_frees (Thm.dest_arg1 tm) fvs) (Thm.dest_arg tm))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   874
    | _ => raise Failure "substitutable_monomial")
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   875
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   876
  fun isolate_variable v th =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   877
    let
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   878
      val w = Thm.dest_arg1 (Thm.cprop_of th)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   879
    in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   880
      if v aconvc w then th
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   881
      else
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   882
        (case Thm.term_of w of
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   883
          \<^term>\<open>(+) :: real \<Rightarrow> _\<close> $ _ $ _ =>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   884
            if Thm.dest_arg1 w aconvc v then shuffle2 th
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   885
            else isolate_variable v (shuffle1 th)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   886
        | _ => error "isolate variable : This should not happen?")
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   887
   end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   888
in
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   889
32268
d50f0cb67578 Functionality for sum of squares to call a remote csdp prover
Philipp Meyer
parents: 32150
diff changeset
   890
fun real_nonlinear_subst_prover prover ctxt =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   891
  let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   892
    val {add = _, mul = real_poly_mul_conv, neg = _, pow = _, sub = _, main = real_poly_conv} =
44453
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   893
      Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   894
        (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
67562
2427d3e72b6e clarified signature;
wenzelm
parents: 67560
diff changeset
   895
        Thm.term_ord
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   896
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   897
    fun make_substitution th =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   898
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   899
        val (c,v) = substitutable_monomial [] (Thm.dest_arg1(concl th))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   900
        val th1 =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   901
          Drule.arg_cong_rule
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 67562
diff changeset
   902
            (Thm.apply \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close> (RealArith.cterm_of_rat (Rat.inv c)))
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   903
            (mk_meta_eq th)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   904
        val th2 = fconv_rule (binop_conv (real_poly_mul_conv ctxt)) th1
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   905
      in fconv_rule (arg_conv (real_poly_conv ctxt)) (isolate_variable v th2) end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   906
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   907
    fun oprconv cv ct =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   908
      let val g = Thm.dest_fun2 ct in
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   909
        if g aconvc \<^cterm>\<open>(\<le>) :: real \<Rightarrow> _\<close> orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   910
        then arg_conv cv ct else arg1_conv cv ct
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   911
      end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   912
    fun mainf cert_choice translator =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   913
      let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   914
        fun substfirst (eqs, les, lts) =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   915
          (let
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   916
              val eth = tryfind make_substitution eqs
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   917
              val modify =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   918
                fconv_rule (arg_conv (oprconv(subst_conv [eth] then_conv (real_poly_conv ctxt))))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   919
            in
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   920
              substfirst
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   921
                (filter_out
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   922
                  (fn t => (Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of) t aconvc \<^cterm>\<open>0::real\<close>)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   923
                  (map modify eqs),
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   924
                  map modify les,
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   925
                  map modify lts)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   926
            end handle Failure  _ =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   927
              real_nonlinear_prover prover ctxt cert_choice translator (rev eqs, rev les, rev lts))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   928
      in substfirst end
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   929
  in mainf end
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   930
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   931
