(* Title: HOL/Library/Sum_of_Squares/sum_of_squares.ML
Author: Amine Chaieb, University of Cambridge
Author: Philipp Meyer, TU Muenchen
A tactic for proving nonlinear inequalities.
*)
signature SUM_OF_SQUARES =
sig
datatype proof_method = Certificate of RealArith.pss_tree | Prover of string -> string
val sos_tac: (RealArith.pss_tree -> unit) -> proof_method -> Proof.context -> int -> tactic
val trace: bool Config.T
exception Failure of string;
end
structure Sum_of_Squares: SUM_OF_SQUARES =
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 max = Integer.max;
val denominator_rat = Rat.quotient_of_rat #> snd #> 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 trace = Attrib.setup_config_bool @{binding sos_trace} (K false);
exception Sanity;
exception Unsolvable;
exception Failure of string;
datatype proof_method =
Certificate of RealArith.pss_tree
| Prover of (string -> string)
(* Turn a rational into a decimal string with d sig digits. *)
local
fun normalize y =
if abs_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 (raw_explode(string_of_int k))) ^
(if e = 0 then "" else "e" ^ string_of_int e)
end
end;
(* Iterations over numbers, and lists indexed by numbers. *)
fun itern k l f a =
(case l of
[] => a
| h::t => itern (k + 1) t f (f h k a));
fun iter (m,n) f a =
if n < m then a
else iter (m + 1, n) f (f m a);
(* The main types. *)
type vector = int * Rat.rat FuncUtil.Intfunc.table;
type matrix = (int * int) * Rat.rat FuncUtil.Intpairfunc.table;
fun iszero (_, r) = r =/ rat_0;
(* Vectors. Conventionally indexed 1..n. *)
fun vector_0 n = (n, FuncUtil.Intfunc.empty): vector;
fun dim (v: vector) = fst v;
fun vector_cmul c (v: vector) =
let val n = dim v in
if c =/ rat_0 then vector_0 n
else (n,FuncUtil.Intfunc.map (fn _ => fn x => c */ x) (snd v))
end;
fun vector_of_list l =
let val n = length l in
(n, fold_rev2 (curry FuncUtil.Intfunc.update) (1 upto n) l FuncUtil.Intfunc.empty): vector
end;
(* Matrices; again rows and columns indexed from 1. *)
fun dimensions (m: matrix) = fst m;
fun row k (m: matrix) : vector =
let val (_, j) = dimensions m in
(j,
FuncUtil.Intpairfunc.fold (fn ((i, j), c) => fn a =>
if i = k then FuncUtil.Intfunc.update (j, c) a else a) (snd m) FuncUtil.Intfunc.empty)
end;
(* Monomials. *)
fun monomial_eval assig m =
FuncUtil.Ctermfunc.fold (fn (x, k) => fn a => a */ rat_pow (FuncUtil.Ctermfunc.apply assig x) k)
m rat_1;
val monomial_1 = FuncUtil.Ctermfunc.empty;
fun monomial_var x = FuncUtil.Ctermfunc.onefunc (x, 1);
val monomial_mul =
FuncUtil.Ctermfunc.combine Integer.add (K false);
fun monomial_multidegree m =
FuncUtil.Ctermfunc.fold (fn (_, k) => fn a => k + a) m 0;
fun monomial_variables m = FuncUtil.Ctermfunc.dom m;
(* Polynomials. *)
fun eval assig p =
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => a +/ c */ monomial_eval assig m) p rat_0;
val poly_0 = FuncUtil.Monomialfunc.empty;
fun poly_isconst p =
FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => FuncUtil.Ctermfunc.is_empty m andalso a)
p true;
fun poly_var x = FuncUtil.Monomialfunc.onefunc (monomial_var x, rat_1);
fun poly_const c =
if c =/ rat_0 then poly_0 else FuncUtil.Monomialfunc.onefunc (monomial_1, c);
fun poly_cmul c p =
if c =/ rat_0 then poly_0
else FuncUtil.Monomialfunc.map (fn _ => fn x => c */ x) p;
fun poly_neg p = FuncUtil.Monomialfunc.map (K Rat.neg) p;
fun poly_add p1 p2 =
FuncUtil.Monomialfunc.combine (curry op +/) (fn x => x =/ rat_0) p1 p2;
fun poly_sub p1 p2 = poly_add p1 (poly_neg p2);
fun poly_cmmul (c,m) p =
