(* Title: HOL/Tools/groebner.ML Author: Amine Chaieb, TU Muenchen
*)
signature GROEBNER = sig val ring_and_ideal_conv:
{idom: thm list, ring: cterm list * thm list, field: cterm list * thm list,
vars: cterm list, semiring: cterm list * thm list, ideal : thm list} ->
(cterm -> Rat.rat) -> (Rat.rat -> cterm) ->
conv -> conv ->
{ring_conv: Proof.context -> conv,
simple_ideal: cterm list -> cterm -> cterm ord -> cterm list,
multi_ideal: cterm list -> cterm list -> cterm list -> (cterm * cterm) list,
poly_eq_ss: simpset, unwind_conv: Proof.context -> conv} val ring_tac: thm list -> thm list -> Proof.context -> int -> tactic val ideal_tac: thm list -> thm list -> Proof.context -> int -> tactic val algebra_tac: thm list -> thm list -> Proof.context -> int -> tactic end
structure Groebner : GROEBNER = struct
val concl = Thm.cprop_of #> Thm.dest_arg;
fun is_binop ct ct' =
(case Thm.term_of ct' of
c $ _ $ _ => Thm.term_of ct aconv c
| _ => false);
fun dest_binary ct ct' = if is_binop ct ct' then Thm.dest_binop ct' elseraise CTERM ("dest_binary: bad binop", [ct, ct'])
val denominator_rat = Rat.dest #> snd #> Rat.of_int; fun int_of_rat a = case Rat.dest a of (i,1) => i | _ => error "int_of_rat: not an int"; val lcm_rat = fn x => fn y => Rat.of_int (Integer.lcm (int_of_rat x) (int_of_rat y));
val (PFalse, PFalse') = letval 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= letfun 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 = Integer.sum m1 val n2 = Integer.sum m2 in
n1 < n2 orelse n1 = n2 andalso lexorder m1 m2 end;
(* Arithmetic on canonical polynomials. *)
fun grob_neg l = map (fn (c,m) => (Rat.neg 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 letval c = c1 + c2 val rest = grob_add o1 o2 in if c = @0 then rest else (c,m1)::rest end elseif morder_lt m2 m1 then (c1,m1)::(grob_add o1 l2) else (c2,m2)::(grob_add l1 o2);
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 = @0 then error "grob_inv: division by zero" else [(@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 = @0 then error "grob_div: division by zero" else grob_cmul (@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" elseif n = 0 then [(@1,map (K 0) vars)] else grob_mul l (grob_pow vars l (n - 1));
(* 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 (_,m1) (_,m2) = (@1, ListPair.map Int.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 => ((letval (c,m) = mdiv cm cm1 in
(grob_cmul (~ c, m) cms,
Mmul ((~ 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 => ((letval (q,hnew) = reduceb cm basis in
reduce basis (grob_add q ptl,Add(hnew,hist)) end) handle (ERROR _) =>
(letval (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),_) => ([],h)
| (_,([],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 (~ (fst cm),snd cm) cm2,his2)));
(* Make a polynomial monic. *)
fun monic (pol,hist) = if null pol then (pol,hist) else letval (c',m') = hd pol in
(map (fn (c,m) => (c / c',m)) pol,
Mmul((@1 / c',map (K 0) m'),hist)) end;
(* The most popular heuristic is to order critical pairs by LCM monomial. *)
fun forder ((_,m1),_) ((_,m2),_) = morder_lt m1 m2;
fun poly_lt p q = case (p,q) of
(_,[]) => false
| ([],_) => 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 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 => iford 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 inif null l then [] else mergepairs [] (map (fn x => [x]) l) end;
fun grobner_basis basis pairs = case pairs of
