--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Thu Aug 06 19:51:59 2009 +0200
@@ -0,0 +1,1512 @@
+(* Title: sum_of_squares.ML
+ Authors: Amine Chaieb, University of Cambridge
+ Philipp Meyer, TU Muenchen
+
+A tactic for proving nonlinear inequalities
+*)
+
+signature SOS =
+sig
+
+ val sos_tac : (string -> string) -> Proof.context -> int -> Tactical.tactic
+
+ val debugging : bool ref;
+
+ exception Failure of string;
+end
+
+structure Sos : SOS =
+struct
+
+val rat_0 = Rat.zero;
+val rat_1 = Rat.one;
+val rat_2 = Rat.two;
+val rat_10 = Rat.rat_of_int 10;
+val rat_1_2 = rat_1 // rat_2;
+val max = curry IntInf.max;
+val min = curry IntInf.min;
+
+val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
+val numerator_rat = Rat.quotient_of_rat #> fst #> Rat.rat_of_int;
+fun int_of_rat a =
+ case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
+fun lcm_rat x y = Rat.rat_of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
+
+fun rat_pow r i =
+ let fun pow r i =
+ if i = 0 then rat_1 else
+ let val d = pow r (i div 2)
+ in d */ d */ (if i mod 2 = 0 then rat_1 else r)
+ end
+ in if i < 0 then pow (Rat.inv r) (~ i) else pow r i end;
+
+fun round_rat r =
+ let val (a,b) = Rat.quotient_of_rat (Rat.abs r)
+ val d = a div b
+ val s = if r </ rat_0 then (Rat.neg o Rat.rat_of_int) else Rat.rat_of_int
+ val x2 = 2 * (a - (b * d))
+ in s (if x2 >= b then d + 1 else d) end
+
+val abs_rat = Rat.abs;
+val pow2 = rat_pow rat_2;
+val pow10 = rat_pow rat_10;
+
+val debugging = ref false;
+
+exception Sanity;
+
+exception Unsolvable;
+
+exception Failure of string;
+
+(* Turn a rational into a decimal string with d sig digits. *)
+
+local
+fun normalize y =
+ if abs_rat y </ (rat_1 // rat_10) then normalize (rat_10 */ y) - 1
+ else if abs_rat y >=/ rat_1 then normalize (y // rat_10) + 1
+ else 0
+ in
+fun decimalize d x =
+ if x =/ rat_0 then "0.0" else
+ let
+ val y = Rat.abs x
+ val e = normalize y
+ val z = pow10(~ e) */ y +/ rat_1
+ val k = int_of_rat (round_rat(pow10 d */ z))
+ in (if x </ rat_0 then "-0." else "0.") ^
+ implode(tl(explode(string_of_int k))) ^
+ (if e = 0 then "" else "e"^string_of_int e)
+ end
+end;
+
+(* Iterations over numbers, and lists indexed by numbers. *)
+
+fun itern k l f a =
+ case l of
+ [] => a
+ | h::t => itern (k + 1) t f (f h k a);
+
+fun iter (m,n) f a =
+ if n < m then a
+ else iter (m+1,n) f (f m a);
+
+(* The main types. *)
+
+fun strict_ord ord (x,y) = case ord (x,y) of LESS => LESS | _ => GREATER
+
+structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
+
+type vector = int* Rat.rat Intfunc.T;
+
+type matrix = (int*int)*(Rat.rat Intpairfunc.T);
+
+type monomial = int Ctermfunc.T;
+
+val cterm_ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t))
+ fun monomial_ord (m1,m2) = list_ord (prod_ord cterm_ord int_ord) (Ctermfunc.graph m1, Ctermfunc.graph m2)
+structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
+
+type poly = Rat.rat Monomialfunc.T;
+
+ fun iszero (k,r) = r =/ rat_0;
+
+fun fold_rev2 f l1 l2 b =
+ case (l1,l2) of
+ ([],[]) => b
+ | (h1::t1,h2::t2) => f h1 h2 (fold_rev2 f t1 t2 b)
+ | _ => error "fold_rev2";
+
+(* Vectors. Conventionally indexed 1..n. *)
+
+fun vector_0 n = (n,Intfunc.undefined):vector;
+
+fun dim (v:vector) = fst v;
+
+fun vector_const c n =
+ if c =/ rat_0 then vector_0 n
+ else (n,fold_rev (fn k => Intfunc.update (k,c)) (1 upto n) Intfunc.undefined) :vector;
+
+val vector_1 = vector_const rat_1;
+
+fun vector_cmul c (v:vector) =
+ let val n = dim v
+ in if c =/ rat_0 then vector_0 n
+ else (n,Intfunc.mapf (fn x => c */ x) (snd v))
+ end;
+
+fun vector_neg (v:vector) = (fst v,Intfunc.mapf Rat.neg (snd v)) :vector;
+
+fun vector_add (v1:vector) (v2:vector) =
+ let val m = dim v1
+ val n = dim v2
+ in if m <> n then error "vector_add: incompatible dimensions"
+ else (n,Intfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd v1) (snd v2)) :vector
+ end;
+
+fun vector_sub v1 v2 = vector_add v1 (vector_neg v2);
+
+fun vector_dot (v1:vector) (v2:vector) =
+ let val m = dim v1
+ val n = dim v2
+ in if m <> n then error "vector_dot: incompatible dimensions"
+ else Intfunc.fold (fn (i,x) => fn a => x +/ a)
+ (Intfunc.combine (curry op */) (fn x => x =/ rat_0) (snd v1) (snd v2)) rat_0
+ end;
+
+fun vector_of_list l =
+ let val n = length l
+ in (n,fold_rev2 (curry Intfunc.update) (1 upto n) l Intfunc.undefined) :vector
+ end;
+
+(* Matrices; again rows and columns indexed from 1. *)
+
+fun matrix_0 (m,n) = ((m,n),Intpairfunc.undefined):matrix;
+
+fun dimensions (m:matrix) = fst m;
+
+fun matrix_const c (mn as (m,n)) =
+ if m <> n then error "matrix_const: needs to be square"
+ else if c =/ rat_0 then matrix_0 mn
+ else (mn,fold_rev (fn k => Intpairfunc.update ((k,k), c)) (1 upto n) Intpairfunc.undefined) :matrix;;
+
+val matrix_1 = matrix_const rat_1;
+
+fun matrix_cmul c (m:matrix) =
+ let val (i,j) = dimensions m
+ in if c =/ rat_0 then matrix_0 (i,j)
+ else ((i,j),Intpairfunc.mapf (fn x => c */ x) (snd m))
+ end;
+
+fun matrix_neg (m:matrix) =
+ (dimensions m, Intpairfunc.mapf Rat.neg (snd m)) :matrix;
+
+fun matrix_add (m1:matrix) (m2:matrix) =
+ let val d1 = dimensions m1
+ val d2 = dimensions m2
+ in if d1 <> d2
+ then error "matrix_add: incompatible dimensions"
+ else (d1,Intpairfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd m1) (snd m2)) :matrix
+ end;;
+
+fun matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);
+
+fun row k (m:matrix) =
+ let val (i,j) = dimensions m
+ in (j,
+ Intpairfunc.fold (fn ((i,j), c) => fn a => if i = k then Intfunc.update (j,c) a else a) (snd m) Intfunc.undefined ) : vector
+ end;
+
+fun column k (m:matrix) =
+ let val (i,j) = dimensions m
+ in (i,
+ Intpairfunc.fold (fn ((i,j), c) => fn a => if j = k then Intfunc.update (i,c) a else a) (snd m) Intfunc.undefined)
+ : vector
+ end;
+
+fun transp (m:matrix) =
+ let val (i,j) = dimensions m
+ in
+ ((j,i),Intpairfunc.fold (fn ((i,j), c) => fn a => Intpairfunc.update ((j,i), c) a) (snd m) Intpairfunc.undefined) :matrix
+ end;
+
+fun diagonal (v:vector) =
+ let val n = dim v
+ in ((n,n),Intfunc.fold (fn (i, c) => fn a => Intpairfunc.update ((i,i), c) a) (snd v) Intpairfunc.undefined) : matrix
+ end;
+
+fun matrix_of_list l =
+ let val m = length l
+ in if m = 0 then matrix_0 (0,0) else
+ let val n = length (hd l)
+ in ((m,n),itern 1 l (fn v => fn i => itern 1 v (fn c => fn j => Intpairfunc.update ((i,j), c))) Intpairfunc.undefined)
+ end
+ end;
+
+(* Monomials. *)
+
+fun monomial_eval assig (m:monomial) =
+ Ctermfunc.fold (fn (x, k) => fn a => a */ rat_pow (Ctermfunc.apply assig x) k)
+ m rat_1;
+val monomial_1 = (Ctermfunc.undefined:monomial);
+
+fun monomial_var x = Ctermfunc.onefunc (x, 1) :monomial;
