--- a/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Thu Oct 01 20:52:18 2009 +0200
+++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML Thu Oct 01 23:03:59 2009 +0200
@@ -1,8 +1,8 @@
-(* Title: sum_of_squares.ML
- Authors: Amine Chaieb, University of Cambridge
- Philipp Meyer, TU Muenchen
+(* Title: HOL/Library/Sum_Of_Squares/sum_of_squares.ML
+ Author: Amine Chaieb, University of Cambridge
+ Author: Philipp Meyer, TU Muenchen
-A tactic for proving nonlinear inequalities
+A tactic for proving nonlinear inequalities.
*)
signature SOS =
@@ -16,11 +16,11 @@
proof_method -> Proof.context -> int -> tactic
val debugging : bool Unsynchronized.ref;
-
+
exception Failure of string;
end
-structure Sos : SOS =
+structure Sos : SOS =
struct
val rat_0 = Rat.zero;
@@ -33,19 +33,19 @@
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 =
+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
+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 =
+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
@@ -74,11 +74,11 @@
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
+ else 0
in
fun decimalize d x =
if x =/ rat_0 then "0.0" else
- let
+ let
val y = Rat.abs x
val e = normalize y
val z = pow10(~ e) */ y +/ rat_1
@@ -113,7 +113,7 @@
([],[]) => 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,FuncUtil.Intfunc.empty):vector;
@@ -127,7 +127,7 @@
val vector_1 = vector_const rat_1;
fun vector_cmul c (v:vector) =
- let val n = dim v
+ let val n = dim v
in if c =/ rat_0 then vector_0 n
else (n,FuncUtil.Intfunc.map (fn x => c */ x) (snd v))
end;
@@ -135,24 +135,24 @@
fun vector_neg (v:vector) = (fst v,FuncUtil.Intfunc.map Rat.neg (snd v)) :vector;
fun vector_add (v1:vector) (v2:vector) =
- let val m = dim v1
- val n = dim v2
+ let val m = dim v1
+ val n = dim v2
in if m <> n then error "vector_add: incompatible dimensions"
- else (n,FuncUtil.Intfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd v1) (snd v2)) :vector
+ else (n,FuncUtil.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 FuncUtil.Intfunc.fold (fn (i,x) => fn a => x +/ a)
+ let val m = dim v1
+ val n = dim v2
+ in if m <> n then error "vector_dot: incompatible dimensions"
+ else FuncUtil.Intfunc.fold (fn (i,x) => fn a => x +/ a)
(FuncUtil.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
+ let val n = length l
in (n,fold_rev2 (curry FuncUtil.Intfunc.update) (1 upto n) l FuncUtil.Intfunc.empty) :vector
end;
@@ -170,18 +170,18 @@
val matrix_1 = matrix_const rat_1;
fun matrix_cmul c (m:matrix) =
- let val (i,j) = dimensions m
+ let val (i,j) = dimensions m
in if c =/ rat_0 then matrix_0 (i,j)
else ((i,j),FuncUtil.Intpairfunc.map (fn x => c */ x) (snd m))
end;
-fun matrix_neg (m:matrix) =
+fun matrix_neg (m:matrix) =
(dimensions m, FuncUtil.Intpairfunc.map 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
+ let val d1 = dimensions m1
+ val d2 = dimensions m2
+ in if d1 <> d2
then error "matrix_add: incompatible dimensions"
else (d1,FuncUtil.Intpairfunc.combine (curry op +/) (fn x => x =/ rat_0) (snd m1) (snd m2)) :matrix
end;;
@@ -189,33 +189,33 @@
fun matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);
fun row k (m:matrix) =
- let val (i,j) = dimensions m
+ let val (i,j) = dimensions m
in (j,
FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if i = k then FuncUtil.Intfunc.update (j,c) a else a) (snd m) FuncUtil.Intfunc.empty ) : vector
end;
fun column k (m:matrix) =
- let val (i,j) = dimensions m
+ let val (i,j) = dimensions m
in (i,
FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if j = k then FuncUtil.Intfunc.update (i,c) a else a) (snd m) FuncUtil.Intfunc.empty)
: vector
end;
fun transp (m:matrix) =
- let val (i,j) = dimensions m
+ let val (i,j) = dimensions m
in
((j,i),FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => FuncUtil.Intpairfunc.update ((j,i), c) a) (snd m) FuncUtil.Intpairfunc.empty) :matrix
end;
fun diagonal (v:vector) =
- let val n = dim v
+ let val n = dim v
in ((n,n),FuncUtil.Intfunc.fold (fn (i, c) => fn a => FuncUtil.Intpairfunc.update ((i,i), c) a) (snd v) FuncUtil.Intpairfunc.empty) : matrix
end;
fun matrix_of_list l =
- let val m = length l
+ let val m = length l
in if m = 0 then matrix_0 (0,0) else
- let val n = length (hd l)
+ let val n = length (hd l)
in ((m,n),itern 1 l (fn v => fn i => itern 1 v (fn c => fn j => FuncUtil.Intpairfunc.update ((i,j), c))) FuncUtil.Intpairfunc.empty)
