Semiring normalization and Groebner Bases.
(* Title: HOL/Tools/Groebner_Basis/groebner.ML
ID: $Id$
Author: Amine Chaieb, TU Muenchen
*)
signature GROEBNER =
sig
val ring_and_ideal_conv :
{idom: thm list, ring: cterm list * thm list, vars: cterm list,
semiring: Thm.cterm list * thm list} ->
(Thm.cterm -> Rat.rat) -> (Rat.rat -> Thm.cterm) ->
Conv.conv -> Conv.conv ->
Conv.conv * (cterm list -> cterm -> (cterm * cterm -> order) -> cterm list)
val ring_conv : Proof.context -> cterm -> thm
end
structure Groebner: GROEBNER =
struct
open Normalizer;
open Misc;
(* FIXME :: Already present in Tools/Presburger/qelim.ML but is much more general!! *)
fun cterm_frees ct =
let fun h acc t =
case (term_of t) of
_$_ => h (h acc (Thm.dest_arg t)) (Thm.dest_fun t)
| Abs(_,_,_) => Thm.dest_abs NONE t ||> h acc |> uncurry (remove (op aconvc))
| Free _ => insert (op aconvc) t acc
| _ => acc
in h [] ct end;
fun assocd x al d = case AList.lookup (op =) al x of SOME y => y | NONE => d;
val rat_0 = Rat.zero;
val rat_1 = Rat.one;
val minus_rat = Rat.neg;
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";
val lcm_rat = fn x => fn y => Rat.rat_of_int (lcm (int_of_rat x, int_of_rat y));
val (eqF_intr, eqF_elim) =
let val [th1,th2] = thms "PFalse"
in (fn th => th COMP th2, fn th => th COMP th1) end;
val (PFalse, PFalse') =
let val PFalse_eq = nth (thms "simp_thms") 13
in (PFalse_eq RS iffD1, PFalse_eq RS iffD2) end;
(* ------------------------------------------------------------------------- *)
(* Type for recording history, i.e. how a polynomial was obtained. *)
(* ------------------------------------------------------------------------- *)
datatype history =
Start of int
| Mmul of (Rat.rat * (int list)) * history
| Add of history * history;
(* Monomial ordering. *)
fun morder_lt m1 m2=
let fun lexorder l1 l2 =
case (l1,l2) of
([],[]) => false
| (x1::o1,x2::o2) => x1 > x2 orelse x1 = x2 andalso lexorder o1 o2
| _ => error "morder: inconsistent monomial lengths"
val n1 = fold (curry op +) m1 0
val n2 = fold (curry op +) m2 0 in
n1 < n2 orelse n1 = n2 andalso lexorder m1 m2
end;
fun morder_le m1 m2 = morder_lt m1 m2 orelse (m1 = m2);
fun morder_gt m1 m2 = morder_lt m2 m1;
(* Arithmetic on canonical polynomials. *)
fun grob_neg l = map (fn (c,m) => (minus_rat c,m)) l;
fun grob_add l1 l2 =
case (l1,l2) of
([],l2) => l2
| (l1,[]) => l1
| ((c1,m1)::o1,(c2,m2)::o2) =>
if m1 = m2 then
let val c = c1+/c2 val rest = grob_add o1 o2 in
if c =/ rat_0 then rest else (c,m1)::rest end
else if morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2)
else (c2,m2)::(grob_add l1 o2);
fun grob_sub l1 l2 = grob_add l1 (grob_neg l2);
fun grob_mmul (c1,m1) (c2,m2) = (c1*/c2,map2 (curry op +) m1 m2);
fun grob_cmul cm pol = map (grob_mmul cm) pol;
fun grob_mul l1 l2 =
case l1 of
[] => []
| (h1::t1) => grob_add (grob_cmul h1 l2) (grob_mul t1 l2);
fun grob_inv l =
case l of
[(c,vs)] => if (forall (fn x => x = 0) vs) then
if (c =/ rat_0) then error "grob_inv: division by zero"
else [(rat_1 // c,vs)]
else error "grob_inv: non-constant divisor polynomial"
| _ => error "grob_inv: non-constant divisor polynomial";
fun grob_div l1 l2 =
case l2 of
[(c,l)] => if (forall (fn x => x = 0) l) then
if c =/ rat_0 then error "grob_div: division by zero"
else grob_cmul (rat_1 // c,l) l1
else error "grob_div: non-constant divisor polynomial"
| _ => error "grob_div: non-constant divisor polynomial";
fun grob_pow vars l n =
