1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Thu Aug 06 19:51:59 2009 +0200
1.3 @@ -0,0 +1,1512 @@
1.4 +(* Title: sum_of_squares.ML
1.5 + Authors: Amine Chaieb, University of Cambridge
1.6 + Philipp Meyer, TU Muenchen
1.7 +
1.8 +A tactic for proving nonlinear inequalities
1.9 +*)
1.10 +
1.11 +signature SOS =
1.12 +sig
1.13 +
1.14 + val sos_tac : (string -> string) -> Proof.context -> int -> Tactical.tactic
1.15 +
1.16 + val debugging : bool ref;
1.17 +
1.18 + exception Failure of string;
1.19 +end
1.20 +
1.21 +structure Sos : SOS =
1.22 +struct
1.23 +
1.24 +val rat_0 = Rat.zero;
1.25 +val rat_1 = Rat.one;
1.26 +val rat_2 = Rat.two;
1.27 +val rat_10 = Rat.rat_of_int 10;
1.28 +val rat_1_2 = rat_1 // rat_2;
1.29 +val max = curry IntInf.max;
1.30 +val min = curry IntInf.min;
1.31 +
1.32 +val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
1.33 +val numerator_rat = Rat.quotient_of_rat #> fst #> Rat.rat_of_int;
1.34 +fun int_of_rat a =
1.35 + case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
1.36 +fun lcm_rat x y = Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
1.37 +
1.38 +fun rat_pow r i =
1.39 + let fun pow r i =
1.40 + if i = 0 then rat_1 else
1.41 + let val d = pow r (i div 2)
1.42 + in d */ d */ (if i mod 2 = 0 then rat_1 else r)
1.43 + end
1.44 + in if i < 0 then pow (Rat.inv r) (~ i) else pow r i end;
1.45 +
1.46 +fun round_rat r =
1.47 + let val (a,b) = Rat.quotient_of_rat (Rat.abs r)
1.48 + val d = a div b
1.49 + val s = if r </ rat_0 then (Rat.neg o Rat.rat_of_int) else Rat.rat_of_int
1.50 + val x2 = 2 * (a - (b * d))
1.51 + in s (if x2 >= b then d + 1 else d) end
1.52 +
1.53 +val abs_rat = Rat.abs;
1.54 +val pow2 = rat_pow rat_2;
1.55 +val pow10 = rat_pow rat_10;
1.56 +
1.57 +val debugging = ref false;
1.58 +
1.59 +exception Sanity;
1.60 +
1.61 +exception Unsolvable;
1.62 +
1.63 +exception Failure of string;
1.64 +
1.65 +(* Turn a rational into a decimal string with d sig digits. *)
1.66 +
1.67 +local
1.68 +fun normalize y =
1.69 + if abs_rat y </ (rat_1 // rat_10) then normalize (rat_10 */ y) - 1
1.70 + else if abs_rat y >=/ rat_1 then normalize (y // rat_10) + 1
1.71 + else 0
1.72 + in
1.73 +fun decimalize d x =
1.74 + if x =/ rat_0 then "0.0" else
1.75 + let
1.76 + val y = Rat.abs x
1.77 + val e = normalize y
1.78 + val z = pow10(~ e) */ y +/ rat_1
1.79 + val k = int_of_rat (round_rat(pow10 d */ z))
1.80 + in (if x </ rat_0 then "-0." else "0.") ^
1.81 + implode(tl(explode(string_of_int k))) ^
1.82 + (if e = 0 then "" else "e"^string_of_int e)
1.83 + end
1.84 +end;
1.85 +
1.86 +(* Iterations over numbers, and lists indexed by numbers. *)
1.87 +
1.88 +fun itern k l f a =
1.89 + case l of
1.90 + [] => a
1.91 + | h::t => itern (k + 1) t f (f h k a);
1.92 +
1.93 +fun iter (m,n) f a =
1.94 + if n < m then a
1.95 + else iter (m+1,n) f (f m a);
1.96 +
1.97 +(* The main types. *)
1.98 +
1.99 +fun strict_ord ord (x,y) = case ord (x,y) of LESS => LESS | _ => GREATER
1.100 +
1.101 +structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
1.102 +
1.103 +type vector = int* Rat.rat Intfunc.T;
1.104 +
1.105 +type matrix = (int*int)*(Rat.rat Intpairfunc.T);
1.106 +
1.107 +type monomial = int Ctermfunc.T;
1.108 +
1.109 +val cterm_ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t))
1.110 + fun monomial_ord (m1,m2) = list_ord (prod_ord cterm_ord int_ord) (Ctermfunc.graph m1, Ctermfunc.graph m2)
1.111 +structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
1.112 +
1.113 +type poly = Rat.rat Monomialfunc.T;
1.114 +
1.115 + fun iszero (k,r) = r =/ rat_0;
1.116 +
1.117 +fun fold_rev2 f l1 l2 b =
1.118 + case (l1,l2) of
1.119 + ([],[]) => b
1.120 + | (h1::t1,h2::t2) => f h1 h2 (fold_rev2 f t1 t2 b)
1.121 + | _ => error "fold_rev2";
1.122 +
1.123 +(* Vectors. Conventionally indexed 1..n. *)
1.124 +
1.125 +fun vector_0 n = (n,Intfunc.undefined):vector;
1.126 +
1.127 +fun dim (v:vector) = fst v;
1.128 +
1.129 +fun vector_const c n =
1.130 + if c =/ rat_0 then vector_0 n
1.131 + else (n,fold_rev (fn k => Intfunc.update (k,c)) (1 upto n) Intfunc.undefined) :vector;
1.132 +
1.133 +val vector_1 = vector_const rat_1;
1.134 +
1.135 +fun vector_cmul c (v:vector) =
1.136 + let val n = dim v
1.137 + in if c =/ rat_0 then vector_0 n
1.138 + else (n,Intfunc.mapf (fn x => c */ x) (snd v))
1.139 + end;
1.140 +
1.141 +fun vector_neg (v:vector) = (fst v,Intfunc.mapf Rat.neg (snd v)) :vector;
1.142 +
1.143 +fun vector_add (v1:vector) (v2:vector) =
1.144 + let val m = dim v1
1.145 + val n = dim v2
1.146 + in if m <> n then error "vector_add: incompatible dimensions"
1.147 + else (n,Intfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd v1) (snd v2)) :vector
1.148 + end;
1.149 +
1.150 +fun vector_sub v1 v2 = vector_add v1 (vector_neg v2);
1.151 +
1.152 +fun vector_dot (v1:vector) (v2:vector) =
1.153 + let val m = dim v1
1.154 + val n = dim v2
1.155 + in if m <> n then error "vector_dot: incompatible dimensions"
1.156 + else Intfunc.fold (fn (i,x) => fn a => x +/ a)
1.157 + (Intfunc.combine (curry op */) (fn x => x =/ rat_0) (snd v1) (snd v2)) rat_0
1.158 + end;
1.159 +
1.160 +fun vector_of_list l =
1.161 + let val n = length l
1.162 + in (n,fold_rev2 (curry Intfunc.update) (1 upto n) l Intfunc.undefined) :vector
1.163 + end;
1.164 +
1.165 +(* Matrices; again rows and columns indexed from 1. *)
1.166 +
1.167 +fun matrix_0 (m,n) = ((m,n),Intpairfunc.undefined):matrix;
1.168 +
1.169 +fun dimensions (m:matrix) = fst m;
1.170 +
1.171 +fun matrix_const c (mn as (m,n)) =
1.172 + if m <> n then error "matrix_const: needs to be square"
1.173 + else if c =/ rat_0 then matrix_0 mn
1.174 + else (mn,fold_rev (fn k => Intpairfunc.update ((k,k), c)) (1 upto n) Intpairfunc.undefined) :matrix;;
1.175 +
1.176 +val matrix_1 = matrix_const rat_1;
1.177 +
1.178 +fun matrix_cmul c (m:matrix) =
1.179 + let val (i,j) = dimensions m
1.180 + in if c =/ rat_0 then matrix_0 (i,j)
1.181 + else ((i,j),Intpairfunc.mapf (fn x => c */ x) (snd m))
1.182 + end;
1.183 +
1.184 +fun matrix_neg (m:matrix) =
1.185 + (dimensions m, Intpairfunc.mapf Rat.neg (snd m)) :matrix;
1.186 +
1.187 +fun matrix_add (m1:matrix) (m2:matrix) =
1.188 + let val d1 = dimensions m1
1.189 + val d2 = dimensions m2
1.190 + in if d1 <> d2
1.191 + then error "matrix_add: incompatible dimensions"
1.192 + else (d1,Intpairfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd m1) (snd m2)) :matrix
1.193 + end;;
1.194 +
1.195 +fun matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);
1.196 +
1.197 +fun row k (m:matrix) =
1.198 + let val (i,j) = dimensions m
1.199 + in (j,
1.200 + Intpairfunc.fold (fn ((i,j), c) => fn a => if i = k then Intfunc.update (j,c) a else a) (snd m) Intfunc.undefined ) : vector
1.201 + end;
1.202 +
1.203 +fun column k (m:matrix) =
1.204 + let val (i,j) = dimensions m
1.205 + in (i,
1.206 + Intpairfunc.fold (fn ((i,j), c) => fn a => if j = k then Intfunc.update (i,c) a else a) (snd m) Intfunc.undefined)
1.207 + : vector
1.208 + end;
1.209 +
1.210 +fun transp (m:matrix) =
1.211 + let val (i,j) = dimensions m
1.212 + in
1.213 + ((j,i),Intpairfunc.fold (fn ((i,j), c) => fn a => Intpairfunc.update ((j,i), c) a) (snd m) Intpairfunc.undefined) :matrix
1.214 + end;
1.215 +
1.216 +fun diagonal (v:vector) =
1.217 + let val n = dim v
1.218 + in ((n,n),Intfunc.fold (fn (i, c) => fn a => Intpairfunc.update ((i,i), c) a) (snd v) Intpairfunc.undefined) : matrix
1.219 + end;
1.220 +
1.221 +fun matrix_of_list l =
1.222 + let val m = length l
1.223 + in if m = 0 then matrix_0 (0,0) else
1.224 + let val n = length (hd l)
1.225 + in ((m,n),itern 1 l (fn v => fn i => itern 1 v (fn c => fn j => Intpairfunc.update ((i,j), c))) Intpairfunc.undefined)
1.226 + end
1.227 + end;
1.228 +
1.229 +(* Monomials. *)
1.230 +
1.231 +fun monomial_eval assig (m:monomial) =
1.232 + Ctermfunc.fold (fn (x, k) => fn a => a */ rat_pow (Ctermfunc.apply assig x) k)
1.233 + m rat_1;
1.234 +val monomial_1 = (Ctermfunc.undefined:monomial);