[] => basis
| (l,(p1,p2))::opairs => letval (sph as (sp,_)) = monic (reduce basis (spoly l p1 p2)) in if null sp orelse criterion2 basis (l,(p1,p2)) opairs then grobner_basis basis opairs elseif 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 => letval p' = reduce (rpols @ ps) p in if null (fst p') then grobner_interreduce rpols ps else grobner_interreduce (p'::rpols) ps end;
(* Overall function. *)
fun grobner pols = letval npols = map_index (fn (n, p) => (p, Start n)) pols val phists = filter (fn (p,_) => not (null p)) npols val bas = grobner_interreduce [] (map monic phists) val prs0 = map_product pair 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. *)
funfind p l = case l of
[] => error "find"
| (h::t) => if p(h) then h elsefind p t;
fun grobner_refute pols = letval gb = grobner pols in
snd(find (fn (p,_) => 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 resolve_proof vars prf = case prf of
Start(~1) => []
| Start m => [(m,[(@1,map (K 0) vars)])]
| Mmul(pol,lin) => letval lis = resolve_proof vars lin in map (fn (n,p) => (n,grob_cmul pol p)) lis end
| Add(lin1,lin2) => letval lis1 = resolve_proof vars lin1 val lis2 = resolve_proof vars lin2 val dom = distinct (op =) (union (op =) (map fst lis1) (map fst lis2)) in map (fn n => letval a = these (AList.lookup (op =) lis1 n) val b = these (AList.lookup (op =) lis2 n) in (n,grob_add a b) end) dom end;
(* Run the procedure and produce Weak Nullstellensatz certificate. *)
fun grobner_weak vars pols = letval cert = resolve_proof vars (grobner_refute pols) val l =
fold_rev (fold_rev (lcm_rat o denominator_rat o fst) o snd) cert @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 = letval (pol',h) = reduce (grobner pols) (grob_neg pol,Start(~1)) in ifnot (null 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 = letval vars' = \<^cterm>\True\::vars val grob_z = [(@1, 1::(map (K 0) vars))] val grob_1 = [(@1, (map (K 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 (fold (Integer.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;
(* Overall parametrized universal procedure for (semi)rings. *) (* We return an ideal_conv and the actual ring prover. *)
fun refute_disj rfn tm = case Thm.term_of tm of
\<^Const_>\<open>disj for _ _\<close> =>
Drule.compose
(refute_disj rfn (Thm.dest_arg tm), 2,
Drule.compose (refute_disj rfn (Thm.dest_arg1 tm), 2, disjE))
| _ => rfn tm ;
fun is_neg t = case Thm.term_of t of
\<^Const_>\<open>Not for _\<close> => true
| _ => false;
fun is_eq t = case Thm.term_of t of
\<^Const_>\<open>HOL.eq _ for _ _\<close> => 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 (Thm.mk_binop b);
val list_dest_binop = fn b => letfun h acc t =
((letval (l,r) = dest_binary b t in h (h acc r) l end) handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *) in h [] end;
val strip_exists = letfun h (acc, t) = case Thm.term_of t of
\<^Const_>\<open>Ex _ for \<open>Abs _\<close>\<close> =>
h (Thm.dest_abs_global (Thm.dest_arg t) |>> (fn v => v::acc))
| _ => (acc,t) in fn t => h ([],t) end;
fun is_forall t = case Thm.term_of t of
\<^Const_>\<open>All _ for \<open>Abs _\<close>\<close> => true
| _ => false;
val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
val list_mk_conj = list_mk_binop \<^cterm>\<open>conj\<close>; val conjs = list_dest_binop \<^cterm>\<open>conj\<close>; val mk_neg = Thm.apply \<^cterm>\<open>Not\<close>;
fun striplist dest = let fun h acc x = casetry dest x of