if c =/ rat_0 then poly_0
else
if FuncUtil.Ctermfunc.is_empty m
then FuncUtil.Monomialfunc.map (fn _ => fn d => c */ d) p
else
FuncUtil.Monomialfunc.fold (fn (m', d) => fn a =>
(FuncUtil.Monomialfunc.update (monomial_mul m m', c */ d) a)) p poly_0;
fun poly_mul p1 p2 =
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a => poly_add (poly_cmmul (c,m) p2) a) p1 poly_0;
fun poly_square p = poly_mul p p;
fun poly_pow p k =
if k = 0 then poly_const 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 multidegree p =
FuncUtil.Monomialfunc.fold (fn (m, _) => fn a => max (monomial_multidegree m) a) p 0;
fun poly_variables p =
sort FuncUtil.cterm_ord
(FuncUtil.Monomialfunc.fold_rev
(fn (m, _) => union (is_equal o FuncUtil.cterm_ord) (monomial_variables m)) p []);
(* 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 poly_of_term tm =
if tm aconvc zero_tm then poly_0
else
if RealArith.is_ratconst tm
then poly_const(RealArith.dest_ratconst tm)
else
(let
val (lop, r) = Thm.dest_comb tm
in
if lop aconvc neg_tm then poly_neg(poly_of_term r)
else if lop aconvc inv_tm then
let val p = poly_of_term r in
if poly_isconst p
then poly_const(Rat.inv (eval FuncUtil.Ctermfunc.empty p))
else error "poly_of_term: inverse of non-constant polyomial"
end
else
(let
val (opr,l) = Thm.dest_comb lop
in
if opr aconvc pow_tm andalso is_numeral r
then poly_pow (poly_of_term l) ((snd o HOLogic.dest_number o term_of) r)
else if opr aconvc add_tm
then poly_add (poly_of_term l) (poly_of_term r)
else if opr aconvc sub_tm
then poly_sub (poly_of_term l) (poly_of_term r)
else if opr aconvc mul_tm
then poly_mul (poly_of_term l) (poly_of_term r)
else if opr aconvc div_tm
then
let
val p = poly_of_term l
val q = poly_of_term r
in
if poly_isconst q
then poly_cmul (Rat.inv (eval FuncUtil.Ctermfunc.empty q)) p
else error "poly_of_term: division by non-constant polynomial"
end
else poly_var tm
end handle CTERM ("dest_comb",_) => poly_var tm)
end handle CTERM ("dest_comb",_) => poly_var tm)
in
val poly_of_term = fn tm =>
if type_of (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 => 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) ((a, (b, c)), (a', (b', c')));
structure Inttriplefunc = FuncFun(type key = int * int * int val ord = triple_int_ord);
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 |> Int.fromString |> the |> Rat.rat_of_int
else
let
val SOME numer = Int.fromString(String.substring(s,0,n))
val SOME den = Int.fromString (String.substring(s,n+1,String.size s - n - 1))
in rat_of_quotient(numer, den) end
end;
fun isnum x = member (op =) ["0", "1", "2", "3", "4", "5", "6", "7", "8", "9"] x;
(* More parser basics. *)
(* FIXME improper use of parser combinators ahead *)
val numeral = Scan.one isnum
val decimalint = Scan.repeat1 numeral >> (rat_of_string o implode)
val decimalfrac = Scan.repeat1 numeral
>> (fn s => rat_of_string(implode s) // pow10 (length s))
val decimalsig =
decimalint -- Scan.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" || $$ "E") |-- signed decimalint;
val decimal = signed decimalsig -- (emptyin rat_0|| exponent)
>> (fn (h, x) => h */ pow10 (int_of_rat x));
fun mkparser p s =
let val (x,rst) = p (raw_explode s)
in if null rst then x else error "mkparser: unparsed input" end;
(* Parse back csdp output. *)
(* FIXME improper use of parser combinators ahead *)
fun ignore _ = ((),[])
fun csdpoutput inp =
((decimal -- Scan.repeat (Scan.$$ " " |-- Scan.option decimal) >>
(fn (h,to) => map_filter I ((SOME h)::to))) --| ignore >> vector_of_list) inp