1.235 +
1.236 +fun monomial_var x = Ctermfunc.onefunc (x, 1) :monomial;
1.237 +
1.238 +val (monomial_mul:monomial->monomial->monomial) =
1.239 + Ctermfunc.combine (curry op +) (K false);
1.240 +
1.241 +fun monomial_pow (m:monomial) k =
1.242 + if k = 0 then monomial_1
1.243 + else Ctermfunc.mapf (fn x => k * x) m;
1.244 +
1.245 +fun monomial_divides (m1:monomial) (m2:monomial) =
1.246 + Ctermfunc.fold (fn (x, k) => fn a => Ctermfunc.tryapplyd m2 x 0 >= k andalso a) m1 true;;
1.247 +
1.248 +fun monomial_div (m1:monomial) (m2:monomial) =
1.249 + let val m = Ctermfunc.combine (curry op +)
1.250 + (fn x => x = 0) m1 (Ctermfunc.mapf (fn x => ~ x) m2)
1.251 + in if Ctermfunc.fold (fn (x, k) => fn a => k >= 0 andalso a) m true then m
1.252 + else error "monomial_div: non-divisible"
1.253 + end;
1.254 +
1.255 +fun monomial_degree x (m:monomial) =
1.256 + Ctermfunc.tryapplyd m x 0;;
1.257 +
1.258 +fun monomial_lcm (m1:monomial) (m2:monomial) =
1.259 + fold_rev (fn x => Ctermfunc.update (x, max (monomial_degree x m1) (monomial_degree x m2)))
1.260 + (gen_union (is_equal o cterm_ord) (Ctermfunc.dom m1, Ctermfunc.dom m2)) (Ctermfunc.undefined :monomial);
1.261 +
1.262 +fun monomial_multidegree (m:monomial) =
1.263 + Ctermfunc.fold (fn (x, k) => fn a => k + a) m 0;;
1.264 +
1.265 +fun monomial_variables m = Ctermfunc.dom m;;
1.266 +
1.267 +(* Polynomials. *)
1.268 +
1.269 +fun eval assig (p:poly) =
1.270 + Monomialfunc.fold (fn (m, c) => fn a => a +/ c */ monomial_eval assig m) p rat_0;
1.271 +
1.272 +val poly_0 = (Monomialfunc.undefined:poly);
1.273 +
1.274 +fun poly_isconst (p:poly) =
1.275 + Monomialfunc.fold (fn (m, c) => fn a => Ctermfunc.is_undefined m andalso a) p true;
1.276 +
1.277 +fun poly_var x = Monomialfunc.onefunc (monomial_var x,rat_1) :poly;
1.278 +
1.279 +fun poly_const c =
1.280 + if c =/ rat_0 then poly_0 else Monomialfunc.onefunc(monomial_1, c);
1.281 +
1.282 +fun poly_cmul c (p:poly) =
1.283 + if c =/ rat_0 then poly_0
1.284 + else Monomialfunc.mapf (fn x => c */ x) p;
1.285 +
1.286 +fun poly_neg (p:poly) = (Monomialfunc.mapf Rat.neg p :poly);;
1.287 +
1.288 +fun poly_add (p1:poly) (p2:poly) =
1.289 + (Monomialfunc.combine (curry op +/) (fn x => x =/ rat_0) p1 p2 :poly);
1.290 +
1.291 +fun poly_sub p1 p2 = poly_add p1 (poly_neg p2);
1.292 +
1.293 +fun poly_cmmul (c,m) (p:poly) =
1.294 + if c =/ rat_0 then poly_0
1.295 + else if Ctermfunc.is_undefined m
1.296 + then Monomialfunc.mapf (fn d => c */ d) p
1.297 + else Monomialfunc.fold (fn (m', d) => fn a => (Monomialfunc.update (monomial_mul m m', c */ d) a)) p poly_0;
1.298 +
1.299 +fun poly_mul (p1:poly) (p2:poly) =
1.300 + Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
1.301 +
1.302 +fun poly_div (p1:poly) (p2:poly) =
1.303 + if not(poly_isconst p2)
1.304 + then error "poly_div: non-constant" else
1.305 + let val c = eval Ctermfunc.undefined p2
1.306 + in if c =/ rat_0 then error "poly_div: division by zero"
1.307 + else poly_cmul (Rat.inv c) p1
1.308 + end;
1.309 +
1.310 +fun poly_square p = poly_mul p p;
1.311 +
1.312 +fun poly_pow p k =
1.313 + if k = 0 then poly_const rat_1
1.314 + else if k = 1 then p
1.315 + else let val q = poly_square(poly_pow p (k div 2)) in
1.316 + if k mod 2 = 1 then poly_mul p q else q end;
1.317 +
1.318 +fun poly_exp p1 p2 =
1.319 + if not(poly_isconst p2)
1.320 + then error "poly_exp: not a constant"
1.321 + else poly_pow p1 (int_of_rat (eval Ctermfunc.undefined p2));
1.322 +
1.323 +fun degree x (p:poly) =
1.324 + Monomialfunc.fold (fn (m,c) => fn a => max (monomial_degree x m) a) p 0;
1.325 +
1.326 +fun multidegree (p:poly) =
1.327 + Monomialfunc.fold (fn (m, c) => fn a => max (monomial_multidegree m) a) p 0;
1.328 +
1.329 +fun poly_variables (p:poly) =
1.330 + sort cterm_ord (Monomialfunc.fold_rev (fn (m, c) => curry (gen_union (is_equal o cterm_ord)) (monomial_variables m)) p []);;
1.331 +
1.332 +(* Order monomials for human presentation. *)
1.333 +
1.334 +fun cterm_ord (t,t') = TermOrd.fast_term_ord (term_of t, term_of t');
1.335 +
1.336 +val humanorder_varpow = prod_ord cterm_ord (rev_order o int_ord);
1.337 +
1.338 +local
1.339 + fun ord (l1,l2) = case (l1,l2) of
1.340 + (_,[]) => LESS
1.341 + | ([],_) => GREATER
1.342 + | (h1::t1,h2::t2) =>
1.343 + (case humanorder_varpow (h1, h2) of
1.344 + LESS => LESS
1.345 + | EQUAL => ord (t1,t2)
1.346 + | GREATER => GREATER)
1.347 +in fun humanorder_monomial m1 m2 =
1.348 + ord (sort humanorder_varpow (Ctermfunc.graph m1),
1.349 + sort humanorder_varpow (Ctermfunc.graph m2))
1.350 +end;
1.351 +
1.352 +fun fold1 f l = case l of
1.353 + [] => error "fold1"
1.354 + | [x] => x
1.355 + | (h::t) => f h (fold1 f t);
1.356 +
1.357 +(* Conversions to strings. *)
1.358 +
1.359 +fun string_of_vector min_size max_size (v:vector) =
1.360 + let val n_raw = dim v
1.361 + in if n_raw = 0 then "[]" else
1.362 + let
1.363 + val n = max min_size (min n_raw max_size)
1.364 + val xs = map (Rat.string_of_rat o (fn i => Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
1.365 + in "[" ^ fold1 (fn s => fn t => s ^ ", " ^ t) xs ^
1.366 + (if n_raw > max_size then ", ...]" else "]")
1.367 + end
1.368 + end;
1.369 +
1.370 +fun string_of_matrix max_size (m:matrix) =
1.371 + let
1.372 + val (i_raw,j_raw) = dimensions m
1.373 + val i = min max_size i_raw
1.374 + val j = min max_size j_raw
1.375 + val rstr = map (fn k => string_of_vector j j (row k m)) (1 upto i)
1.376 + in "["^ fold1 (fn s => fn t => s^";\n "^t) rstr ^
1.377 + (if j > max_size then "\n ...]" else "]")
1.378 + end;
1.379 +
1.380 +fun string_of_term t =
1.381 + case t of
1.382 + a$b => "("^(string_of_term a)^" "^(string_of_term b)^")"
1.383 + | Abs x =>
1.384 + let val (xn, b) = Term.dest_abs x
1.385 + in "(\\"^xn^"."^(string_of_term b)^")"
1.386 + end
1.387 + | Const(s,_) => s
1.388 + | Free (s,_) => s
1.389 + | Var((s,_),_) => s
1.390 + | _ => error "string_of_term";
1.391 +
1.392 +val string_of_cterm = string_of_term o term_of;
1.393 +
1.394 +fun string_of_varpow x k =
1.395 + if k = 1 then string_of_cterm x
1.396 + else string_of_cterm x^"^"^string_of_int k;
1.397 +
1.398 +fun string_of_monomial m =
1.399 + if Ctermfunc.is_undefined m then "1" else
1.400 + let val vps = fold_rev (fn (x,k) => fn a => string_of_varpow x k :: a)
1.401 + (sort humanorder_varpow (Ctermfunc.graph m)) []
1.402 + in fold1 (fn s => fn t => s^"*"^t) vps
1.403 + end;
1.404 +
1.405 +fun string_of_cmonomial (c,m) =
1.406 + if Ctermfunc.is_undefined m then Rat.string_of_rat c
1.407 + else if c =/ rat_1 then string_of_monomial m
1.408 + else Rat.string_of_rat c ^ "*" ^ string_of_monomial m;;
1.409 +
1.410 +fun string_of_poly (p:poly) =
1.411 + if Monomialfunc.is_undefined p then "<<0>>" else
1.412 + let
1.413 + val cms = sort (fn ((m1,_),(m2,_)) => humanorder_monomial m1 m2) (Monomialfunc.graph p)
1.414 + val s = fold (fn (m,c) => fn a =>
1.415 + if c </ rat_0 then a ^ " - " ^ string_of_cmonomial(Rat.neg c,m)
1.416 + else a ^ " + " ^ string_of_cmonomial(c,m))
1.417 + cms ""
1.418 + val s1 = String.substring (s, 0, 3)
1.419 + val s2 = String.substring (s, 3, String.size s - 3)
1.420 + in "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>"
1.421 + end;
1.422 +
1.423 +(* Conversion from HOL term. *)
1.424 +
1.425 +local
1.426 + val neg_tm = @{cterm "uminus :: real => _"}
1.427 + val add_tm = @{cterm "op + :: real => _"}
1.428 + val sub_tm = @{cterm "op - :: real => _"}
1.429 + val mul_tm = @{cterm "op * :: real => _"}
1.430 + val inv_tm = @{cterm "inverse :: real => _"}
1.431 + val div_tm = @{cterm "op / :: real => _"}
1.432 + val pow_tm = @{cterm "op ^ :: real => _"}
1.433 + val zero_tm = @{cterm "0:: real"}
1.434 + val is_numeral = can (HOLogic.dest_number o term_of)
1.435 + fun is_comb t = case t of _$_ => true | _ => false
1.436 + fun poly_of_term tm =
1.437 + if tm aconvc zero_tm then poly_0
1.438 + else if RealArith.is_ratconst tm
1.439 + then poly_const(RealArith.dest_ratconst tm)
1.440 + else
