(* 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 * cterm -> order) -> 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'
else raise 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 (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 = 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
let val c = c1 + c2 val rest = grob_add o1 o2 in
if c = @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, ListPair.map (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 = @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"
else if 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 => ((let val (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 => ((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),_) => ([],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
let val (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 poly_eq p1 p2 =
eq_list (fn ((c1, m1), (c2, m2)) => c1 = c2 andalso (m1: int list) = m2) (p1, p2);
fun memx ((p1,_),(p2,_)) 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 null l then [] else mergepairs [] (map (fn x => [x]) l)
end;
fun grobner_basis basis pairs =
case pairs of
[] => basis
| (l,(p1,p2))::opairs =>
let val (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
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 null (fst p') then grobner_interreduce rpols ps
else grobner_interreduce (p'::rpols) ps end;
(* Overall function. *)
fun grobner pols =
let val 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. *)
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,_) => 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) =>
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 =) (union (op =) (map fst lis1) (map fst lis2))
in
map (fn n => let val 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 =
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 @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 not (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 =
let val 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(@{const_name HOL.disj},_)$_$_ =>
Drule.compose
(refute_disj rfn (Thm.dest_arg tm), 2,
Drule.compose (refute_disj rfn (Thm.dest_arg1 tm), 2, disjE))
| _ => rfn tm ;
val notnotD = @{thm notnotD};
fun mk_binop ct x y = Thm.apply (Thm.apply ct x) y
fun is_neg t =
case Thm.term_of t of
(Const(@{const_name Not},_)$_) => true
| _ => false;
fun is_eq t =
case Thm.term_of t of
(Const(@{const_name HOL.eq},_)$_$_) => 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_binary 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 Thm.term_of t of
Const (@{const_name Ex}, _) $ Abs _ =>
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 Thm.term_of t of
(Const(@{const_name All},_)$Abs(_,_,_)) => true
| _ => false;
val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
val nnf_simps = @{thms nnf_simps};
fun weak_dnf_conv ctxt =
Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms weak_dnf_simps});
val initial_ss =
simpset_of (put_simpset HOL_basic_ss @{context}
addsimps nnf_simps
addsimps [not_all, not_ex]
addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));
fun initial_conv ctxt =
Simplifier.rewrite (put_simpset initial_ss ctxt);
val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
val cTrp = @{cterm "Trueprop"};
val cConj = @{cterm HOL.conj};
val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
val assume_Trueprop = Thm.apply cTrp #> Thm.assume;
val list_mk_conj = list_mk_binop cConj;
val conjs = list_dest_binop cConj;
val mk_neg = Thm.apply cNot;
fun striplist dest =
let
fun h acc x = case try dest x of
SOME (a,b) => h (h acc b) a
| NONE => x::acc
in h [] end;
fun list_mk_binop b = foldr1 (fn (s,t) => Thm.apply (Thm.apply b s) t);
val eq_commute = mk_meta_eq @{thm eq_commute};
fun sym_conv eq =
let val (l,r) = Thm.dest_binop eq
in Thm.instantiate' [SOME (Thm.ctyp_of_cterm l)] [SOME l, SOME r] eq_commute
end;
(* FIXME : copied from cqe.ML -- complex QE*)
fun conjuncts ct =
case Thm.term_of ct of
@{term HOL.conj}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
| _ => [ct];
fun fold1 f = foldr1 (uncurry f);
fun mk_conj_tab th =
let fun h acc th =
case Thm.prop_of th of
@{term "Trueprop"}$(@{term HOL.conj}$_$_) =>
h (h acc (th RS conjunct2)) (th RS conjunct1)
| @{term "Trueprop"}$p => (p,th)::acc
in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
fun is_conj (@{term HOL.conj}$_$_) = 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 (Thm.apply @{cterm Trueprop} l))
val ctabr = mk_conj_tab (Thm.assume (Thm.apply @{cterm Trueprop} 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 (@{const_name Ex}, (T --> @{typ bool}) --> @{typ bool}))
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
@{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
let
val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
val T = (hd o Thm.dest_ctyp o 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 Trueprop} (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 ((Thm.apply @{cterm Trueprop} 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 (Thm.apply @{cterm Trueprop} 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 "op == :: bool => _"} 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 addsimps @{thms simp_thms(39)})
fun frees t = Drule.cterm_add_frees t [];
fun free_in v t = member op aconvc (frees 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 "True"}, @{cterm "True"}));
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 "True"}, @{cterm "True"}));
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 =
let val (l,r) = Thm.dest_comb t
in if Term.could_unify(Thm.term_of l, Thm.term_of ring_neg_tm) then r
else raise CTERM ("ring_dest_neg", [t])
end
fun field_dest_inv t =
let val (l,r) = Thm.dest_comb t in
if Term.could_unify (Thm.term_of l, Thm.term_of field_inv_tm) then r
else raise CTERM ("field_dest_inv", [t])
end
val ring_dest_add = dest_binary ring_add_tm;
val ring_mk_add = 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 = 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 = 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 field_dest_inv tm