val parse_csdpoutput = mkparser csdpoutput
(* Try some apparently sensible scaling first. Note that this is purely to *)
(* get a cleaner translation to floating-point, and doesn't affect any of *)
(* the results, in principle. In practice it seems a lot better when there *)
(* are extreme numbers in the original problem. *)
(* Version for (int*int*int) keys *)
local
fun max_rat x y = if x </ y then y else x
fun common_denominator fld amat acc =
fld (fn (_,c) => fn a => lcm_rat (denominator_rat c) a) amat acc
fun maximal_element fld amat acc =
fld (fn (_,c) => fn maxa => max_rat maxa (abs_rat c)) amat acc
fun float_of_rat x =
let val (a,b) = Rat.quotient_of_rat x
in Real.fromInt a / Real.fromInt b end;
fun int_of_float x = (trunc x handle Overflow => 0 | Domain => 0)
in
fun tri_scale_then solver (obj:vector) mats =
let
val cd1 = fold_rev (common_denominator Inttriplefunc.fold) mats (rat_1)
val cd2 = common_denominator FuncUtil.Intfunc.fold (snd obj) (rat_1)
val mats' = map (Inttriplefunc.map (fn _ => fn x => cd1 */ x)) mats
val obj' = vector_cmul cd2 obj
val max1 = fold_rev (maximal_element Inttriplefunc.fold) mats' (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 mats'' = map (Inttriplefunc.map (fn _ => fn x => x */ scal1)) mats'
val obj'' = vector_cmul scal2 obj'
in solver obj'' mats'' end
end;
(* Round a vector to "nice" rationals. *)
fun nice_rational n x = round_rat (n */ x) // n;
fun nice_vector n ((d,v) : vector) =
(d, FuncUtil.Intfunc.fold (fn (i,c) => fn a =>
let val y = nice_rational n c in
if c =/ 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);
(* Stuff for "equations" ((int*int*int)->num functions). *)
fun tri_equation_cmul c eq =
if c =/ rat_0 then Inttriplefunc.empty
else Inttriplefunc.map (fn _ => fn d => c */ d) eq;
fun tri_equation_add eq1 eq2 =
Inttriplefunc.combine (curry op +/) (fn x => x =/ 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 all variables, in an essentially arbitrary order. *)
fun tri_eliminate_all_equations one =
let
fun choose_variable eq =
let val (v,_) = Inttriplefunc.choose eq
in
if is_equal (triple_int_ord(v,one)) then
let
val eq' = Inttriplefunc.delete_safe v eq
in
if Inttriplefunc.is_empty eq' then error "choose_variable"
else fst (Inttriplefunc.choose eq')
end
else v
end
fun eliminate dun eqs =
(case eqs of
[] => dun
| eq :: oeqs =>
if Inttriplefunc.is_empty eq then eliminate dun oeqs
else
let
val v = choose_variable eq
val a = Inttriplefunc.apply eq v
val eq' =
tri_equation_cmul ((Rat.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
else tri_equation_add e (tri_equation_cmul (Rat.neg b // a) eq)
end
in
eliminate (Inttriplefunc.update(v, eq') (Inttriplefunc.map (K elim) dun))
(map elim oeqs)
end)
in
fn eqs =>
let
val assig = eliminate Inttriplefunc.empty eqs
val vs = Inttriplefunc.fold (fn (_, f) => fn a =>
remove (dest_ord triple_int_ord) one (Inttriplefunc.dom f) @ a) assig []
in (distinct (dest_ord triple_int_ord) vs,assig) end
end;
(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
fun tri_epoly_pmul p q acc =
FuncUtil.Monomialfunc.fold (fn (m1, c) => fn a =>
FuncUtil.Monomialfunc.fold (fn (m2, e) => fn b =>
let
val m = monomial_mul m1 m2
val es = FuncUtil.Monomialfunc.tryapplyd b m Inttriplefunc.empty
in
FuncUtil.Monomialfunc.update (m,tri_equation_add (tri_equation_cmul c e) es) b
end) q a) p acc;
(* Hence produce the "relevant" monomials: those whose squares lie in the *)
(* Newton polytope of the monomials in the input. (This is enough according *)
(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal, *)