1.441 + (let val (lop,r) = Thm.dest_comb tm
1.442 + in if lop aconvc neg_tm then poly_neg(poly_of_term r)
1.443 + else if lop aconvc inv_tm then
1.444 + let val p = poly_of_term r
1.445 + in if poly_isconst p
1.446 + then poly_const(Rat.inv (eval Ctermfunc.undefined p))
1.447 + else error "poly_of_term: inverse of non-constant polyomial"
1.448 + end
1.449 + else (let val (opr,l) = Thm.dest_comb lop
1.450 + in
1.451 + if opr aconvc pow_tm andalso is_numeral r
1.452 + then poly_pow (poly_of_term l) ((snd o HOLogic.dest_number o term_of) r)
1.453 + else if opr aconvc add_tm
1.454 + then poly_add (poly_of_term l) (poly_of_term r)
1.455 + else if opr aconvc sub_tm
1.456 + then poly_sub (poly_of_term l) (poly_of_term r)
1.457 + else if opr aconvc mul_tm
1.458 + then poly_mul (poly_of_term l) (poly_of_term r)
1.459 + else if opr aconvc div_tm
1.460 + then let
1.461 + val p = poly_of_term l
1.462 + val q = poly_of_term r
1.463 + in if poly_isconst q then poly_cmul (Rat.inv (eval Ctermfunc.undefined q)) p
1.464 + else error "poly_of_term: division by non-constant polynomial"
1.465 + end
1.466 + else poly_var tm
1.467 +
1.468 + end
1.469 + handle CTERM ("dest_comb",_) => poly_var tm)
1.470 + end
1.471 + handle CTERM ("dest_comb",_) => poly_var tm)
1.472 +in
1.473 +val poly_of_term = fn tm =>
1.474 + if type_of (term_of tm) = @{typ real} then poly_of_term tm
1.475 + else error "poly_of_term: term does not have real type"
1.476 +end;
1.477 +
1.478 +(* String of vector (just a list of space-separated numbers). *)
1.479 +
1.480 +fun sdpa_of_vector (v:vector) =
1.481 + let
1.482 + val n = dim v
1.483 + val strs = map (decimalize 20 o (fn i => Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
1.484 + in fold1 (fn x => fn y => x ^ " " ^ y) strs ^ "\n"
1.485 + end;
1.486 +
1.487 +fun increasing f ord (x,y) = ord (f x, f y);
1.488 +fun triple_int_ord ((a,b,c),(a',b',c')) =
1.489 + prod_ord int_ord (prod_ord int_ord int_ord)
1.490 + ((a,(b,c)),(a',(b',c')));
1.491 +structure Inttriplefunc = FuncFun(type key = int*int*int val ord = triple_int_ord);
1.492 +
1.493 +(* String for block diagonal matrix numbered k. *)
1.494 +
1.495 +fun sdpa_of_blockdiagonal k m =
1.496 + let
1.497 + val pfx = string_of_int k ^" "
1.498 + val ents =
1.499 + Inttriplefunc.fold (fn ((b,i,j), c) => fn a => if i > j then a else ((b,i,j),c)::a) m []
1.500 + val entss = sort (increasing fst triple_int_ord ) ents
1.501 + in fold_rev (fn ((b,i,j),c) => fn a =>
1.502 + pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
1.503 + " " ^ decimalize 20 c ^ "\n" ^ a) entss ""
1.504 + end;
1.505 +
1.506 +(* String for a matrix numbered k, in SDPA sparse format. *)
1.507 +
1.508 +fun sdpa_of_matrix k (m:matrix) =
1.509 + let
1.510 + val pfx = string_of_int k ^ " 1 "
1.511 + val ms = Intpairfunc.fold (fn ((i,j), c) => fn a => if i > j then a else ((i,j),c)::a) (snd m) []
1.512 + val mss = sort (increasing fst (prod_ord int_ord int_ord)) ms
1.513 + in fold_rev (fn ((i,j),c) => fn a =>
1.514 + pfx ^ string_of_int i ^ " " ^ string_of_int j ^
1.515 + " " ^ decimalize 20 c ^ "\n" ^ a) mss ""
1.516 + end;;
1.517 +
1.518 +(* ------------------------------------------------------------------------- *)
1.519 +(* String in SDPA sparse format for standard SDP problem: *)
1.520 +(* *)
1.521 +(* X = v_1 * [M_1] + ... + v_m * [M_m] - [M_0] must be PSD *)
1.522 +(* Minimize obj_1 * v_1 + ... obj_m * v_m *)
1.523 +(* ------------------------------------------------------------------------- *)
1.524 +
1.525 +fun sdpa_of_problem obj mats =
1.526 + let
1.527 + val m = length mats - 1
1.528 + val (n,_) = dimensions (hd mats)
1.529 + in
1.530 + string_of_int m ^ "\n" ^
1.531 + "1\n" ^
1.532 + string_of_int n ^ "\n" ^
1.533 + sdpa_of_vector obj ^
1.534 + fold_rev2 (fn k => fn m => fn a => sdpa_of_matrix (k - 1) m ^ a) (1 upto length mats) mats ""
1.535 + end;
1.536 +
1.537 +fun index_char str chr pos =
1.538 + if pos >= String.size str then ~1
1.539 + else if String.sub(str,pos) = chr then pos
1.540 + else index_char str chr (pos + 1);
1.541 +fun rat_of_quotient (a,b) = if b = 0 then rat_0 else Rat.rat_of_quotient (a,b);
1.542 +fun rat_of_string s =
1.543 + let val n = index_char s #"/" 0 in
1.544 + if n = ~1 then s |> IntInf.fromString |> valOf |> Rat.rat_of_int
1.545 + else
1.546 + let val SOME numer = IntInf.fromString(String.substring(s,0,n))
1.547 + val SOME den = IntInf.fromString (String.substring(s,n+1,String.size s - n - 1))
1.548 + in rat_of_quotient(numer, den)
1.549 + end
1.550 + end;
1.551 +
1.552 +fun isspace x = x = " " ;
1.553 +fun isnum x = x mem_string ["0","1","2","3","4","5","6","7","8","9"]
1.554 +
1.555 +(* More parser basics. *)
1.556 +
1.557 +local
1.558 + open Scan
1.559 +in
1.560 + val word = this_string
1.561 + fun token s =
1.562 + repeat ($$ " ") |-- word s --| repeat ($$ " ")
1.563 + val numeral = one isnum
1.564 + val decimalint = bulk numeral >> (rat_of_string o implode)
1.565 + val decimalfrac = bulk numeral
1.566 + >> (fn s => rat_of_string(implode s) // pow10 (length s))
1.567 + val decimalsig =
1.568 + decimalint -- option (Scan.$$ "." |-- decimalfrac)
1.569 + >> (fn (h,NONE) => h | (h,SOME x) => h +/ x)
1.570 + fun signed prs =
1.571 + $$ "-" |-- prs >> Rat.neg
1.572 + || $$ "+" |-- prs
1.573 + || prs;
1.574 +
1.575 +fun emptyin def xs = if null xs then (def,xs) else Scan.fail xs
1.576 +
1.577 + val exponent = ($$ "e" || $$ "E") |-- signed decimalint;
1.578 +
1.579 + val decimal = signed decimalsig -- (emptyin rat_0|| exponent)
1.580 + >> (fn (h, x) => h */ pow10 (int_of_rat x));
1.581 +end;
1.582 +
1.583 + fun mkparser p s =
1.584 + let val (x,rst) = p (explode s)
1.585 + in if null rst then x
1.586 + else error "mkparser: unparsed input"
1.587 + end;;
1.588 +
1.589 +(* Parse back csdp output. *)
1.590 +
1.591 + fun ignore inp = ((),[])
1.592 + fun csdpoutput inp = ((decimal -- Scan.bulk (Scan.$$ " " |-- Scan.option decimal) >> (fn (h,to) => map_filter I ((SOME h)::to))) --| ignore >> vector_of_list) inp
1.593 + val parse_csdpoutput = mkparser csdpoutput
1.594 +
1.595 +(* Run prover on a problem in linear form. *)
1.596 +
1.597 +fun run_problem prover obj mats =
1.598 + parse_csdpoutput (prover (sdpa_of_problem obj mats))
1.599 +
1.600 +(* Try some apparently sensible scaling first. Note that this is purely to *)
1.601 +(* get a cleaner translation to floating-point, and doesn't affect any of *)
1.602 +(* the results, in principle. In practice it seems a lot better when there *)
1.603 +(* are extreme numbers in the original problem. *)
1.604 +
1.605 + (* Version for (int*int) keys *)
1.606 +local
1.607 + fun max_rat x y = if x </ y then y else x
1.608 + fun common_denominator fld amat acc =
1.609 + fld (fn (m,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
1.610 + fun maximal_element fld amat acc =
1.611 + fld (fn (m,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
1.612 +fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x
1.613 + in Real.fromLargeInt a / Real.fromLargeInt b end;
1.614 +in
1.615 +
1.616 +fun pi_scale_then solver (obj:vector) mats =
1.617 + let
1.618 + val cd1 = fold_rev (common_denominator Intpairfunc.fold) mats (rat_1)
1.619 + val cd2 = common_denominator Intfunc.fold (snd obj) (rat_1)
1.620 + val mats' = map (Intpairfunc.mapf (fn x => cd1 */ x)) mats
1.621 + val obj' = vector_cmul cd2 obj
1.622 + val max1 = fold_rev (maximal_element Intpairfunc.fold) mats' (rat_0)
1.623 + val max2 = maximal_element Intfunc.fold (snd obj') (rat_0)
1.624 + val scal1 = pow2 (20 - trunc(Math.ln (float_of_rat max1) / Math.ln 2.0))
1.625 + val scal2 = pow2 (20 - trunc(Math.ln (float_of_rat max2) / Math.ln 2.0))
1.626 + val mats'' = map (Intpairfunc.mapf (fn x => x */ scal1)) mats'
1.627 + val obj'' = vector_cmul scal2 obj'
1.628 + in solver obj'' mats''
1.629 + end
1.630 +end;
1.631 +
1.632 +(* Try some apparently sensible scaling first. Note that this is purely to *)