SOME (a,b) => h (h acc b) a
| NONE => x::acc in h [] end; fun list_mk_binop b = foldr1 (fn (s,t) => Thm.apply (Thm.apply b s) t);
val eq_commute = mk_meta_eq @{thm eq_commute};
fun sym_conv eq = letval (l,r) = Thm.dest_binop eq in Thm.instantiate' [SOME (Thm.ctyp_of_cterm l)] [SOME l, SOME r] eq_commute end;
(* FIXME : copied from cqe.ML -- complex QE*) fun conjuncts ct = case Thm.term_of ct of
\<^Const_>\<open>conj for _ _\<close> => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
| _ => [ct];
fun fold1 f = foldr1 (uncurry f);
fun mk_conj_tab th = letfun h acc th = case Thm.prop_of th of
\<^Const_>\<open>Trueprop for \<^Const_>\<open>conj for _ _\<close>\<close> =>
h (h acc (th RS conjunct2)) (th RS conjunct1)
| \<^Const_>\<open>Trueprop for p\<close> => (p,th)::acc in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
fun is_conj \<^Const_>\<open>conj for _ _\<close> = true
| is_conj _ = false;
fun prove_conj tab cjs = case cjs of
[c] => if is_conj (Thm.term_of c) then prove_conj tab (conjuncts c) else tab c
| c::cs => conjI OF [prove_conj tab [c], prove_conj tab cs];
fun conj_ac_rule eq = let val (l,r) = Thm.dest_equals eq val ctabl = mk_conj_tab (Thm.assume (HOLogic.mk_judgment l)) val ctabr = mk_conj_tab (Thm.assume (HOLogic.mk_judgment r)) fun tabl c = the (Termtab.lookup ctabl (Thm.term_of c)) fun tabr c = the (Termtab.lookup ctabr (Thm.term_of c)) val thl = prove_conj tabl (conjuncts r) |> implies_intr_hyps val thr = prove_conj tabr (conjuncts l) |> implies_intr_hyps val eqI = Thm.instantiate' [] [SOME l, SOME r] @{thm iffI} in Thm.implies_elim (Thm.implies_elim eqI thl) thr |> mk_meta_eq end;
(* END FIXME.*)
(* Conversion for the equivalence of existential statements where
EX quantifiers are rearranged differently *) fun ext ctxt T = Thm.cterm_of ctxt \<^Const>\<open>Ex T\<close> fun mk_ex ctxt v t = Thm.apply (ext ctxt (Thm.typ_of_cterm v)) (Thm.lambda v t)
fun choose v th th' = case Thm.concl_of th of
\<^Const_>\<open>Trueprop for \<^Const_>\<open>Ex _ for _\<close>\<close> => let val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p) val th0 = Conv.fconv_rule (Thm.beta_conversion true)
(Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE) val pv = (Thm.rhs_of o Thm.beta_conversion true)
(Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v)) val th1 = Thm.forall_intr v (Thm.implies_intr pv th') in Thm.implies_elim (Thm.implies_elim th0 th) th1 end
| _ => error ""(* FIXME ? *)
fun simple_choose ctxt v th =
choose v (Thm.assume ((HOLogic.mk_judgment o mk_ex ctxt v)
(Thm.dest_arg (hd (Thm.chyps_of th))))) th
fun mkexi v th = let val p = Thm.lambda v (Thm.dest_arg (Thm.cprop_of th)) in Thm.implies_elim
(Conv.fconv_rule (Thm.beta_conversion true)
(Thm.instantiate' [SOME (Thm.ctyp_of_cterm v)] [SOME p, SOME v] @{thm exI}))
th end fun ex_eq_conv ctxt t = let val (p0,q0) = Thm.dest_binop t val (vs',P) = strip_exists p0 val (vs,_) = strip_exists q0 val th = Thm.assume (HOLogic.mk_judgment P) val th1 = implies_intr_hyps (fold (simple_choose ctxt) vs' (fold mkexi vs th)) val th2 = implies_intr_hyps (fold (simple_choose ctxt) vs (fold mkexi vs' th)) val p = (Thm.dest_arg o Thm.dest_arg1 o Thm.cprop_of) th1 val q = (Thm.dest_arg o Thm.dest_arg o Thm.cprop_of) th1 in Thm.implies_elim (Thm.implies_elim (Thm.instantiate' [] [SOME p, SOME q] iffI) th1) th2
|> mk_meta_eq end;
fun getname v = case Thm.term_of v of
Free(s,_) => s
