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(* Title: HOL/Tools/Groebner_Basis/groebner.ML
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ID: $Id$
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Author: Amine Chaieb, TU Muenchen
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*)
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signature GROEBNER =
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sig
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val ring_and_ideal_conv :
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{idom: thm list, ring: cterm list * thm list, vars: cterm list,
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semiring: Thm.cterm list * thm list} ->
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(Thm.cterm -> Rat.rat) -> (Rat.rat -> Thm.cterm) ->
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Conv.conv -> Conv.conv ->
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Conv.conv * (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list)
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val ring_conv : Proof.context -> cterm -> thm
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end
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structure Groebner: GROEBNER =
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struct
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open Normalizer;
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open Misc;
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(* FIXME :: Already present in Tools/Presburger/qelim.ML but is much more general!! *)
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fun cterm_frees ct =
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let fun h acc t =
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case (term_of t) of
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_$_ => h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
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| Abs(_,_,_) => Thm.dest_abs NONE t ||> h acc |> uncurry (remove (op aconvc))
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| Free _ => insert (op aconvc) t acc
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| _ => acc
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in h [] ct end;
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fun assocd x al d = case AList.lookup (op =) al x of SOME y => y | NONE => d;
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val rat_0 = Rat.zero;
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val rat_1 = Rat.one;
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val minus_rat = Rat.neg;
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val denominator_rat = Rat.quotient_of_rat #> snd #> Rat.rat_of_int;
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fun int_of_rat a =
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case Rat.quotient_of_rat a of (i,1) => i | _ => error "int_of_rat: not an int";
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val lcm_rat = fn x => fn y => Rat.rat_of_int (lcm (int_of_rat x, int_of_rat y));
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val (eqF_intr, eqF_elim) =
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let val [th1,th2] = thms "PFalse"
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in (fn th => th COMP th2, fn th => th COMP th1) end;
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val (PFalse, PFalse') =
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let val PFalse_eq = nth (thms "simp_thms") 13
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in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
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(* ------------------------------------------------------------------------- *)
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(* Type for recording history, i.e. how a polynomial was obtained. *)
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(* ------------------------------------------------------------------------- *)
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datatype history =
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Start of integer
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| Mmul of (Rat.rat * (integer list)) * history
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| Add of history * history;
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(* Monomial ordering. *)
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fun morder_lt m1 m2=
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let fun lexorder l1 l2 =
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case (l1,l2) of
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([],[]) => false
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| (x1::o1,x2::o2) => x1 > x2 orelse x1 = x2 andalso lexorder o1 o2
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| _ => error "morder: inconsistent monomial lengths"
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val n1 = fold Integer.add m1 0
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val n2 = fold Integer.add m2 0 in
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n1 < n2 orelse n1 = n2 andalso lexorder m1 m2
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end;
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fun morder_le m1 m2 = morder_lt m1 m2 orelse (m1 = m2);