if n < 0 then error "grob_pow: negative power"
else if n = 0 then [(rat_1,map (fn v => 0) vars)]
else grob_mul l (grob_pow vars l (n - 1));
val max = fn x => fn y => if x < y then y else x;
fun degree vn p =
case p of
[] => error "Zero polynomial"
| [(c,ns)] => nth ns vn
| (c,ns)::p' => max (nth ns vn) (degree vn p');
fun head_deg vn p = let val d = degree vn p in
(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;
val is_zerop = forall (fn (c,ns) => c =/ rat_0 andalso forall (curry (op =) 0) ns);
val grob_pdiv =
let fun pdiv_aux vn (n,a) p k s =
if is_zerop s then (k,s) else
let val (m,b) = head_deg vn s
in if m < n then (k,s) else
let val p' = grob_mul p [(rat_1, map_index (fn (i,v) => if i = vn then m - n else 0)
(snd (hd s)))]
in if a = b then pdiv_aux vn (n,a) p k (grob_sub s p')
else pdiv_aux vn (n,a) p (k + 1) (grob_sub (grob_mul a s) (grob_mul b p'))
end
end
in fn vn => fn s => fn p => pdiv_aux vn (head_deg vn p) p 0 s
end;
(* Monomial division operation. *)
fun mdiv (c1,m1) (c2,m2) =
(c1//c2,
map2 (fn n1 => fn n2 => if n1 < n2 then error "mdiv" else n1-n2) m1 m2);
(* Lowest common multiple of two monomials. *)
fun mlcm (c1,m1) (c2,m2) = (rat_1,map2 max m1 m2);
(* Reduce monomial cm by polynomial pol, returning replacement for cm. *)
fun reduce1 cm (pol,hpol) =
case pol of
[] => error "reduce1"
| cm1::cms => ((let val (c,m) = mdiv cm cm1 in
(grob_cmul (minus_rat c,m) cms,
Mmul((minus_rat c,m),hpol)) end)
handle ERROR _ => error "reduce1");
(* Try this for all polynomials in a basis. *)
fun tryfind f l =
case l of
[] => error "tryfind"
| (h::t) => ((f h) handle ERROR _ => tryfind f t);
fun reduceb cm basis = tryfind (fn p => reduce1 cm p) basis;
(* Reduction of a polynomial (always picking largest monomial possible). *)
fun reduce basis (pol,hist) =
case pol of
[] => (pol,hist)
| cm::ptl => ((let val (q,hnew) = reduceb cm basis in
reduce basis (grob_add q ptl,Add(hnew,hist)) end)
handle (ERROR _) =>
(let val (q,hist') = reduce basis (ptl,hist) in
(cm::q,hist') end));
(* Check for orthogonality w.r.t. LCM. *)
fun orthogonal l p1 p2 =
snd l = snd(grob_mmul (hd p1) (hd p2));
(* Compute S-polynomial of two polynomials. *)
fun spoly cm ph1 ph2 =
case (ph1,ph2) of
(([],h),p) => ([],h)
| (p,([],h)) => ([],h)
| ((cm1::ptl1,his1),(cm2::ptl2,his2)) =>
(grob_sub (grob_cmul (mdiv cm cm1) ptl1)
(grob_cmul (mdiv cm cm2) ptl2),
Add(Mmul(mdiv cm cm1,his1),
Mmul(mdiv (minus_rat(fst cm),snd cm) cm2,his2)));
(* Make a polynomial monic. *)
fun monic (pol,hist) =
if pol = [] then (pol,hist) else
let val (c',m') = hd pol in
(map (fn (c,m) => (c//c',m)) pol,
Mmul((rat_1 // c',map (K 0) m'),hist)) end;
(* The most popular heuristic is to order critical pairs by LCM monomial. *)
fun forder ((c1,m1),_) ((c2,m2),_) = morder_lt m1 m2;
fun poly_lt p q =
case (p,q) of
(p,[]) => false
| ([],q) => true
| ((c1,m1)::o1,(c2,m2)::o2) =>
c1 </ c2 orelse
c1 =/ c2 andalso ((morder_lt m1 m2) orelse m1 = m2 andalso poly_lt o1 o2);
fun align ((p,hp),(q,hq)) =
if poly_lt p q then ((p,hp),(q,hq)) else ((q,hq),(p,hp));
fun forall2 p l1 l2 =
case (l1,l2) of
([],[]) => true
| (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
| _ => false;
fun poly_eq p1 p2 =
forall2 (fn (c1,m1) => fn (c2,m2) => c1 =/ c2 andalso m1 = m2) p1 p2;
fun memx ((p1,h1),(p2,h2)) ppairs =
not (exists (fn ((q1,_),(q2,_)) => poly_eq p1 q1 andalso poly_eq p2 q2) ppairs);
(* Buchberger's second criterion. *)
fun criterion2 basis (lcm,((p1,h1),(p2,h2))) opairs =