end
end;
@@ -240,20 +240,20 @@
FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => FuncUtil.Ctermfunc.tryapplyd m2 x 0 >= k andalso a) m1 true;;
fun monomial_div m1 m2 =
- let val m = FuncUtil.Ctermfunc.combine (curry op +)
- (fn x => x = 0) m1 (FuncUtil.Ctermfunc.map (fn x => ~ x) m2)
+ let val m = FuncUtil.Ctermfunc.combine (curry op +)
+ (fn x => x = 0) m1 (FuncUtil.Ctermfunc.map (fn x => ~ x) m2)
in if FuncUtil.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 =
+fun monomial_degree x m =
FuncUtil.Ctermfunc.tryapplyd m x 0;;
fun monomial_lcm m1 m2 =
fold_rev (fn x => FuncUtil.Ctermfunc.update (x, max (monomial_degree x m1) (monomial_degree x m2)))
(gen_union (is_equal o FuncUtil.cterm_ord) (FuncUtil.Ctermfunc.dom m1, FuncUtil.Ctermfunc.dom m2)) (FuncUtil.Ctermfunc.empty);
-fun monomial_multidegree m =
+fun monomial_multidegree m =
FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => k + a) m 0;;
fun monomial_variables m = FuncUtil.Ctermfunc.dom m;;
@@ -265,7 +265,7 @@
val poly_0 = FuncUtil.Monomialfunc.empty;
-fun poly_isconst p =
+fun poly_isconst p =
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a) p true;
fun poly_var x = FuncUtil.Monomialfunc.onefunc (monomial_var x,rat_1);
@@ -286,7 +286,7 @@
fun poly_cmmul (c,m) p =
if c =/ rat_0 then poly_0
- else if FuncUtil.Ctermfunc.is_empty m
+ else if FuncUtil.Ctermfunc.is_empty m
then FuncUtil.Monomialfunc.map (fn d => c */ d) p
else FuncUtil.Monomialfunc.fold (fn (m', d) => fn a => (FuncUtil.Monomialfunc.update (monomial_mul m m', c */ d) a)) p poly_0;
@@ -294,9 +294,9 @@
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
fun poly_div p1 p2 =
- if not(poly_isconst p2)
+ if not(poly_isconst p2)
then error "poly_div: non-constant" else
- let val c = eval FuncUtil.Ctermfunc.empty p2
+ let val c = eval FuncUtil.Ctermfunc.empty p2
in if c =/ rat_0 then error "poly_div: division by zero"
else poly_cmul (Rat.inv c) p1
end;
@@ -310,11 +310,11 @@
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"
+ if not(poly_isconst p2)
+ then error "poly_exp: not a constant"
else poly_pow p1 (int_of_rat (eval FuncUtil.Ctermfunc.empty p2));
-fun degree x p =
+fun degree x p =
FuncUtil.Monomialfunc.fold (fn (m,c) => fn a => max (monomial_degree x m) a) p 0;
fun multidegree p =
@@ -329,14 +329,14 @@
local
fun ord (l1,l2) = case (l1,l2) of
- (_,[]) => LESS
+ (_,[]) => LESS
| ([],_) => GREATER
- | (h1::t1,h2::t2) =>
- (case humanorder_varpow (h1, h2) of
+ | (h1::t1,h2::t2) =>
+ (case humanorder_varpow (h1, h2) of
LESS => LESS
| EQUAL => ord (t1,t2)
| GREATER => GREATER)
-in fun humanorder_monomial m1 m2 =
+in fun humanorder_monomial m1 m2 =
ord (sort humanorder_varpow (FuncUtil.Ctermfunc.dest m1),
sort humanorder_varpow (FuncUtil.Ctermfunc.dest m2))
end;
@@ -344,30 +344,30 @@
(* Conversions to strings. *)
fun string_of_vector min_size max_size (v:vector) =
- let val n_raw = dim v
+ 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 => FuncUtil.Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
+ let
+ val n = max min_size (min n_raw max_size)
+ val xs = map (Rat.string_of_rat o (fn i => FuncUtil.Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
in "[" ^ space_implode ", " xs ^
(if n_raw > max_size then ", ...]" else "]")
end
end;
fun string_of_matrix max_size (m:matrix) =
- let
+ let
val (i_raw,j_raw) = dimensions m
- val i = min max_size i_raw
+ 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)
+ val rstr = map (fn k => string_of_vector j j (row k m)) (1 upto i)
in "["^ space_implode ";\n " rstr ^
(if j > max_size then "\n ...]" else "]")
end;
-fun string_of_term t =
+fun string_of_term t =
case t of
a$b => "("^(string_of_term a)^" "^(string_of_term b)^")"
- | Abs x =>
+ | Abs x =>
let val (xn, b) = Term.dest_abs x
in "(\\"^xn^"."^(string_of_term b)^")"
end
@@ -379,13 +379,13 @@
val string_of_cterm = string_of_term o term_of;
fun string_of_varpow x k =
- if k = 1 then string_of_cterm x
+ if k = 1 then string_of_cterm x
else string_of_cterm x^"^"^string_of_int k;
fun string_of_monomial m =
if FuncUtil.Ctermfunc.is_empty m then "1" else
let val vps = fold_rev (fn (x,k) => fn a => string_of_varpow x k :: a)
- (sort humanorder_varpow (FuncUtil.Ctermfunc.dest m)) []
+ (sort humanorder_varpow (FuncUtil.Ctermfunc.dest m)) []
in space_implode "*" vps
end;
@@ -396,14 +396,14 @@
fun string_of_poly p =
if FuncUtil.Monomialfunc.is_empty p then "<<0>>" else