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fun morder_gt m1 m2 = morder_lt m2 m1;
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(* Arithmetic on canonical polynomials. *)
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fun grob_neg l = map (fn (c,m) => (minus_rat c,m)) l;
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fun grob_add l1 l2 =
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case (l1,l2) of
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([],l2) => l2
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| (l1,[]) => l1
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| ((c1,m1)::o1,(c2,m2)::o2) =>
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if m1 = m2 then
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let val c = c1+/c2 val rest = grob_add o1 o2 in
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if c =/ rat_0 then rest else (c,m1)::rest end
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else if morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2)
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else (c2,m2)::(grob_add l1 o2);
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fun grob_sub l1 l2 = grob_add l1 (grob_neg l2);
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fun grob_mmul (c1,m1) (c2,m2) = (c1*/c2,map2 Integer.add m1 m2);
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fun grob_cmul cm pol = map (grob_mmul cm) pol;
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fun grob_mul l1 l2 =
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case l1 of
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[] => []
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| (h1::t1) => grob_add (grob_cmul h1 l2) (grob_mul t1 l2);
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fun grob_inv l =
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case l of
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[(c,vs)] => if (forall (fn x => x = 0) vs) then
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if (c =/ rat_0) then error "grob_inv: division by zero"
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else [(rat_1 // c,vs)]
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else error "grob_inv: non-constant divisor polynomial"
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| _ => error "grob_inv: non-constant divisor polynomial";
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fun grob_div l1 l2 =
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case l2 of
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[(c,l)] => if (forall (fn x => x = 0) l) then
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if c =/ rat_0 then error "grob_div: division by zero"
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else grob_cmul (rat_1 // c,l) l1
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else error "grob_div: non-constant divisor polynomial"
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| _ => error "grob_div: non-constant divisor polynomial";
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fun grob_pow vars l n =
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if n < 0 then error "grob_pow: negative power"
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else if n = 0 then [(rat_1,map (fn v => 0) vars)]
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else grob_mul l (grob_pow vars l (n -% 1));
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val max = fn (x: integer) => fn y => if x < y then y else x;
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fun degree vn p =
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case p of
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[] => error "Zero polynomial"
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| [(c,ns)] => nth ns vn
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| (c,ns)::p' => max (nth ns vn) (degree vn p');
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fun head_deg vn p = let val d = degree vn p in
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(d,fold (fn (c,r) => fn q => grob_add q [(c, map_index (fn (i,n) => if i = vn then 0 else n) r)]) (filter (fn (c,ns) => c <>/ rat_0 andalso nth ns vn = d) p) []) end;
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val is_zerop = forall (fn (c,ns) => c =/ rat_0 andalso forall (curry (op =) (0: integer)) ns);
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val grob_pdiv =
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let fun pdiv_aux vn (n,a) p k s =
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if is_zerop s then (k,s) else
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let val (m,b) = head_deg vn s
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in if m < n then (k,s) else
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let val p' = grob_mul p [(rat_1, map_index (fn (i,v) => if i = vn then m -% n else 0)
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(snd (hd s)))]
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in if a = b then pdiv_aux vn (n,a) p k (grob_sub s p')
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else pdiv_aux vn (n,a) p (k +% 1) (grob_sub (grob_mul a s) (grob_mul b p'))
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end
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end
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in fn vn => fn s => fn p => pdiv_aux vn (head_deg vn p) p 0 s