exists (fn g => not(poly_eq (fst g) p1) andalso not(poly_eq (fst g) p2) andalso
can (mdiv lcm) (hd(fst g)) andalso
not(memx (align (g,(p1,h1))) (map snd opairs)) andalso
not(memx (align (g,(p2,h2))) (map snd opairs))) basis;
(* Test for hitting constant polynomial. *)
fun constant_poly p =
length p = 1 andalso forall (fn x=>x=0) (snd(hd p));
(* ------------------------------------------------------------------------- *)
(* Grobner basis algorithm. *)
(* ------------------------------------------------------------------------- *)
(* FIXME: try to get rid of mergesort? *)
fun merge ord l1 l2 =
case l1 of
[] => l2
| h1::t1 =>
case l2 of
[] => l1
| h2::t2 => if ord h1 h2 then h1::(merge ord t1 l2)
else h2::(merge ord l1 t2);
fun mergesort ord l =
let
fun mergepairs l1 l2 =
case (l1,l2) of
([s],[]) => s
| (l,[]) => mergepairs [] l
| (l,[s1]) => mergepairs (s1::l) []
| (l,(s1::s2::ss)) => mergepairs ((merge ord s1 s2)::l) ss
in if l = [] then [] else mergepairs [] (map (fn x => [x]) l)
end;
fun grobner_basis basis pairs =
(writeln (Int.toString(length basis)^" basis elements and "^
Int.toString(length pairs)^" critical pairs");
case pairs of
[] => basis
| (l,(p1,p2))::opairs =>
let val (sph as (sp,hist)) = monic (reduce basis (spoly l p1 p2))
in if sp = [] orelse criterion2 basis (l,(p1,p2)) opairs
then grobner_basis basis opairs
else if constant_poly sp then grobner_basis (sph::basis) []
else let val rawcps = map (fn p => (mlcm (hd(fst p)) (hd sp),align(p,sph)))
basis
val newcps = filter
(fn (l,(p,q)) => not(orthogonal l (fst p) (fst q)))
rawcps
in grobner_basis (sph::basis)
(merge forder opairs (mergesort forder newcps))
end
end);
(* ------------------------------------------------------------------------- *)
(* Interreduce initial polynomials. *)
(* ------------------------------------------------------------------------- *)
fun grobner_interreduce rpols ipols =
case ipols of
[] => map monic (rev rpols)
| p::ps => let val p' = reduce (rpols @ ps) p in
if fst p' = [] then grobner_interreduce rpols ps
else grobner_interreduce (p'::rpols) ps end;
(* ------------------------------------------------------------------------- *)
(* Overall function. *)
(* ------------------------------------------------------------------------- *)
fun grobner pols =
let val npols = map2 (fn p => fn n => (p,Start n)) pols (0 upto (length pols - 1))
val phists = filter (fn (p,_) => p <> []) npols
val bas = grobner_interreduce [] (map monic phists)
val prs0 = product bas bas
val prs1 = filter (fn ((x,_),(y,_)) => poly_lt x y) prs0
val prs2 = map (fn (p,q) => (mlcm (hd(fst p)) (hd(fst q)),(p,q))) prs1
val prs3 =
filter (fn (l,(p,q)) => not(orthogonal l (fst p) (fst q))) prs2 in
grobner_basis bas (mergesort forder prs3) end;
(* ------------------------------------------------------------------------- *)
(* Get proof of contradiction from Grobner basis. *)
(* ------------------------------------------------------------------------- *)
fun find p l =
case l of
[] => error "find"
| (h::t) => if p(h) then h else find p t;
fun grobner_refute pols =
let val gb = grobner pols in
snd(find (fn (p,h) => length p = 1 andalso forall (fn x=> x=0) (snd(hd p))) gb)
end;
(* ------------------------------------------------------------------------- *)
(* Turn proof into a certificate as sum of multipliers. *)
(* *)
(* In principle this is very inefficient: in a heavily shared proof it may *)
