--- a/src/HOL/Groebner_Basis.thy Fri May 07 15:05:52 2010 +0200
+++ b/src/HOL/Groebner_Basis.thy Fri May 07 16:12:25 2010 +0200
@@ -7,7 +7,7 @@
theory Groebner_Basis
imports Semiring_Normalization
uses
- ("Tools/Groebner_Basis/groebner.ML")
+ ("Tools/groebner.ML")
begin
subsection {* Groebner Bases *}
@@ -40,7 +40,7 @@
setup Algebra_Simplification.setup
-use "Tools/Groebner_Basis/groebner.ML"
+use "Tools/groebner.ML"
method_setup algebra = Groebner.algebra_method
"solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
--- a/src/HOL/Tools/Groebner_Basis/groebner.ML Fri May 07 15:05:52 2010 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1045 +0,0 @@
-(* Title: HOL/Tools/Groebner_Basis/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 : 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 : 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
- val algebra_method: (Proof.context -> Method.method) context_parser
-end
-
-structure Groebner : GROEBNER =
-struct
-
-open Conv Drule Thm;
-
-fun is_comb ct =
- (case Thm.term_of ct of
- _ $ _ => true
- | _ => false);
-
-val concl = Thm.cprop_of #> Thm.dest_arg;
-
-fun is_binop ct ct' =
- (case Thm.term_of ct' of
- c $ _ $ _ => 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'])
-
-fun inst_thm inst = Thm.instantiate ([], inst);
-
-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 (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;
-
-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, 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 =/ 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));
-
-fun degree vn p =
- case p of
- [] => error "Zero polynomial"
-| [(c,ns)] => nth ns vn
-| (c,ns)::p' => Int.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, 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 (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 null 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: int list) = 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 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,hist)) = 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,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 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 =) (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 (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 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 = [(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 (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 term_of tm of
- Const("op |",_)$l$r =>
- compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
- | _ => rfn tm ;
-
-val notnotD = @{thm notnotD};
-fun mk_binop ct x y = capply (capply ct x) y
-
-val mk_comb = capply;
-fun is_neg t =
- case term_of t of
- (Const("Not",_)$p) => true
- | _ => false;
-fun is_eq t =
- case term_of t of
- (Const("op =",_)$_$_) => true
-| _ => false;
-
-fun end_itlist f l =
- case l of
- [] => error "end_itlist"
- | [x] => x
- | (h::t) => f h (end_itlist f t);
-
-val list_mk_binop = fn b => end_itlist (mk_binop b);
-
-val list_dest_binop = fn b =>
- let fun h acc t =
- ((let val (l,r) = dest_binary b t in h (h acc r) l end)
- handle CTERM _ => (t::acc)) (* Why had I handle _ => ? *)
- in h []
- end;
-
-val strip_exists =
- let fun h (acc, t) =
- case (term_of t) of
- Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (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",_)$Abs(_,_,_)) => true
-| _ => false;
-
-val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq;
-val bool_simps = @{thms bool_simps};
-val nnf_simps = @{thms nnf_simps};
-val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps)
-val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms weak_dnf_simps});
-val initial_conv =
- Simplifier.rewrite
- (HOL_basic_ss addsimps nnf_simps
- addsimps [not_all, not_ex]
- addsimps map (fn th => th RS sym) (@{thms ex_simps} @ @{thms all_simps}));
-
-val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
-
-val cTrp = @{cterm "Trueprop"};
-val cConj = @{cterm "op &"};
-val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"});