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end;
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(* Monomial division operation. *)
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fun mdiv (c1,m1) (c2,m2) =
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(c1//c2,
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map2 (fn n1 => fn n2 => if n1 < n2 then error "mdiv" else n1 -% n2) m1 m2);
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(* Lowest common multiple of two monomials. *)
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fun mlcm (c1,m1) (c2,m2) = (rat_1,map2 max m1 m2);
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(* Reduce monomial cm by polynomial pol, returning replacement for cm. *)
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fun reduce1 cm (pol,hpol) =
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case pol of
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[] => error "reduce1"
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| cm1::cms => ((let val (c,m) = mdiv cm cm1 in
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(grob_cmul (minus_rat c,m) cms,
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Mmul((minus_rat c,m),hpol)) end)
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handle ERROR _ => error "reduce1");
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(* Try this for all polynomials in a basis. *)
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fun tryfind f l =
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case l of
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[] => error "tryfind"
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| (h::t) => ((f h) handle ERROR _ => tryfind f t);
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fun reduceb cm basis = tryfind (fn p => reduce1 cm p) basis;
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(* Reduction of a polynomial (always picking largest monomial possible). *)
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fun reduce basis (pol,hist) =
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case pol of
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[] => (pol,hist)
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| cm::ptl => ((let val (q,hnew) = reduceb cm basis in
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reduce basis (grob_add q ptl,Add(hnew,hist)) end)
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handle (ERROR _) =>
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(let val (q,hist') = reduce basis (ptl,hist) in
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(cm::q,hist') end));
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(* Check for orthogonality w.r.t. LCM. *)
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fun orthogonal l p1 p2 =
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snd l = snd(grob_mmul (hd p1) (hd p2));
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(* Compute S-polynomial of two polynomials. *)
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fun spoly cm ph1 ph2 =
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case (ph1,ph2) of
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(([],h),p) => ([],h)
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| (p,([],h)) => ([],h)
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| ((cm1::ptl1,his1),(cm2::ptl2,his2)) =>
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(grob_sub (grob_cmul (mdiv cm cm1) ptl1)
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(grob_cmul (mdiv cm cm2) ptl2),
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Add(Mmul(mdiv cm cm1,his1),
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Mmul(mdiv (minus_rat(fst cm),snd cm) cm2,his2)));
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(* Make a polynomial monic. *)
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fun monic (pol,hist) =
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if pol = [] then (pol,hist) else
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let val (c',m') = hd pol in
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(map (fn (c,m) => (c//c',m)) pol,
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Mmul((rat_1 // c',map (K 0) m'),hist)) end;
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(* The most popular heuristic is to order critical pairs by LCM monomial. *)
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fun forder ((c1,m1),_) ((c2,m2),_) = morder_lt m1 m2;
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fun poly_lt p q =
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case (p,q) of
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(p,[]) => false
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| ([],q) => true
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| ((c1,m1)::o1,(c2,m2)::o2) =>
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c1 </ c2 orelse
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c1 =/ c2 andalso ((morder_lt m1 m2) orelse m1 = m2 andalso poly_lt o1 o2);
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fun align ((p,hp),(q,hq)) =
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if poly_lt p q then ((p,hp),(q,hq)) else ((q,hq),(p,hp));
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fun forall2 p l1 l2 =
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case (l1,l2) of
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([],[]) => true