(* Overall function. *)
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   932
32645
1cc5b24f5a01 sos method generates and uses proof certificates
Philipp Meyer
parents: 32332
diff changeset
   933
fun real_sos prover ctxt =
32828
ad76967c703d removed opening of structures
Philipp Meyer
parents: 32740
diff changeset
   934
  RealArith.gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt)
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   935
31119
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   936
end;
2532bb2d65c7 A decision method for universal multivariate real arithmetic with add
chaieb
parents:
diff changeset
   937
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   938
val known_sos_constants =
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   939
  [\<^term>\<open>(\<Longrightarrow>)\<close>, \<^term>\<open>Trueprop\<close>,
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   940
   \<^term>\<open>HOL.False\<close>, \<^term>\<open>HOL.implies\<close>, \<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>,
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   941
   \<^term>\<open>Not\<close>, \<^term>\<open>(=) :: bool \<Rightarrow> _\<close>,
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   942
   \<^term>\<open>All :: (real \<Rightarrow> _) \<Rightarrow> _\<close>, \<^term>\<open>Ex :: (real \<Rightarrow> _) \<Rightarrow> _\<close>,
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   943
   \<^term>\<open>(=) :: real \<Rightarrow> _\<close>, \<^term>\<open>(<) :: real \<Rightarrow> _\<close>,
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   944
   \<^term>\<open>(\<le>) :: real \<Rightarrow> _\<close>,
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   945
   \<^term>\<open>(+) :: real \<Rightarrow> _\<close>, \<^term>\<open>(-) :: real \<Rightarrow> _\<close>,
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 67562
diff changeset
   946
   \<^term>\<open>(*) :: real \<Rightarrow> _\<close>, \<^term>\<open>uminus :: real \<Rightarrow> _\<close>,
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   947
   \<^term>\<open>(/) :: real \<Rightarrow> _\<close>, \<^term>\<open>inverse :: real \<Rightarrow> _\<close>,
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   948
   \<^term>\<open>(^) :: real \<Rightarrow> _\<close>, \<^term>\<open>abs :: real \<Rightarrow> _\<close>,
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   949
   \<^term>\<open>min :: real \<Rightarrow> _\<close>, \<^term>\<open>max :: real \<Rightarrow> _\<close>,
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   950
   \<^term>\<open>0::real\<close>, \<^term>\<open>1::real\<close>,
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   951
   \<^term>\<open>numeral :: num \<Rightarrow> nat\<close>,
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   952
   \<^term>\<open>numeral :: num \<Rightarrow> real\<close>,
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   953
   \<^term>\<open>Num.Bit0\<close>, \<^term>\<open>Num.Bit1\<close>, \<^term>\<open>Num.One\<close>];
31512
27118561c2e0 Tuned sos tactic to reject non SOS goals
chaieb
parents: 31131
diff changeset
   954
32839
a007a7cd8c2f tuned header;
wenzelm
parents: 32838
diff changeset
   955
fun check_sos kcts ct =
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   956
  let
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   957
    val t = Thm.term_of ct
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   958
    val _ =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   959
      if not (null (Term.add_tfrees t []) andalso null (Term.add_tvars t []))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   960
      then error "SOS: not sos. Additional type varables"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   961
      else ()
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   962
    val fs = Term.add_frees t []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   963
    val _ =
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   964
      if exists (fn ((_,T)) => not (T = \<^typ>\<open>real\<close>)) fs
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   965
      then error "SOS: not sos. Variables with type not real"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   966
      else ()
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   967
    val vs = Term.add_vars t []
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   968
    val _ =
67271
48ef58c6cf4c more symbols;
wenzelm
parents: 67091
diff changeset
   969
      if exists (fn ((_,T)) => not (T = \<^typ>\<open>real\<close>)) vs
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   970
      then error "SOS: not sos. Variables with type not real"
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   971
      else ()
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   972
    val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   973
    val _ =
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   974
      if null ukcs then ()
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   975
      else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   976
  in () end
31512
27118561c2e0 Tuned sos tactic to reject non SOS goals
chaieb
parents: 31131
diff changeset
   977
60752
b48830b670a1 prefer tactics with explicit context;
wenzelm
parents: 60329
diff changeset
   978
fun core_sos_tac print_cert prover = SUBPROOF (fn {concl, context = ctxt, ...} =>
32831
1352736e5727 changed core_sos_tac to use SUBPROOF
Philipp Meyer
parents: 32830
diff changeset
   979
  let
32838
d9dfd30af9ae core_sos_tac: SUBPROOF body operates on subgoal 1;
wenzelm
parents: 32831
diff changeset
   980
    val _ = check_sos known_sos_constants concl