- let
+ let
val cms = sort (fn ((m1,_),(m2,_)) => humanorder_monomial m1 m2) (FuncUtil.Monomialfunc.dest p)
val s = fold (fn (m,c) => fn a =>
if c </ rat_0 then a ^ " - " ^ string_of_cmonomial(Rat.neg c,m)
else a ^ " + " ^ string_of_cmonomial(c,m))
cms ""
val s1 = String.substring (s, 0, 3)
- val s2 = String.substring (s, 3, String.size s - 3)
+ val s2 = String.substring (s, 3, String.size s - 3)
in "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>"
end;
@@ -422,36 +422,36 @@
fun is_comb t = case t of _$_ => true | _ => false
fun poly_of_term tm =
if tm aconvc zero_tm then poly_0
- else if RealArith.is_ratconst tm
+ else if RealArith.is_ratconst tm
then poly_const(RealArith.dest_ratconst tm)
- else
+ 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
+ let val p = poly_of_term r
+ in if poly_isconst p
then poly_const(Rat.inv (eval FuncUtil.Ctermfunc.empty p))
else error "poly_of_term: inverse of non-constant polyomial"
end
else (let val (opr,l) = Thm.dest_comb lop
- in
- if opr aconvc pow_tm andalso is_numeral r
+ 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
+ else if opr aconvc add_tm
then poly_add (poly_of_term l) (poly_of_term r)
- else if opr aconvc sub_tm
+ else if opr aconvc sub_tm
then poly_sub (poly_of_term l) (poly_of_term r)
- else if opr aconvc mul_tm
+ 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
+ else if opr aconvc div_tm
+ then let
+ val p = poly_of_term l
+ val q = poly_of_term r
in if poly_isconst q then poly_cmul (Rat.inv (eval FuncUtil.Ctermfunc.empty q)) p
else error "poly_of_term: division by non-constant polynomial"
end
else poly_var tm
-
+
end
handle CTERM ("dest_comb",_) => poly_var tm)
end
@@ -465,21 +465,21 @@
(* String of vector (just a list of space-separated numbers). *)
fun sdpa_of_vector (v:vector) =
- let
+ let
val n = dim v
- val strs = map (decimalize 20 o (fn i => FuncUtil.Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
+ val strs = map (decimalize 20 o (fn i => FuncUtil.Intfunc.tryapplyd (snd v) i rat_0)) (1 upto n)
in space_implode " " strs ^ "\n"
end;
-fun triple_int_ord ((a,b,c),(a',b',c')) =
- prod_ord int_ord (prod_ord int_ord int_ord)
+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
+ 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 []
@@ -492,10 +492,10 @@
(* String for a matrix numbered k, in SDPA sparse format. *)
fun sdpa_of_matrix k (m:matrix) =
- let
+ let
val pfx = string_of_int k ^ " 1 "
- val ms = FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if i > j then a else ((i,j),c)::a) (snd m) []
- val mss = sort ((prod_ord int_ord int_ord) o pairself fst) ms
+ val ms = FuncUtil.Intpairfunc.fold (fn ((i,j), c) => fn a => if i > j then a else ((i,j),c)::a) (snd m) []
+ val mss = sort ((prod_ord int_ord int_ord) o pairself fst) 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 ""
@@ -509,9 +509,9 @@
(* ------------------------------------------------------------------------- *)
fun sdpa_of_problem obj mats =
- let
+ let
val m = length mats - 1
- val (n,_) = dimensions (hd mats)
+ val (n,_) = dimensions (hd mats)
in
string_of_int m ^ "\n" ^
"1\n" ^
@@ -525,10 +525,10 @@
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 =
+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
+ 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)
@@ -551,7 +551,7 @@
decimalint -- Scan.option (Scan.$$ "." |-- decimalfrac)
>> (fn (h,NONE) => h | (h,SOME x) => h +/ x)
fun signed prs =
- $$ "-" |-- prs >> Rat.neg
+ $$ "-" |-- prs >> Rat.neg
|| $$ "+" |-- prs
|| prs;
@@ -563,8 +563,8 @@
>> (fn (h, x) => h */ pow10 (int_of_rat x));
fun mkparser p s =
- let val (x,rst) = p (explode s)
- in if null rst then x
+ let val (x,rst) = p (explode s)
+ in if null rst then x
else error "mkparser: unparsed input"
end;;
@@ -590,23 +590,23 @@
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
+ 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
+ let
val cd1 = fold_rev (common_denominator FuncUtil.Intpairfunc.fold) mats (rat_1)
- val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) (rat_1)
+ val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) (rat_1)
val mats' = map (FuncUtil.Intpairfunc.map (fn x => cd1 */ x)) mats
val obj' = vector_cmul cd2 obj
val max1 = fold_rev (maximal_element FuncUtil.Intpairfunc.fold) mats' (rat_0)
- val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') (rat_0)