1.633 +(* get a cleaner translation to floating-point, and doesn't affect any of *)
1.634 +(* the results, in principle. In practice it seems a lot better when there *)
1.635 +(* are extreme numbers in the original problem. *)
1.636 +
1.637 + (* Version for (int*int*int) keys *)
1.638 +local
1.639 + fun max_rat x y = if x </ y then y else x
1.640 + fun common_denominator fld amat acc =
1.641 + fld (fn (m,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
1.642 + fun maximal_element fld amat acc =
1.643 + fld (fn (m,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
1.644 +fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x
1.645 + in Real.fromLargeInt a / Real.fromLargeInt b end;
1.646 +fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)
1.647 +in
1.648 +
1.649 +fun tri_scale_then solver (obj:vector) mats =
1.650 + let
1.651 + val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats (rat_1)
1.652 + val cd2 = common_denominator Intfunc.fold (snd obj) (rat_1)
1.653 + val mats' = map (Inttriplefunc.mapf (fn x => cd1 */ x)) mats
1.654 + val obj' = vector_cmul cd2 obj
1.655 + val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' (rat_0)
1.656 + val max2 = maximal_element Intfunc.fold (snd obj') (rat_0)
1.657 + val scal1 = pow2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))
1.658 + val scal2 = pow2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))
1.659 + val mats'' = map (Inttriplefunc.mapf (fn x => x */ scal1)) mats'
1.660 + val obj'' = vector_cmul scal2 obj'
1.661 + in solver obj'' mats''
1.662 + end
1.663 +end;
1.664 +
1.665 +(* Round a vector to "nice" rationals. *)
1.666 +
1.667 +fun nice_rational n x = round_rat (n */ x) // n;;
1.668 +fun nice_vector n ((d,v) : vector) =
1.669 + (d, Intfunc.fold (fn (i,c) => fn a =>
1.670 + let val y = nice_rational n c
1.671 + in if c =/ rat_0 then a
1.672 + else Intfunc.update (i,y) a end) v Intfunc.undefined):vector
1.673 +
1.674 +fun dest_ord f x = is_equal (f x);
1.675 +
1.676 +(* Stuff for "equations" ((int*int*int)->num functions). *)
1.677 +
1.678 +fun tri_equation_cmul c eq =
1.679 + if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (fn d => c */ d) eq;
1.680 +
1.681 +fun tri_equation_add eq1 eq2 = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0) eq1 eq2;
1.682 +
1.683 +fun tri_equation_eval assig eq =
1.684 + let fun value v = Inttriplefunc.apply assig v
1.685 + in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
1.686 + end;
1.687 +
1.688 +(* Eliminate among linear equations: return unconstrained variables and *)
1.689 +(* assignments for the others in terms of them. We give one pseudo-variable *)
1.690 +(* "one" that's used for a constant term. *)
1.691 +
1.692 +local
1.693 + fun extract_first p l = case l of (* FIXME : use find_first instead *)
1.694 + [] => error "extract_first"
1.695 + | h::t => if p h then (h,t) else
1.696 + let val (k,s) = extract_first p t in (k,h::s) end
1.697 +fun eliminate vars dun eqs = case vars of
1.698 + [] => if forall Inttriplefunc.is_undefined eqs then dun
1.699 + else raise Unsolvable
1.700 + | v::vs =>
1.701 + ((let
1.702 + val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
1.703 + val a = Inttriplefunc.apply eq v
1.704 + val eq' = tri_equation_cmul ((Rat.neg rat_1) // a) (Inttriplefunc.undefine v eq)
1.705 + fun elim e =
1.706 + let val b = Inttriplefunc.tryapplyd e v rat_0
1.707 + in if b =/ rat_0 then e else
1.708 + tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
1.709 + end
1.710 + in eliminate vs (Inttriplefunc.update (v,eq') (Inttriplefunc.mapf elim dun)) (map elim oeqs)
1.711 + end)
1.712 + handle Failure _ => eliminate vs dun eqs)
1.713 +in
1.714 +fun tri_eliminate_equations one vars eqs =
1.715 + let
1.716 + val assig = eliminate vars Inttriplefunc.undefined eqs
1.717 + val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
1.718 + in (distinct (dest_ord triple_int_ord) vs, assig)
1.719 + end
1.720 +end;
1.721 +
1.722 +(* Eliminate all variables, in an essentially arbitrary order. *)
1.723 +
1.724 +fun tri_eliminate_all_equations one =
1.725 + let
1.726 + fun choose_variable eq =
1.727 + let val (v,_) = Inttriplefunc.choose eq
1.728 + in if is_equal (triple_int_ord(v,one)) then
1.729 + let val eq' = Inttriplefunc.undefine v eq
1.730 + in if Inttriplefunc.is_undefined eq' then error "choose_variable"
1.731 + else fst (Inttriplefunc.choose eq')
1.732 + end
1.733 + else v
1.734 + end
1.735 + fun eliminate dun eqs = case eqs of
1.736 + [] => dun
1.737 + | eq::oeqs =>
1.738 + if Inttriplefunc.is_undefined eq then eliminate dun oeqs else
1.739 + let val v = choose_variable eq
1.740 + val a = Inttriplefunc.apply eq v
1.741 + val eq' = tri_equation_cmul ((Rat.rat_of_int ~1) // a)
1.742 + (Inttriplefunc.undefine v eq)
1.743 + fun elim e =
1.744 + let val b = Inttriplefunc.tryapplyd e v rat_0
1.745 + in if b =/ rat_0 then e
1.746 + else tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
1.747 + end
1.748 + in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.mapf elim dun))
1.749 + (map elim oeqs)
1.750 + end
1.751 +in fn eqs =>
1.752 + let
1.753 + val assig = eliminate Inttriplefunc.undefined eqs
1.754 + val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
1.755 + in (distinct (dest_ord triple_int_ord) vs,assig)
1.756 + end
1.757 +end;
1.758 +
1.759 +(* Solve equations by assigning arbitrary numbers. *)
1.760 +
1.761 +fun tri_solve_equations one eqs =
1.762 + let
1.763 + val (vars,assigs) = tri_eliminate_all_equations one eqs
1.764 + val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
1.765 + (Inttriplefunc.onefunc(one, Rat.rat_of_int ~1))
1.766 + val ass =
1.767 + Inttriplefunc.combine (curry op +/) (K false)
1.768 + (Inttriplefunc.mapf (tri_equation_eval vfn) assigs) vfn
1.769 + in if forall (fn e => tri_equation_eval ass e =/ rat_0) eqs
1.770 + then Inttriplefunc.undefine one ass else raise Sanity
1.771 + end;
1.772 +
1.773 +(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
1.774 +
1.775 +fun tri_epoly_pmul p q acc =
1.776 + Monomialfunc.fold (fn (m1, c) => fn a =>
1.777 + Monomialfunc.fold (fn (m2,e) => fn b =>
1.778 + let val m = monomial_mul m1 m2
1.779 + val es = Monomialfunc.tryapplyd b m Inttriplefunc.undefined
1.780 + in Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
1.781 + end) q a) p acc ;
1.782 +
1.783 +(* Usual operations on equation-parametrized poly. *)
1.784 +
1.785 +fun tri_epoly_cmul c l =
1.786 + if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (tri_equation_cmul c) l;;
1.787 +
1.788 +val tri_epoly_neg = tri_epoly_cmul (Rat.rat_of_int ~1);
1.789 +
1.790 +val tri_epoly_add = Inttriplefunc.combine tri_equation_add Inttriplefunc.is_undefined;
1.791 +
1.792 +fun tri_epoly_sub p q = tri_epoly_add p (tri_epoly_neg q);;
1.793 +
1.794 +(* Stuff for "equations" ((int*int)->num functions). *)
1.795 +
1.796 +fun pi_equation_cmul c eq =
1.797 + if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (fn d => c */ d) eq;
1.798 +
1.799 +fun pi_equation_add eq1 eq2 = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0) eq1 eq2;
1.800 +
1.801 +fun pi_equation_eval assig eq =
1.802 + let fun value v = Inttriplefunc.apply assig v
1.803 + in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
1.804 + end;
1.805 +
1.806 +(* Eliminate among linear equations: return unconstrained variables and *)
1.807 +(* assignments for the others in terms of them. We give one pseudo-variable *)
1.808 +(* "one" that's used for a constant term. *)
1.809 +
1.810 +local
1.811 +fun extract_first p l = case l of
1.812 + [] => error "extract_first"
1.813 + | h::t => if p h then (h,t) else
1.814 + let val (k,s) = extract_first p t in (k,h::s) end
1.815 +fun eliminate vars dun eqs = case vars of
1.816 + [] => if forall Inttriplefunc.is_undefined eqs then dun
1.817 + else raise Unsolvable
1.818 + | v::vs =>
1.819 + let
1.820 + val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
1.821 + val a = Inttriplefunc.apply eq v