-val assume_Trueprop = mk_comb cTrp #> assume;
-val list_mk_conj = list_mk_binop cConj;
-val conjs = list_dest_binop cConj;
-val mk_neg = mk_comb cNot;
-
-fun striplist dest =
- let
- fun h acc x = case try dest x of
- SOME (a,b) => h (h acc b) a
- | NONE => x::acc
- in h [] end;
-fun list_mk_binop b = foldr1 (fn (s,t) => Thm.capply (Thm.capply b s) t);
-
-val eq_commute = mk_meta_eq @{thm eq_commute};
-
-fun sym_conv eq =
- let val (l,r) = Thm.dest_binop eq
- in instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute
- end;
-
- (* FIXME : copied from cqe.ML -- complex QE*)
-fun conjuncts ct =
- case term_of ct of
- @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
-| _ => [ct];
-
-fun fold1 f = foldr1 (uncurry f);
-
-val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ;
-
-fun mk_conj_tab th =
- let fun h acc th =
- case prop_of th of
- @{term "Trueprop"}$(@{term "op &"}$p$q) =>
- h (h acc (th RS conjunct2)) (th RS conjunct1)
- | @{term "Trueprop"}$p => (p,th)::acc
-in fold (Termtab.insert Thm.eq_thm) (h [] th) Termtab.empty end;
-
-fun is_conj (@{term "op &"}$_$_) = true
- | is_conj _ = false;
-
-fun prove_conj tab cjs =
- case cjs of
- [c] => if is_conj (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 (assume (Thm.capply @{cterm Trueprop} l))
- val ctabr = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} r))
- fun tabl c = the (Termtab.lookup ctabl (term_of c))
- fun tabr c = the (Termtab.lookup ctabr (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 = instantiate' [] [SOME l, SOME r] @{thm iffI}
- in implies_elim (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 T = cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
- fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
-
-fun choose v th th' = case concl_of th of
- @{term Trueprop} $ (Const("Ex",_)$_) =>
- let
- val p = (funpow 2 Thm.dest_arg o cprop_of) th
- val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
- val th0 = fconv_rule (Thm.beta_conversion true)
- (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
- val pv = (Thm.rhs_of o Thm.beta_conversion true)
- (Thm.capply @{cterm Trueprop} (Thm.capply p v))
- val th1 = forall_intr v (implies_intr pv th')
- in implies_elim (implies_elim th0 th) th1 end
-| _ => error ""
-
-fun simple_choose v th =
- choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
-
-
- fun mkexi v th =
- let
- val p = Thm.cabs v (Thm.dest_arg (Thm.cprop_of th))
- in implies_elim
- (fconv_rule (Thm.beta_conversion true) (instantiate' [SOME (ctyp_of_term v)] [SOME p, SOME v] @{thm exI}))
- th
- end
- fun ex_eq_conv t =
- let
- val (p0,q0) = Thm.dest_binop t
- val (vs',P) = strip_exists p0
- val (vs,_) = strip_exists q0
- val th = assume (Thm.capply @{cterm Trueprop} P)
- val th1 = implies_intr_hyps (fold simple_choose vs' (fold mkexi vs th))
- val th2 = implies_intr_hyps (fold simple_choose vs (fold mkexi vs' th))
- val p = (Thm.dest_arg o Thm.dest_arg1 o cprop_of) th1
- val q = (Thm.dest_arg o Thm.dest_arg o cprop_of) th1
- in implies_elim (implies_elim (instantiate' [] [SOME p, SOME q] iffI) th1) th2
- |> mk_meta_eq
- end;
-
-
- fun getname v = case term_of v of
- Free(s,_) => s
- | Var ((s,_),_) => s
- | _ => "x"
- fun mk_eq s t = Thm.capply (Thm.capply @{cterm "op == :: bool => _"} s) t
- fun mkeq s t = Thm.capply @{cterm Trueprop} (Thm.capply (Thm.capply @{cterm "op = :: bool => _"} s) t)
- fun mk_exists v th = arg_cong_rule (ext (ctyp_of_term v))
- (Thm.abstract_rule (getname v) v th)
- val simp_ex_conv =
- Simplifier.rewrite (HOL_basic_ss addsimps @{thms simp_thms(39)})
-
-fun frees t = Thm.add_cterm_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.capply (Thm.cabs v tm) t)
-in fold vsubst end;
-
-
-(** main **)
-
-fun ring_and_ideal_conv
- {vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules),
- field = (f_ops, f_rules), 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 dest_fun2 [add_pat, mul_pat, pow_pat];
-
- val (ring_sub_tm, ring_neg_tm) =
- (case r_ops of
- [sub_pat, neg_pat] => (dest_fun2 sub_pat, dest_fun neg_pat)