+ val max2 = maximal_element FuncUtil.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 scal2 = pow2 (20 - trunc(Math.ln (float_of_rat max2) / Math.ln 2.0))
val mats'' = map (FuncUtil.Intpairfunc.map (fn x => x */ scal1)) mats'
- val obj'' = vector_cmul scal2 obj'
+ val obj'' = vector_cmul scal2 obj'
in solver obj'' mats''
end
end;
@@ -622,24 +622,24 @@
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
+ 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
+ let
val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats (rat_1)
- val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) (rat_1)
+ val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) (rat_1)
val mats' = map (Inttriplefunc.map (fn x => cd1 */ x)) mats
val obj' = vector_cmul cd2 obj
val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' (rat_0)
- val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') (rat_0)
+ val max2 = maximal_element FuncUtil.Intfunc.fold (snd obj') (rat_0)
val scal1 = pow2 (20 - int_of_float(Math.ln (float_of_rat max1) / Math.ln 2.0))
- val scal2 = pow2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))
+ val scal2 = pow2 (20 - int_of_float(Math.ln (float_of_rat max2) / Math.ln 2.0))
val mats'' = map (Inttriplefunc.map (fn x => x */ scal1)) mats'
- val obj'' = vector_cmul scal2 obj'
+ val obj'' = vector_cmul scal2 obj'
in solver obj'' mats''
end
end;
@@ -647,10 +647,10 @@
(* Round a vector to "nice" rationals. *)
fun nice_rational n x = round_rat (n */ x) // n;;
-fun nice_vector n ((d,v) : vector) =
- (d, FuncUtil.Intfunc.fold (fn (i,c) => fn a =>
- let val y = nice_rational n c
- in if c =/ rat_0 then a
+fun nice_vector n ((d,v) : vector) =
+ (d, FuncUtil.Intfunc.fold (fn (i,c) => fn a =>
+ let val y = nice_rational n c
+ in if c =/ rat_0 then a
else FuncUtil.Intfunc.update (i,y) a end) v FuncUtil.Intfunc.empty):vector
fun dest_ord f x = is_equal (f x);
@@ -663,7 +663,7 @@
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
+ let fun value v = Inttriplefunc.apply assig v
in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
end;
@@ -676,16 +676,16 @@
[] => 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
+fun eliminate vars dun eqs = case vars of
[] => if forall Inttriplefunc.is_empty eqs then dun
else raise Unsolvable
| v::vs =>
- ((let
- val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
+ ((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.delete_safe v eq)
fun elim e =
- let val b = Inttriplefunc.tryapplyd e v rat_0
+ 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
@@ -694,7 +694,7 @@
handle Failure _ => eliminate vs dun eqs)
in
fun tri_eliminate_equations one vars eqs =
- let
+ let
val assig = eliminate vars Inttriplefunc.empty 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)
@@ -704,50 +704,50 @@
(* Eliminate all variables, in an essentially arbitrary order. *)
fun tri_eliminate_all_equations one =
- let
+ let
fun choose_variable eq =
- let val (v,_) = Inttriplefunc.choose eq
+ let val (v,_) = Inttriplefunc.choose eq
in if is_equal (triple_int_ord(v,one)) then
- let val eq' = Inttriplefunc.delete_safe v eq
- in if Inttriplefunc.is_empty eq' then error "choose_variable"
+ let val eq' = Inttriplefunc.delete_safe v eq
+ in if Inttriplefunc.is_empty eq' then error "choose_variable"
else fst (Inttriplefunc.choose eq')
end
- else v
+ else v
end
- fun eliminate dun eqs = case eqs of
+ fun eliminate dun eqs = case eqs of
[] => dun
| eq::oeqs =>
if Inttriplefunc.is_empty eq then eliminate dun oeqs else
let val v = choose_variable eq
val a = Inttriplefunc.apply eq v
- val eq' = tri_equation_cmul ((Rat.rat_of_int ~1) // a)
+ val eq' = tri_equation_cmul ((Rat.rat_of_int ~1) // a)
(Inttriplefunc.delete_safe v eq)
fun elim e =
- let val b = Inttriplefunc.tryapplyd e v rat_0
- in if b =/ rat_0 then e
+ let val b = Inttriplefunc.tryapplyd e v rat_0
+ in if b =/ rat_0 then e
else tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
end
- in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map elim dun))
- (map elim oeqs)
+ in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map elim dun))
+ (map elim oeqs)
end
in fn eqs =>
- let
+ let
val assig = eliminate Inttriplefunc.empty 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
+ let
val (vars,assigs) = tri_eliminate_all_equations one eqs
- val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
+ 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.map (tri_equation_eval vfn) assigs) vfn