1.822 + val eq' = pi_equation_cmul ((Rat.neg rat_1) // a) (Inttriplefunc.undefine v eq)
1.823 + fun elim e =
1.824 + let val b = Inttriplefunc.tryapplyd e v rat_0
1.825 + in if b =/ rat_0 then e else
1.826 + pi_equation_add e (pi_equation_cmul (Rat.neg b // a) eq)
1.827 + end
1.828 + in eliminate vs (Inttriplefunc.update (v,eq') (Inttriplefunc.mapf elim dun)) (map elim oeqs)
1.829 + end
1.830 + handle Failure _ => eliminate vs dun eqs
1.831 +in
1.832 +fun pi_eliminate_equations one vars eqs =
1.833 + let
1.834 + val assig = eliminate vars Inttriplefunc.undefined eqs
1.835 + val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
1.836 + in (distinct (dest_ord triple_int_ord) vs, assig)
1.837 + end
1.838 +end;
1.839 +
1.840 +(* Eliminate all variables, in an essentially arbitrary order. *)
1.841 +
1.842 +fun pi_eliminate_all_equations one =
1.843 + let
1.844 + fun choose_variable eq =
1.845 + let val (v,_) = Inttriplefunc.choose eq
1.846 + in if is_equal (triple_int_ord(v,one)) then
1.847 + let val eq' = Inttriplefunc.undefine v eq
1.848 + in if Inttriplefunc.is_undefined eq' then error "choose_variable"
1.849 + else fst (Inttriplefunc.choose eq')
1.850 + end
1.851 + else v
1.852 + end
1.853 + fun eliminate dun eqs = case eqs of
1.854 + [] => dun
1.855 + | eq::oeqs =>
1.856 + if Inttriplefunc.is_undefined eq then eliminate dun oeqs else
1.857 + let val v = choose_variable eq
1.858 + val a = Inttriplefunc.apply eq v
1.859 + val eq' = pi_equation_cmul ((Rat.rat_of_int ~1) // a)
1.860 + (Inttriplefunc.undefine v eq)
1.861 + fun elim e =
1.862 + let val b = Inttriplefunc.tryapplyd e v rat_0
1.863 + in if b =/ rat_0 then e
1.864 + else pi_equation_add e (pi_equation_cmul (Rat.neg b // a) eq)
1.865 + end
1.866 + in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.mapf elim dun))
1.867 + (map elim oeqs)
1.868 + end
1.869 +in fn eqs =>
1.870 + let
1.871 + val assig = eliminate Inttriplefunc.undefined eqs
1.872 + val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
1.873 + in (distinct (dest_ord triple_int_ord) vs,assig)
1.874 + end
1.875 +end;
1.876 +
1.877 +(* Solve equations by assigning arbitrary numbers. *)
1.878 +
1.879 +fun pi_solve_equations one eqs =
1.880 + let
1.881 + val (vars,assigs) = pi_eliminate_all_equations one eqs
1.882 + val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
1.883 + (Inttriplefunc.onefunc(one, Rat.rat_of_int ~1))
1.884 + val ass =
1.885 + Inttriplefunc.combine (curry op +/) (K false)
1.886 + (Inttriplefunc.mapf (pi_equation_eval vfn) assigs) vfn
1.887 + in if forall (fn e => pi_equation_eval ass e =/ rat_0) eqs
1.888 + then Inttriplefunc.undefine one ass else raise Sanity
1.889 + end;
1.890 +
1.891 +(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
1.892 +
1.893 +fun pi_epoly_pmul p q acc =
1.894 + Monomialfunc.fold (fn (m1, c) => fn a =>
1.895 + Monomialfunc.fold (fn (m2,e) => fn b =>
1.896 + let val m = monomial_mul m1 m2
1.897 + val es = Monomialfunc.tryapplyd b m Inttriplefunc.undefined
1.898 + in Monomialfunc.update (m,pi_equation_add (pi_equation_cmul c e) es) b
1.899 + end) q a) p acc ;
1.900 +
1.901 +(* Usual operations on equation-parametrized poly. *)
1.902 +
1.903 +fun pi_epoly_cmul c l =
1.904 + if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (pi_equation_cmul c) l;;
1.905 +
1.906 +val pi_epoly_neg = pi_epoly_cmul (Rat.rat_of_int ~1);
1.907 +
1.908 +val pi_epoly_add = Inttriplefunc.combine pi_equation_add Inttriplefunc.is_undefined;
1.909 +
1.910 +fun pi_epoly_sub p q = pi_epoly_add p (pi_epoly_neg q);;
1.911 +
1.912 +fun allpairs f l1 l2 = fold_rev (fn x => (curry (op @)) (map (f x) l2)) l1 [];
1.913 +
1.914 +(* Hence produce the "relevant" monomials: those whose squares lie in the *)
1.915 +(* Newton polytope of the monomials in the input. (This is enough according *)
1.916 +(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal, *)
1.917 +(* vol 45, pp. 363--374, 1978. *)
1.918 +(* *)
1.919 +(* These are ordered in sort of decreasing degree. In particular the *)
1.920 +(* constant monomial is last; this gives an order in diagonalization of the *)
1.921 +(* quadratic form that will tend to display constants. *)
1.922 +
1.923 +(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form. *)
1.924 +
1.925 +local
1.926 +fun diagonalize n i m =
1.927 + if Intpairfunc.is_undefined (snd m) then []
1.928 + else
1.929 + let val a11 = Intpairfunc.tryapplyd (snd m) (i,i) rat_0
1.930 + in if a11 </ rat_0 then raise Failure "diagonalize: not PSD"
1.931 + else if a11 =/ rat_0 then
1.932 + if Intfunc.is_undefined (snd (row i m)) then diagonalize n (i + 1) m
1.933 + else raise Failure "diagonalize: not PSD ___ "
1.934 + else
1.935 + let
1.936 + val v = row i m
1.937 + val v' = (fst v, Intfunc.fold (fn (i, c) => fn a =>
1.938 + let val y = c // a11
1.939 + in if y = rat_0 then a else Intfunc.update (i,y) a
1.940 + end) (snd v) Intfunc.undefined)
1.941 + fun upt0 x y a = if y = rat_0 then a else Intpairfunc.update (x,y) a
1.942 + val m' =
1.943 + ((n,n),
1.944 + iter (i+1,n) (fn j =>
1.945 + iter (i+1,n) (fn k =>
1.946 + (upt0 (j,k) (Intpairfunc.tryapplyd (snd m) (j,k) rat_0 -/ Intfunc.tryapplyd (snd v) j rat_0 */ Intfunc.tryapplyd (snd v') k rat_0))))
1.947 + Intpairfunc.undefined)
1.948 + in (a11,v')::diagonalize n (i + 1) m'
1.949 + end
1.950 + end
1.951 +in
1.952 +fun diag m =
1.953 + let
1.954 + val nn = dimensions m
1.955 + val n = fst nn
1.956 + in if snd nn <> n then error "diagonalize: non-square matrix"
1.957 + else diagonalize n 1 m
1.958 + end
1.959 +end;
1.960 +
1.961 +fun gcd_rat a b = Rat.rat_of_int (Integer.gcd (int_of_rat a) (int_of_rat b));
1.962 +
1.963 +(* Adjust a diagonalization to collect rationals at the start. *)
1.964 + (* FIXME : Potentially polymorphic keys, but here only: integers!! *)
1.965 +local
1.966 + fun upd0 x y a = if y =/ rat_0 then a else Intfunc.update(x,y) a;
1.967 + fun mapa f (d,v) =
1.968 + (d, Intfunc.fold (fn (i,c) => fn a => upd0 i (f c) a) v Intfunc.undefined)
1.969 + fun adj (c,l) =
1.970 + let val a =
1.971 + Intfunc.fold (fn (i,c) => fn a => lcm_rat a (denominator_rat c))
1.972 + (snd l) rat_1 //
1.973 + Intfunc.fold (fn (i,c) => fn a => gcd_rat a (numerator_rat c))
1.974 + (snd l) rat_0
1.975 + in ((c // (a */ a)),mapa (fn x => a */ x) l)
1.976 + end
1.977 +in
1.978 +fun deration d = if null d then (rat_0,d) else
1.979 + let val d' = map adj d
1.980 + val a = fold (lcm_rat o denominator_rat o fst) d' rat_1 //
1.981 + fold (gcd_rat o numerator_rat o fst) d' rat_0
1.982 + in ((rat_1 // a),map (fn (c,l) => (a */ c,l)) d')
1.983 + end
1.984 +end;
1.985 +
1.986 +(* Enumeration of monomials with given multidegree bound. *)
1.987 +
1.988 +fun enumerate_monomials d vars =
1.989 + if d < 0 then []
1.990 + else if d = 0 then [Ctermfunc.undefined]
1.991 + else if null vars then [monomial_1] else
1.992 + let val alts =
1.993 + map (fn k => let val oths = enumerate_monomials (d - k) (tl vars)
1.994 + in map (fn ks => if k = 0 then ks else Ctermfunc.update (hd vars, k) ks) oths end) (0 upto d)
1.995 + in fold1 (curry op @) alts
1.996 + end;
1.997 +
1.998 +(* Enumerate products of distinct input polys with degree <= d. *)
1.999 +(* We ignore any constant input polynomials. *)
1.1000 +(* Give the output polynomial and a record of how it was derived. *)
1.1001 +
1.1002 +local
1.1003 + open RealArith
1.1004 +in
1.1005 +fun enumerate_products d pols =
1.1006 +if d = 0 then [(poly_const rat_1,Rational_lt rat_1)]
1.1007 +else if d < 0 then [] else
1.1008 +case pols of
1.1009 + [] => [(poly_const rat_1,Rational_lt rat_1)]
1.1010 + | (p,b)::ps =>
1.1011 + let val e = multidegree p
1.1012 + in if e = 0 then enumerate_products d ps else
1.1013 + enumerate_products d ps @
1.1014 + map (fn (q,c) => (poly_mul p q,Product(b,c)))
1.1015 + (enumerate_products (d - e) ps)
1.1016 + end
1.1017 +end;
1.1018 +
1.1019 +(* Convert regular polynomial. Note that we treat (0,0,0) as -1. *)
1.1020 +
1.1021 +fun epoly_of_poly p =