then
let val gvs = grobvars (Thm.dest_arg tm) []
in if null gvs then acc else tm::acc
end
else if can field_dest_div tm then
let val lvs = grobvars (Thm.dest_arg1 tm) acc
val gvs = grobvars (Thm.dest_arg tm) []
in if null 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
[(@1, map (fn i => if i aconvc tm then 1 else 0) vars)])
handle CTERM _ =>
((let val x = dest_const tm
in if 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 _ =>
((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) = field_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) ((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 =
let val (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 (fn (x, y) => Term_Ord.term_ord (Thm.term_of x, Thm.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 (Numeral.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 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 addsimps [idom_thm]);
fun prove_nz n = eqF_elim
(ring_eq_conv(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 tm aconvc false_tm then assume_Trueprop tm else
((let
val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims ctxt (assume_Trueprop 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 ctxt 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 (Thm.apply cTrp tm)
(Thm.equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th2))
(Thm.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 ctxt (HOLogic.conj_intr ctxt 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 ctxt 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 ctxt (mk_add (Thm.symmetric th1) th2 |> mk_object_eq) 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 (Thm.apply cTrp tm)
(Thm.equal_elim (Drule.arg_cong_rule cTrp (eqF_intr th4))
(Thm.reflexive l |> mk_object_eq))
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;
val all = Thm.cterm_of ctxt (Const (@{const_name All}, (T --> @{typ bool}) --> @{typ bool}))
in Thm.apply all (Thm.lambda x p) end
val avs = Drule.cterm_add_frees tm []
val P' = fold mk_forall avs tm
val th1 = initial_conv ctxt (mk_neg P')
val (evs,bod) = strip_exists(concl th1) in
if is_forall bod then raise 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 = [mk_object_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 addsimps [not_ex RS sym])
(th2 |> Thm.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) => (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 =
let val (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 =
let val th = poly_eq_conv ct
in if Thm.is_reflexive th then NONE else SOME th end
in
Simplifier.cert_simproc (Thm.theory_of_thm idl_sub) "poly_eq_simproc"
{lhss = [Thm.term_of (Thm.lhs_of idl_sub)],
proc = fn _ => fn _ => proc}
end;
val poly_eq_ss =
simpset_of (put_simpset HOL_basic_ss @{context}
addsimps @{thms simp_thms}
addsimprocs [poly_eq_simproc])
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(member (op aconvc) (Drule.cterm_add_frees 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 HOL.conj}) bod
val th1 = (the (get_first (try (isolate_variable vars)) cjs)
handle Option.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 HOL.conj} 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
in if 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
in if null mons2 then zero_tm else fold1 ring_mk_add mons2
end
fun isolate_monomials vars tm =
let
val (cmons,vmons) =
List.partition (fn m => null (inter (op aconvc) vars (frees m)))
(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
(fn (s,t) => Term_Ord.term_ord (Thm.term_of s, Thm.term_of t))
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;
in {ring_conv = ring, simple_ideal = ideal, multi_ideal = solve_idealism,
poly_eq_ss = poly_eq_ss, unwind_conv = unwind_polys_conv}
end;
fun find_term bounds tm =
(case Thm.term_of tm of
Const (@{const_name HOL.eq}, T) $ _ $ _ =>
if domain_type T = HOLogic.boolT then find_args bounds tm
else Thm.dest_arg tm
| Const (@{const_name Not}, _) $ _ => find_term bounds (Thm.dest_arg tm)
| Const (@{const_name All}, _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const (@{const_name Ex}, _) $ _ => find_body bounds (Thm.dest_arg tm)
| Const (@{const_name HOL.conj}, _) $ _ $ _ => find_args bounds tm
| Const (@{const_name HOL.disj}, _) $ _ $ _ => find_args bounds tm
| Const (@{const_name HOL.implies}, _) $ _ $ _ => find_args bounds tm
| @{term "op ==>"} $_$_ => find_args bounds tm
| Const("op ==",_)$_$_ => find_args bounds tm
| @{term Trueprop}$_ => find_term bounds (Thm.dest_arg tm)
| _ => raise TERM ("find_term", []))
and find_args bounds tm =
let val (t, u) = Thm.dest_binop tm
in (find_term bounds t handle TERM _ => find_term bounds u) end
and find_body bounds b =
let val (_, b') = Thm.dest_abs (SOME (Name.bound bounds)) b
in find_term (bounds + 1) b' end;
fun get_ring_ideal_convs ctxt form =
case try (find_term 0) form of
NONE => NONE
| SOME tm =>
(case Semiring_Normalizer.match ctxt tm of
NONE => NONE
| SOME (res as (theory, {is_const = _, dest_const,
mk_const, conv = ring_eq_conv})) =>
SOME (ring_and_ideal_conv theory
dest_const (mk_const (Thm.ctyp_of_cterm tm)) (ring_eq_conv ctxt)
(Semiring_Normalizer.semiring_normalize_wrapper ctxt res)))
fun presimplify ctxt add_thms del_thms =
asm_full_simp_tac (put_simpset HOL_basic_ss ctxt
addsimps (Named_Theorems.get ctxt @{named_theorems algebra})
delsimps del_thms addsimps add_thms);
fun ring_tac add_ths del_ths ctxt =
Object_Logic.full_atomize_tac ctxt
THEN' presimplify ctxt add_ths del_ths
THEN' CSUBGOAL (fn (p, i) =>
resolve_tac ctxt [let val form = Object_Logic.dest_judgment ctxt p
in case get_ring_ideal_convs ctxt form of
NONE => Thm.reflexive form
| SOME thy => #ring_conv thy ctxt form
end] i
handle TERM _ => no_tac
| CTERM _ => no_tac
| THM _ => no_tac);
local
fun lhs t = case Thm.term_of t of
Const(@{const_name HOL.eq},_)$_$_ => Thm.dest_arg1 t
| _=> raise CTERM ("ideal_tac - lhs",[t])
fun exitac _ NONE = no_tac
| exitac ctxt (SOME y) =
resolve_tac ctxt [Thm.instantiate' [SOME (Thm.ctyp_of_cterm y)] [NONE,SOME y] exI] 1
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;