+ Inttriplefunc.combine (curry op +/) (K false)
+ (Inttriplefunc.map (tri_equation_eval vfn) assigs) vfn
in if forall (fn e => tri_equation_eval ass e =/ rat_0) eqs
then Inttriplefunc.delete_safe one ass else raise Sanity
end;
@@ -758,8 +758,8 @@
FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>
FuncUtil.Monomialfunc.fold (fn (m2,e) => fn b =>
let val m = monomial_mul m1 m2
- val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
- in FuncUtil.Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
+ val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
+ in FuncUtil.Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
end) q a) p acc ;
(* Usual operations on equation-parametrized poly. *)
@@ -781,7 +781,7 @@
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
+ let fun value v = Inttriplefunc.apply assig v
in Inttriplefunc.fold (fn (v, c) => fn a => a +/ value v */ c) eq rat_0
end;
@@ -790,20 +790,20 @@
(* "one" that's used for a constant term. *)
local
-fun extract_first p l = case l of
+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
+fun eliminate vars dun eqs = case vars of
[] => if forall Inttriplefunc.is_empty eqs then dun
else raise Unsolvable
| v::vs =>
- let
- val (eq,oeqs) = extract_first (fn e => Inttriplefunc.defined e v) eqs
+ 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.delete_safe v eq)
fun elim e =
- let val b = Inttriplefunc.tryapplyd e v rat_0
+ 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
@@ -812,7 +812,7 @@
handle Failure _ => eliminate vs dun eqs
in
fun pi_eliminate_equations one vars eqs =
- let
+ let
val assig = eliminate vars Inttriplefunc.empty 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)
@@ -822,50 +822,50 @@
(* Eliminate all variables, in an essentially arbitrary order. *)
fun pi_eliminate_all_equations one =
- let
+ let
fun choose_variable eq =
- let val (v,_) = Inttriplefunc.choose eq
+ let val (v,_) = Inttriplefunc.choose eq
in if is_equal (triple_int_ord(v,one)) then
- let val eq' = Inttriplefunc.delete_safe v eq
- in if Inttriplefunc.is_empty eq' then error "choose_variable"
+ let val eq' = Inttriplefunc.delete_safe v eq
+ in if Inttriplefunc.is_empty eq' then error "choose_variable"
else fst (Inttriplefunc.choose eq')
end
- else v
+ else v
end
- fun eliminate dun eqs = case eqs of
+ fun eliminate dun eqs = case eqs of
[] => dun
| eq::oeqs =>
if Inttriplefunc.is_empty eq then eliminate dun oeqs else
let val v = choose_variable eq
val a = Inttriplefunc.apply eq v
- val eq' = pi_equation_cmul ((Rat.rat_of_int ~1) // a)
+ val eq' = pi_equation_cmul ((Rat.rat_of_int ~1) // a)
(Inttriplefunc.delete_safe v eq)
fun elim e =
- let val b = Inttriplefunc.tryapplyd e v rat_0
- in if b =/ rat_0 then e
+ 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.map elim dun))
- (map elim oeqs)
+ in eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map elim dun))
+ (map elim oeqs)
end
in fn eqs =>
- let
+ let
val assig = eliminate Inttriplefunc.empty 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
+ let
val (vars,assigs) = pi_eliminate_all_equations one eqs
- val vfn = fold_rev (fn v => Inttriplefunc.update(v,rat_0)) vars
+ 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.map (pi_equation_eval vfn) assigs) vfn
+ Inttriplefunc.combine (curry op +/) (K false)
+ (Inttriplefunc.map (pi_equation_eval vfn) assigs) vfn
in if forall (fn e => pi_equation_eval ass e =/ rat_0) eqs
then Inttriplefunc.delete_safe one ass else raise Sanity
end;
@@ -876,8 +876,8 @@
FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>
FuncUtil.Monomialfunc.fold (fn (m2,e) => fn b =>
let val m = monomial_mul m1 m2
- val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
- in FuncUtil.Monomialfunc.update (m,pi_equation_add (pi_equation_cmul c e) es) b
+ val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
+ in FuncUtil.Monomialfunc.update (m,pi_equation_add (pi_equation_cmul c e) es) b
end) q a) p acc ;
(* Usual operations on equation-parametrized poly. *)
@@ -906,19 +906,19 @@
local
fun diagonalize n i m =
- if FuncUtil.Intpairfunc.is_empty (snd m) then []
+ if FuncUtil.Intpairfunc.is_empty (snd m) then []
else
- let val a11 = FuncUtil.Intpairfunc.tryapplyd (snd m) (i,i) rat_0
+ let val a11 = FuncUtil.Intpairfunc.tryapplyd (snd m) (i,i) rat_0
in if a11 </ rat_0 then raise Failure "diagonalize: not PSD"
else if a11 =/ rat_0 then