1.1022 + Monomialfunc.fold (fn (m,c) => fn a => Monomialfunc.update (m, Inttriplefunc.onefunc ((0,0,0), Rat.neg c)) a) p Monomialfunc.undefined;
1.1023 +
1.1024 +(* String for block diagonal matrix numbered k. *)
1.1025 +
1.1026 +fun sdpa_of_blockdiagonal k m =
1.1027 + let
1.1028 + val pfx = string_of_int k ^" "
1.1029 + val ents =
1.1030 + Inttriplefunc.fold
1.1031 + (fn ((b,i,j),c) => fn a => if i > j then a else ((b,i,j),c)::a)
1.1032 + m []
1.1033 + val entss = sort (increasing fst triple_int_ord) ents
1.1034 + in fold_rev (fn ((b,i,j),c) => fn a =>
1.1035 + pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
1.1036 + " " ^ decimalize 20 c ^ "\n" ^ a) entss ""
1.1037 + end;
1.1038 +
1.1039 +(* SDPA for problem using block diagonal (i.e. multiple SDPs) *)
1.1040 +
1.1041 +fun sdpa_of_blockproblem nblocks blocksizes obj mats =
1.1042 + let val m = length mats - 1
1.1043 + in
1.1044 + string_of_int m ^ "\n" ^
1.1045 + string_of_int nblocks ^ "\n" ^
1.1046 + (fold1 (fn s => fn t => s^" "^t) (map string_of_int blocksizes)) ^
1.1047 + "\n" ^
1.1048 + sdpa_of_vector obj ^
1.1049 + fold_rev2 (fn k => fn m => fn a => sdpa_of_blockdiagonal (k - 1) m ^ a)
1.1050 + (1 upto length mats) mats ""
1.1051 + end;
1.1052 +
1.1053 +(* Run prover on a problem in block diagonal form. *)
1.1054 +
1.1055 +fun run_blockproblem prover nblocks blocksizes obj mats=
1.1056 + parse_csdpoutput (prover (sdpa_of_blockproblem nblocks blocksizes obj mats))
1.1057 +
1.1058 +(* 3D versions of matrix operations to consider blocks separately. *)
1.1059 +
1.1060 +val bmatrix_add = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0);
1.1061 +fun bmatrix_cmul c bm =
1.1062 + if c =/ rat_0 then Inttriplefunc.undefined
1.1063 + else Inttriplefunc.mapf (fn x => c */ x) bm;
1.1064 +
1.1065 +val bmatrix_neg = bmatrix_cmul (Rat.rat_of_int ~1);
1.1066 +fun bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;
1.1067 +
1.1068 +(* Smash a block matrix into components. *)
1.1069 +
1.1070 +fun blocks blocksizes bm =
1.1071 + map (fn (bs,b0) =>
1.1072 + let val m = Inttriplefunc.fold
1.1073 + (fn ((b,i,j),c) => fn a => if b = b0 then Intpairfunc.update ((i,j),c) a else a) bm Intpairfunc.undefined
1.1074 + val d = Intpairfunc.fold (fn ((i,j),c) => fn a => max a (max i j)) m 0
1.1075 + in (((bs,bs),m):matrix) end)
1.1076 + (blocksizes ~~ (1 upto length blocksizes));;
1.1077 +
1.1078 +(* FIXME : Get rid of this !!!*)
1.1079 +local
1.1080 + fun tryfind_with msg f [] = raise Failure msg
1.1081 + | tryfind_with msg f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
1.1082 +in
1.1083 + fun tryfind f = tryfind_with "tryfind" f
1.1084 +end
1.1085 +
1.1086 +(*
1.1087 +fun tryfind f [] = error "tryfind"
1.1088 + | tryfind f (x::xs) = (f x handle ERROR _ => tryfind f xs);
1.1089 +*)
1.1090 +
1.1091 +(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
1.1092 +
1.1093 +
1.1094 +local
1.1095 + open RealArith
1.1096 +in
1.1097 +fun real_positivnullstellensatz_general prover linf d eqs leqs pol =
1.1098 +let
1.1099 + val vars = fold_rev (curry (gen_union (op aconvc)) o poly_variables)
1.1100 + (pol::eqs @ map fst leqs) []
1.1101 + val monoid = if linf then
1.1102 + (poly_const rat_1,Rational_lt rat_1)::
1.1103 + (filter (fn (p,c) => multidegree p <= d) leqs)
1.1104 + else enumerate_products d leqs
1.1105 + val nblocks = length monoid
1.1106 + fun mk_idmultiplier k p =
1.1107 + let
1.1108 + val e = d - multidegree p
1.1109 + val mons = enumerate_monomials e vars
1.1110 + val nons = mons ~~ (1 upto length mons)
1.1111 + in (mons,
1.1112 + fold_rev (fn (m,n) => Monomialfunc.update(m,Inttriplefunc.onefunc((~k,~n,n),rat_1))) nons Monomialfunc.undefined)
1.1113 + end
1.1114 +
1.1115 + fun mk_sqmultiplier k (p,c) =
1.1116 + let
1.1117 + val e = (d - multidegree p) div 2
1.1118 + val mons = enumerate_monomials e vars
1.1119 + val nons = mons ~~ (1 upto length mons)
1.1120 + in (mons,
1.1121 + fold_rev (fn (m1,n1) =>
1.1122 + fold_rev (fn (m2,n2) => fn a =>
1.1123 + let val m = monomial_mul m1 m2
1.1124 + in if n1 > n2 then a else
1.1125 + let val c = if n1 = n2 then rat_1 else rat_2
1.1126 + val e = Monomialfunc.tryapplyd a m Inttriplefunc.undefined
1.1127 + in Monomialfunc.update(m, tri_equation_add (Inttriplefunc.onefunc((k,n1,n2), c)) e) a
1.1128 + end
1.1129 + end) nons)
1.1130 + nons Monomialfunc.undefined)
1.1131 + end
1.1132 +
1.1133 + val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)
1.1134 + val (idmonlist,ids) = split_list(map2 mk_idmultiplier (1 upto length eqs) eqs)
1.1135 + val blocksizes = map length sqmonlist
1.1136 + val bigsum =
1.1137 + fold_rev2 (fn p => fn q => fn a => tri_epoly_pmul p q a) eqs ids
1.1138 + (fold_rev2 (fn (p,c) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs
1.1139 + (epoly_of_poly(poly_neg pol)))
1.1140 + val eqns = Monomialfunc.fold (fn (m,e) => fn a => e::a) bigsum []
1.1141 + val (pvs,assig) = tri_eliminate_all_equations (0,0,0) eqns
1.1142 + val qvars = (0,0,0)::pvs
1.1143 + val allassig = fold_rev (fn v => Inttriplefunc.update(v,(Inttriplefunc.onefunc(v,rat_1)))) pvs assig
1.1144 + fun mk_matrix v =
1.1145 + Inttriplefunc.fold (fn ((b,i,j), ass) => fn m =>
1.1146 + if b < 0 then m else
1.1147 + let val c = Inttriplefunc.tryapplyd ass v rat_0
1.1148 + in if c = rat_0 then m else
1.1149 + Inttriplefunc.update ((b,j,i), c) (Inttriplefunc.update ((b,i,j), c) m)
1.1150 + end)
1.1151 + allassig Inttriplefunc.undefined
1.1152 + val diagents = Inttriplefunc.fold
1.1153 + (fn ((b,i,j), e) => fn a => if b > 0 andalso i = j then tri_equation_add e a else a)
1.1154 + allassig Inttriplefunc.undefined
1.1155 +
1.1156 + val mats = map mk_matrix qvars
1.1157 + val obj = (length pvs,
1.1158 + itern 1 pvs (fn v => fn i => Intfunc.updatep iszero (i,Inttriplefunc.tryapplyd diagents v rat_0))
1.1159 + Intfunc.undefined)
1.1160 + val raw_vec = if null pvs then vector_0 0
1.1161 + else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats
1.1162 + fun int_element (d,v) i = Intfunc.tryapplyd v i rat_0
1.1163 + fun cterm_element (d,v) i = Ctermfunc.tryapplyd v i rat_0
1.1164 +
1.1165 + fun find_rounding d =
1.1166 + let
1.1167 + val _ = if !debugging
1.1168 + then writeln ("Trying rounding with limit "^Rat.string_of_rat d ^ "\n")
1.1169 + else ()
1.1170 + val vec = nice_vector d raw_vec
1.1171 + val blockmat = iter (1,dim vec)
1.1172 + (fn i => fn a => bmatrix_add (bmatrix_cmul (int_element vec i) (nth mats i)) a)
1.1173 + (bmatrix_neg (nth mats 0))
1.1174 + val allmats = blocks blocksizes blockmat
1.1175 + in (vec,map diag allmats)
1.1176 + end
1.1177 + val (vec,ratdias) =
1.1178 + if null pvs then find_rounding rat_1
1.1179 + else tryfind find_rounding (map Rat.rat_of_int (1 upto 31) @
1.1180 + map pow2 (5 upto 66))
1.1181 + val newassigs =
1.1182 + fold_rev (fn k => Inttriplefunc.update (nth pvs (k - 1), int_element vec k))
1.1183 + (1 upto dim vec) (Inttriplefunc.onefunc ((0,0,0), Rat.rat_of_int ~1))
1.1184 + val finalassigs =
1.1185 + Inttriplefunc.fold (fn (v,e) => fn a => Inttriplefunc.update(v, tri_equation_eval newassigs e) a) allassig newassigs
1.1186 + fun poly_of_epoly p =
1.1187 + Monomialfunc.fold (fn (v,e) => fn a => Monomialfunc.updatep iszero (v,tri_equation_eval finalassigs e) a)
1.1188 + p Monomialfunc.undefined
1.1189 + fun mk_sos mons =
1.1190 + let fun mk_sq (c,m) =
1.1191 + (c,fold_rev (fn k=> fn a => Monomialfunc.updatep iszero (nth mons (k - 1), int_element m k) a)
1.1192 + (1 upto length mons) Monomialfunc.undefined)
1.1193 + in map mk_sq
1.1194 + end
1.1195 + val sqs = map2 mk_sos sqmonlist ratdias
1.1196 + val cfs = map poly_of_epoly ids
1.1197 + val msq = filter (fn (a,b) => not (null b)) (map2 pair monoid sqs)
1.1198 + fun eval_sq sqs = fold_rev (fn (c,q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0
1.1199 + val sanity =
1.1200 + fold_rev (fn ((p,c),s) => poly_add (poly_mul p (eval_sq s))) msq
1.1201 + (fold_rev2 (fn p => fn q => poly_add (poly_mul p q)) cfs eqs