- |_ => (@{cterm "True"}, @{cterm "True"}));
-
- val (field_div_tm, field_inv_tm) =
- (case f_ops of
- [div_pat, inv_pat] => (dest_fun2 div_pat, 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) = dest_comb t
- in if Term.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 field_dest_inv t =
- let val (l,r) = dest_comb t in
- if Term.could_unify(term_of l, 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_mk_sub = mk_binop 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 field_mk_div = mk_binop 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 (dest_arg tm) acc
- else if can ring_dest_pow tm then grobvars (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 (dest_arg1 tm) (grobvars (dest_arg tm) acc)
- else if can field_dest_inv tm
- then
- let val gvs = grobvars (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 (dest_arg1 tm) acc
- val gvs = grobvars (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
- [(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 (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) ((term_of #> HOLogic.dest_number #> snd) r)
- end)
- handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
-val eq_tm = idom_thm |> concl |> dest_arg |> dest_arg |> 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 (dest_arg1 eq) (grobvars (dest_arg eq) a)) cjs []
- val vars = sort (fn (x, y) => Term_Ord.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 (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 (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(arg_cong_rule ring_add_tm th1);
-
-fun refute tm =
- if tm aconvc false_tm then assume_Trueprop tm else
- ((let
- val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims (assume_Trueprop tm))
- val nths = filter (is_eq o 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(HOLogic.conj_intr th1 th2)) nths
- val th2 = Conv.fconv_rule
- ((arg_conv #> arg_conv)
- (binop_conv ring_normalize_conv)) th1
- val conc = th2 |> concl |> dest_arg
- val (l,r) = conc |> dest_eq
- in implies_intr (mk_comb cTrp tm)
- (equal_elim (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(HOLogic.conj_intr th1 th2)) nths
- val (vars,pol::pols) =
- grobify_equations(list_mk_conj(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 HOLogic.conj_intr th1) neq_01
- in (vars,l,cert,th2)
- end)
- 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) => 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 = HOLogic.conj_intr(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(dest_arg(concl th4))
- in implies_intr (mk_comb cTrp tm)
- (equal_elim (arg_cong_rule cTrp (eqF_intr th4))
- (reflexive l |> mk_object_eq))
- end
- end) handle ERROR _ => raise CTERM ("Gorbner-refute: unable to refute",[tm]))
-
-fun ring tm =
- let
- fun mk_forall x p =
- mk_comb (cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (cabs x p)
- val avs = add_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 raise CTERM("ring: non-universal formula",[tm])
- 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_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 fconv_rule (arg_conv (arg1_conv ring_normalize_conv))
- (instantiate' [] [SOME a, SOME b] idl_sub)
- end
- val poly_eq_simproc =
- let
- fun proc phi ss t =
- let val th = poly_eq_conv t
- in if Thm.is_reflexive th then NONE else SOME th
- end
- in make_simproc {lhss = [Thm.lhs_of idl_sub],
- name = "poly_eq_simproc", proc = proc, identifier = []}
- end;
- val poly_eq_ss = HOL_basic_ss 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) (Thm.add_cterm_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 = transitive th1
- (instantiate' [] [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v]
- idl_add0)
- in fconv_rule(funpow 2 arg_conv ring_normalize_conv) th2
- end
- in
- fun unwind_polys_conv tm =
- let
- val (vars,bod) = strip_exists tm
- val cjs = striplist (dest_binary @{cterm "op &"}) bod
- val th1 = (the (get_first (try (isolate_variable vars)) cjs)
- handle Option => raise CTERM ("unwind_polys_conv",[tm]))
- val eq = Thm.lhs_of th1
- val bod' = list_mk_binop @{cterm "op &"} (eq::(remove op aconvc eq cjs))