| Var ((s,_),_) => s
| _ => "x" fun mk_eq s t = Thm.apply (Thm.apply \<^cterm>\<open>(\<equiv>) :: bool \<Rightarrow> _\<close> s) t fun mk_exists ctxt v th = Drule.arg_cong_rule (ext ctxt (Thm.typ_of_cterm v))
(Thm.abstract_rule (getname v) v th) fun simp_ex_conv ctxt =
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt
|> Simplifier.add_simps @{thms simp_thms(39)})
fun free_in v t = Cterms.defined (Cterms.build (Drule.add_frees_cterm t)) v;
val vsubst = let fun vsubst (t,v) tm =
(Thm.rhs_of o Thm.beta_conversion false) (Thm.apply (Thm.lambda v tm) t) in fold vsubst end;
(** main **)
fun ring_and_ideal_conv
{vars = _, semiring = (sr_ops, _), ring = (r_ops, _),
field = (f_ops, _), idom, ideal}
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_fun2 [add_pat, mul_pat, pow_pat];
val (ring_sub_tm, ring_neg_tm) =
(case r_ops of
[sub_pat, neg_pat] => (Thm.dest_fun2 sub_pat, Thm.dest_fun neg_pat)
|_ => (\<^cterm>\<open>True\<close>, \<^cterm>\<open>True\<close>));
val (field_div_tm, field_inv_tm) =
(case f_ops of
[div_pat, inv_pat] => (Thm.dest_fun2 div_pat, Thm.dest_fun inv_pat)
| _ => (\<^cterm>\<open>True\<close>, \<^cterm>\<open>True\<close>));
val [idom_thm, neq_thm] = idom; val [idl_sub, idl_add0] = if length ideal = 2 then ideal else [eq_commute, eq_commute] fun ring_dest_neg t = letval (l,r) = Thm.dest_comb t inif Term.could_unify(Thm.term_of l, Thm.term_of ring_neg_tm) then r elseraise CTERM ("ring_dest_neg", [t]) end
fun field_dest_inv t = letval (l,r) = Thm.dest_comb t in if Term.could_unify (Thm.term_of l, Thm.term_of field_inv_tm) then r elseraise CTERM ("field_dest_inv", [t]) end val ring_dest_add = dest_binary ring_add_tm; val ring_mk_add = Thm.mk_binop ring_add_tm; val ring_dest_sub = dest_binary ring_sub_tm; val ring_dest_mul = dest_binary ring_mul_tm; val ring_mk_mul = Thm.mk_binop ring_mul_tm; val field_dest_div = dest_binary field_div_tm; val ring_dest_pow = dest_binary ring_pow_tm; val ring_mk_pow = Thm.mk_binop ring_pow_tm ; fun grobvars tm acc = if can dest_const tm then acc elseif can ring_dest_neg tm then grobvars (Thm.dest_arg tm) acc elseif can ring_dest_pow tm then grobvars (Thm.dest_arg1 tm) acc elseif 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) elseif can field_dest_inv tm then letval gvs = grobvars (Thm.dest_arg tm) [] inif null gvs then acc else tm::acc end elseif can field_dest_div tm then letval lvs = grobvars (Thm.dest_arg1 tm) acc val gvs = grobvars (Thm.dest_arg tm) [] inif null gvs then lvs else tm::acc end else tm::acc ;
fun grobify_term vars tm =
((ifnot (member (op aconvc) vars tm) thenraise CTERM ("Not a variable", [tm]) else
[(@1, map (fn i => if i aconvc tm then 1 else 0) vars)]) handle CTERM _ =>
((letval x = dest_const tm inif x = @0 then [] else [(x,map (K 0) vars)] end) handle ERROR _ =>
((grob_neg(grobify_term vars (ring_dest_neg tm))) handle CTERM _ =>
(
(grob_inv(grobify_term vars (field_dest_inv tm))) handle CTERM _ =>
((letval (l,r) = ring_dest_add tm in grob_add (grobify_term vars l) (grobify_term vars r) end) handle CTERM _ =>
((letval (l,r) = ring_dest_sub tm in grob_sub (grobify_term vars l) (grobify_term vars r) end) handle CTERM _ =>
((letval (l,r) = ring_dest_mul tm in grob_mul (grobify_term vars l) (grobify_term vars r) end) handle CTERM _ =>
( (letval (l,r) = field_dest_div tm in grob_div (grobify_term vars l) (grobify_term vars r) end) handle CTERM _ =>