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| (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
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| _ => false;
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fun poly_eq p1 p2 =
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forall2 (fn (c1,m1) => fn (c2,m2) => c1 =/ c2 andalso m1 = m2) p1 p2;
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fun memx ((p1,h1),(p2,h2)) ppairs =
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not (exists (fn ((q1,_),(q2,_)) => poly_eq p1 q1 andalso poly_eq p2 q2) ppairs);
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(* Buchberger's second criterion. *)
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fun criterion2 basis (lcm,((p1,h1),(p2,h2))) opairs =
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exists (fn g => not(poly_eq (fst g) p1) andalso not(poly_eq (fst g) p2) andalso
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can (mdiv lcm) (hd(fst g)) andalso
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not(memx (align (g,(p1,h1))) (map snd opairs)) andalso
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not(memx (align (g,(p2,h2))) (map snd opairs))) basis;
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(* Test for hitting constant polynomial. *)
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fun constant_poly p =
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length p = 1 andalso forall (fn x => x = (0: integer)) (snd(hd p));
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(* ------------------------------------------------------------------------- *)
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(* Grobner basis algorithm. *)
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(* ------------------------------------------------------------------------- *)
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(* FIXME: try to get rid of mergesort? *)
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fun merge ord l1 l2 =
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case l1 of
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[] => l2
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| h1::t1 =>
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case l2 of
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[] => l1
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| h2::t2 => if ord h1 h2 then h1::(merge ord t1 l2)
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else h2::(merge ord l1 t2);
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fun mergesort ord l =
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let
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fun mergepairs l1 l2 =
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case (l1,l2) of
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([s],[]) => s
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| (l,[]) => mergepairs [] l
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| (l,[s1]) => mergepairs (s1::l) []
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| (l,(s1::s2::ss)) => mergepairs ((merge ord s1 s2)::l) ss
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in if l = [] then [] else mergepairs [] (map (fn x => [x]) l)
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end;
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fun grobner_basis basis pairs =
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(writeln (Int.toString(length basis)^" basis elements and "^
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Int.toString(length pairs)^" critical pairs");
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case pairs of
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[] => basis
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| (l,(p1,p2))::opairs =>
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let val (sph as (sp,hist)) = monic (reduce basis (spoly l p1 p2))
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in if sp = [] orelse criterion2 basis (l,(p1,p2)) opairs
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then grobner_basis basis opairs
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else if constant_poly sp then grobner_basis (sph::basis) []
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else let val rawcps = map (fn p => (mlcm (hd(fst p)) (hd sp),align(p,sph)))
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basis
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val newcps = filter
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(fn (l,(p,q)) => not(orthogonal l (fst p) (fst q)))
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rawcps
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in grobner_basis (sph::basis)
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(merge forder opairs (mergesort forder newcps))
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end
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end);
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(* ------------------------------------------------------------------------- *)
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(* Interreduce initial polynomials. *)
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(* ------------------------------------------------------------------------- *)
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fun grobner_interreduce rpols ipols =