- val th2 = conj_ac_rule (mk_eq bod bod')
- val th3 = transitive th2
- (Drule.binop_cong_rule @{cterm "op &"} th1
- (reflexive (Thm.dest_arg (Thm.rhs_of th2))))
- val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))
- val vars' = (remove op aconvc v vars) @ [v]
- val th4 = fconv_rule (arg_conv simp_ex_conv) (mk_exists v th3)
- val th5 = ex_eq_conv (mk_eq tm (fold mk_ex (remove op aconvc v vars) (Thm.lhs_of th4)))
- in transitive th5 (fold mk_exists (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.capply 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 (term_of s, 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 term_of tm of
- Const ("op =", T) $ _ $ _ =>
- if domain_type T = HOLogic.boolT then find_args bounds tm
- else dest_arg tm
- | Const ("Not", _) $ _ => find_term bounds (dest_arg tm)
- | Const ("All", _) $ _ => find_body bounds (dest_arg tm)
- | Const ("Ex", _) $ _ => find_body bounds (dest_arg tm)
- | Const ("op &", _) $ _ $ _ => find_args bounds tm
- | Const ("op |", _) $ _ $ _ => find_args bounds tm
- | Const ("op -->", _) $ _ $ _ => find_args bounds tm
- | @{term "op ==>"} $_$_ => find_args bounds tm
- | Const("op ==",_)$_$_ => find_args bounds tm
- | @{term Trueprop}$_ => find_term bounds (dest_arg tm)
- | _ => raise TERM ("find_term", []))
-and find_args bounds tm =
- let val (t, u) = Thm.dest_binop tm
- in (find_term bounds t handle TERM _ => find_term bounds u) end
-and find_body bounds b =
- let val (_, b') = dest_abs (SOME (Name.bound bounds)) b
- in find_term (bounds + 1) b' end;
-
-
-fun get_ring_ideal_convs ctxt form =
- case try (find_term 0) form of
- NONE => NONE
-| SOME tm =>
- (case Normalizer.match ctxt tm of
- NONE => NONE
- | SOME (res as (theory, {is_const, dest_const,
- mk_const, conv = ring_eq_conv})) =>
- SOME (ring_and_ideal_conv theory
- dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
- (Normalizer.semiring_normalize_wrapper ctxt res)))
-
-fun ring_solve ctxt form =
- (case try (find_term 0 (* FIXME !? *)) form of
- NONE => reflexive form
- | SOME tm =>
- (case Normalizer.match ctxt tm of
- NONE => reflexive form
- | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
- #ring_conv (ring_and_ideal_conv theory
- dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
- (Normalizer.semiring_normalize_wrapper ctxt res)) form));
-
-fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
- (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
-
-fun ring_tac add_ths del_ths ctxt =
- Object_Logic.full_atomize_tac
- THEN' presimplify ctxt add_ths del_ths
- THEN' CSUBGOAL (fn (p, i) =>
- rtac (let val form = Object_Logic.dest_judgment p
- in case get_ring_ideal_convs ctxt form of
- NONE => reflexive form
- | SOME thy => #ring_conv thy form
- end) i
- handle TERM _ => no_tac
- | CTERM _ => no_tac
- | THM _ => no_tac);
-
-local
- fun lhs t = case term_of t of
- Const("op =",_)$_$_ => Thm.dest_arg1 t
- | _=> raise CTERM ("ideal_tac - lhs",[t])
- fun exitac NONE = no_tac
- | exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
-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 = params, context = ctxt, schematics = scs} =
- 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 o AList.lookup op aconvc cfs) evs
- in EVERY (rev ws) THEN Method.insert_tac prems 1
- THEN ring_tac add_ths del_ths ctxt 1
- end
- in
- clarify_tac @{claset} i
- THEN Object_Logic.full_atomize_tac i
- THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i
- THEN clarify_tac @{claset} i
- THEN (REPEAT (CONVERSION (#unwind_conv thy) 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
-
-local
-
-fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
-val addN = "add"
-val delN = "del"
-val any_keyword = keyword addN || keyword delN
-val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-
-in
-
-val algebra_method = ((Scan.optional (keyword addN |-- thms) []) --
- (Scan.optional (keyword delN |-- thms) [])) >>
- (fn (add_ths, del_ths) => fn ctxt =>
- SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
-
-end;
-