((letval (l,r) = ring_dest_pow tm in grob_pow vars (grobify_term vars l) ((Thm.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_fun2; val dest_eq = dest_binary eq_tm;
fun grobify_equation vars tm = letval (l,r) = dest_binary 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 Thm.term_ord (distinct (op aconvc) rawvars) in (vars,map (grobify_equation vars) cjs) end;
val holify_polynomial = letfun holify_varpow (v,n) = if n = 1 then v else ring_mk_pow v (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> n) (* FIXME *) fun holify_monomial vars (c,m) = letval 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 null p then mk_const @0 else end_itlist ring_mk_add (map (holify_monomial vars) p) in holify_polynomial end ;
fun idom_rule ctxt = simplify (put_simpset HOL_basic_ss ctxt |> Simplifier.add_simp idom_thm); fun prove_nz n = eqF_elim (ring_eq_conv (Thm.mk_binop eq_tm (mk_const n) (mk_const @0))); val neq_01 = prove_nz @1; fun neq_rule n th = [prove_nz n, th] MRS neq_thm; fun mk_add th1 = Thm.combination (Drule.arg_cong_rule ring_add_tm th1);
fun refute ctxt tm = if Thm.term_of tm aconv \<^Const>\<open>False\<close> then Thm.assume (HOLogic.mk_judgment tm) else
((let val (nths0,eths0) = List.partition (is_neg o concl)
(HOLogic.conj_elims (Thm.assume (HOLogic.mk_judgment 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 ctxt (HOLogic.conj_intr th1 th2)) nths val th2 =
Conv.fconv_rule
((Conv.arg_conv #> Conv.arg_conv) (Conv.binop_conv ring_normalize_conv)) th1 val conc = th2 |> concl |> Thm.dest_arg val (l,_) = conc |> dest_eq in Thm.implies_intr (HOLogic.mk_judgment tm)
(Thm.equal_elim (Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (eqF_intr th2))
(HOLogic.mk_obj_eq (Thm.reflexive l))) end else let val (vars,l,cert,noteqth) =( if null nths then letval (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 ctxt (HOLogic.conj_intr 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 ((Conv.arg_conv o Conv.arg_conv) (Conv.binop_conv ring_normalize_conv)) nth val th2 = funpow deg (idom_rule ctxt o HOLogic.conj_intr th1) neq_01 in (vars,l,cert,th2) end) val cert_pos = map (fn (i,p) => (i,filter (fn (c,_) => c > @0) p)) cert val cert_neg = map (fn (i,p) => (i,map (fn (c,m) => (~ c,m))
(filter (fn (c,_) => c < @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 Thm.reflexive(mk_const @0) else
end_itlist mk_add
(map (fn (i,p) => Drule.arg_cong_rule (Thm.apply 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 = HOLogic.conj_intr (HOLogic.mk_obj_eq (mk_add (Thm.symmetric th1) th2)) noteqth val th4 =
Conv.fconv_rule ((Conv.arg_conv o Conv.arg_conv o Conv.binop_conv) ring_normalize_conv)
(neq_rule l th3) val (l, _) = dest_eq(Thm.dest_arg(concl th4)) in Thm.implies_intr (HOLogic.mk_judgment tm)
(Thm.equal_elim (Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (eqF_intr th4))
(HOLogic.mk_obj_eq (Thm.reflexive l))) end end) handle ERROR _ => raise CTERM ("Groebner-refute: unable to refute",[tm]))
fun ring ctxt tm = let fun mk_forall x p = let val T = Thm.typ_of_cterm x; valall = Thm.cterm_of ctxt \<^Const>\<open>All T\<close> in Thm.apply all (Thm.lambda x p) end val avs = Cterms.build (Drule.add_frees_cterm tm) val P' = fold mk_forall (Cterms.list_set_rev avs) tm val th1 = initial_conv ctxt (mk_neg P') val (evs,bod) = strip_exists(concl th1) in if is_forall bod thenraise CTERM("ring: non-universal formula",[tm]) else let val th1a = weak_dnf_conv ctxt bod val boda = concl th1a val th2a = refute_disj (refute ctxt) boda val th2b = [HOLogic.mk_obj_eq th1a, (th2a COMP notI) COMP PFalse'] MRS trans val th2 = fold (fn v => fn th => (Thm.forall_intr v th) COMP allI) evs (th2b RS PFalse) val th3 =