+
+val (monomial_mul:monomial->monomial->monomial) =
+ Ctermfunc.combine (curry op +) (K false);
+
+fun monomial_pow (m:monomial) k =
+ if k = 0 then monomial_1
+ else Ctermfunc.mapf (fn x => k * x) m;
+
+fun monomial_divides (m1:monomial) (m2:monomial) =
+ Ctermfunc.fold (fn (x, k) => fn a => Ctermfunc.tryapplyd m2 x 0 >= k andalso a) m1 true;;
+
+fun monomial_div (m1:monomial) (m2:monomial) =
+ let val m = Ctermfunc.combine (curry op +)
+ (fn x => x = 0) m1 (Ctermfunc.mapf (fn x => ~ x) m2)
+ in if Ctermfunc.fold (fn (x, k) => fn a => k >= 0 andalso a) m true then m
+ else error "monomial_div: non-divisible"
+ end;
+
+fun monomial_degree x (m:monomial) =
+ Ctermfunc.tryapplyd m x 0;;
+
+fun monomial_lcm (m1:monomial) (m2:monomial) =
+ fold_rev (fn x => Ctermfunc.update (x, max (monomial_degree x m1) (monomial_degree x m2)))
+ (gen_union (is_equal o cterm_ord) (Ctermfunc.dom m1, Ctermfunc.dom m2)) (Ctermfunc.undefined :monomial);
+
+fun monomial_multidegree (m:monomial) =
+ Ctermfunc.fold (fn (x, k) => fn a => k + a) m 0;;
+
+fun monomial_variables m = Ctermfunc.dom m;;
+
+(* Polynomials. *)
+
+fun eval assig (p:poly) =
+ Monomialfunc.fold (fn (m, c) => fn a => a +/ c */ monomial_eval assig m) p rat_0;
+
+val poly_0 = (Monomialfunc.undefined:poly);
+
+fun poly_isconst (p:poly) =
+ Monomialfunc.fold (fn (m, c) => fn a => Ctermfunc.is_undefined m andalso a) p true;
+
+fun poly_var x = Monomialfunc.onefunc (monomial_var x,rat_1) :poly;
+
+fun poly_const c =
+ if c =/ rat_0 then poly_0 else Monomialfunc.onefunc(monomial_1, c);
+
+fun poly_cmul c (p:poly) =
+ if c =/ rat_0 then poly_0
+ else Monomialfunc.mapf (fn x => c */ x) p;
+
+fun poly_neg (p:poly) = (Monomialfunc.mapf Rat.neg p :poly);;
+
+fun poly_add (p1:poly) (p2:poly) =
+ (Monomialfunc.combine (curry op +/) (fn x => x =/ rat_0) p1 p2 :poly);
+
+fun poly_sub p1 p2 = poly_add p1 (poly_neg p2);
+
+fun poly_cmmul (c,m) (p:poly) =
+ if c =/ rat_0 then poly_0
+ else if Ctermfunc.is_undefined m
+ then Monomialfunc.mapf (fn d => c */ d) p
+ else Monomialfunc.fold (fn (m', d) => fn a => (Monomialfunc.update (monomial_mul m m', c */ d) a)) p poly_0;
+
+fun poly_mul (p1:poly) (p2:poly) =
+ Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
+
+fun poly_div (p1:poly) (p2:poly) =
+ if not(poly_isconst p2)
+ then error "poly_div: non-constant" else
+ let val c = eval Ctermfunc.undefined p2
+ in if c =/ rat_0 then error "poly_div: division by zero"
+ else poly_cmul (Rat.inv c) p1
+ end;
+
+fun poly_square p = poly_mul p p;
+
+fun poly_pow p k =
+ if k = 0 then poly_const rat_1
+ else if k = 1 then p
+ else let val q = poly_square(poly_pow p (k div 2)) in
+ if k mod 2 = 1 then poly_mul p q else q end;
+
+fun poly_exp p1 p2 =
+ if not(poly_isconst p2)
+ then error "poly_exp: not a constant"
+ else poly_pow p1 (int_of_rat (eval Ctermfunc.undefined p2));
+
+fun degree x (p:poly) =
+ Monomialfunc.fold (fn (m,c) => fn a => max (monomial_degree x m) a) p 0;
+
+fun multidegree (p:poly) =
+ Monomialfunc.fold (fn (m, c) => fn a => max (monomial_multidegree m) a) p 0;
+
+fun poly_variables (p:poly) =
+ sort cterm_ord (Monomialfunc.fold_rev (fn (m, c) => curry (gen_union (is_equal o cterm_ord)) (monomial_variables m)) p []);;
+
+(* Order monomials for human presentation. *)
+
+fun cterm_ord (t,t') = TermOrd.fast_term_ord (term_of t, term_of t');
+
+val humanorder_varpow = prod_ord cterm_ord (rev_order o int_ord);
+
+local
+ fun ord (l1,l2) = case (l1,l2) of
+ (_,[]) => LESS
+ | ([],_) => GREATER
+ | (h1::t1,h2::t2) =>
+ (case humanorder_varpow (h1, h2) of
+ LESS => LESS
+ | EQUAL => ord (t1,t2)
+ | GREATER => GREATER)
+in fun humanorder_monomial m1 m2 =
+ ord (sort humanorder_varpow (Ctermfunc.graph m1),
+ sort humanorder_varpow (Ctermfunc.graph m2))
+end;
+
+fun fold1 f l = case l of
+ [] => error "fold1"
+ | [x] => x
+ | (h::t) => f h (fold1 f t);
+
+(* Conversions to strings. *)
+
+fun string_of_vector min_size max_size (v:vector) =
+ let val n_raw = dim v
+ in if n_raw = 0 then "[]" else
+ let
+ val n = max min_size (min n_raw max_size)
+ val xs = map (Rat.string_of_rat o (fn i => Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
+ in "[" ^ fold1 (fn s => fn t => s ^ ", " ^ t) xs ^
+ (if n_raw > max_size then ", ...]" else "]")
+ end
+ end;
+
+fun string_of_matrix max_size (m:matrix) =
+ let
+ val (i_raw,j_raw) = dimensions m
+ val i = min max_size i_raw
+ val j = min max_size j_raw
+ val rstr = map (fn k => string_of_vector j j (row k m)) (1 upto i)
+ in "["^ fold1 (fn s => fn t => s^";\n "^t) rstr ^
+ (if j > max_size then "\n ...]" else "]")
+ end;
+
+fun string_of_term t =
+ case t of
+ a$b => "("^(string_of_term a)^" "^(string_of_term b)^")"
+ | Abs x =>
+ let val (xn, b) = Term.dest_abs x
+ in "(\\"^xn^"."^(string_of_term b)^")"
+ end
+ | Const(s,_) => s
+ | Free (s,_) => s
+ | Var((s,_),_) => s
+ | _ => error "string_of_term";
+
+val string_of_cterm = string_of_term o term_of;
+
+fun string_of_varpow x k =
+ if k = 1 then string_of_cterm x
+ else string_of_cterm x^"^"^string_of_int k;
+
+fun string_of_monomial m =
+ if Ctermfunc.is_undefined m then "1" else
+ let val vps = fold_rev (fn (x,k) => fn a => string_of_varpow x k :: a)
+ (sort humanorder_varpow (Ctermfunc.graph m)) []
+ in fold1 (fn s => fn t => s^"*"^t) vps
+ end;
+
+fun string_of_cmonomial (c,m) =
+ if Ctermfunc.is_undefined m then Rat.string_of_rat c
+ else if c =/ rat_1 then string_of_monomial m
+ else Rat.string_of_rat c ^ "*" ^ string_of_monomial m;;
+
+fun string_of_poly (p:poly) =
+ if Monomialfunc.is_undefined p then "<<0>>" else
+ let
+ val cms = sort (fn ((m1,_),(m2,_)) => humanorder_monomial m1 m2) (Monomialfunc.graph p)
+ val s = fold (fn (m,c) => fn a =>
+ if c </ rat_0 then a ^ " - " ^ string_of_cmonomial(Rat.neg c,m)
+ else a ^ " + " ^ string_of_cmonomial(c,m))
+ cms ""
+ val s1 = String.substring (s, 0, 3)
+ val s2 = String.substring (s, 3, String.size s - 3)
+ in "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>"
+ end;
+
+(* Conversion from HOL term. *)
+
+local
+ val neg_tm = @{cterm "uminus :: real => _"}
+ val add_tm = @{cterm "op + :: real => _"}
+ val sub_tm = @{cterm "op - :: real => _"}
+ val mul_tm = @{cterm "op * :: real => _"}
+ val inv_tm = @{cterm "inverse :: real => _"}
+ val div_tm = @{cterm "op / :: real => _"}
+ val pow_tm = @{cterm "op ^ :: real => _"}
+ val zero_tm = @{cterm "0:: real"}
+ val is_numeral = can (HOLogic.dest_number o term_of)
+ fun is_comb t = case t of _$_ => true | _ => false
+ fun poly_of_term tm =
+ if tm aconvc zero_tm then poly_0
+ else if RealArith.is_ratconst tm
+ then poly_const(RealArith.dest_ratconst tm)
+ else
+ (let val (lop,r) = Thm.dest_comb tm