-end;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/groebner.ML Fri May 07 16:12:25 2010 +0200
@@ -0,0 +1,1045 @@
+(* Title: HOL/Tools/Groebner_Basis/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 : 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 : 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
+ val algebra_method: (Proof.context -> Method.method) context_parser
+end
+
+structure Groebner : GROEBNER =
+struct
+
+open Conv Drule Thm;
+
+fun is_comb ct =
+ (case Thm.term_of ct of
+ _ $ _ => true
+ | _ => false);
+
+val concl = Thm.cprop_of #> Thm.dest_arg;
+
+fun is_binop ct ct' =
+ (case Thm.term_of ct' of
+ c $ _ $ _ => 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'])
+
+fun inst_thm inst = Thm.instantiate ([], inst);
+
+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 (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;
+
+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, 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 =/ 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));
+
+fun degree vn p =
+ case p of
+ [] => error "Zero polynomial"
+| [(c,ns)] => nth ns vn
+| (c,ns)::p' => Int.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, 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 (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 null 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: int list) = 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 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,hist)) = 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,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 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 =) (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 (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 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 = [(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 (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 term_of tm of
+ Const("op |",_)$l$r =>
+ compose_single(refute_disj rfn (dest_arg tm),2,compose_single(refute_disj rfn (dest_arg1 tm),2,disjE))
+ | _ => rfn tm ;
+
+val notnotD = @{thm notnotD};
+fun mk_binop ct x y = capply (capply ct x) y
+
+val mk_comb = 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_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 (term_of t) of
+ Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (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) (@{thms ex_simps} @ @{thms 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_Trueprop = mk_comb cTrp #> assume;
+val list_mk_conj = list_mk_binop cConj;
+val conjs = list_dest_binop cConj;
+val mk_neg = mk_comb 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.capply (Thm.capply 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 instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute
+ end;
+
+ (* FIXME : copied from cqe.ML -- complex QE*)
+fun conjuncts ct =
+ case term_of ct of
+ @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct))
+| _ => [ct];
+
+fun fold1 f = foldr1 (uncurry f);
+
+val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ;
+
+fun mk_conj_tab th =
+ let fun h acc th =
+ case prop_of th of
+ @{term "Trueprop"}$(@{term "op &"}$p$q) =>
+ 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 "op &"}$_$_) = true
+ | is_conj _ = false;
+
+fun prove_conj tab cjs =
+ case cjs of
+ [c] => if is_conj (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 (assume (Thm.capply @{cterm Trueprop} l))
+ val ctabr = mk_conj_tab (assume (Thm.capply @{cterm Trueprop} r))
+ fun tabl c = the (Termtab.lookup ctabl (term_of c))
+ fun tabr c = the (Termtab.lookup ctabr (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 = instantiate' [] [SOME l, SOME r] @{thm iffI}
+ in implies_elim (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 T = cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
+ fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
+
+fun choose v th th' = case concl_of th of
+ @{term Trueprop} $ (Const("Ex",_)$_) =>
+ let
+ val p = (funpow 2 Thm.dest_arg o cprop_of) th