(* vol 45, pp. 363--374, 1978. *)
(* *)
(* These are ordered in sort of decreasing degree. In particular the *)
(* constant monomial is last; this gives an order in diagonalization of the *)
(* quadratic form that will tend to display constants. *)
(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form. *)
local
fun diagonalize n i m =
if FuncUtil.Intpairfunc.is_empty (snd m) then []
else
let
val a11 = FuncUtil.Intpairfunc.tryapplyd (snd m) (i,i) 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
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
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' =
((n, n),
iter (i + 1, n) (fn j =>
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' end
end
in
fun diag m =
let
val nn = dimensions m
val n = fst nn
in
if snd nn <> n then error "diagonalize: non-square matrix"
else diagonalize n 1 m
end
end;
(* Enumeration of monomials with given multidegree bound. *)
fun enumerate_monomials d vars =
if d < 0 then []
else if d = 0 then [FuncUtil.Ctermfunc.empty]
else if null vars then [monomial_1]
else
let val alts =
map_range (fn k =>
let
val oths = enumerate_monomials (d - k) (tl vars)
in map (fn ks => if k = 0 then ks else FuncUtil.Ctermfunc.update (hd vars, k) ks) oths end)
(d + 1)
in flat alts end;
(* Enumerate products of distinct input polys with degree <= d. *)
(* We ignore any constant input polynomials. *)
(* Give the output polynomial and a record of how it was derived. *)
fun enumerate_products d pols =
if d = 0 then [(poly_const rat_1,RealArith.Rational_lt rat_1)]
else if d < 0 then []
else
(case pols of
[] => [(poly_const rat_1, RealArith.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, RealArith.Product (b, c)))
(enumerate_products (d - e) ps)
end)
(* Convert regular polynomial. Note that we treat (0,0,0) as -1. *)
fun epoly_of_poly p =
FuncUtil.Monomialfunc.fold (fn (m, c) => fn a =>
FuncUtil.Monomialfunc.update (m, Inttriplefunc.onefunc ((0, 0, 0), Rat.neg c)) a)
p FuncUtil.Monomialfunc.empty;
(* String for block diagonal matrix numbered k. *)
fun sdpa_of_blockdiagonal k m =
let
val pfx = string_of_int k ^" "
val ents =
Inttriplefunc.fold
(fn ((b, i, j), c) => fn a => if i > j then a else ((b, i, j), c) :: a)
m []
val entss = sort (triple_int_ord o 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 ""
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" ^
(space_implode " " (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.empty
else Inttriplefunc.map (fn _ => fn x => c */ x) bm;
val bmatrix_neg = bmatrix_cmul (Rat.rat_of_int ~1);
(* Smash a block matrix into components. *)
fun blocks blocksizes bm =
map (fn (bs, b0) =>
let
val m =
Inttriplefunc.fold
(fn ((b, i, j), c) => fn a =>
if b = b0 then FuncUtil.Intpairfunc.update ((i, j), c) a else a)
bm FuncUtil.Intpairfunc.empty
val _ = FuncUtil.Intpairfunc.fold (fn ((i, j), _) => fn a => max a (max i j)) m 0
in (((bs, bs), m): matrix) end)
(blocksizes ~~ (1 upto length blocksizes));
(* FIXME : Get rid of this !!!*)
local
fun tryfind_with msg _ [] = raise Failure msg
| tryfind_with _ f (x::xs) = (f x handle Failure s => tryfind_with s f xs);
in
fun tryfind f = tryfind_with "tryfind" f
end
(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
fun real_positivnullstellensatz_general ctxt prover linf d eqs leqs pol =
let
val vars =
fold_rev (union (op aconvc) o poly_variables)
(pol :: eqs @ map fst leqs) []
val monoid =
if linf then
(poly_const rat_1,RealArith.Rational_lt rat_1)::