1.1202 + (poly_neg pol))
1.1203 +
1.1204 +in if not(Monomialfunc.is_undefined sanity) then raise Sanity else
1.1205 + (cfs,map (fn (a,b) => (snd a,b)) msq)
1.1206 + end
1.1207 +
1.1208 +
1.1209 +end;
1.1210 +
1.1211 +(* Iterative deepening. *)
1.1212 +
1.1213 +fun deepen f n =
1.1214 + (writeln ("Searching with depth limit " ^ string_of_int n) ; (f n handle Failure s => (writeln ("failed with message: " ^ s) ; deepen f (n+1))))
1.1215 +
1.1216 +(* The ordering so we can create canonical HOL polynomials. *)
1.1217 +
1.1218 +fun dest_monomial mon = sort (increasing fst cterm_ord) (Ctermfunc.graph mon);
1.1219 +
1.1220 +fun monomial_order (m1,m2) =
1.1221 + if Ctermfunc.is_undefined m2 then LESS
1.1222 + else if Ctermfunc.is_undefined m1 then GREATER
1.1223 + else
1.1224 + let val mon1 = dest_monomial m1
1.1225 + val mon2 = dest_monomial m2
1.1226 + val deg1 = fold (curry op + o snd) mon1 0
1.1227 + val deg2 = fold (curry op + o snd) mon2 0
1.1228 + in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
1.1229 + else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
1.1230 + end;
1.1231 +
1.1232 +fun dest_poly p =
1.1233 + map (fn (m,c) => (c,dest_monomial m))
1.1234 + (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p));
1.1235 +
1.1236 +(* Map back polynomials and their composites to HOL. *)
1.1237 +
1.1238 +local
1.1239 + open Thm Numeral RealArith
1.1240 +in
1.1241 +
1.1242 +fun cterm_of_varpow x k = if k = 1 then x else capply (capply @{cterm "op ^ :: real => _"} x)
1.1243 + (mk_cnumber @{ctyp nat} k)
1.1244 +
1.1245 +fun cterm_of_monomial m =
1.1246 + if Ctermfunc.is_undefined m then @{cterm "1::real"}
1.1247 + else
1.1248 + let
1.1249 + val m' = dest_monomial m
1.1250 + val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
1.1251 + in fold1 (fn s => fn t => capply (capply @{cterm "op * :: real => _"} s) t) vps
1.1252 + end
1.1253 +
1.1254 +fun cterm_of_cmonomial (m,c) = if Ctermfunc.is_undefined m then cterm_of_rat c
1.1255 + else if c = Rat.one then cterm_of_monomial m
1.1256 + else capply (capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
1.1257 +
1.1258 +fun cterm_of_poly p =
1.1259 + if Monomialfunc.is_undefined p then @{cterm "0::real"}
1.1260 + else
1.1261 + let
1.1262 + val cms = map cterm_of_cmonomial
1.1263 + (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p))
1.1264 + in fold1 (fn t1 => fn t2 => capply(capply @{cterm "op + :: real => _"} t1) t2) cms
1.1265 + end;
1.1266 +
1.1267 +fun cterm_of_sqterm (c,p) = Product(Rational_lt c,Square(cterm_of_poly p));
1.1268 +
1.1269 +fun cterm_of_sos (pr,sqs) = if null sqs then pr
1.1270 + else Product(pr,fold1 (fn a => fn b => Sum(a,b)) (map cterm_of_sqterm sqs));
1.1271 +
1.1272 +end
1.1273 +
1.1274 +(* Interface to HOL. *)
1.1275 +local
1.1276 + open Thm Conv RealArith
1.1277 + val concl = dest_arg o cprop_of
1.1278 + fun simple_cterm_ord t u = TermOrd.fast_term_ord (term_of t, term_of u) = LESS
1.1279 +in
1.1280 + (* FIXME: Replace tryfind by get_first !! *)
1.1281 +fun real_nonlinear_prover prover ctxt =
1.1282 + let
1.1283 + val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
1.1284 + (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
1.1285 + simple_cterm_ord
1.1286 + val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
1.1287 + real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
1.1288 + fun mainf translator (eqs,les,lts) =
1.1289 + let
1.1290 + val eq0 = map (poly_of_term o dest_arg1 o concl) eqs
1.1291 + val le0 = map (poly_of_term o dest_arg o concl) les
1.1292 + val lt0 = map (poly_of_term o dest_arg o concl) lts
1.1293 + val eqp0 = map (fn (t,i) => (t,Axiom_eq i)) (eq0 ~~ (0 upto (length eq0 - 1)))
1.1294 + val lep0 = map (fn (t,i) => (t,Axiom_le i)) (le0 ~~ (0 upto (length le0 - 1)))
1.1295 + val ltp0 = map (fn (t,i) => (t,Axiom_lt i)) (lt0 ~~ (0 upto (length lt0 - 1)))
1.1296 + val (keq,eq) = List.partition (fn (p,_) => multidegree p = 0) eqp0
1.1297 + val (klep,lep) = List.partition (fn (p,_) => multidegree p = 0) lep0
1.1298 + val (kltp,ltp) = List.partition (fn (p,_) => multidegree p = 0) ltp0
1.1299 + fun trivial_axiom (p,ax) =
1.1300 + case ax of
1.1301 + Axiom_eq n => if eval Ctermfunc.undefined p <>/ Rat.zero then nth eqs n
1.1302 + else raise Failure "trivial_axiom: Not a trivial axiom"
1.1303 + | Axiom_le n => if eval Ctermfunc.undefined p </ Rat.zero then nth les n
1.1304 + else raise Failure "trivial_axiom: Not a trivial axiom"
1.1305 + | Axiom_lt n => if eval Ctermfunc.undefined p <=/ Rat.zero then nth lts n
1.1306 + else raise Failure "trivial_axiom: Not a trivial axiom"
1.1307 + | _ => error "trivial_axiom: Not a trivial axiom"
1.1308 + in
1.1309 + ((let val th = tryfind trivial_axiom (keq @ klep @ kltp)
1.1310 + in fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv field_comp_conv) th end)
1.1311 + handle Failure _ => (
1.1312 + let
1.1313 + val pol = fold_rev poly_mul (map fst ltp) (poly_const Rat.one)
1.1314 + val leq = lep @ ltp
1.1315 + fun tryall d =
1.1316 + let val e = multidegree pol
1.1317 + val k = if e = 0 then 0 else d div e
1.1318 + val eq' = map fst eq
1.1319 + in tryfind (fn i => (d,i,real_positivnullstellensatz_general prover false d eq' leq
1.1320 + (poly_neg(poly_pow pol i))))
1.1321 + (0 upto k)
1.1322 + end
1.1323 + val (d,i,(cert_ideal,cert_cone)) = deepen tryall 0
1.1324 + val proofs_ideal =
1.1325 + map2 (fn q => fn (p,ax) => Eqmul(cterm_of_poly q,ax)) cert_ideal eq
1.1326 + val proofs_cone = map cterm_of_sos cert_cone
1.1327 + val proof_ne = if null ltp then Rational_lt Rat.one else
1.1328 + let val p = fold1 (fn s => fn t => Product(s,t)) (map snd ltp)
1.1329 + in funpow i (fn q => Product(p,q)) (Rational_lt Rat.one)
1.1330 + end
1.1331 + val proof = fold1 (fn s => fn t => Sum(s,t))
1.1332 + (proof_ne :: proofs_ideal @ proofs_cone)
1.1333 + in writeln "Translating proof certificate to HOL";
1.1334 + translator (eqs,les,lts) proof
1.1335 + end))
1.1336 + end
1.1337 + in mainf end
1.1338 +end
1.1339 +
1.1340 +fun C f x y = f y x;
1.1341 + (* FIXME : This is very bad!!!*)
1.1342 +fun subst_conv eqs t =
1.1343 + let
1.1344 + val t' = fold (Thm.cabs o Thm.lhs_of) eqs t
1.1345 + in Conv.fconv_rule (Thm.beta_conversion true) (fold (C combination) eqs (reflexive t'))
1.1346 + end
1.1347 +
1.1348 +(* A wrapper that tries to substitute away variables first. *)
1.1349 +
1.1350 +local
1.1351 + open Thm Conv RealArith
1.1352 + fun simple_cterm_ord t u = TermOrd.fast_term_ord (term_of t, term_of u) = LESS
1.1353 + val concl = dest_arg o cprop_of
1.1354 + val shuffle1 =
1.1355 + fconv_rule (rewr_conv @{lemma "(a + x == y) == (x == y - (a::real))" by (atomize (full)) (simp add: ring_simps) })
1.1356 + val shuffle2 =
1.1357 + fconv_rule (rewr_conv @{lemma "(x + a == y) == (x == y - (a::real))" by (atomize (full)) (simp add: ring_simps)})
1.1358 + fun substitutable_monomial fvs tm = case term_of tm of
1.1359 + Free(_,@{typ real}) => if not (member (op aconvc) fvs tm) then (Rat.one,tm)
1.1360 + else raise Failure "substitutable_monomial"
1.1361 + | @{term "op * :: real => _"}$c$(t as Free _ ) =>
1.1362 + if is_ratconst (dest_arg1 tm) andalso not (member (op aconvc) fvs (dest_arg tm))
1.1363 + then (dest_ratconst (dest_arg1 tm),dest_arg tm) else raise Failure "substitutable_monomial"
1.1364 + | @{term "op + :: real => _"}$s$t =>
1.1365 + (substitutable_monomial (add_cterm_frees (dest_arg tm) fvs) (dest_arg1 tm)
1.1366 + handle Failure _ => substitutable_monomial (add_cterm_frees (dest_arg1 tm) fvs) (dest_arg tm))
1.1367 + | _ => raise Failure "substitutable_monomial"
1.1368 +
1.1369 + fun isolate_variable v th =
1.1370 + let val w = dest_arg1 (cprop_of th)
1.1371 + in if v aconvc w then th
1.1372 + else case term_of w of
1.1373 + @{term "op + :: real => _"}$s$t =>
1.1374 + if dest_arg1 w aconvc v then shuffle2 th
1.1375 + else isolate_variable v (shuffle1 th)
1.1376 + | _ => error "isolate variable : This should not happen?"