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case ipols of
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[] => map monic (rev rpols)
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| p::ps => let val p' = reduce (rpols @ ps) p in
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if fst p' = [] then grobner_interreduce rpols ps
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else grobner_interreduce (p'::rpols) ps end;
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(* ------------------------------------------------------------------------- *)
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(* Overall function. *)
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(* ------------------------------------------------------------------------- *)
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fun grobner pols =
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let val npols = map2 (fn p => fn n => (p,Start n)) pols
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(map Integer.int (0 upto (length pols - 1)))
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val phists = filter (fn (p,_) => p <> []) npols
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val bas = grobner_interreduce [] (map monic phists)
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val prs0 = product bas bas
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val prs1 = filter (fn ((x,_),(y,_)) => poly_lt x y) prs0
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val prs2 = map (fn (p,q) => (mlcm (hd(fst p)) (hd(fst q)),(p,q))) prs1
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val prs3 =
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filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q))) prs2 in
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grobner_basis bas (mergesort forder prs3) end;
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(* ------------------------------------------------------------------------- *)
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(* Get proof of contradiction from Grobner basis. *)
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(* ------------------------------------------------------------------------- *)
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fun find p l =
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case l of
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[] => error "find"
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| (h::t) => if p(h) then h else find p t;
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fun grobner_refute pols =
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let val gb = grobner pols in
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snd(find (fn (p,h) => length p = 1 andalso forall (fn x=> x=0) (snd(hd p))) gb)
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end;
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(* ------------------------------------------------------------------------- *)
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(* Turn proof into a certificate as sum of multipliers. *)
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(* *)
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(* In principle this is very inefficient: in a heavily shared proof it may *)
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(* make the same calculation many times. Could put in a cache or something. *)
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(* ------------------------------------------------------------------------- *)
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fun assoc x l = snd(find (fn p => fst p = x) l);
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fun resolve_proof vars prf =
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case prf of
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Start(~1) => []
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| Start m => [(m,[(rat_1,map (K 0) vars)])]
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| Mmul(pol,lin) =>
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let val lis = resolve_proof vars lin in
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map (fn (n,p) => (n,grob_cmul pol p)) lis end
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352 |
| Add(lin1,lin2) =>
|
|
353 |
let val lis1 = resolve_proof vars lin1
|
|
354 |
val lis2 = resolve_proof vars lin2
|
|
355 |
val dom = distinct (op =) ((map fst lis1) union (map fst lis2))
|
|
356 |
in
|
|
357 |
map (fn n => let val a = ((assoc n lis1) handle _ => []) (* FIXME *)
|
|
358 |
val b = ((assoc n lis2) handle _ => []) in (* FIXME *)
|
|
359 |
(n,grob_add a b) end) dom end;
|
|
360 |
|
|
361 |
(* ------------------------------------------------------------------------- *)
|
|
362 |
(* Run the procedure and produce Weak Nullstellensatz certificate. *)
|
|
363 |
(* ------------------------------------------------------------------------- *)
|
|
364 |
fun grobner_weak vars pols =
|
|
365 |
let val cert = resolve_proof vars (grobner_refute pols)
|
|
366 |
val l =
|
|
367 |
fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert (rat_1) in
|
|
368 |
(l,map (fn (i,p) => (i,map (fn (d,m) => (l*/d,m)) p)) cert) end;
|
|
369 |
|
|
370 |
(* ------------------------------------------------------------------------- *)
|
|
371 |