(filter (fn (p,_) => multidegree p <= d) leqs)
else enumerate_products d leqs
val nblocks = length monoid
fun mk_idmultiplier k p =
let
val e = d - multidegree p
val mons = enumerate_monomials e vars
val nons = mons ~~ (1 upto length mons)
in
(mons,
fold_rev (fn (m, n) =>
FuncUtil.Monomialfunc.update (m, Inttriplefunc.onefunc ((~k, ~n, n), rat_1)))
nons FuncUtil.Monomialfunc.empty)
end
fun mk_sqmultiplier k (p,_) =
let
val e = (d - multidegree p) div 2
val mons = enumerate_monomials e vars
val nons = mons ~~ (1 upto length mons)
in
(mons,
fold_rev (fn (m1, n1) =>
fold_rev (fn (m2, n2) => fn a =>
let val m = monomial_mul m1 m2 in
if n1 > n2 then a
else
let
val c = if n1 = n2 then rat_1 else rat_2
val e = FuncUtil.Monomialfunc.tryapplyd a m Inttriplefunc.empty
in
FuncUtil.Monomialfunc.update
(m, tri_equation_add (Inttriplefunc.onefunc ((k, n1, n2), c)) e) a
end
end) nons) nons FuncUtil.Monomialfunc.empty)
end
val (sqmonlist,sqs) = split_list (map2 mk_sqmultiplier (1 upto length monoid) monoid)
val (_(*idmonlist*),ids) = split_list (map2 mk_idmultiplier (1 upto length eqs) eqs)
val blocksizes = map length sqmonlist
val bigsum =
fold_rev2 (fn p => fn q => fn a => tri_epoly_pmul p q a) eqs ids
(fold_rev2 (fn (p,_) => fn s => fn a => tri_epoly_pmul p s a) monoid sqs
(epoly_of_poly(poly_neg pol)))
val eqns = FuncUtil.Monomialfunc.fold (fn (_, e) => fn a => e :: a) bigsum []
val (pvs, assig) = tri_eliminate_all_equations (0, 0, 0) eqns
val qvars = (0, 0, 0) :: pvs
val allassig =
fold_rev (fn v => Inttriplefunc.update (v, (Inttriplefunc.onefunc (v, 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.empty
val diagents =
Inttriplefunc.fold
(fn ((b, i, j), e) => fn a => if b > 0 andalso i = j then tri_equation_add e a else a)
allassig Inttriplefunc.empty
val mats = map mk_matrix qvars
val obj =
(length pvs,
itern 1 pvs (fn v => fn i =>
FuncUtil.Intfunc.updatep iszero (i,Inttriplefunc.tryapplyd diagents v rat_0))
FuncUtil.Intfunc.empty)
val raw_vec =
if null pvs then vector_0 0
else tri_scale_then (run_blockproblem prover nblocks blocksizes) obj mats
fun int_element (_, v) i = FuncUtil.Intfunc.tryapplyd v i rat_0
fun find_rounding d =
let
val _ =
if Config.get ctxt trace
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 =
FuncUtil.Monomialfunc.fold (fn (v, e) => fn a =>
FuncUtil.Monomialfunc.updatep iszero (v, tri_equation_eval finalassigs e) a)
p FuncUtil.Monomialfunc.empty
fun mk_sos mons =
let
fun mk_sq (c, m) =
(c, fold_rev (fn k => fn a =>
FuncUtil.Monomialfunc.updatep iszero (nth mons (k - 1), int_element m k) a)
(1 upto length mons) FuncUtil.Monomialfunc.empty)
in map mk_sq end
val sqs = map2 mk_sos sqmonlist ratdias
val cfs = map poly_of_epoly ids
val msq = filter (fn (_, b) => not (null b)) (map2 pair monoid sqs)
fun eval_sq sqs = fold_rev (fn (c, q) => poly_add (poly_cmul c (poly_mul q q))) sqs poly_0
val sanity =
fold_rev (fn ((p, _), s) => poly_add (poly_mul p (eval_sq s))) msq
(fold_rev2 (fn p => fn q => poly_add (poly_mul p q)) cfs eqs (poly_neg pol))
in
if not(FuncUtil.Monomialfunc.is_empty sanity) then raise Sanity
else (cfs, map (fn (a, b) => (snd a, b)) msq)
end
(* Iterative deepening. *)
fun deepen 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))));
(* Map back polynomials and their composites to a positivstellensatz. *)
fun cterm_of_sqterm (c, p) = RealArith.Product (RealArith.Rational_lt c, RealArith.Square p);
fun cterm_of_sos (pr,sqs) =
if null sqs then pr
else RealArith.Product (pr, foldr1 RealArith.Sum (map cterm_of_sqterm sqs));
(* Interface to HOL. *)