if FuncUtil.Intfunc.is_empty (snd (row i m)) then diagonalize n (i + 1) m
else raise Failure "diagonalize: not PSD ___ "
else
- let
+ let
val v = row i m
- val v' = (fst v, FuncUtil.Intfunc.fold (fn (i, c) => fn a =>
- let val y = c // a11
- in if y = rat_0 then a else FuncUtil.Intfunc.update (i,y) a
+ val v' = (fst v, FuncUtil.Intfunc.fold (fn (i, c) => fn a =>
+ let val y = c // a11
+ in if y = rat_0 then a else FuncUtil.Intfunc.update (i,y) a
end) (snd v) FuncUtil.Intfunc.empty)
fun upt0 x y a = if y = rat_0 then a else FuncUtil.Intpairfunc.update (x,y) a
val m' =
@@ -927,15 +927,15 @@
iter (i+1,n) (fn k =>
(upt0 (j,k) (FuncUtil.Intpairfunc.tryapplyd (snd m) (j,k) rat_0 -/ FuncUtil.Intfunc.tryapplyd (snd v) j rat_0 */ FuncUtil.Intfunc.tryapplyd (snd v') k rat_0))))
FuncUtil.Intpairfunc.empty)
- in (a11,v')::diagonalize n (i + 1) m'
+ 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"
+ 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;
@@ -946,13 +946,13 @@
(* FIXME : Potentially polymorphic keys, but here only: integers!! *)
local
fun upd0 x y a = if y =/ rat_0 then a else FuncUtil.Intfunc.update(x,y) a;
- fun mapa f (d,v) =
+ fun mapa f (d,v) =
(d, FuncUtil.Intfunc.fold (fn (i,c) => fn a => upd0 i (f c) a) v FuncUtil.Intfunc.empty)
fun adj (c,l) =
- let val a =
- FuncUtil.Intfunc.fold (fn (i,c) => fn a => lcm_rat a (denominator_rat c))
+ let val a =
+ FuncUtil.Intfunc.fold (fn (i,c) => fn a => lcm_rat a (denominator_rat c))
(snd l) rat_1 //
- FuncUtil.Intfunc.fold (fn (i,c) => fn a => gcd_rat a (numerator_rat c))
+ FuncUtil.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
@@ -960,20 +960,20 @@
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
+ 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 =
+fun enumerate_monomials d vars =
if d < 0 then []
else if d = 0 then [FuncUtil.Ctermfunc.empty]
else if null vars then [monomial_1] else
let val alts =
- map (fn k => let val oths = enumerate_monomials (d - k) (tl vars)
- in map (fn ks => if k = 0 then ks else FuncUtil.Ctermfunc.update (hd vars, k) ks) oths end) (0 upto d)
+ map (fn k => let val oths = enumerate_monomials (d - k) (tl vars)
+ in map (fn ks => if k = 0 then ks else FuncUtil.Ctermfunc.update (hd vars, k) ks) oths end) (0 upto d)
in flat alts
end;
@@ -982,12 +982,12 @@
(* Give the output polynomial and a record of how it was derived. *)
fun enumerate_products d pols =
-if d = 0 then [(poly_const rat_1,RealArith.Rational_lt rat_1)]
+if d = 0 then [(poly_const rat_1,RealArith.Rational_lt rat_1)]
else if d < 0 then [] else
-case pols of
+case pols of
[] => [(poly_const rat_1,RealArith.Rational_lt rat_1)]
- | (p,b)::ps =>
- let val e = multidegree p
+ | (p,b)::ps =>
+ let val e = multidegree p
in if e = 0 then enumerate_products d ps else
enumerate_products d ps @
map (fn (q,c) => (poly_mul p q,RealArith.Product(b,c)))
@@ -1002,13 +1002,13 @@
(* String for block diagonal matrix numbered k. *)
fun sdpa_of_blockdiagonal k m =
- let
+ let
val pfx = string_of_int k ^" "
val ents =
- Inttriplefunc.fold
- (fn ((b,i,j),c) => fn a => if i > j then a else ((b,i,j),c)::a)
- m []
- val entss = sort (triple_int_ord o pairself fst) ents
+ Inttriplefunc.fold
+ (fn ((b,i,j),c) => fn a => if i > j then a else ((b,i,j),c)::a)
+ m []
+ val entss = sort (triple_int_ord o pairself fst) ents
in fold_rev (fn ((b,i,j),c) => fn a =>
pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
" " ^ decimalize 20 c ^ "\n" ^ a) entss ""
@@ -1017,7 +1017,7 @@
(* SDPA for problem using block diagonal (i.e. multiple SDPs) *)
fun sdpa_of_blockproblem nblocks blocksizes obj mats =
- let val m = length mats - 1
+ let val m = length mats - 1
in
string_of_int m ^ "\n" ^
string_of_int nblocks ^ "\n" ^
@@ -1049,7 +1049,7 @@
map (fn (bs,b0) =>
let val m = Inttriplefunc.fold
(fn ((b,i,j),c) => fn a => if b = b0 then FuncUtil.Intpairfunc.update ((i,j),c) a else a) bm FuncUtil.Intpairfunc.empty
- val d = FuncUtil.Intpairfunc.fold (fn ((i,j),c) => fn a => max a (max i j)) m 0
+ val d = FuncUtil.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));;
@@ -1057,43 +1057,43 @@
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
+in
fun tryfind f = tryfind_with "tryfind" f
end
(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
-
+
fun real_positivnullstellensatz_general prover linf d eqs leqs pol =
-let
- val vars = fold_rev (curry (gen_union (op aconvc)) o poly_variables)