+ in if lop aconvc neg_tm then poly_neg(poly_of_term r)
+ else if lop aconvc inv_tm then
+ let val p = poly_of_term r
+ in if poly_isconst p
+ then poly_const(Rat.inv (eval Ctermfunc.undefined p))
+ else error "poly_of_term: inverse of non-constant polyomial"
+ end
+ else (let val (opr,l) = Thm.dest_comb lop
+ in
+ if opr aconvc pow_tm andalso is_numeral r
+ then poly_pow (poly_of_term l) ((snd o HOLogic.dest_number o term_of) r)
+ else if opr aconvc add_tm
+ then poly_add (poly_of_term l) (poly_of_term r)
+ else if opr aconvc sub_tm
+ then poly_sub (poly_of_term l) (poly_of_term r)
+ else if opr aconvc mul_tm
+ then poly_mul (poly_of_term l) (poly_of_term r)
+ else if opr aconvc div_tm
+ then let
+ val p = poly_of_term l
+ val q = poly_of_term r
+ in if poly_isconst q then poly_cmul (Rat.inv (eval Ctermfunc.undefined q)) p
+ else error "poly_of_term: division by non-constant polynomial"
+ end
+ else poly_var tm
+
+ end
+ handle CTERM ("dest_comb",_) => poly_var tm)
+ end
+ handle CTERM ("dest_comb",_) => poly_var tm)
+in
+val poly_of_term = fn tm =>
+ if type_of (term_of tm) = @{typ real} then poly_of_term tm
+ else error "poly_of_term: term does not have real type"
+end;
+
+(* String of vector (just a list of space-separated numbers). *)
+
+fun sdpa_of_vector (v:vector) =
+ let
+ val n = dim v
+ val strs = map (decimalize 20 o (fn i => Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
+ in fold1 (fn x => fn y => x ^ " " ^ y) strs ^ "\n"
+ end;
+
+fun increasing f ord (x,y) = ord (f x, f y);
+fun triple_int_ord ((a,b,c),(a',b',c')) =
+ prod_ord int_ord (prod_ord int_ord int_ord)
+ ((a,(b,c)),(a',(b',c')));
+structure Inttriplefunc = FuncFun(type key = int*int*int val ord = triple_int_ord);
+
+(* String for block diagonal matrix numbered k. *)
+
+fun sdpa_of_blockdiagonal k m =
+ let
+ val pfx = string_of_int k ^" "
+ val ents =
+ Inttriplefunc.fold (fn ((b,i,j), c) => fn a => if i > j then a else ((b,i,j),c)::a) m []
+ val entss = sort (increasing fst triple_int_ord ) ents
+ in fold_rev (fn ((b,i,j),c) => fn a =>
+ pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) entss ""
+ end;
+
+(* String for a matrix numbered k, in SDPA sparse format. *)
+
+fun sdpa_of_matrix k (m:matrix) =
+ let
+ val pfx = string_of_int k ^ " 1 "
+ val ms = Intpairfunc.fold (fn ((i,j), c) => fn a => if i > j then a else ((i,j),c)::a) (snd m) []
+ val mss = sort (increasing fst (prod_ord int_ord int_ord)) ms
+ in fold_rev (fn ((i,j),c) => fn a =>
+ pfx ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) mss ""
+ end;;
+
+(* ------------------------------------------------------------------------- *)
+(* String in SDPA sparse format for standard SDP problem: *)
+(* *)
+(* X = v_1 * [M_1] + ... + v_m * [M_m] - [M_0] must be PSD *)
+(* Minimize obj_1 * v_1 + ... obj_m * v_m *)
+(* ------------------------------------------------------------------------- *)
+
+fun sdpa_of_problem obj mats =
+ let
+ val m = length mats - 1
+ val (n,_) = dimensions (hd mats)
+ in
+ string_of_int m ^ "\n" ^
+ "1\n" ^
+ string_of_int n ^ "\n" ^
+ sdpa_of_vector obj ^
+ fold_rev2 (fn k => fn m => fn a => sdpa_of_matrix (k - 1) m ^ a) (1 upto length mats) mats ""
+ end;
+
+fun index_char str chr pos =
+ if pos >= String.size str then ~1
+ else if String.sub(str,pos) = chr then pos
+ else index_char str chr (pos + 1);
+fun rat_of_quotient (a,b) = if b = 0 then rat_0 else Rat.rat_of_quotient (a,b);
+fun rat_of_string s =
+ let val n = index_char s #"/" 0 in
+ if n = ~1 then s |> IntInf.fromString |> valOf |> Rat.rat_of_int
+ else
+ let val SOME numer = IntInf.fromString(String.substring(s,0,n))
+ val SOME den = IntInf.fromString (String.substring(s,n+1,String.size s - n - 1))
+ in rat_of_quotient(numer, den)
+ end
+ end;
+
+fun isspace x = x = " " ;
+fun isnum x = x mem_string ["0","1","2","3","4","5","6","7","8","9"]
+
+(* More parser basics. *)
+
+local
+ open Scan
+in
+ val word = this_string
+ fun token s =
+ repeat ($$ " ") |-- word s --| repeat ($$ " ")
+ val numeral = one isnum
+ val decimalint = bulk numeral >> (rat_of_string o implode)
+ val decimalfrac = bulk numeral
+ >> (fn s => rat_of_string(implode s) // pow10 (length s))
+ val decimalsig =
+ decimalint -- option (Scan.$$ "." |-- decimalfrac)
+ >> (fn (h,NONE) => h | (h,SOME x) => h +/ x)
+ fun signed prs =
+ $$ "-" |-- prs >> Rat.neg
+ || $$ "+" |-- prs
+ || prs;
+
+fun emptyin def xs = if null xs then (def,xs) else Scan.fail xs
+
+ val exponent = ($$ "e" || $$ "E") |-- signed decimalint;
+
+ val decimal = signed decimalsig -- (emptyin rat_0|| exponent)
+ >> (fn (h, x) => h */ pow10 (int_of_rat x));
+end;
+
+ fun mkparser p s =
+ let val (x,rst) = p (explode s)
+ in if null rst then x
+ else error "mkparser: unparsed input"
+ end;;
+
+(* Parse back csdp output. *)
+
+ fun ignore inp = ((),[])
+ fun csdpoutput inp = ((decimal -- Scan.bulk (Scan.$$ " " |-- Scan.option decimal) >> (fn (h,to) => map_filter I ((SOME h)::to))) --| ignore >> vector_of_list) inp
+ val parse_csdpoutput = mkparser csdpoutput
+
+(* Run prover on a problem in linear form. *)
+
+fun run_problem prover obj mats =
+ parse_csdpoutput (prover (sdpa_of_problem obj mats))
+
+(* Try some apparently sensible scaling first. Note that this is purely to *)
+(* get a cleaner translation to floating-point, and doesn't affect any of *)
+(* the results, in principle. In practice it seems a lot better when there *)
+(* are extreme numbers in the original problem. *)
+
+ (* Version for (int*int) keys *)
+local
+ fun max_rat x y = if x </ y then y else x
+ fun common_denominator fld amat acc =
+ fld (fn (m,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
+ fun maximal_element fld amat acc =
+ fld (fn (m,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
+fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x
+ in Real.fromLargeInt a / Real.fromLargeInt b end;
+in
+
+fun pi_scale_then solver (obj:vector) mats =
+ let
+ val cd1 = fold_rev (common_denominator Intpairfunc.fold) mats (rat_1)
+ val cd2 = common_denominator Intfunc.fold (snd obj) (rat_1)
+ val mats' = map (Intpairfunc.mapf (fn x => cd1 */ x)) mats
+ val obj' = vector_cmul cd2 obj
+ val max1 = fold_rev (maximal_element Intpairfunc.fold) mats' (rat_0)
+ val max2 = maximal_element Intfunc.fold (snd obj') (rat_0)
+ val scal1 = pow2 (20 - trunc(Math.ln (float_of_rat max1) / Math.ln 2.0))
+ val scal2 = pow2 (20 - trunc(Math.ln (float_of_rat max2) / Math.ln 2.0))
+ val mats'' = map (Intpairfunc.mapf (fn x => x */ scal1)) mats'
+ val obj'' = vector_cmul scal2 obj'
+ in solver obj'' mats''
+ end
+end;
+
+(* Try some apparently sensible scaling first. Note that this is purely to *)
+(* get a cleaner translation to floating-point, and doesn't affect any of *)