local
open Conv
val concl = Thm.dest_arg o cprop_of
fun simple_cterm_ord t u = Term_Ord.fast_term_ord (term_of t, term_of u) = LESS
in
(* FIXME: Replace tryfind by get_first !! *)
fun real_nonlinear_prover proof_method ctxt =
let
val {add = _, mul = _, neg = _, pow = _, sub = _, main = real_poly_conv} =
Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
fun mainf cert_choice translator (eqs, les, lts) =
let
val eq0 = map (poly_of_term o Thm.dest_arg1 o concl) eqs
val le0 = map (poly_of_term o Thm.dest_arg o concl) les
val lt0 = map (poly_of_term o Thm.dest_arg o concl) lts
val eqp0 = map_index (fn (i, t) => (t,RealArith.Axiom_eq i)) eq0
val lep0 = map_index (fn (i, t) => (t,RealArith.Axiom_le i)) le0
val ltp0 = map_index (fn (i, t) => (t,RealArith.Axiom_lt i)) lt0
val (keq,eq) = List.partition (fn (p, _) => multidegree p = 0) eqp0
val (klep,lep) = List.partition (fn (p, _) => multidegree p = 0) lep0
val (kltp,ltp) = List.partition (fn (p, _) => multidegree p = 0) ltp0
fun trivial_axiom (p, ax) =
(case ax of
RealArith.Axiom_eq n =>
if eval FuncUtil.Ctermfunc.empty p <>/ 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
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
else raise Failure "trivial_axiom: Not a trivial axiom"
| _ => error "trivial_axiom: Not a trivial axiom")
in
let val th = tryfind trivial_axiom (keq @ klep @ kltp) in
(fconv_rule (arg_conv (arg1_conv (real_poly_conv ctxt))
then_conv Numeral_Simprocs.field_comp_conv ctxt) th,
RealArith.Trivial)
end handle Failure _ =>
let
val proof =
(case proof_method of
Certificate certs =>
(* choose certificate *)
let
fun chose_cert [] (RealArith.Cert c) = c
| chose_cert (RealArith.Left::s) (RealArith.Branch (l, _)) = chose_cert s l
| chose_cert (RealArith.Right::s) (RealArith.Branch (_, r)) = chose_cert s r
| chose_cert _ _ = error "certificate tree in invalid form"
in
chose_cert cert_choice certs
end
| Prover prover =>
(* call prover *)
let
val pol = fold_rev poly_mul (map fst ltp) (poly_const 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 ctxt prover false d eq' leq
(poly_neg(poly_pow pol i))))
(0 upto k)
end
val (_,i,(cert_ideal,cert_cone)) = deepen tryall 0
val proofs_ideal =
map2 (fn q => fn (_,ax) => RealArith.Eqmul(q,ax)) cert_ideal eq
val proofs_cone = map cterm_of_sos cert_cone
val proof_ne =
if null ltp then RealArith.Rational_lt Rat.one
else
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)
end)
in
(translator (eqs,les,lts) proof, RealArith.Cert 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.lambda o Thm.lhs_of) eqs t
in
Conv.fconv_rule (Thm.beta_conversion true) (fold (C Thm.combination) eqs (Thm.reflexive t'))
end
(* A wrapper that tries to substitute away variables first. *)
local
open Conv
fun simple_cterm_ord t u = Term_Ord.fast_term_ord (term_of t, term_of u) = LESS
val concl = Thm.dest_arg o cprop_of
val shuffle1 =
fconv_rule (rewr_conv @{lemma "(a + x == y) == (x == y - (a::real))"
by (atomize (full)) (simp add: field_simps)})
val shuffle2 =
fconv_rule (rewr_conv @{lemma "(x + a == y) == (x == y - (a::real))"
by (atomize (full)) (simp add: field_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 => _"} $ _ $ (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 => _"}$_$_ =>
(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 =
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 => _"} $ _ $ _ =>
if Thm.dest_arg1 w aconvc v then shuffle2 th
else isolate_variable v (shuffle1 th)
| _ => error "isolate variable : This should not happen?")