(* Prove a polynomial is in ideal generated by others, using Grobner basis. *)
|
|
372 |
(* ------------------------------------------------------------------------- *)
|
|
373 |
|
|
374 |
fun grobner_ideal vars pols pol =
|
|
375 |
let val (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in
|
|
376 |
if pol <> [] then error "grobner_ideal: not in the ideal" else
|
|
377 |
resolve_proof vars h end;
|
|
378 |
|
|
379 |
(* ------------------------------------------------------------------------- *)
|
|
380 |
(* Produce Strong Nullstellensatz certificate for a power of pol. *)
|
|
381 |
(* ------------------------------------------------------------------------- *)
|
|
382 |
|
|
383 |
fun grobner_strong vars pols pol =
|
|
384 |
let val vars' = @{cterm "True"}::vars
|
|
385 |
val grob_z = [(rat_1,1::(map (fn x => 0) vars))]
|
|
386 |
val grob_1 = [(rat_1,(map (fn x => 0) vars'))]
|
|
387 |
fun augment p= map (fn (c,m) => (c,0::m)) p
|
|
388 |
val pols' = map augment pols
|
|
389 |
val pol' = augment pol
|
|
390 |
val allpols = (grob_sub (grob_mul grob_z pol') grob_1)::pols'
|
|
391 |
val (l,cert) = grobner_weak vars' allpols
|
|
392 |
val d = fold_rev (fold_rev (max o hd o snd) o snd) cert 0
|
|
393 |
fun transform_monomial (c,m) =
|
|
394 |
grob_cmul (c,tl m) (grob_pow vars pol (d - hd m))
|
|
395 |
fun transform_polynomial q = fold_rev (grob_add o transform_monomial) q []
|
|
396 |
val cert' = map (fn (c,q) => (c-1,transform_polynomial q))
|
|
397 |
(filter (fn (k,_) => k <> 0) cert) in
|
|
398 |
(d,l,cert') end;
|
|
399 |
|
|
400 |
fun string_of_pol vars pol =
|
|
401 |
foldl (fn ((c,m),s) => ((Rat.string_of_rat c)
|
|
402 |
^ "*(" ^
|
|
403 |
(snd (foldl
|
|
404 |
(fn (e,(i,s)) =>
|
|
405 |
(i+ 1,
|
|
406 |
(nth vars i
|
23259
|
407 |
|>cterm_of (the_context()) (* FIXME *)
|
23252
|
408 |
|> string_of_cterm)^ "^"
|
|
409 |
^ (Int.toString e) ^" * " ^ s)) (0,"0") m))
|
|
410 |
^ ") + ") ^ s) "" pol;
|
|
411 |
|
|
412 |
|
|
413 |
(* ------------------------------------------------------------------------- *)
|
|
414 |
(* Overall parametrized universal procedure for (semi)rings. *)
|
|
415 |
(* We return an ideal_conv and the actual ring prover. *)
|
|
416 |
(* ------------------------------------------------------------------------- *)
|
|
417 |
fun refute_disj rfn tm =
|
|
418 |
case term_of tm of
|
|
419 |
Const("op |",_)$l$r =>
|
|
420 |
Drule.compose_single(refute_disj rfn (Thm.dest_arg tm),2,Drule.compose_single(refute_disj rfn (Thm.dest_arg1 tm),2,disjE))
|
|
421 |
| _ => rfn tm ;
|
|
422 |
|
|
423 |
val notnotD = @{thm "notnotD"};
|
|
424 |
fun mk_binop ct x y =
|
|
425 |
Thm.capply (Thm.capply ct x) y
|
|
426 |
|
|
427 |
val mk_comb = Thm.capply;
|
|
428 |
fun is_neg t =
|
|
429 |
case term_of t of
|
|
430 |
(Const("Not",_)$p) => true
|
|
431 |
| _ => false;
|
|
432 |
fun is_eq t =
|
|
433 |
case term_of t of
|
|
434 |
(Const("op =",_)$_$_) => true
|
|
435 |
| _ => false;
|
|
436 |
|
|
437 |
fun end_itlist f l =
|
|
438 |
case l of
|
|
439 |
[] => error "end_itlist"
|
|
440 |
| [x] => x
|
|
441 |
| (h::t) => f h (end_itlist f t);
|
|
442 |
|
|
443 |
val list_mk_binop = fn b => end_itlist (mk_binop b);
|
|
444 |
|
|
445 |
val list_dest_binop = fn b =>
|
|
446 |
let fun h acc t =
|
|
447 |
((let val (l,r) = dest_binop b t in h (h acc r) l end)
|
|
448 |
handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
|
|
449 |
in h []
|
|
450 |
end;
|
|
451 |
|
|
452 |
val strip_exists =
|
|
453 |
let fun h (acc, t) =
|
|
454 |
case (term_of t) of
|
|
455 |
Const("Ex",_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
|
|
456 |
| _ => (acc,t)
|
|
457 |
in fn t => h ([],t)
|
|
458 |
end;
|
|
459 |
|
|
460 |
fun is_forall t =
|
|
461 |
case term_of t of
|
|
462 |
(Const("All",_)$Abs(_,_,_)) => true
|
|
463 |
| _ => false;
|
|
464 |
|
|
465 |
val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
|
|
466 |
val bool_simps = @{thms "bool_simps"};
|
|
467 |
val nnf_simps = @{thms "nnf_simps"};
|
|
468 |
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
|
|
469 |
val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps (@{thms "weak_dnf_simps"}));
|
|
470 |
val initial_conv =
|
|
471 |
Simplifier.rewrite
|
|
472 |
(HOL_basic_ss addsimps nnf_simps
|
|
473 |
addsimps [not_all, not_ex] addsimps map (fn th => th RS sym) (ex_simps @ all_simps));
|
|
474 |
|
|
475 |
val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
|
|
476 |
|
|
477 |
val cTrp = @{cterm "Trueprop"};
|
|
478 |
val cConj = @{cterm "op &"};
|
|
479 |
val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
|
|
480 |
val ASSUME = mk_comb cTrp #> assume;
|
|
481 |
val list_mk_conj = list_mk_binop cConj;
|
|
482 |
val conjs = list_dest_binop cConj;
|
|
483 |
val mk_neg = mk_comb cNot;
|
|
484 |
|
|
485 |
|
|
486 |
|
|
487 |
(** main **)
|
|
488 |
|
|
489 |
fun ring_and_ideal_conv
|
|
490 |
{vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), idom}
|
|
491 |
dest_const mk_const ring_eq_conv ring_normalize_conv =
|
|
492 |
let
|
|
493 |
val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
|
|
494 |
val [ring_add_tm, ring_mul_tm, ring_pow_tm] =
|
|
495 |
map (Thm.dest_fun o Thm.dest_fun) [add_pat, mul_pat, pow_pat];
|
|
496 |
|
|
497 |
val (ring_sub_tm, ring_neg_tm) =
|
|
498 |
(case r_ops of
|
|
499 |
[] => (@{cterm "True"}, @{cterm "True"})
|
|
500 |
| [sub_pat, neg_pat] => (Thm.dest_fun (Thm.dest_fun sub_pat), Thm.dest_fun neg_pat));
|
|
501 |
|
|
502 |
val [idom_thm, neq_thm] = idom;
|
|
503 |
|
|
504 |