+(* the results, in principle. In practice it seems a lot better when there *)
+(* are extreme numbers in the original problem. *)
+
+ (* Version for (int*int*int) keys *)
+local
+ fun max_rat x y = if x </ y then y else x
+ fun common_denominator fld amat acc =
+ fld (fn (m,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
+ fun maximal_element fld amat acc =
+ fld (fn (m,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
+fun float_of_rat x = let val (a,b) = Rat.quotient_of_rat x
+ in Real.fromLargeInt a / Real.fromLargeInt b end;
+fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)
+in
+
+fun tri_scale_then solver (obj:vector) mats =
+ let
+ val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats (rat_1)
+ val cd2 = common_denominator Intfunc.fold (snd obj) (rat_1)
+ val mats' = map (Inttriplefunc.mapf (fn x => cd1 */ x)) mats
+ val obj' = vector_cmul cd2 obj
+ val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' (rat_0)
+ val max2 = maximal_element Intfunc.fold (snd obj') (rat_0)
+ val scal1 = pow2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))
+ val scal2 = pow2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))
+ val mats'' = map (Inttriplefunc.mapf (fn x => x */ scal1)) mats'
+ val obj'' = vector_cmul scal2 obj'
+ in solver obj'' mats''
+ end
+end;
+
+(* Round a vector to "nice" rationals. *)
+
+fun nice_rational n x = round_rat (n */ x) // n;;
+fun nice_vector n ((d,v) : vector) =
+ (d, Intfunc.fold (fn (i,c) => fn a =>
+ let val y = nice_rational n c
+ in if c =/ rat_0 then a
+ else Intfunc.update (i,y) a end) v Intfunc.undefined):vector
+
+fun dest_ord f x = is_equal (f x);
+
+(* Stuff for "equations" ((int*int*int)->num functions). *)
+
+fun tri_equation_cmul c eq =
+ if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (fn d => c */ d) eq;
+
+fun tri_equation_add eq1 eq2 = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0) eq1 eq2;
+
+fun tri_equation_eval assig eq =
+ let fun value v = Inttriplefunc.apply assig v
+ in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
+ end;
+
+(* Eliminate among linear equations: return unconstrained variables and *)
+(* assignments for the others in terms of them. We give one pseudo-variable *)
+(* "one" that's used for a constant term. *)
+
+local
+ fun extract_first p l = case l of (* FIXME : use find_first instead *)
+ [] => error "extract_first"
+ | h::t => if p h then (h,t) else
+ let val (k,s) = extract_first p t in (k,h::s) end
+fun eliminate vars dun eqs = case vars of
+ [] => if forall Inttriplefunc.is_undefined eqs then dun
+ else raise Unsolvable
+ | v::vs =>
+ ((let
+ val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
+ val a = Inttriplefunc.apply eq v
+ val eq' = tri_equation_cmul ((Rat.neg rat_1) // a) (Inttriplefunc.undefine v eq)
+ fun elim e =
+ let val b = Inttriplefunc.tryapplyd e v rat_0
+ in if b =/ rat_0 then e else
+ tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
+ end
+ in eliminate vs (Inttriplefunc.update (v,eq') (Inttriplefunc.mapf elim dun)) (map elim oeqs)
+ end)
+ handle Failure _ => eliminate vs dun eqs)
+in
+fun tri_eliminate_equations one vars eqs =
+ let
+ val assig = eliminate vars Inttriplefunc.undefined eqs
+ val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ in (distinct (dest_ord triple_int_ord) vs, assig)
+ end
+end;
+
+(* Eliminate all variables, in an essentially arbitrary order. *)
+
+fun tri_eliminate_all_equations one =
+ let
+ fun choose_variable eq =
+ let val (v,_) = Inttriplefunc.choose eq
+ in if is_equal (triple_int_ord(v,one)) then
+ let val eq' = Inttriplefunc.undefine v eq
+ in if Inttriplefunc.is_undefined eq' then error "choose_variable"
+ else fst (Inttriplefunc.choose eq')
+ end
+ else v
+ end
+ fun eliminate dun eqs = case eqs of
+ [] => dun
+ | eq::oeqs =>
+ if Inttriplefunc.is_undefined eq then eliminate dun oeqs else
+ let val v = choose_variable eq
+ val a = Inttriplefunc.apply eq v
+ val eq' = tri_equation_cmul ((Rat.rat_of_int ~1) // a)
+ (Inttriplefunc.undefine v eq)
+ fun elim e =
+ let val b = Inttriplefunc.tryapplyd e v rat_0
+ in if b =/ rat_0 then e
+ else tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
+ end
+ in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.mapf elim dun))
+ (map elim oeqs)
+ end
+in fn eqs =>
+ let
+ val assig = eliminate Inttriplefunc.undefined eqs
+ val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ in (distinct (dest_ord triple_int_ord) vs,assig)
+ end
+end;
+
+(* Solve equations by assigning arbitrary numbers. *)
+
+fun tri_solve_equations one eqs =
+ let
+ val (vars,assigs) = tri_eliminate_all_equations one eqs
+ val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
+ (Inttriplefunc.onefunc(one, Rat.rat_of_int ~1))
+ val ass =
+ Inttriplefunc.combine (curry op +/) (K false)
+ (Inttriplefunc.mapf (tri_equation_eval vfn) assigs) vfn
+ in if forall (fn e => tri_equation_eval ass e =/ rat_0) eqs
+ then Inttriplefunc.undefine one ass else raise Sanity
+ end;
+
+(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
+
+fun tri_epoly_pmul p q acc =
+ Monomialfunc.fold (fn (m1, c) => fn a =>
+ Monomialfunc.fold (fn (m2,e) => fn b =>
+ let val m = monomial_mul m1 m2
+ val es = Monomialfunc.tryapplyd b m Inttriplefunc.undefined
+ in Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
+ end) q a) p acc ;
+
+(* Usual operations on equation-parametrized poly. *)
+
+fun tri_epoly_cmul c l =
+ if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (tri_equation_cmul c) l;;
+
+val tri_epoly_neg = tri_epoly_cmul (Rat.rat_of_int ~1);
+
+val tri_epoly_add = Inttriplefunc.combine tri_equation_add Inttriplefunc.is_undefined;
+
+fun tri_epoly_sub p q = tri_epoly_add p (tri_epoly_neg q);;
+
+(* Stuff for "equations" ((int*int)->num functions). *)
+
+fun pi_equation_cmul c eq =
+ if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (fn d => c */ d) eq;
+
+fun pi_equation_add eq1 eq2 = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0) eq1 eq2;
+
+fun pi_equation_eval assig eq =
+ let fun value v = Inttriplefunc.apply assig v
+ in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
+ end;
+
+(* Eliminate among linear equations: return unconstrained variables and *)
+(* assignments for the others in terms of them. We give one pseudo-variable *)
+(* "one" that's used for a constant term. *)
+
+local
+fun extract_first p l = case l of
+ [] => error "extract_first"
+ | h::t => if p h then (h,t) else
+ let val (k,s) = extract_first p t in (k,h::s) end
+fun eliminate vars dun eqs = case vars of
+ [] => if forall Inttriplefunc.is_undefined eqs then dun
+ else raise Unsolvable
+ | v::vs =>
+ let
+ val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