60752
b48830b670a1 prefer tactics with explicit context;
wenzelm
parents: 60329
diff changeset
   981
    val (th, certificates) = real_sos prover ctxt (Thm.dest_arg concl)
32838
d9dfd30af9ae core_sos_tac: SUBPROOF body operates on subgoal 1;
wenzelm
parents: 32831
diff changeset
   982
    val _ = print_cert certificates
60752
b48830b670a1 prefer tactics with explicit context;
wenzelm
parents: 60329
diff changeset
   983
  in resolve_tac ctxt [th] 1 end);
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   984
44453
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   985
fun default_SOME _ NONE v = SOME v
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   986
  | default_SOME _ (SOME v) _ = SOME v;
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   987
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   988
fun lift_SOME f NONE a = f a
44453
082edd97ffe1 comment out dead code to avoid compiler warnings
huffman
parents: 42616
diff changeset
   989
  | lift_SOME _ (SOME a) _ = SOME a;
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   990
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   991
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   992
local
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   993
  val is_numeral = can (HOLogic.dest_number o Thm.term_of)
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
   994
in
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   995
  fun get_denom b ct =
59582
0fbed69ff081 tuned signature -- prefer qualified names;
wenzelm
parents: 59058
diff changeset
   996
    (case Thm.term_of ct of
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
   997
      \<^term>\<open>(/) :: real \<Rightarrow> _\<close> $ _ $ _ =>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   998
        if is_numeral (Thm.dest_arg ct)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
   999
        then get_denom b (Thm.dest_arg1 ct)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1000
        else default_SOME (get_denom b) (get_denom b (Thm.dest_arg ct)) (Thm.dest_arg ct, b)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
  1001
    | \<^term>\<open>(<) :: real \<Rightarrow> _\<close> $ _ $ _ =>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1002
        lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67271
diff changeset
  1003
    | \<^term>\<open>(\<le>) :: real \<Rightarrow> _\<close> $ _ $ _ =>
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1004
        lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1005
    | _ $ _ => lift_SOME (get_denom b) (get_denom b (Thm.dest_fun ct)) (Thm.dest_arg ct)
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1006
    | _ => NONE)
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1007
end;
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1008
70334
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1009
val sos_ss = @{context}
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1010
  |> fold Simplifier.add_simp @{thms field_simps}
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1011
  |> Simplifier.del_simp @{thm mult_nonneg_nonneg}
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1012
  |> Simplifier.simpset_of;
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1013
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1014
fun elim_one_denom_tac ctxt = CSUBGOAL (fn (P, i) =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1015
  (case get_denom false P of
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1016
    NONE => no_tac
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1017
  | SOME (d, ord) =>
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1018
      let
70334
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1019
        val simp_ctxt = ctxt
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1020
          |> Simplifier.put_simpset sos_ss
55508
90c42b130652 tuned whitespace;
wenzelm
parents: 54742
diff changeset
  1021
        val th =
60801
7664e0916eec tuned signature;
wenzelm
parents: 60752
diff changeset
  1022
          Thm.instantiate' [] [SOME d, SOME (Thm.dest_arg P)]
67091
1393c2340eec more symbols;
wenzelm
parents: 63523
diff changeset
  1023
            (if ord then @{lemma "(d=0 \<longrightarrow> P) \<and> (d>0 \<longrightarrow> P) \<and> (d<(0::real) \<longrightarrow> P) \<Longrightarrow> P" by auto}
1393c2340eec more symbols;
wenzelm
parents: 63523
diff changeset
  1024
             else @{lemma "(d=0 \<longrightarrow> P) \<and> (d \<noteq> (0::real) \<longrightarrow> P) \<Longrightarrow> P" by blast})
60752
b48830b670a1 prefer tactics with explicit context;
wenzelm
parents: 60329
diff changeset
  1025
      in resolve_tac ctxt [th] i THEN Simplifier.asm_full_simp_tac simp_ctxt i end));
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1026
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1027
fun elim_denom_tac ctxt i = REPEAT (elim_one_denom_tac ctxt i);
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1028
32949
aa6c470a962a eliminated slightly odd get/set operations in favour of Unsynchronized.ref;
wenzelm
parents: 32839
diff changeset
  1029
fun sos_tac print_cert prover ctxt =
70334
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1030
  Object_Logic.full_atomize_tac ctxt THEN'
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1031
  elim_denom_tac ctxt THEN'
bba46912379e prefer fixed simpset for proof procedure
haftmann
parents: 69064
diff changeset
  1032
  core_sos_tac print_cert prover ctxt;
31131
d9752181691a Now deals with division
chaieb
parents: 31119
diff changeset
  1033
31512
27118561c2e0 Tuned sos tactic to reject non SOS goals
chaieb
parents: 31131
diff changeset
  1034
end;