+ val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
+ val th0 = fconv_rule (Thm.beta_conversion true)
+ (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
+ val pv = (Thm.rhs_of o Thm.beta_conversion true)
+ (Thm.capply @{cterm Trueprop} (Thm.capply p v))
+ val th1 = forall_intr v (implies_intr pv th')
+ in implies_elim (implies_elim th0 th) th1 end
+| _ => error ""
+
+fun simple_choose v th =
+ choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
+
+
+ fun mkexi v th =
+ let
+ val p = Thm.cabs v (Thm.dest_arg (Thm.cprop_of th))
+ in implies_elim
+ (fconv_rule (Thm.beta_conversion true) (instantiate' [SOME (ctyp_of_term v)] [SOME p, SOME v] @{thm exI}))
+ th
+ end
+ fun ex_eq_conv t =
+ let
+ val (p0,q0) = Thm.dest_binop t
+ val (vs',P) = strip_exists p0
+ val (vs,_) = strip_exists q0
+ val th = assume (Thm.capply @{cterm Trueprop} P)
+ val th1 = implies_intr_hyps (fold simple_choose vs' (fold mkexi vs th))
+ val th2 = implies_intr_hyps (fold simple_choose vs (fold mkexi vs' th))
+ val p = (Thm.dest_arg o Thm.dest_arg1 o cprop_of) th1
+ val q = (Thm.dest_arg o Thm.dest_arg o cprop_of) th1
+ in implies_elim (implies_elim (instantiate' [] [SOME p, SOME q] iffI) th1) th2
+ |> mk_meta_eq
+ end;
+
+
+ fun getname v = case term_of v of
+ Free(s,_) => s
+ | Var ((s,_),_) => s
+ | _ => "x"
+ fun mk_eq s t = Thm.capply (Thm.capply @{cterm "op == :: bool => _"} s) t
+ fun mkeq s t = Thm.capply @{cterm Trueprop} (Thm.capply (Thm.capply @{cterm "op = :: bool => _"} s) t)
+ fun mk_exists v th = arg_cong_rule (ext (ctyp_of_term v))
+ (Thm.abstract_rule (getname v) v th)
+ val simp_ex_conv =
+ Simplifier.rewrite (HOL_basic_ss addsimps @{thms simp_thms(39)})
+
+fun frees t = Thm.add_cterm_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.capply (Thm.cabs v tm) t)
+in fold vsubst end;
+
+
+(** main **)
+
+fun ring_and_ideal_conv
+ {vars, semiring = (sr_ops, sr_rules), ring = (r_ops, r_rules),
+ field = (f_ops, f_rules), 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 dest_fun2 [add_pat, mul_pat, pow_pat];
+
+ val (ring_sub_tm, ring_neg_tm) =
+ (case r_ops of
+ [sub_pat, neg_pat] => (dest_fun2 sub_pat, dest_fun neg_pat)
+ |_ => (@{cterm "True"}, @{cterm "True"}));
+
+ val (field_div_tm, field_inv_tm) =
+ (case f_ops of
+ [div_pat, inv_pat] => (dest_fun2 div_pat, 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) = dest_comb t
+ in if Term.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 field_dest_inv t =
+ let val (l,r) = dest_comb t in
+ if Term.could_unify(term_of l, 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_mk_sub = mk_binop 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 field_mk_div = mk_binop 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 (dest_arg tm) acc
+ else if can ring_dest_pow tm then grobvars (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 (dest_arg1 tm) (grobvars (dest_arg tm) acc)
+ else if can field_dest_inv tm
+ then
+ let val gvs = grobvars (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 (dest_arg1 tm) acc
+ val gvs = grobvars (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
+ [(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 (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) ((term_of #> HOLogic.dest_number #> snd) r)
+ end)
+ handle CTERM _ => error "grobify_term: unknown or invalid term")))))))));
+val eq_tm = idom_thm |> concl |> dest_arg |> dest_arg |> 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 (dest_arg1 eq) (grobvars (dest_arg eq) a)) cjs []
+ val vars = sort (fn (x, y) => Term_Ord.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 (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 (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(arg_cong_rule ring_add_tm th1);
+
+fun refute tm =
+ if tm aconvc false_tm then assume_Trueprop tm else
+ ((let
+ val (nths0,eths0) = List.partition (is_neg o concl) (HOLogic.conj_elims (assume_Trueprop tm))
+ val nths = filter (is_eq o 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(HOLogic.conj_intr th1 th2)) nths
+ val th2 = Conv.fconv_rule
+ ((arg_conv #> arg_conv)