+ val a = Inttriplefunc.apply eq v
+ val eq' = pi_equation_cmul ((Rat.neg rat_1) // a) (Inttriplefunc.undefine v eq)
+ fun elim e =
+ let val b = Inttriplefunc.tryapplyd e v rat_0
+ in if b =/ rat_0 then e else
+ pi_equation_add e (pi_equation_cmul (Rat.neg b // a) eq)
+ end
+ in eliminate vs (Inttriplefunc.update (v,eq') (Inttriplefunc.mapf elim dun)) (map elim oeqs)
+ end
+ handle Failure _ => eliminate vs dun eqs
+in
+fun pi_eliminate_equations one vars eqs =
+ let
+ val assig = eliminate vars Inttriplefunc.undefined eqs
+ val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ in (distinct (dest_ord triple_int_ord) vs, assig)
+ end
+end;
+
+(* Eliminate all variables, in an essentially arbitrary order. *)
+
+fun pi_eliminate_all_equations one =
+ let
+ fun choose_variable eq =
+ let val (v,_) = Inttriplefunc.choose eq
+ in if is_equal (triple_int_ord(v,one)) then
+ let val eq' = Inttriplefunc.undefine v eq
+ in if Inttriplefunc.is_undefined eq' then error "choose_variable"
+ else fst (Inttriplefunc.choose eq')
+ end
+ else v
+ end
+ fun eliminate dun eqs = case eqs of
+ [] => dun
+ | eq::oeqs =>
+ if Inttriplefunc.is_undefined eq then eliminate dun oeqs else
+ let val v = choose_variable eq
+ val a = Inttriplefunc.apply eq v
+ val eq' = pi_equation_cmul ((Rat.rat_of_int ~1) // a)
+ (Inttriplefunc.undefine v eq)
+ fun elim e =
+ let val b = Inttriplefunc.tryapplyd e v rat_0
+ in if b =/ rat_0 then e
+ else pi_equation_add e (pi_equation_cmul (Rat.neg b // a) eq)
+ end
+ in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.mapf elim dun))
+ (map elim oeqs)
+ end
+in fn eqs =>
+ let
+ val assig = eliminate Inttriplefunc.undefined eqs
+ val vs = Inttriplefunc.fold (fn (x, f) => fn a => remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
+ in (distinct (dest_ord triple_int_ord) vs,assig)
+ end
+end;
+
+(* Solve equations by assigning arbitrary numbers. *)
+
+fun pi_solve_equations one eqs =
+ let
+ val (vars,assigs) = pi_eliminate_all_equations one eqs
+ val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
+ (Inttriplefunc.onefunc(one, Rat.rat_of_int ~1))
+ val ass =
+ Inttriplefunc.combine (curry op +/) (K false)
+ (Inttriplefunc.mapf (pi_equation_eval vfn) assigs) vfn
+ in if forall (fn e => pi_equation_eval ass e =/ rat_0) eqs
+ then Inttriplefunc.undefine one ass else raise Sanity
+ end;
+
+(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
+
+fun pi_epoly_pmul p q acc =
+ Monomialfunc.fold (fn (m1, c) => fn a =>
+ Monomialfunc.fold (fn (m2,e) => fn b =>
+ let val m = monomial_mul m1 m2
+ val es = Monomialfunc.tryapplyd b m Inttriplefunc.undefined
+ in Monomialfunc.update (m,pi_equation_add (pi_equation_cmul c e) es) b
+ end) q a) p acc ;
+
+(* Usual operations on equation-parametrized poly. *)
+
+fun pi_epoly_cmul c l =
+ if c =/ rat_0 then Inttriplefunc.undefined else Inttriplefunc.mapf (pi_equation_cmul c) l;;
+
+val pi_epoly_neg = pi_epoly_cmul (Rat.rat_of_int ~1);
+
+val pi_epoly_add = Inttriplefunc.combine pi_equation_add Inttriplefunc.is_undefined;
+
+fun pi_epoly_sub p q = pi_epoly_add p (pi_epoly_neg q);;
+
+fun allpairs f l1 l2 = fold_rev (fn x => (curry (op @)) (map (f x) l2)) l1 [];
+
+(* Hence produce the "relevant" monomials: those whose squares lie in the *)
+(* Newton polytope of the monomials in the input. (This is enough according *)
+(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal, *)
+(* vol 45, pp. 363--374, 1978. *)
+(* *)
+(* These are ordered in sort of decreasing degree. In particular the *)
+(* constant monomial is last; this gives an order in diagonalization of the *)
+(* quadratic form that will tend to display constants. *)
+
+(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form. *)
+
+local
+fun diagonalize n i m =
+ if Intpairfunc.is_undefined (snd m) then []
+ else
+ let val a11 = Intpairfunc.tryapplyd (snd m) (i,i) rat_0
+ in if a11 </ rat_0 then raise Failure "diagonalize: not PSD"
+ else if a11 =/ rat_0 then
+ if Intfunc.is_undefined (snd (row i m)) then diagonalize n (i + 1) m
+ else raise Failure "diagonalize: not PSD ___ "
+ else
+ let
+ val v = row i m
+ val v' = (fst v, Intfunc.fold (fn (i, c) => fn a =>
+ let val y = c // a11
+ in if y = rat_0 then a else Intfunc.update (i,y) a
+ end) (snd v) Intfunc.undefined)
+ fun upt0 x y a = if y = rat_0 then a else Intpairfunc.update (x,y) a
+ val m' =
+ ((n,n),
+ iter (i+1,n) (fn j =>
+ iter (i+1,n) (fn k =>
+ (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))))
+ Intpairfunc.undefined)
+ in (a11,v')::diagonalize n (i + 1) m'
+ end
+ end
+in
+fun diag m =
+ let
+ val nn = dimensions m
+ val n = fst nn
+ in if snd nn <> n then error "diagonalize: non-square matrix"
+ else diagonalize n 1 m
+ end
+end;
+
+fun gcd_rat a b = Rat.rat_of_int (Integer.gcd (int_of_rat a) (int_of_rat b));
+
+(* Adjust a diagonalization to collect rationals at the start. *)
+ (* FIXME : Potentially polymorphic keys, but here only: integers!! *)
+local
+ fun upd0 x y a = if y =/ rat_0 then a else Intfunc.update(x,y) a;
+ fun mapa f (d,v) =
+ (d, Intfunc.fold (fn (i,c) => fn a => upd0 i (f c) a) v Intfunc.undefined)
+ fun adj (c,l) =
+ let val a =
+ Intfunc.fold (fn (i,c) => fn a => lcm_rat a (denominator_rat c))
+ (snd l) rat_1 //
+ Intfunc.fold (fn (i,c) => fn a => gcd_rat a (numerator_rat c))
+ (snd l) rat_0
+ in ((c // (a */ a)),mapa (fn x => a */ x) l)
+ end
+in
+fun deration d = if null d then (rat_0,d) else
+ let val d' = map adj d
+ val a = fold (lcm_rat o denominator_rat o fst) d' rat_1 //
+ fold (gcd_rat o numerator_rat o fst) d' rat_0
+ in ((rat_1 // a),map (fn (c,l) => (a */ c,l)) d')
+ end
+end;
+
+(* Enumeration of monomials with given multidegree bound. *)
+
+fun enumerate_monomials d vars =
+ if d < 0 then []
+ else if d = 0 then [Ctermfunc.undefined]
+ else if null vars then [monomial_1] else
+ let val alts =
+ map (fn k => let val oths = enumerate_monomials (d - k) (tl vars)
+ in map (fn ks => if k = 0 then ks else Ctermfunc.update (hd vars, k) ks) oths end) (0 upto d)
+ in fold1 (curry op @) alts
+ end;
+
+(* Enumerate products of distinct input polys with degree <= d. *)
+(* We ignore any constant input polynomials. *)
+(* Give the output polynomial and a record of how it was derived. *)
+
+local
+ open RealArith
+in
+fun enumerate_products d pols =
+if d = 0 then [(poly_const rat_1,Rational_lt rat_1)]
+else if d < 0 then [] else
+case pols of
+ [] => [(poly_const rat_1,Rational_lt rat_1)]
+ | (p,b)::ps =>
+ let val e = multidegree p
+ in if e = 0 then enumerate_products d ps else
+ enumerate_products d ps @
+ map (fn (q,c) => (poly_mul p q,Product(b,c)))
+ (enumerate_products (d - e) ps)
+ end
+end;
+
+(* Convert regular polynomial. Note that we treat (0,0,0) as -1. *)