Thm.equal_elim
(Simplifier.rewrite (put_simpset HOL_basic_ss ctxt |> Simplifier.add_simp (not_ex RS sym))
(th2 |> Thm.cprop_of)) th2 in specl (Cterms.list_set_rev avs)
([[[HOLogic.mk_obj_eq th1, th3 RS PFalse'] MRS trans] MRS PFalse] MRS @{thm notnotD}) end end fun ideal tms tm ord = let val rawvars = fold_rev grobvars (tm::tms) [] val vars = sort ord (distinct (fn (x,y) => (Thm.term_of x) aconv (Thm.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_range (fn n => these (AList.lookup (op =) cert n) |> holify_polynomial vars)
(length pols) end
fun poly_eq_conv t = letval (a,b) = Thm.dest_binop t in Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv ring_normalize_conv))
(Thm.instantiate' [] [SOME a, SOME b] idl_sub) end
val poly_eq_simproc = let fun proc ct = letval th = poly_eq_conv ct inif Thm.is_reflexive th then NONE else SOME th end in
Simplifier.cert_simproc (Thm.theory_of_thm idl_sub)
{name = "poly_eq_simproc",
kind = Simproc,
lhss = [Thm.term_of (Thm.lhs_of idl_sub)],
proc = Morphism.entity (fn _ => fn _ => proc),
identifier = []} end;
local fun is_defined v t = let val mons = striplist(dest_binary ring_add_tm) t in member (op aconvc) mons v andalso
forall (fn m => v aconvc m
orelse not(Cterms.defined (Cterms.build (Drule.add_frees_cterm m)) v)) mons end
fun isolate_variable vars tm = let val th = poly_eq_conv tm val th' = (sym_conv then_conv poly_eq_conv) tm val (v,th1) = case find_first(fn v=> is_defined v (Thm.dest_arg1 (Thm.rhs_of th))) vars of
SOME v => (v,th')
| NONE => (the (find_first
(fn v => is_defined v (Thm.dest_arg1 (Thm.rhs_of th'))) vars) ,th) val th2 = Thm.transitive th1
(Thm.instantiate' [] [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v]
idl_add0) in Conv.fconv_rule(funpow 2 Conv.arg_conv ring_normalize_conv) th2 end in fun unwind_polys_conv ctxt tm = let val (vars,bod) = strip_exists tm val cjs = striplist (dest_binary \<^cterm>\<open>HOL.conj\<close>) bod val th1 = (the (get_first (try (isolate_variable vars)) cjs) handleOption.Option => raise CTERM ("unwind_polys_conv",[tm])) val eq = Thm.lhs_of th1 val bod' = list_mk_binop \<^cterm>\HOL.conj\ (eq::(remove (op aconvc) eq cjs)) val th2 = conj_ac_rule (mk_eq bod bod') val th3 =
Thm.transitive th2
(Drule.binop_cong_rule \<^cterm>\<open>HOL.conj\<close> th1
(Thm.reflexive (Thm.dest_arg (Thm.rhs_of th2)))) val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3)) val th4 = Conv.fconv_rule (Conv.arg_conv (simp_ex_conv ctxt)) (mk_exists ctxt v th3) val th5 = ex_eq_conv ctxt (mk_eq tm (fold (mk_ex ctxt) (remove op aconvc v vars) (Thm.lhs_of th4))) in Thm.transitive th5 (fold (mk_exists ctxt) (remove op aconvc v vars) th4) end; end
local fun scrub_var v m = let val ps = striplist ring_dest_mul m val ps' = remove op aconvc v ps inif null ps' then one_tm else fold1 ring_mk_mul ps' end fun find_multipliers v mons = let val mons1 = filter (fn m => free_in v m) mons val mons2 = map (scrub_var v) mons1 inif null mons2 then zero_tm else fold1 ring_mk_add mons2 end
fun isolate_monomials vars tm = let val (vmons, cmons) = List.partition (fn m => letval frees = Cterms.build (Drule.add_frees_cterm m) inexists (Cterms.defined frees) vars end) (striplist ring_dest_add tm) val cofactors = map (fn v => find_multipliers v vmons) vars val cnc = if null cmons then zero_tm else Thm.apply ring_neg_tm