(* make the same calculation many times. Could put in a cache or something. *)
(* ------------------------------------------------------------------------- *)
fun assoc x l = snd(find (fn p => fst p = x) l);
fun resolve_proof vars prf =
case prf of
Start(~1) => []
| Start m => [(m,[(rat_1,map (K 0) vars)])]
| Mmul(pol,lin) =>
let val lis = resolve_proof vars lin in
map (fn (n,p) => (n,grob_cmul pol p)) lis end
| Add(lin1,lin2) =>
let val lis1 = resolve_proof vars lin1
val lis2 = resolve_proof vars lin2
val dom = distinct (op =) ((map fst lis1) union (map fst lis2))
in
map (fn n => let val a = ((assoc n lis1) handle _ => []) (* FIXME *)
val b = ((assoc n lis2) handle _ => []) in (* FIXME *)
(n,grob_add a b) end) dom end;
(* ------------------------------------------------------------------------- *)
(* Run the procedure and produce Weak Nullstellensatz certificate. *)
(* ------------------------------------------------------------------------- *)
fun grobner_weak vars pols =
let val cert = resolve_proof vars (grobner_refute pols)
val l =
fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert (rat_1) in
(l,map (fn (i,p) => (i,map (fn (d,m) => (l*/d,m)) p)) cert) end;
(* ------------------------------------------------------------------------- *)
(* Prove a polynomial is in ideal generated by others, using Grobner basis. *)
(* ------------------------------------------------------------------------- *)
fun grobner_ideal vars pols pol =
let val (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in
if pol <> [] then error "grobner_ideal: not in the ideal" else
resolve_proof vars h end;
(* ------------------------------------------------------------------------- *)
(* Produce Strong Nullstellensatz certificate for a power of pol. *)
(* ------------------------------------------------------------------------- *)
fun grobner_strong vars pols pol =
let val vars' = @{cterm "True"}::vars
val grob_z = [(rat_1,1::(map (fn x => 0) vars))]
val grob_1 = [(rat_1,(map (fn x => 0) vars'))]
fun augment p= map (fn (c,m) => (c,0::m)) p
val pols' = map augment pols
val pol' = augment pol
val allpols = (grob_sub (grob_mul grob_z pol') grob_1)::pols'
val (l,cert) = grobner_weak vars' allpols
val d = fold_rev (fold_rev (max o hd o snd) o snd) cert 0
fun transform_monomial (c,m) =
grob_cmul (c,tl m) (grob_pow vars pol (d - hd m))
fun transform_polynomial q = fold_rev (grob_add o transform_monomial) q []
val cert' = map (fn (c,q) => (c-1,transform_polynomial q))
(filter (fn (k,_) => k <> 0) cert) in
(d,l,cert') end;
fun string_of_pol vars pol =
foldl (fn ((c,m),s) => ((Rat.string_of_rat c)
^ "*(" ^
(snd (foldl
(fn (e,(i,s)) =>
(i+ 1,
(nth vars i
|>cterm_of (the_context())
|> string_of_cterm)^ "^"
^ (Int.toString e) ^" * " ^ s)) (0,"0") m))
^ ") + ") ^ s) "" pol;
(* ------------------------------------------------------------------------- *)
(* Overall parametrized universal procedure for (semi)rings. *)
(* We return an ideal_conv and the actual ring prover. *)
(* ------------------------------------------------------------------------- *)
fun refute_disj rfn tm =
case term_of tm of
Const("op |",_)$l$r =>
Drule.compose_single(refute_disj rfn (Thm.dest_arg tm),2,Drule.compose_single(refute_disj rfn (Thm.dest_arg1 tm),2,disjE))
| _ => rfn tm ;
val notnotD = @{thm "notnotD"};
fun mk_binop ct x y =
Thm.capply (Thm.capply ct x) y
val mk_comb = Thm.capply;
fun is_neg t =
case term_of t of
(Const("Not",_)$p) => true