+
+fun epoly_of_poly p =
+ Monomialfunc.fold (fn (m,c) => fn a => Monomialfunc.update (m, Inttriplefunc.onefunc ((0,0,0), Rat.neg c)) a) p Monomialfunc.undefined;
+
+(* String for block diagonal matrix numbered k. *)
+
+fun sdpa_of_blockdiagonal k m =
+ let
+ val pfx = string_of_int k ^" "
+ val ents =
+ Inttriplefunc.fold
+ (fn ((b,i,j),c) => fn a => if i > j then a else ((b,i,j),c)::a)
+ m []
+ val entss = sort (increasing fst triple_int_ord) ents
+ in fold_rev (fn ((b,i,j),c) => fn a =>
+ pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) entss ""
+ end;
+
+(* SDPA for problem using block diagonal (i.e. multiple SDPs) *)
+
+fun sdpa_of_blockproblem nblocks blocksizes obj mats =
+ let val m = length mats - 1
+ in
+ string_of_int m ^ "\n" ^
+ string_of_int nblocks ^ "\n" ^
+ (fold1 (fn s => fn t => s^" "^t) (map string_of_int blocksizes)) ^
+ "\n" ^
+ sdpa_of_vector obj ^
+ fold_rev2 (fn k => fn m => fn a => sdpa_of_blockdiagonal (k - 1) m ^ a)
+ (1 upto length mats) mats ""
+ end;
+
+(* Run prover on a problem in block diagonal form. *)
+
+fun run_blockproblem prover nblocks blocksizes obj mats=
+ parse_csdpoutput (prover (sdpa_of_blockproblem nblocks blocksizes obj mats))
+
+(* 3D versions of matrix operations to consider blocks separately. *)
+
+val bmatrix_add = Inttriplefunc.combine (curry op +/) (fn x => x =/ rat_0);
+fun bmatrix_cmul c bm =
+ if c =/ rat_0 then Inttriplefunc.undefined
+ else Inttriplefunc.mapf (fn x => c */ x) bm;
+
+val bmatrix_neg = bmatrix_cmul (Rat.rat_of_int ~1);
+fun bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;
+
+(* Smash a block matrix into components. *)
+
+fun blocks blocksizes bm =
+ map (fn (bs,b0) =>
+ let val m = Inttriplefunc.fold
+ (fn ((b,i,j),c) => fn a => if b = b0 then Intpairfunc.update ((i,j),c) a else a) bm Intpairfunc.undefined
+ val d = Intpairfunc.fold (fn ((i,j),c) => fn a => max a (max i j)) m 0
+ in (((bs,bs),m):matrix) end)
+ (blocksizes ~~ (1 upto length blocksizes));;
+
+(* FIXME : Get rid of this !!!*)
+local
+ fun tryfind_with msg f [] = raise Failure msg
+ | tryfind_with msg f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
+in
+ fun tryfind f = tryfind_with "tryfind" f
+end
+
+(*
+fun tryfind f [] = error "tryfind"
+ | tryfind f (x::xs) = (f x handle ERROR _ => tryfind f xs);
+*)
+
+(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
+
+
+local
+ open RealArith
+in
+fun real_positivnullstellensatz_general prover linf d eqs leqs pol =
+let
+ val vars = fold_rev (curry (gen_union (op aconvc)) o poly_variables)
+ (pol::eqs @ map fst leqs) []
+ val monoid = if linf then
+ (poly_const rat_1,Rational_lt rat_1)::
+ (filter (fn (p,c) => multidegree p <= d) leqs)
+ else enumerate_products d leqs
+ val nblocks = length monoid
+ fun mk_idmultiplier k p =
+ let
+ val e = d - multidegree p
+ val mons = enumerate_monomials e vars
+ val nons = mons ~~ (1 upto length mons)
+ in (mons,
+ fold_rev (fn (m,n) => Monomialfunc.update(m,Inttriplefunc.onefunc((~k,~n,n),rat_1))) nons Monomialfunc.undefined)
+ end
+
+ fun mk_sqmultiplier k (p,c) =
+ let
+ val e = (d - multidegree p) div 2
+ val mons = enumerate_monomials e vars
+ val nons = mons ~~ (1 upto length mons)
+ in (mons,
+ fold_rev (fn (m1,n1) =>
+ fold_rev (fn (m2,n2) => fn a =>
+ let val m = monomial_mul m1 m2
+ in if n1 > n2 then a else
+ let val c = if n1 = n2 then rat_1 else rat_2
+ val e = Monomialfunc.tryapplyd a m Inttriplefunc.undefined
+ in Monomialfunc.update(m, tri_equation_add (Inttriplefunc.onefunc((k,n1,n2), c)) e) a
+ end
+ end) nons)
+ nons Monomialfunc.undefined)
+ end
+
+ val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)
+ val (idmonlist,ids) = split_list(map2 mk_idmultiplier (1 upto length eqs) eqs)
+ val blocksizes = map length sqmonlist
+ val bigsum =
+ fold_rev2 (fn p => fn q => fn a => tri_epoly_pmul p q a) eqs ids
+ (fold_rev2 (fn (p,c) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs
+ (epoly_of_poly(poly_neg pol)))
+ val eqns = Monomialfunc.fold (fn (m,e) => fn a => e::a) bigsum []
+ val (pvs,assig) = tri_eliminate_all_equations (0,0,0) eqns
+ val qvars = (0,0,0)::pvs
+ val allassig = fold_rev (fn v => Inttriplefunc.update(v,(Inttriplefunc.onefunc(v,rat_1)))) pvs assig
+ fun mk_matrix v =
+ Inttriplefunc.fold (fn ((b,i,j), ass) => fn m =>
+ if b < 0 then m else
+ let val c = Inttriplefunc.tryapplyd ass v rat_0
+ in if c = rat_0 then m else
+ Inttriplefunc.update ((b,j,i), c) (Inttriplefunc.update ((b,i,j), c) m)
+ end)
+ allassig Inttriplefunc.undefined
+ val diagents = Inttriplefunc.fold
+ (fn ((b,i,j), e) => fn a => if b > 0 andalso i = j then tri_equation_add e a else a)
+ allassig Inttriplefunc.undefined
+
+ val mats = map mk_matrix qvars
+ val obj = (length pvs,
+ itern 1 pvs (fn v => fn i => Intfunc.updatep iszero (i,Inttriplefunc.tryapplyd diagents v rat_0))
+ Intfunc.undefined)
+ val raw_vec = if null pvs then vector_0 0
+ else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats
+ fun int_element (d,v) i = Intfunc.tryapplyd v i rat_0
+ fun cterm_element (d,v) i = Ctermfunc.tryapplyd v i rat_0
+
+ fun find_rounding d =
+ let
+ val _ = if !debugging
+ then writeln ("Trying rounding with limit "^Rat.string_of_rat d ^ "\n")
+ else ()
+ val vec = nice_vector d raw_vec
+ val blockmat = iter (1,dim vec)
+ (fn i => fn a => bmatrix_add (bmatrix_cmul (int_element vec i) (nth mats i)) a)
+ (bmatrix_neg (nth mats 0))
+ val allmats = blocks blocksizes blockmat
+ in (vec,map diag allmats)
+ end
+ val (vec,ratdias) =
+ if null pvs then find_rounding rat_1
+ else tryfind find_rounding (map Rat.rat_of_int (1 upto 31) @
+ map pow2 (5 upto 66))
+ val newassigs =
+ fold_rev (fn k => Inttriplefunc.update (nth pvs (k - 1), int_element vec k))
+ (1 upto dim vec) (Inttriplefunc.onefunc ((0,0,0), Rat.rat_of_int ~1))
+ val finalassigs =
+ Inttriplefunc.fold (fn (v,e) => fn a => Inttriplefunc.update(v, tri_equation_eval newassigs e) a) allassig newassigs
+ fun poly_of_epoly p =
+ Monomialfunc.fold (fn (v,e) => fn a => Monomialfunc.updatep iszero (v,tri_equation_eval finalassigs e) a)
+ p Monomialfunc.undefined
+ fun mk_sos mons =
+ let fun mk_sq (c,m) =
+ (c,fold_rev (fn k=> fn a => Monomialfunc.updatep iszero (nth mons (k - 1), int_element m k) a)
+ (1 upto length mons) Monomialfunc.undefined)
+ in map mk_sq
+ end
+ val sqs = map2 mk_sos sqmonlist ratdias
+ val cfs = map poly_of_epoly ids
+ val msq = filter (fn (a,b) => not (null b)) (map2 pair monoid sqs)
+ fun eval_sq sqs = fold_rev (fn (c,q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0
+ val sanity =
+ fold_rev (fn ((p,c),s) => poly_add (poly_mul p (eval_sq s))) msq
+ (fold_rev2 (fn p => fn q => poly_add (poly_mul p q)) cfs eqs