val ring_dest_neg =
|
|
505 |
fn t => let val (l,r) = Thm.dest_comb t in
|
|
506 |
if could_unify(term_of l,term_of ring_neg_tm) then r else raise CTERM ("ring_dest_neg", [t])
|
|
507 |
end
|
|
508 |
|
|
509 |
val ring_mk_neg = fn tm => mk_comb (ring_neg_tm) (tm);
|
|
510 |
(*
|
|
511 |
fun ring_dest_inv t =
|
|
512 |
let val (l,r) = Thm.dest_comb t in
|
|
513 |
if could_unify(term_of l, term_of ring_inv_tm) then r else raise CTERM "ring_dest_inv"
|
|
514 |
end
|
|
515 |
*)
|
|
516 |
val ring_dest_add = dest_binop ring_add_tm;
|
|
517 |
val ring_mk_add = mk_binop ring_add_tm;
|
|
518 |
val ring_dest_sub = dest_binop ring_sub_tm;
|
|
519 |
val ring_mk_sub = mk_binop ring_sub_tm;
|
|
520 |
val ring_dest_mul = dest_binop ring_mul_tm;
|
|
521 |
val ring_mk_mul = mk_binop ring_mul_tm;
|
|
522 |
(* val ring_dest_div = dest_binop ring_div_tm;
|
|
523 |
val ring_mk_div = mk_binop ring_div_tm;*)
|
|
524 |
val ring_dest_pow = dest_binop ring_pow_tm;
|
|
525 |
val ring_mk_pow = mk_binop ring_pow_tm ;
|
|
526 |
fun grobvars tm acc =
|
|
527 |
if can dest_const tm then acc
|
|
528 |
else if can ring_dest_neg tm then grobvars (Thm.dest_arg tm) acc
|
|
529 |
else if can ring_dest_pow tm then grobvars (Thm.dest_arg1 tm) acc
|
|
530 |
else if can ring_dest_add tm orelse can ring_dest_sub tm
|
|
531 |
orelse can ring_dest_mul tm
|
|
532 |
then grobvars (Thm.dest_arg1 tm) (grobvars (Thm.dest_arg tm) acc)
|
|
533 |
(* else if can ring_dest_inv tm
|
|
534 |
then
|
|
535 |
let val gvs = grobvars (Thm.dest_arg tm) [] in
|
|
536 |
if gvs = [] then acc else tm::acc
|
|
537 |
end
|
|
538 |
else if can ring_dest_div tm then
|
|
539 |
let val lvs = grobvars (Thm.dest_arg1 tm) acc
|
|
540 |
val gvs = grobvars (Thm.dest_arg tm) []
|
|
541 |
in if gvs = [] then lvs else tm::acc
|
|
542 |
end *)
|
|
543 |
else tm::acc ;
|
|
544 |
|
|
545 |
fun grobify_term vars tm =
|
|
546 |
((if not (member (op aconvc) vars tm) then raise CTERM ("Not a variable", [tm]) else
|
|
547 |
[(rat_1,map (fn i => if i aconvc tm then 1 else 0) vars)])
|
|
548 |
handle CTERM _ =>
|
|
549 |
((let val x = dest_const tm
|
|
550 |
in if x =/ rat_0 then [] else [(x,map (fn v => 0) vars)]
|
|
551 |
end)
|
|
552 |
handle ERROR _ =>
|
|
553 |
((grob_neg(grobify_term vars (ring_dest_neg tm)))
|
|
554 |
handle CTERM _ =>
|
|
555 |
(
|
|
556 |
(* (grob_inv(grobify_term vars (ring_dest_inv tm)))
|
|
557 |
handle CTERM _ => *)
|
|
558 |
((let val (l,r) = ring_dest_add tm
|
|
559 |
in grob_add (grobify_term vars l) (grobify_term vars r)
|
|
560 |
end)
|
|
561 |
handle CTERM _ =>
|
|
562 |
((let val (l,r) = ring_dest_sub tm
|
|
563 |
in grob_sub (grobify_term vars l) (grobify_term vars r)
|
|
564 |
end)
|
|
565 |
handle CTERM _ =>
|
|
566 |
((let val (l,r) = ring_dest_mul tm
|
|
567 |
in grob_mul (grobify_term vars l) (grobify_term vars r)
|
|
568 |
end)
|
|
569 |
handle CTERM _ =>
|
|
570 |
(
|
|
571 |
(* (let val (l,r) = ring_dest_div tm
|
|
572 |
in grob_div (grobify_term vars l) (grobify_term vars r)
|
|
573 |
end)
|
|
574 |
handle CTERM _ => *)
|
|
575 |
((let val (l,r) = ring_dest_pow tm
|
|
576 |
in grob_pow vars (grobify_term vars l) ((term_of #> HOLogic.dest_number #> snd) r)
|
|
577 |
end)
|
|
578 |
handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
|
|
579 |
val eq_tm = idom_thm |> concl |> Thm.dest_arg |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_fun ;
|
|
580 |
(*ring_integral |> hd |> concl |> Thm.dest_arg
|
|
581 |
|> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_fun; *)
|
|
582 |
val dest_eq = dest_binop eq_tm;
|
|
583 |
|
|
584 |
fun grobify_equation vars tm =
|
|
585 |
let val (l,r) = dest_binop eq_tm tm
|
|
586 |
in grob_sub (grobify_term vars l) (grobify_term vars r)
|
|
587 |
end;
|
|
588 |
|
|
589 |
fun grobify_equations tm =
|
|
590 |
let
|
|
591 |
val cjs = conjs tm
|
|
592 |
val rawvars = fold_rev (fn eq => fn a =>
|
|
593 |
grobvars (Thm.dest_arg1 eq) (grobvars (Thm.dest_arg eq) a)) cjs []
|
|
594 |
val vars = sort (fn (x, y) => Term.term_ord(term_of x,term_of y))
|
|
595 |
(distinct (op aconvc) rawvars)
|
|
596 |
in (vars,map (grobify_equation vars) cjs)
|
|
597 |
end;
|
|
598 |
|
|
599 |
val holify_polynomial =
|
|
600 |
let fun holify_varpow (v,n) =
|
|
601 |
if n = 1 then v else ring_mk_pow v (mk_cnumber @{ctyp "nat"} n) (* FIXME *)
|
|
602 |
fun holify_monomial vars (c,m) =
|
23259
|
603 |
let val xps = map holify_varpow (filter (fn (_,n) => n <> (0: integer)) (vars ~~ m))
|
23252
|
604 |
in end_itlist ring_mk_mul (mk_const c :: xps)
|
|
605 |
end
|
|
606 |
fun holify_polynomial vars p =
|
|
607 |
if p = [] then mk_const (rat_0)
|
|
608 |
else end_itlist ring_mk_add (map (holify_monomial vars) p)
|
|
609 |
in holify_polynomial
|
|
610 |
end ;
|
|
611 |
val idom_rule = simplify (HOL_basic_ss addsimps [idom_thm]);
|
|
612 |
fun prove_nz n = eqF_elim
|
|
613 |
(ring_eq_conv(mk_binop eq_tm (mk_const n) (mk_const(rat_0))));
|
|
614 |
val neq_01 = prove_nz (rat_1);
|
|
615 |
fun neq_rule n th = [prove_nz n, th] MRS neq_thm;
|
|
616 |
fun mk_add th1 = combination(Drule.arg_cong_rule ring_add_tm th1);
|
|
617 |
|
|
618 |
fun refute tm =
|
|
619 |
if tm aconvc false_tm then ASSUME tm else
|
|
620 |
let
|
|
621 |
val (nths0,eths0) = List.partition (is_neg o concl) (conjuncts(ASSUME tm))
|
|
622 |
val nths = filter (is_eq o Thm.dest_arg o concl) nths0
|
|
623 |
val eths = filter (is_eq o concl) eths0
|
|
624 |
in
|
|
625 |
if null eths then
|
|
626 |
let
|
|
627 |