| _ => false;
fun is_eq t =
case term_of t of
(Const("op =",_)$_$_) => true
| _ => false;
fun end_itlist f l =
case l of
[] => error "end_itlist"
| [x] => x
| (h::t) => f h (end_itlist f t);
val list_mk_binop = fn b => end_itlist (mk_binop b);
val list_dest_binop = fn b =>
let fun h acc t =
((let val (l,r) = dest_binop b t in h (h acc r) l end)
handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
in h []
end;
val strip_exists =
let fun h (acc, t) =
case (term_of t) of
Const("Ex",_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t)
in fn t => h ([],t)
end;
fun is_forall t =
case term_of t of
(Const("All",_)$Abs(_,_,_)) => true
| _ => false;
val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
val bool_simps = @{thms "bool_simps"};
val nnf_simps = @{thms "nnf_simps"};
val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps (@{thms "weak_dnf_simps"}));
val initial_conv =
Simplifier.rewrite
(HOL_basic_ss addsimps nnf_simps
addsimps [not_all, not_ex] addsimps map (fn th => th RS sym) (ex_simps @ all_simps));
val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
val cTrp = @{cterm "Trueprop"};
val cConj = @{cterm "op &"};
val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
val ASSUME = mk_comb cTrp #> assume;
val list_mk_conj = list_mk_binop cConj;
val conjs = list_dest_binop cConj;
val mk_neg = mk_comb cNot;
(** main **)
fun ring_and_ideal_conv
{vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules), idom}
dest_const mk_const ring_eq_conv ring_normalize_conv =
let
val [add_pat, mul_pat, pow_pat, zero_tm, one_tm] = sr_ops;
val [ring_add_tm, ring_mul_tm, ring_pow_tm] =
map (Thm.dest_fun o Thm.dest_fun) [add_pat, mul_pat, pow_pat];
val (ring_sub_tm, ring_neg_tm) =
(case r_ops of
[] => (@{cterm "True"}, @{cterm "True"})
| [sub_pat, neg_pat] => (Thm.dest_fun (Thm.dest_fun sub_pat), Thm.dest_fun neg_pat));
val [idom_thm, neq_thm] = idom;
val ring_dest_neg =
fn t => let val (l,r) = Thm.dest_comb t in
if could_unify(term_of l,term_of ring_neg_tm) then r else raise CTERM ("ring_dest_neg", [t])
end
val ring_mk_neg = fn tm => mk_comb (ring_neg_tm) (tm);
(*
fun ring_dest_inv t =
let val (l,r) = Thm.dest_comb t in
if could_unify(term_of l, term_of ring_inv_tm) then r else raise CTERM "ring_dest_inv"
end
*)
val ring_dest_add = dest_binop ring_add_tm;
val ring_mk_add = mk_binop ring_add_tm;
val ring_dest_sub = dest_binop ring_sub_tm;
val ring_mk_sub = mk_binop ring_sub_tm;
val ring_dest_mul = dest_binop ring_mul_tm;
val ring_mk_mul = mk_binop ring_mul_tm;
(* val ring_dest_div = dest_binop ring_div_tm;
val ring_mk_div = mk_binop ring_div_tm;*)
val ring_dest_pow = dest_binop ring_pow_tm;
val ring_mk_pow = mk_binop ring_pow_tm ;
fun grobvars tm acc =
if can dest_const tm then acc
else if can ring_dest_neg tm then grobvars (Thm.dest_arg tm) acc
else if can ring_dest_pow tm then grobvars (Thm.dest_arg1 tm) acc
else if can ring_dest_add tm orelse can ring_dest_sub tm
orelse can ring_dest_mul tm
then grobvars (Thm.dest_arg1 tm) (grobvars (Thm.dest_arg tm) acc)
(* else if can ring_dest_inv tm
then
let val gvs = grobvars (Thm.dest_arg tm) [] in
if gvs = [] then acc else tm::acc
end
else if can ring_dest_div tm then
let val lvs = grobvars (Thm.dest_arg1 tm) acc
val gvs = grobvars (Thm.dest_arg tm) []
in if gvs = [] then lvs else tm::acc
end *)
else tm::acc ;
fun grobify_term vars tm =
((if not (member (op aconvc) vars tm) then raise CTERM ("Not a variable", [tm]) else