+ (poly_neg pol))
+
+in if not(Monomialfunc.is_undefined sanity) then raise Sanity else
+ (cfs,map (fn (a,b) => (snd a,b)) msq)
+ end
+
+
+end;
+
+(* Iterative deepening. *)
+
+fun deepen f n =
+ (writeln ("Searching with depth limit " ^ string_of_int n) ; (f n handle Failure s => (writeln ("failed with message: " ^ s) ; deepen f (n+1))))
+
+(* The ordering so we can create canonical HOL polynomials. *)
+
+fun dest_monomial mon = sort (increasing fst cterm_ord) (Ctermfunc.graph mon);
+
+fun monomial_order (m1,m2) =
+ if Ctermfunc.is_undefined m2 then LESS
+ else if Ctermfunc.is_undefined m1 then GREATER
+ else
+ let val mon1 = dest_monomial m1
+ val mon2 = dest_monomial m2
+ val deg1 = fold (curry op + o snd) mon1 0
+ val deg2 = fold (curry op + o snd) mon2 0
+ in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
+ else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
+ end;
+
+fun dest_poly p =
+ map (fn (m,c) => (c,dest_monomial m))
+ (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p));
+
+(* Map back polynomials and their composites to HOL. *)
+
+local
+ open Thm Numeral RealArith
+in
+
+fun cterm_of_varpow x k = if k = 1 then x else capply (capply @{cterm "op ^ :: real => _"} x)
+ (mk_cnumber @{ctyp nat} k)
+
+fun cterm_of_monomial m =
+ if Ctermfunc.is_undefined m then @{cterm "1::real"}
+ else
+ let
+ val m' = dest_monomial m
+ val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
+ in fold1 (fn s => fn t => capply (capply @{cterm "op * :: real => _"} s) t) vps
+ end
+
+fun cterm_of_cmonomial (m,c) = if Ctermfunc.is_undefined m then cterm_of_rat c
+ else if c = Rat.one then cterm_of_monomial m
+ else capply (capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
+
+fun cterm_of_poly p =
+ if Monomialfunc.is_undefined p then @{cterm "0::real"}
+ else
+ let
+ val cms = map cterm_of_cmonomial
+ (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p))
+ in fold1 (fn t1 => fn t2 => capply(capply @{cterm "op + :: real => _"} t1) t2) cms
+ end;
+
+fun cterm_of_sqterm (c,p) = Product(Rational_lt c,Square(cterm_of_poly p));
+
+fun cterm_of_sos (pr,sqs) = if null sqs then pr
+ else Product(pr,fold1 (fn a => fn b => Sum(a,b)) (map cterm_of_sqterm sqs));
+
+end
+
+(* Interface to HOL. *)
+local
+ open Thm Conv RealArith
+ val concl = dest_arg o cprop_of
+ fun simple_cterm_ord t u = TermOrd.fast_term_ord (term_of t, term_of u) = LESS
+in
+ (* FIXME: Replace tryfind by get_first !! *)
+fun real_nonlinear_prover prover ctxt =
+ let
+ val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
+ (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ simple_cterm_ord
+ val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
+ real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
+ fun mainf translator (eqs,les,lts) =
+ let
+ val eq0 = map (poly_of_term o dest_arg1 o concl) eqs
+ val le0 = map (poly_of_term o dest_arg o concl) les
+ val lt0 = map (poly_of_term o dest_arg o concl) lts
+ val eqp0 = map (fn (t,i) => (t,Axiom_eq i)) (eq0 ~~ (0 upto (length eq0 - 1)))
+ val lep0 = map (fn (t,i) => (t,Axiom_le i)) (le0 ~~ (0 upto (length le0 - 1)))
+ val ltp0 = map (fn (t,i) => (t,Axiom_lt i)) (lt0 ~~ (0 upto (length lt0 - 1)))
+ val (keq,eq) = List.partition (fn (p,_) => multidegree p = 0) eqp0
+ val (klep,lep) = List.partition (fn (p,_) => multidegree p = 0) lep0
+ val (kltp,ltp) = List.partition (fn (p,_) => multidegree p = 0) ltp0
+ fun trivial_axiom (p,ax) =
+ case ax of
+ Axiom_eq n => if eval Ctermfunc.undefined p <>/ Rat.zero then nth eqs n
+ else raise Failure "trivial_axiom: Not a trivial axiom"
+ | Axiom_le n => if eval Ctermfunc.undefined p </ Rat.zero then nth les n
+ else raise Failure "trivial_axiom: Not a trivial axiom"
+ | Axiom_lt n => if eval Ctermfunc.undefined p <=/ Rat.zero then nth lts n
+ else raise Failure "trivial_axiom: Not a trivial axiom"
+ | _ => error "trivial_axiom: Not a trivial axiom"
+ in
+ ((let val th = tryfind trivial_axiom (keq @ klep @ kltp)
+ in fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv field_comp_conv) th end)
+ handle Failure _ => (
+ let
+ val pol = fold_rev poly_mul (map fst ltp) (poly_const Rat.one)
+ val leq = lep @ ltp
+ fun tryall d =
+ let val e = multidegree pol
+ val k = if e = 0 then 0 else d div e
+ val eq' = map fst eq
+ in tryfind (fn i => (d,i,real_positivnullstellensatz_general prover false d eq' leq
+ (poly_neg(poly_pow pol i))))
+ (0 upto k)
+ end
+ val (d,i,(cert_ideal,cert_cone)) = deepen tryall 0
+ val proofs_ideal =
+ map2 (fn q => fn (p,ax) => Eqmul(cterm_of_poly q,ax)) cert_ideal eq
+ val proofs_cone = map cterm_of_sos cert_cone
+ val proof_ne = if null ltp then Rational_lt Rat.one else
+ let val p = fold1 (fn s => fn t => Product(s,t)) (map snd ltp)
+ in funpow i (fn q => Product(p,q)) (Rational_lt Rat.one)
+ end
+ val proof = fold1 (fn s => fn t => Sum(s,t))
+ (proof_ne :: proofs_ideal @ proofs_cone)
+ in writeln "Translating proof certificate to HOL";
+ translator (eqs,les,lts) proof
+ end))
+ end
+ in mainf end
+end
+
+fun C f x y = f y x;
+ (* FIXME : This is very bad!!!*)
+fun subst_conv eqs t =
+ let
+ val t' = fold (Thm.cabs o Thm.lhs_of) eqs t
+ in Conv.fconv_rule (Thm.beta_conversion true) (fold (C combination) eqs (reflexive t'))
+ end
+
+(* A wrapper that tries to substitute away variables first. *)
+
+local
+ open Thm Conv RealArith
+ fun simple_cterm_ord t u = TermOrd.fast_term_ord (term_of t, term_of u) = LESS
+ val concl = dest_arg o cprop_of
+ val shuffle1 =
+ fconv_rule (rewr_conv @{lemma "(a + x == y) == (x == y - (a::real))" by (atomize (full)) (simp add: ring_simps) })
+ val shuffle2 =
+ fconv_rule (rewr_conv @{lemma "(x + a == y) == (x == y - (a::real))" by (atomize (full)) (simp add: ring_simps)})
+ fun substitutable_monomial fvs tm = case term_of tm of
+ Free(_,@{typ real}) => if not (member (op aconvc) fvs tm) then (Rat.one,tm)
+ else raise Failure "substitutable_monomial"
+ | @{term "op * :: real => _"}$c$(t as Free _ ) =>
+ if is_ratconst (dest_arg1 tm) andalso not (member (op aconvc) fvs (dest_arg tm))
+ then (dest_ratconst (dest_arg1 tm),dest_arg tm) else raise Failure "substitutable_monomial"
+ | @{term "op + :: real => _"}$s$t =>
+ (substitutable_monomial (add_cterm_frees (dest_arg tm) fvs) (dest_arg1 tm)
+ handle Failure _ => substitutable_monomial (add_cterm_frees (dest_arg1 tm) fvs) (dest_arg tm))
+ | _ => raise Failure "substitutable_monomial"
+
+ fun isolate_variable v th =
+ let val w = dest_arg1 (cprop_of th)
+ in if v aconvc w then th
+ else case term_of w of
+ @{term "op + :: real => _"}$s$t =>
+ if dest_arg1 w aconvc v then shuffle2 th
+ else isolate_variable v (shuffle1 th)
+ | _ => error "isolate variable : This should not happen?"