end
in
fun real_nonlinear_subst_prover prover ctxt =
let
val {add = _, mul = real_poly_mul_conv, neg = _, pow = _, sub = _, main = real_poly_conv} =
Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
(the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
simple_cterm_ord
fun make_substitution th =
let
val (c,v) = substitutable_monomial [] (Thm.dest_arg1(concl th))
val th1 =
Drule.arg_cong_rule
(Thm.apply @{cterm "op * :: real => _"} (RealArith.cterm_of_rat (Rat.inv c)))
(mk_meta_eq th)
val th2 = fconv_rule (binop_conv (real_poly_mul_conv ctxt)) th1
in fconv_rule (arg_conv (real_poly_conv ctxt)) (isolate_variable v th2) end
fun oprconv cv ct =
let val g = Thm.dest_fun2 ct in
if g aconvc @{cterm "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
fun substfirst (eqs, les, lts) =
(let
val eth = tryfind make_substitution eqs
val modify =
fconv_rule (arg_conv (oprconv(subst_conv [eth] then_conv (real_poly_conv ctxt))))
in
substfirst
(filter_out
(fn t => (Thm.dest_arg1 o Thm.dest_arg o cprop_of) t aconvc @{cterm "0::real"})
(map modify eqs),
map modify les,
map modify lts)
end handle Failure _ =>
real_nonlinear_prover prover ctxt cert_choice translator (rev eqs, rev les, rev lts))
in substfirst end
in mainf end
(* Overall function. *)
fun real_sos prover ctxt =
RealArith.gen_prover_real_arith ctxt (real_nonlinear_subst_prover prover ctxt)
end;
val known_sos_constants =
[@{term "op ==>"}, @{term "Trueprop"},
@{term HOL.implies}, @{term HOL.conj}, @{term HOL.disj},
@{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 "numeral :: num => nat"},
@{term "numeral :: num => real"},
@{term "Num.Bit0"}, @{term "Num.Bit1"}, @{term "Num.One"}];
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"})) vs
then error "SOS: not sos. Variables with type not real"
else ()
val ukcs = subtract (fn (t,p) => Const p aconv t) kcts (Term.add_consts t [])
val _ =
if null ukcs then ()
else error ("SOSO: Unknown constants in Subgoal:" ^ commas (map fst ukcs))
in () end
fun core_sos_tac print_cert prover = SUBPROOF (fn {concl, context, ...} =>
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 1 end);
fun default_SOME _ NONE v = SOME v
| default_SOME _ (SOME v) _ = SOME v;
fun lift_SOME f NONE a = f a
| lift_SOME _ (SOME a) _ = SOME a;
local
val is_numeral = can (HOLogic.dest_number o 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 simp_ctxt =
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 simp_ctxt i end));
fun elim_denom_tac ctxt i = REPEAT (elim_one_denom_tac ctxt i);
fun sos_tac print_cert prover ctxt =
(* The SOS prover breaks if mult_nonneg_nonneg is in the simpset *)
let val ctxt' = ctxt delsimps [@{thm mult_nonneg_nonneg}]
in Object_Logic.full_atomize_tac ctxt' THEN'
elim_denom_tac ctxt' THEN'
core_sos_tac print_cert prover ctxt'
end;
end;