+let
+ val vars = fold_rev (curry (gen_union (op aconvc)) o poly_variables)
(pol::eqs @ map fst leqs) []
- val monoid = if linf then
+ val monoid = if linf then
(poly_const rat_1,RealArith.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
+ let
val e = d - multidegree p
val mons = enumerate_monomials e vars
- val nons = mons ~~ (1 upto length mons)
+ val nons = mons ~~ (1 upto length mons)
in (mons,
fold_rev (fn (m,n) => FuncUtil.Monomialfunc.update(m,Inttriplefunc.onefunc((~k,~n,n),rat_1))) nons FuncUtil.Monomialfunc.empty)
end
fun mk_sqmultiplier k (p,c) =
- let
+ let
val e = (d - multidegree p) div 2
val mons = enumerate_monomials e vars
- val nons = mons ~~ (1 upto length mons)
- in (mons,
+ 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
+ let val m = monomial_mul m1 m2
in if n1 > n2 then a else
let val c = if n1 = n2 then rat_1 else rat_2
- val e = FuncUtil.Monomialfunc.tryapplyd a m Inttriplefunc.empty
+ val e = FuncUtil.Monomialfunc.tryapplyd a m Inttriplefunc.empty
in FuncUtil.Monomialfunc.update(m, tri_equation_add (Inttriplefunc.onefunc((k,n1,n2), c)) e) a
end
end) nons)
@@ -1112,7 +1112,7 @@
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 =>
+ 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
@@ -1133,15 +1133,15 @@
fun cterm_element (d,v) i = FuncUtil.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")
+ 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
+ val allmats = blocks blocksizes blockmat
in (vec,map diag allmats)
end
val (vec,ratdias) =
@@ -1178,7 +1178,7 @@
(* Iterative deepening. *)
-fun deepen f n =
+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))))
@@ -1197,14 +1197,14 @@
in
(* FIXME: Replace tryfind by get_first !! *)
fun real_nonlinear_prover proof_method ctxt =
- let
+ let
val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
- (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (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 cert_choice translator (eqs,les,lts) =
- let
+ fun mainf cert_choice translator (eqs,les,lts) =
+ let
val eq0 = map (poly_of_term o Thm.dest_arg1 o concl) eqs
val le0 = map (poly_of_term o Thm.dest_arg o concl) les
val lt0 = map (poly_of_term o Thm.dest_arg o concl) lts
@@ -1216,19 +1216,19 @@
val (kltp,ltp) = List.partition (fn (p,_) => multidegree p = 0) ltp0
fun trivial_axiom (p,ax) =
case ax of
- RealArith.Axiom_eq n => if eval FuncUtil.Ctermfunc.empty p <>/ Rat.zero then nth eqs n
+ RealArith.Axiom_eq n => if eval FuncUtil.Ctermfunc.empty p <>/ Rat.zero then nth eqs n
else raise Failure "trivial_axiom: Not a trivial axiom"
- | RealArith.Axiom_le n => if eval FuncUtil.Ctermfunc.empty p </ Rat.zero then nth les n
+ | RealArith.Axiom_le n => if eval FuncUtil.Ctermfunc.empty p </ Rat.zero then nth les n
else raise Failure "trivial_axiom: Not a trivial axiom"
- | RealArith.Axiom_lt n => if eval FuncUtil.Ctermfunc.empty p <=/ Rat.zero then nth lts n
+ | RealArith.Axiom_lt n => if eval FuncUtil.Ctermfunc.empty p <=/ Rat.zero then nth lts n
else raise Failure "trivial_axiom: Not a trivial axiom"
| _ => error "trivial_axiom: Not a trivial axiom"
- in
+ 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, RealArith.Trivial)
end)
- handle Failure _ =>
+ handle Failure _ =>
(let val proof =
(case proof_method of Certificate certs =>
(* choose certificate *)
@@ -1242,13 +1242,13 @@
end
| Prover prover =>
(* call prover *)
- let
+ 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
+ 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)
@@ -1258,11 +1258,11 @@
map2 (fn q => fn (p,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq
val proofs_cone = map cterm_of_sos cert_cone
val proof_ne = if null ltp then RealArith.Rational_lt Rat.one else
- let val p = foldr1 RealArith.Product (map snd ltp)
+ let val p = foldr1 RealArith.Product (map snd ltp)
in funpow i (fn q => RealArith.Product(p,q)) (RealArith.Rational_lt Rat.one)
end
- in
- foldr1 RealArith.Sum (proof_ne :: proofs_ideal @ proofs_cone)
+ in
+ foldr1 RealArith.Sum (proof_ne :: proofs_ideal @ proofs_cone)
end)
in
(translator (eqs,les,lts) proof, RealArith.Cert proof)
@@ -1273,8 +1273,8 @@
fun C f x y = f y x;
(* FIXME : This is very bad!!!*)
-fun subst_conv eqs t =
- let
+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
@@ -1285,62 +1285,62 @@
open Conv
fun simple_cterm_ord t u = TermOrd.fast_term_ord (term_of t, term_of u) = LESS