+ end
+in
+
+fun real_nonlinear_subst_prover prover ctxt =
+ let
+ val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
+ (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ simple_cterm_ord
+
+ val (real_poly_add_conv,real_poly_mul_conv,real_poly_neg_conv,
+ real_poly_pow_conv,real_poly_sub_conv,real_poly_conv) = (add,mul,neg,pow,sub,main)
+
+ fun make_substitution th =
+ let
+ val (c,v) = substitutable_monomial [] (dest_arg1(concl th))
+ val th1 = Drule.arg_cong_rule (capply @{cterm "op * :: real => _"} (cterm_of_rat (Rat.inv c))) (mk_meta_eq th)
+ val th2 = fconv_rule (binop_conv real_poly_mul_conv) th1
+ in fconv_rule (arg_conv real_poly_conv) (isolate_variable v th2)
+ end
+ fun oprconv cv ct =
+ let val g = Thm.dest_fun2 ct
+ in if g aconvc @{cterm "op <= :: real => _"}
+ orelse g aconvc @{cterm "op < :: real => _"}
+ then arg_conv cv ct else arg1_conv cv ct
+ end
+ fun mainf translator =
+ let
+ fun substfirst(eqs,les,lts) =
+ ((let
+ val eth = tryfind make_substitution eqs
+ val modify = fconv_rule (arg_conv (oprconv(subst_conv [eth] then_conv real_poly_conv)))
+ in substfirst
+ (filter_out (fn t => (Thm.dest_arg1 o Thm.dest_arg o cprop_of) t
+ aconvc @{cterm "0::real"}) (map modify eqs),
+ map modify les,map modify lts)
+ end)
+ handle Failure _ => real_nonlinear_prover prover ctxt translator (rev eqs, rev les, rev lts))
+ in substfirst
+ end
+
+
+ in mainf
+ end
+
+(* Overall function. *)
+
+fun real_sos prover ctxt t = gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt) t;
+end;
+
+(* A tactic *)
+fun strip_all ct =
+ case term_of ct of
+ Const("all",_) $ Abs (xn,xT,p) =>
+ let val (a,(v,t')) = (apsnd (Thm.dest_abs (SOME xn)) o Thm.dest_comb) ct
+ in apfst (cons v) (strip_all t')
+ end
+| _ => ([],ct)
+
+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})
+
+val known_sos_constants =
+ [@{term "op ==>"}, @{term "Trueprop"},
+ @{term "op -->"}, @{term "op &"}, @{term "op |"},
+ @{term "Not"}, @{term "op = :: bool => _"},
+ @{term "All :: (real => _) => _"}, @{term "Ex :: (real => _) => _"},
+ @{term "op = :: real => _"}, @{term "op < :: real => _"},
+ @{term "op <= :: real => _"},
+ @{term "op + :: real => _"}, @{term "op - :: real => _"},
+ @{term "op * :: real => _"}, @{term "uminus :: real => _"},
+ @{term "op / :: real => _"}, @{term "inverse :: real => _"},
+ @{term "op ^ :: real => _"}, @{term "abs :: real => _"},
+ @{term "min :: real => _"}, @{term "max :: real => _"},
+ @{term "0::real"}, @{term "1::real"}, @{term "number_of :: int => real"},
+ @{term "number_of :: int => nat"},
+ @{term "Int.Bit0"}, @{term "Int.Bit1"},
+ @{term "Int.Pls"}, @{term "Int.Min"}];
+
+fun check_sos kcts ct =
+ let
+ val t = term_of ct
+ val _ = if not (null (Term.add_tfrees t [])
+ andalso null (Term.add_tvars t []))
+ then error "SOS: not sos. Additional type varables" else ()
+ val fs = Term.add_frees t []
+ val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs
+ then error "SOS: not sos. Variables with type not real" else ()
+ val vs = Term.add_vars t []
+ val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs
+ then error "SOS: not sos. Variables with type not real" else ()
+ val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])
+ val _ = if null ukcs then ()
+ else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))
+in () end
+
+fun core_sos_tac prover ctxt = CSUBGOAL (fn (ct, i) =>
+ let val _ = check_sos known_sos_constants ct
+ val (avs, p) = strip_all ct
+ val th = standard (fold_rev forall_intr avs (real_sos prover ctxt (Thm.dest_arg p)))
+ in rtac th i end);
+
+fun default_SOME f NONE v = SOME v
+ | default_SOME f (SOME v) _ = SOME v;
+
+fun lift_SOME f NONE a = f a
+ | lift_SOME f (SOME a) _ = SOME a;
+
+
+local
+ val is_numeral = can (HOLogic.dest_number o term_of)
+in
+fun get_denom b ct = case term_of ct of
+ @{term "op / :: real => _"} $ _ $ _ =>
+ if is_numeral (Thm.dest_arg ct) then get_denom b (Thm.dest_arg1 ct)
+ else default_SOME (get_denom b) (get_denom b (Thm.dest_arg ct)) (Thm.dest_arg ct, b)
+ | @{term "op < :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
+ | @{term "op <= :: real => _"} $ _ $ _ => lift_SOME (get_denom true) (get_denom true (Thm.dest_arg ct)) (Thm.dest_arg1 ct)
+ | _ $ _ => lift_SOME (get_denom b) (get_denom b (Thm.dest_fun ct)) (Thm.dest_arg ct)
+ | _ => NONE
+end;
+
+fun elim_one_denom_tac ctxt =
+CSUBGOAL (fn (P,i) =>
+ case get_denom false P of
+ NONE => no_tac
+ | SOME (d,ord) =>
+ let
+ val ss = simpset_of ctxt addsimps @{thms field_simps}
+ addsimps [@{thm nonzero_power_divide}, @{thm power_divide}]
+ val th = instantiate' [] [SOME d, SOME (Thm.dest_arg P)]
+ (if ord then @{lemma "(d=0 --> P) & (d>0 --> P) & (d<(0::real) --> P) ==> P" by auto}
+ else @{lemma "(d=0 --> P) & (d ~= (0::real) --> P) ==> P" by blast})
+ in (rtac th i THEN Simplifier.asm_full_simp_tac ss i) end);
+
+fun elim_denom_tac ctxt i = REPEAT (elim_one_denom_tac ctxt i);
+
+fun sos_tac prover ctxt = ObjectLogic.full_atomize_tac THEN' elim_denom_tac ctxt THEN' core_sos_tac prover ctxt
+
+
+end;