1.1377 + end
1.1378 +in
1.1379 +
1.1380 +fun real_nonlinear_subst_prover prover ctxt =
1.1381 + let
1.1382 + val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
1.1383 + (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
1.1384 + simple_cterm_ord
1.1385 +
1.1386 + val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
1.1387 + real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
1.1388 +
1.1389 + fun make_substitution th =
1.1390 + let
1.1391 + val (c,v) = substitutable_monomial [] (dest_arg1(concl th))
1.1392 + val th1 = Drule.arg_cong_rule (capply @{cterm "op * :: real => _"} (cterm_of_rat (Rat.inv c))) (mk_meta_eq th)
1.1393 + val th2 = fconv_rule (binop_conv real_poly_mul_conv) th1
1.1394 + in fconv_rule (arg_conv real_poly_conv) (isolate_variable v th2)
1.1395 + end
1.1396 + fun oprconv cv ct =
1.1397 + let val g = Thm.dest_fun2 ct
1.1398 + in if g aconvc @{cterm "op <= :: real => _"}
1.1399 + orelse g aconvc @{cterm "op < :: real => _"}
1.1400 + then arg_conv cv ct else arg1_conv cv ct
1.1401 + end
1.1402 + fun mainf translator =
1.1403 + let
1.1404 + fun substfirst(eqs,les,lts) =
1.1405 + ((let
1.1406 + val eth = tryfind make_substitution eqs
1.1407 + val modify = fconv_rule (arg_conv (oprconv(subst_conv [eth] then_conv real_poly_conv)))
1.1408 + in substfirst
1.1409 + (filter_out (fn t => (Thm.dest_arg1 o Thm.dest_arg o cprop_of) t
1.1410 + aconvc @{cterm "0::real"}) (map modify eqs),
1.1411 + map modify les,map modify lts)
1.1412 + end)
1.1413 + handle Failure _ => real_nonlinear_prover prover ctxt translator (rev eqs, rev les, rev lts))
1.1414 + in substfirst
1.1415 + end
1.1416 +
1.1417 +
1.1418 + in mainf
1.1419 + end
1.1420 +
1.1421 +(* Overall function. *)
1.1422 +
1.1423 +fun real_sos prover ctxt t = gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt) t;
1.1424 +end;
1.1425 +
1.1426 +(* A tactic *)
1.1427 +fun strip_all ct =
1.1428 + case term_of ct of
1.1429 + Const("all",_) $ Abs (xn,xT,p) =>
1.1430 + let val (a,(v,t')) = (apsnd (Thm.dest_abs (SOME xn)) o Thm.dest_comb) ct
1.1431 + in apfst (cons v) (strip_all t')
1.1432 + end
1.1433 +| _ => ([],ct)
1.1434 +
1.1435 +fun core_sos_conv prover ctxt t = Drule.arg_cong_rule @{cterm Trueprop} (real_sos prover ctxt (Thm.dest_arg t) RS @{thm Eq_TrueI})
1.1436 +
1.1437 +val known_sos_constants =
1.1438 + [@{term "op ==>"}, @{term "Trueprop"},
1.1439 + @{term "op -->"}, @{term "op &"}, @{term "op |"},
1.1440 + @{term "Not"}, @{term "op = :: bool => _"},
1.1441 + @{term "All :: (real => _) => _"}, @{term "Ex :: (real => _) => _"},
1.1442 + @{term "op = :: real => _"}, @{term "op < :: real => _"},
1.1443 + @{term "op <= :: real => _"},
1.1444 + @{term "op + :: real => _"}, @{term "op - :: real => _"},
1.1445 + @{term "op * :: real => _"}, @{term "uminus :: real => _"},
1.1446 + @{term "op / :: real => _"}, @{term "inverse :: real => _"},
1.1447 + @{term "op ^ :: real => _"}, @{term "abs :: real => _"},
1.1448 + @{term "min :: real => _"}, @{term "max :: real => _"},
1.1449 + @{term "0::real"}, @{term "1::real"}, @{term "number_of :: int => real"},
1.1450 + @{term "number_of :: int => nat"},
1.1451 + @{term "Int.Bit0"}, @{term "Int.Bit1"},
1.1452 + @{term "Int.Pls"}, @{term "Int.Min"}];
1.1453 +
1.1454 +fun check_sos kcts ct =
1.1455 + let
1.1456 + val t = term_of ct
1.1457 + val _ = if not (null (Term.add_tfrees t [])
1.1458 + andalso null (Term.add_tvars t []))
1.1459 + then error "SOS: not sos. Additional type varables" else ()
1.1460 + val fs = Term.add_frees t []
1.1461 + val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs
1.1462 + then error "SOS: not sos. Variables with type not real" else ()
1.1463 + val vs = Term.add_vars t []
1.1464 + val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs
1.1465 + then error "SOS: not sos. Variables with type not real" else ()
1.1466 + val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])
1.1467 + val _ = if null ukcs then ()
1.1468 + else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))
1.1469 +in () end
1.1470 +
1.1471 +fun core_sos_tac prover ctxt = CSUBGOAL (fn (ct, i) =>
1.1472 + let val _ = check_sos known_sos_constants ct
1.1473 + val (avs, p) = strip_all ct
1.1474 + val th = standard (fold_rev forall_intr avs (real_sos prover ctxt (Thm.dest_arg p)))
1.1475 + in rtac th i end);
1.1476 +
1.1477 +fun default_SOME f NONE v = SOME v
1.1478 + | default_SOME f (SOME v) _ = SOME v;
1.1479 +
1.1480 +fun lift_SOME f NONE a = f a
1.1481 + | lift_SOME f (SOME a) _ = SOME a;
1.1482 +
1.1483 +
1.1484 +local
1.1485 + val is_numeral = can (HOLogic.dest_number o term_of)
1.1486 +in
1.1487 +fun get_denom b ct = case term_of ct of
1.1488 + @{term "op / :: real => _"} $ _ $ _ =>
1.1489 + if is_numeral (Thm.dest_arg ct) then get_denom b (Thm.dest_arg1 ct)
1.1490 + else default_SOME (get_denom b) (get_denom b (Thm.dest_arg ct)) (Thm.dest_arg ct, b)
1.1491 + | @{term "op < :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
1.1492 + | @{term "op <= :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
1.1493 + | _ $ _ => lift_SOME (get_denom b) (get_denom b (Thm.dest_fun ct)) (Thm.dest_arg ct)
1.1494 + | _ => NONE
1.1495 +end;
1.1496 +
1.1497 +fun elim_one_denom_tac ctxt =
1.1498 +CSUBGOAL (fn (P,i) =>
1.1499 + case get_denom false P of
1.1500 + NONE => no_tac
1.1501 + | SOME (d,ord) =>
1.1502 + let
1.1503 + val ss = simpset_of ctxt addsimps @{thms field_simps}
1.1504 + addsimps [@{thm nonzero_power_divide}, @{thm power_divide}]
1.1505 + val th = instantiate' [] [SOME d, SOME (Thm.dest_arg P)]
1.1506 + (if ord then @{lemma "(d=0 --> P) & (d>0 --> P) & (d<(0::real) --> P) ==> P" by auto}
1.1507 + else @{lemma "(d=0 --> P) & (d ~= (0::real) --> P) ==> P" by blast})
1.1508 + in (rtac th i THEN Simplifier.asm_full_simp_tac ss i) end);
1.1509 +
1.1510 +fun elim_denom_tac ctxt i = REPEAT (elim_one_denom_tac ctxt i);
1.1511 +
1.1512 +fun sos_tac prover ctxt = ObjectLogic.full_atomize_tac THEN' elim_denom_tac ctxt THEN' core_sos_tac prover ctxt
1.1513 +
1.1514 +
1.1515 +end;