[(rat_1,map (fn i => if i aconvc tm then 1 else 0) vars)])
handle CTERM _ =>
((let val x = dest_const tm
in if x =/ rat_0 then [] else [(x,map (fn v => 0) vars)]
end)
handle ERROR _ =>
((grob_neg(grobify_term vars (ring_dest_neg tm)))
handle CTERM _ =>
(
(* (grob_inv(grobify_term vars (ring_dest_inv tm)))
handle CTERM _ => *)
((let val (l,r) = ring_dest_add tm
in grob_add (grobify_term vars l) (grobify_term vars r)
end)
handle CTERM _ =>
((let val (l,r) = ring_dest_sub tm
in grob_sub (grobify_term vars l) (grobify_term vars r)
end)
handle CTERM _ =>
((let val (l,r) = ring_dest_mul tm
in grob_mul (grobify_term vars l) (grobify_term vars r)
end)
handle CTERM _ =>
(
(* (let val (l,r) = ring_dest_div tm
in grob_div (grobify_term vars l) (grobify_term vars r)
end)
handle CTERM _ => *)
((let val (l,r) = ring_dest_pow tm
in grob_pow vars (grobify_term vars l) ((term_of #> HOLogic.dest_number #> snd) r)
end)
handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
val eq_tm = idom_thm |> concl |> Thm.dest_arg |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_fun ;
(*ring_integral |> hd |> concl |> Thm.dest_arg
|> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_fun; *)
val dest_eq = dest_binop eq_tm;
fun grobify_equation vars tm =
let val (l,r) = dest_binop eq_tm tm
in grob_sub (grobify_term vars l) (grobify_term vars r)
end;
fun grobify_equations tm =
let
val cjs = conjs tm
val rawvars = fold_rev (fn eq => fn a =>
grobvars (Thm.dest_arg1 eq) (grobvars (Thm.dest_arg eq) a)) cjs []
val vars = sort (fn (x, y) => Term.term_ord(term_of x,term_of y))
(distinct (op aconvc) rawvars)
in (vars,map (grobify_equation vars) cjs)
end;
val holify_polynomial =
let fun holify_varpow (v,n) =
if n = 1 then v else ring_mk_pow v (mk_cnumber @{ctyp "nat"} n) (* FIXME *)
fun holify_monomial vars (c,m) =
let val xps = map holify_varpow (filter (fn (_,n) => n <> 0) (vars ~~ m))
in end_itlist ring_mk_mul (mk_const c :: xps)
end
fun holify_polynomial vars p =
if p = [] then mk_const (rat_0)
else end_itlist ring_mk_add (map (holify_monomial vars) p)
in holify_polynomial
end ;
val idom_rule = simplify (HOL_basic_ss addsimps [idom_thm]);
fun prove_nz n = eqF_elim
(ring_eq_conv(mk_binop eq_tm (mk_const n) (mk_const(rat_0))));
val neq_01 = prove_nz (rat_1);
fun neq_rule n th = [prove_nz n, th] MRS neq_thm;
fun mk_add th1 = combination(Drule.arg_cong_rule ring_add_tm th1);
fun refute tm =
if tm aconvc false_tm then ASSUME tm else
let
val (nths0,eths0) = List.partition (is_neg o concl) (conjuncts(ASSUME tm))
val nths = filter (is_eq o Thm.dest_arg o concl) nths0
val eths = filter (is_eq o concl) eths0
in
if null eths then
let
val th1 = end_itlist (fn th1 => fn th2 => idom_rule(conji th1 th2)) nths
val th2 = Conv.fconv_rule
((arg_conv #> arg_conv)
(binop_conv ring_normalize_conv)) th1
val conc = th2 |> concl |> Thm.dest_arg
val (l,r) = conc |> dest_eq
in implies_intr (mk_comb cTrp tm)
(equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th2))
(reflexive l |> mk_object_eq))
end
else
let
val (vars,l,cert,noteqth) =(
if null nths then
let val (vars,pols) = grobify_equations(list_mk_conj(map concl eths))
val (l,cert) = grobner_weak vars pols
in (vars,l,cert,neq_01)
end
else
let
val nth = end_itlist (fn th1 => fn th2 => idom_rule(conji th1 th2)) nths
val (vars,pol::pols) =
grobify_equations(list_mk_conj(Thm.dest_arg(concl nth)::map concl eths))
val (deg,l,cert) = grobner_strong vars pols pol