(list_mk_binop ring_add_tm cmons) in (cofactors,cnc) end;
fun isolate_variables evs ps eq = let val vars = filter (fn v => free_in v eq) evs val (qs,p) = isolate_monomials vars eq val rs = ideal (qs @ ps) p Thm.term_ord in (eq, take (length qs) rs ~~ vars) end; fun subst_in_poly i p = Thm.rhs_of (ring_normalize_conv (vsubst i p)); in fun solve_idealism evs ps eqs = if null evs then [] else let val (eq,cfs) = get_first (try (isolate_variables evs ps)) eqs |> the val evs' = subtract op aconvc evs (map snd cfs) val eqs' = map (subst_in_poly cfs) (remove op aconvc eq eqs) in cfs @ solve_idealism evs' ps eqs' end; end;
fun find_term tm ctxt =
(case Thm.term_of tm of
\<^Const_>\<open>HOL.eq T for _ _\<close> => if T = \<^Type>\<open>bool\<close> then find_args tm ctxt else (Thm.dest_arg tm, ctxt)
| \<^Const_>\<open>Not for _\<close> => find_term (Thm.dest_arg tm) ctxt
| \<^Const_>\<open>All _ for _\<close> => find_body (Thm.dest_arg tm) ctxt
| \<^Const_>\<open>Ex _ for _\<close> => find_body (Thm.dest_arg tm) ctxt
| \<^Const_>\<open>conj for _ _\<close> => find_args tm ctxt
| \<^Const_>\<open>disj for _ _\<close> => find_args tm ctxt
| \<^Const_>\<open>implies for _ _\<close> => find_args tm ctxt
| \<^Const_>\<open>Pure.imp for _ _\<close> => find_args tm ctxt
| \<^Const_>\<open>Pure.eq _ for _ _\<close> => find_args tm ctxt
| \<^Const_>\<open>Trueprop for _\<close> => find_term (Thm.dest_arg tm) ctxt
| _ => raise TERM ("find_term", [])) and find_args tm ctxt = letval (t, u) = Thm.dest_binop tm in (find_term t ctxt handle TERM _ => find_term u ctxt) end and find_body b ctxt = letval ((_, b'), ctxt') = Variable.dest_abs_cterm b ctxt in find_term b' ctxt'end;
fun get_ring_ideal_convs ctxt form = case \<^try>\<open>find_term form ctxt\<close> of
NONE => NONE
| SOME (tm, ctxt') =>
(case Semiring_Normalizer.match ctxt' tm of
NONE => NONE
| SOME (res as (theory, {is_const = _, dest_const,
mk_const, conv = ring_eq_conv})) =>
SOME (ring_and_ideal_conv theory
dest_const (mk_const (Thm.ctyp_of_cterm tm)) (ring_eq_conv ctxt')
(Semiring_Normalizer.semiring_normalize_wrapper ctxt' res)))
val claset = claset_of \<^context> in fun ideal_tac add_ths del_ths ctxt =
presimplify ctxt add_ths del_ths THEN'
CSUBGOAL (fn (p, i) => case get_ring_ideal_convs ctxt p of
NONE => no_tac
| SOME thy => let fun poly_exists_tac {asms = asms, concl = concl, prems = prems,
params = _, context = ctxt, schematics = _} = let val (evs,bod) = strip_exists (Thm.dest_arg concl) val ps = map_filter (try (lhs o Thm.dest_arg)) asms val cfs = (map swap o #multi_ideal thy evs ps)
(map Thm.dest_arg1 (conjuncts bod)) val ws = map (exitac ctxt o AList.lookup op aconvc cfs) evs in EVERY (rev ws) THEN Method.insert_tac ctxt prems 1 THEN ring_tac add_ths del_ths ctxt 1 end in
clarify_tac (put_claset claset ctxt) i THEN Object_Logic.full_atomize_tac ctxt i THEN asm_full_simp_tac (put_simpset (#poly_eq_ss thy) ctxt) i THEN clarify_tac (put_claset claset ctxt) i THEN (REPEAT (CONVERSION (#unwind_conv thy ctxt) i)) THEN SUBPROOF poly_exists_tac ctxt i end handle TERM _ => no_tac
| CTERM _ => no_tac
| THM _ => no_tac); end;
fun algebra_tac add_ths del_ths ctxt i =
ring_tac add_ths del_ths ctxt i ORELSE ideal_tac add_ths del_ths ctxt i
end;
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