val th1 = end_itlist (fn th1 => fn th2 => idom_rule(conji th1 th2)) nths
|
|
628 |
val th2 = Conv.fconv_rule
|
|
629 |
((arg_conv #> arg_conv)
|
|
630 |
(binop_conv ring_normalize_conv)) th1
|
|
631 |
val conc = th2 |> concl |> Thm.dest_arg
|
|
632 |
val (l,r) = conc |> dest_eq
|
|
633 |
in implies_intr (mk_comb cTrp tm)
|
|
634 |
(equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th2))
|
|
635 |
(reflexive l |> mk_object_eq))
|
|
636 |
end
|
|
637 |
else
|
|
638 |
let
|
|
639 |
val (vars,l,cert,noteqth) =(
|
|
640 |
if null nths then
|
|
641 |
let val (vars,pols) = grobify_equations(list_mk_conj(map concl eths))
|
|
642 |
val (l,cert) = grobner_weak vars pols
|
|
643 |
in (vars,l,cert,neq_01)
|
|
644 |
end
|
|
645 |
else
|
|
646 |
let
|
|
647 |
val nth = end_itlist (fn th1 => fn th2 => idom_rule(conji th1 th2)) nths
|
|
648 |
val (vars,pol::pols) =
|
|
649 |
grobify_equations(list_mk_conj(Thm.dest_arg(concl nth)::map concl eths))
|
|
650 |
val (deg,l,cert) = grobner_strong vars pols pol
|
|
651 |
val th1 = Conv.fconv_rule((arg_conv o arg_conv)(binop_conv ring_normalize_conv)) nth
|
23259
|
652 |
val th2 = funpow (Integer.machine_int deg) (idom_rule o conji th1) neq_01
|
23252
|
653 |
in (vars,l,cert,th2)
|
|
654 |
end)
|
|
655 |
val _ = writeln ("Translating certificate to HOL inferences")
|
|
656 |
val cert_pos = map (fn (i,p) => (i,filter (fn (c,m) => c >/ rat_0) p)) cert
|
|
657 |
val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))
|
|
658 |
(filter (fn (c,m) => c </ rat_0) p))) cert
|
|
659 |
val herts_pos = map (fn (i,p) => (i,holify_polynomial vars p)) cert_pos
|
|
660 |
val herts_neg = map (fn (i,p) => (i,holify_polynomial vars p)) cert_neg
|
|
661 |
fun thm_fn pols =
|
|
662 |
if null pols then reflexive(mk_const rat_0) else
|
|
663 |
end_itlist mk_add
|
23259
|
664 |
(map (fn (i,p) => Drule.arg_cong_rule (mk_comb ring_mul_tm p)
|
|
665 |
(nth eths (Integer.machine_int i) |> mk_meta_eq)) pols)
|
23252
|
666 |
val th1 = thm_fn herts_pos
|
|
667 |
val th2 = thm_fn herts_neg
|
|
668 |
val th3 = conji(mk_add (symmetric th1) th2 |> mk_object_eq) noteqth
|
|
669 |
val th4 = Conv.fconv_rule ((arg_conv o arg_conv o binop_conv) ring_normalize_conv)
|
|
670 |
(neq_rule l th3)
|
|
671 |
val (l,r) = dest_eq(Thm.dest_arg(concl th4))
|
|
672 |
in implies_intr (mk_comb cTrp tm)
|
|
673 |
(equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th4))
|
|
674 |
(reflexive l |> mk_object_eq))
|
|
675 |
end
|
|
676 |
end
|
|
677 |
|
|
678 |
fun ring tm =
|
|
679 |
let
|
|
680 |
fun mk_forall x p =
|
|
681 |
mk_comb (Drule.cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (Thm.cabs x p)
|
|
682 |
val avs = cterm_frees tm
|
|
683 |
val P' = fold mk_forall avs tm
|
|
684 |
val th1 = initial_conv(mk_neg P')
|
|
685 |
val (evs,bod) = strip_exists(concl th1) in
|
|
686 |
if is_forall bod then error "ring: non-universal formula"
|
|
687 |
else
|
|
688 |
let
|
|
689 |
val th1a = weak_dnf_conv bod
|
|
690 |
val boda = concl th1a
|
|
691 |
val th2a = refute_disj refute boda
|
|
692 |
val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans
|
|
693 |
val th2 = fold (fn v => fn th => (forall_intr v th) COMP allI) evs (th2b RS PFalse)
|
|
694 |
val th3 = equal_elim
|
|
695 |
(Simplifier.rewrite (HOL_basic_ss addsimps [not_ex RS sym])
|
|
696 |
(th2 |> cprop_of)) th2
|
|
697 |
in specl avs
|
|
698 |
([[[mk_object_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS notnotD)
|
|
699 |
end
|
|
700 |
end
|
|
701 |
fun ideal tms tm ord =
|
|
702 |
let
|
|
703 |
val rawvars = fold_rev grobvars (tm::tms) []
|
|
704 |
val vars = sort ord (distinct (fn (x,y) => (term_of x) aconv (term_of y)) rawvars)
|
|
705 |
val pols = map (grobify_term vars) tms
|
|
706 |
val pol = grobify_term vars tm
|
|
707 |
val cert = grobner_ideal vars pols pol
|
|
708 |
in map (fn n => let val p = assocd n cert [] in holify_polynomial vars p end)
|
23259
|
709 |
(map Integer.int (0 upto (length pols - 1)))
|
23252
|
710 |
end
|
|
711 |
in (ring,ideal)
|
|
712 |
end;
|
|
713 |
|
|
714 |
|
|
715 |
fun find_term bounds tm =
|
|
716 |
(case term_of tm of
|
|
717 |
Const ("op =", T) $ _ $ _ =>
|
|
718 |
if domain_type T = HOLogic.boolT then find_args bounds tm
|
|
719 |
else Thm.dest_arg tm
|
|
720 |
| Const ("Not", _) $ _ => find_term bounds (Thm.dest_arg tm)
|
|
721 |
| Const ("All", _) $ _ => find_body bounds (Thm.dest_arg tm)
|
|
722 |
| Const ("Ex", _) $ _ => find_body bounds (Thm.dest_arg tm)
|
|
723 |
| Const ("op &", _) $ _ $ _ => find_args bounds tm
|
|
724 |
| Const ("op |", _) $ _ $ _ => find_args bounds tm
|
|
725 |
| Const ("op -->", _) $ _ $ _ => find_args bounds tm
|
|
726 |
| _ => raise TERM ("find_term", []))
|
|
727 |
and find_args bounds tm =
|
|
728 |
let val (t, u) = Thm.dest_binop tm
|
|
729 |
in (find_term bounds t handle TERM _ => find_term bounds u) end
|
|
730 |
and find_body bounds b =
|
|
731 |
let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b
|
|
732 |
in find_term (bounds + 1) b' end;
|
|
733 |
|
|
734 |
fun ring_conv ctxt form =
|
|
735 |
(case try (find_term 0 (* FIXME !? *)) form of
|
|
736 |
NONE => reflexive form
|
|
737 |
| SOME tm =>
|
|
738 |
(case NormalizerData.match ctxt tm of
|
|
739 |
NONE => reflexive form
|
|
740 |
| SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
|
|
741 |
fst (ring_and_ideal_conv theory
|
|
742 |
dest_const (mk_const (Thm.ctyp_of_term tm)) ring_eq_conv
|
|
743 |
(semiring_normalize_wrapper res)) form));
|
|
744 |
|
|
745 |
end;
|