val concl = Thm.dest_arg o cprop_of
- val shuffle1 =
+ 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)
+ 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 _ ) =>
+ | @{term "op * :: real => _"}$c$(t as Free _ ) =>
if RealArith.is_ratconst (Thm.dest_arg1 tm) andalso not (member (op aconvc) fvs (Thm.dest_arg tm))
then (RealArith.dest_ratconst (Thm.dest_arg1 tm),Thm.dest_arg tm) else raise Failure "substitutable_monomial"
- | @{term "op + :: real => _"}$s$t =>
+ | @{term "op + :: real => _"}$s$t =>
(substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg tm) fvs) (Thm.dest_arg1 tm)
handle Failure _ => substitutable_monomial (Thm.add_cterm_frees (Thm.dest_arg1 tm) fvs) (Thm.dest_arg tm))
| _ => raise Failure "substitutable_monomial"
- fun isolate_variable v th =
+ fun isolate_variable v th =
let val w = Thm.dest_arg1 (cprop_of th)
in if v aconvc w then th
else case term_of w of
- @{term "op + :: real => _"}$s$t =>
- if Thm.dest_arg1 w aconvc v then shuffle2 th
+ @{term "op + :: real => _"}$s$t =>
+ if Thm.dest_arg1 w aconvc v then shuffle2 th
else isolate_variable v (shuffle1 th)
| _ => error "isolate variable : This should not happen?"
- end
+ end
in
fun real_nonlinear_subst_prover prover ctxt =
- let
+ let
val {add,mul,neg,pow,sub,main} = Normalizer.semiring_normalizers_ord_wrapper ctxt
- (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
+ (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
+ let
val (c,v) = substitutable_monomial [] (Thm.dest_arg1(concl th))
val th1 = Drule.arg_cong_rule (Thm.capply @{cterm "op * :: real => _"} (RealArith.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 =
+ fun oprconv cv ct =
let val g = Thm.dest_fun2 ct
- in if g aconvc @{cterm "op <= :: real => _"}
- orelse g aconvc @{cterm "op < :: real => _"}
+ in if g aconvc @{cterm "op <= :: real => _"}
+ orelse g aconvc @{cterm "op < :: real => _"}
then arg_conv cv ct else arg1_conv cv ct
end
fun mainf cert_choice translator =
- let
+ let
fun substfirst(eqs,les,lts) =
- ((let
+ ((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
+ (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)
@@ -1358,51 +1358,46 @@
RealArith.gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt)
end;
-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 => _"},
+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 "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.Bit0"}, @{term "Int.Bit1"},
@{term "Int.Pls"}, @{term "Int.Min"}];
-fun check_sos kcts ct =
+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 []))
+ 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
+ val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) fs
then error "SOS: not sos. Variables with type not real" else ()
val vs = Term.add_vars t []
- val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) vs
+ val _ = if exists (fn ((_,T)) => not (T = @{typ "real"})) vs
then error "SOS: not sos. Variables with type not real" else ()
val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])
- val _ = if null ukcs then ()
+ val _ = if null ukcs then ()
else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))
in () end
-fun core_sos_tac print_cert prover ctxt i =
+fun core_sos_tac print_cert prover = SUBPROOF (fn {concl, context, ...} =>
let
- fun core_tac {concl, context, ...} =
- let
- val _ = check_sos known_sos_constants concl
- val (ths, certificates) = real_sos prover context (Thm.dest_arg concl)
- val _ = print_cert certificates
- in rtac ths i end
- in
- SUBPROOF core_tac ctxt i
- end
+ val _ = check_sos known_sos_constants concl
+ val (ths, certificates) = real_sos prover context (Thm.dest_arg concl)
+ val _ = print_cert certificates
+ in rtac ths 1 end)
fun default_SOME f NONE v = SOME v
| default_SOME f (SOME v) _ = SOME v;
@@ -1415,7 +1410,7 @@
val is_numeral = can (HOLogic.dest_number o term_of)
in
fun get_denom b ct = case term_of ct of
- @{term "op / :: real => _"} $ _ $ _ =>
+ @{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)
@@ -1424,15 +1419,15 @@
| _ => NONE
end;
-fun elim_one_denom_tac ctxt =
-CSUBGOAL (fn (P,i) =>
- case get_denom false P of
+fun elim_one_denom_tac ctxt =
+CSUBGOAL (fn (P,i) =>
+ case get_denom false P of
NONE => no_tac
- | SOME (d,ord) =>
- let
- val ss = simpset_of ctxt addsimps @{thms field_simps}
+ | 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)]
+ 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);