val th1 = Conv.fconv_rule((arg_conv o arg_conv)(binop_conv ring_normalize_conv)) nth
val th2 = funpow deg (idom_rule o conji th1) neq_01
in (vars,l,cert,th2)
end)
val _ = writeln ("Translating certificate to HOL inferences")
val cert_pos = map (fn (i,p) => (i,filter (fn (c,m) => c >/ rat_0) p)) cert
val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (minus_rat c,m))
(filter (fn (c,m) => c </ rat_0) p))) cert
val herts_pos = map (fn (i,p) => (i,holify_polynomial vars p)) cert_pos
val herts_neg = map (fn (i,p) => (i,holify_polynomial vars p)) cert_neg
fun thm_fn pols =
if null pols then reflexive(mk_const rat_0) else
end_itlist mk_add
(map (fn (i,p) => Drule.arg_cong_rule (mk_comb ring_mul_tm p) (nth eths i |> mk_meta_eq)) pols)
val th1 = thm_fn herts_pos
val th2 = thm_fn herts_neg
val th3 = conji(mk_add (symmetric th1) th2 |> mk_object_eq) noteqth
val th4 = Conv.fconv_rule ((arg_conv o arg_conv o binop_conv) ring_normalize_conv)
(neq_rule l th3)
val (l,r) = dest_eq(Thm.dest_arg(concl th4))
in implies_intr (mk_comb cTrp tm)
(equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th4))
(reflexive l |> mk_object_eq))
end
end
fun ring tm =
let
fun mk_forall x p =
mk_comb (Drule.cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (Thm.cabs x p)
val avs = cterm_frees tm
val P' = fold mk_forall avs tm
val th1 = initial_conv(mk_neg P')
val (evs,bod) = strip_exists(concl th1) in
if is_forall bod then error "ring: non-universal formula"
else
let
val th1a = weak_dnf_conv bod
val boda = concl th1a
val th2a = refute_disj refute boda
val th2b = [mk_object_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans
val th2 = fold (fn v => fn th => (forall_intr v th) COMP allI) evs (th2b RS PFalse)
val th3 = equal_elim
(Simplifier.rewrite (HOL_basic_ss addsimps [not_ex RS sym])
(th2 |> cprop_of)) th2
in specl avs
([[[mk_object_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS notnotD)
end
end
fun ideal tms tm ord =
let
val rawvars = fold_rev grobvars (tm::tms) []
val vars = sort ord (distinct (fn (x,y) => (term_of x) aconv (term_of y)) rawvars)
val pols = map (grobify_term vars) tms
val pol = grobify_term vars tm
val cert = grobner_ideal vars pols pol
in map (fn n => let val p = assocd n cert [] in holify_polynomial vars p end)
(0 upto (length pols-1))
end
in (ring,ideal)
end;
fun find_term bounds tm =
(case term_of tm of
Const ("op =", T) $ _ $ _ =>
if domain_type T = HOLogic.boolT then find_args bounds tm
else Thm.dest_arg tm
| Const ("Not", _) $ _ => find_term bounds (Thm.dest_arg tm)
| Const ("All", _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const ("Ex", _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const ("op &", _) $ _ $ _ => find_args bounds tm
| Const ("op |", _) $ _ $ _ => find_args bounds tm
| Const ("op -->", _) $ _ $ _ => find_args bounds tm
| _ => raise TERM ("find_term", []))
and find_args bounds tm =
let val (t, u) = Thm.dest_binop tm
in (find_term bounds t handle TERM _ => find_term bounds u) end
and find_body bounds b =
let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b
in find_term (bounds + 1) b' end;
fun ring_conv ctxt form =
(case try (find_term 0 (* FIXME !? *)) form of
NONE => reflexive form
| SOME tm =>
(case NormalizerData.match ctxt tm of
NONE => reflexive form
| SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
fst (ring_and_ideal_conv theory
dest_const (mk_const (Thm.ctyp_of_term tm)) ring_eq_conv
(semiring_normalize_wrapper res)) form));
end;