+ (binop_conv ring_normalize_conv)) th1
+ val conc = th2 |> concl |> dest_arg
+ val (l,r) = conc |> dest_eq
+ in implies_intr (mk_comb cTrp tm)
+ (equal_elim (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(HOLogic.conj_intr th1 th2)) nths
+ val (vars,pol::pols) =
+ grobify_equations(list_mk_conj(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 HOLogic.conj_intr th1) neq_01
+ in (vars,l,cert,th2)
+ end)
+ 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) => 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 = HOLogic.conj_intr(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(dest_arg(concl th4))
+ in implies_intr (mk_comb cTrp tm)
+ (equal_elim (arg_cong_rule cTrp (eqF_intr th4))
+ (reflexive l |> mk_object_eq))
+ end
+ end) handle ERROR _ => raise CTERM ("Gorbner-refute: unable to refute",[tm]))
+
+fun ring tm =
+ let
+ fun mk_forall x p =
+ mk_comb (cterm_rule (instantiate' [SOME (ctyp_of_term x)] []) @{cpat "All:: (?'a => bool) => _"}) (cabs x p)
+ val avs = add_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 raise CTERM("ring: non-universal formula",[tm])
+ 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_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 fconv_rule (arg_conv (arg1_conv ring_normalize_conv))
+ (instantiate' [] [SOME a, SOME b] idl_sub)
+ end
+ val poly_eq_simproc =
+ let
+ fun proc phi ss t =
+ let val th = poly_eq_conv t
+ in if Thm.is_reflexive th then NONE else SOME th
+ end
+ in make_simproc {lhss = [Thm.lhs_of idl_sub],
+ name = "poly_eq_simproc", proc = proc, identifier = []}
+ end;
+ val poly_eq_ss = HOL_basic_ss 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) (Thm.add_cterm_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 = transitive th1
+ (instantiate' [] [(SOME o Thm.dest_arg1 o Thm.rhs_of) th1, SOME v]
+ idl_add0)
+ in fconv_rule(funpow 2 arg_conv ring_normalize_conv) th2
+ end
+ in
+ fun unwind_polys_conv tm =
+ let
+ val (vars,bod) = strip_exists tm
+ val cjs = striplist (dest_binary @{cterm "op &"}) bod
+ val th1 = (the (get_first (try (isolate_variable vars)) cjs)
+ handle Option => raise CTERM ("unwind_polys_conv",[tm]))
+ val eq = Thm.lhs_of th1
+ val bod' = list_mk_binop @{cterm "op &"} (eq::(remove op aconvc eq cjs))
+ val th2 = conj_ac_rule (mk_eq bod bod')
+ val th3 = transitive th2
+ (Drule.binop_cong_rule @{cterm "op &"} th1
+ (reflexive (Thm.dest_arg (Thm.rhs_of th2))))
+ val v = Thm.dest_arg1(Thm.dest_arg1(Thm.rhs_of th3))
+ val vars' = (remove op aconvc v vars) @ [v]
+ val th4 = fconv_rule (arg_conv simp_ex_conv) (mk_exists v th3)
+ val th5 = ex_eq_conv (mk_eq tm (fold mk_ex (remove op aconvc v vars) (Thm.lhs_of th4)))
+ in transitive th5 (fold mk_exists (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.capply 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 (term_of s, 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 term_of tm of
+ Const ("op =", T) $ _ $ _ =>
+ if domain_type T = HOLogic.boolT then find_args bounds tm
+ else dest_arg tm
+ | Const ("Not", _) $ _ => find_term bounds (dest_arg tm)
+ | Const ("All", _) $ _ => find_body bounds (dest_arg tm)
+ | Const ("Ex", _) $ _ => find_body bounds (dest_arg tm)
+ | Const ("op &", _) $ _ $ _ => find_args bounds tm
+ | Const ("op |", _) $ _ $ _ => find_args bounds tm
+ | Const ("op -->", _) $ _ $ _ => find_args bounds tm
+ | @{term "op ==>"} $_$_ => find_args bounds tm
+ | Const("op ==",_)$_$_ => find_args bounds tm
+ | @{term Trueprop}$_ => find_term bounds (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') = 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 (ctyp_of_term tm)) (ring_eq_conv ctxt)
+ (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)))
+
+fun ring_solve ctxt form =
+ (case try (find_term 0 (* FIXME !? *)) form of
+ NONE => reflexive form
+ | SOME tm =>
+ (case Semiring_Normalizer.match ctxt tm of
+ NONE => reflexive form
+ | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) =>
+ #ring_conv (ring_and_ideal_conv theory
+ dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt)
+ (Semiring_Normalizer.semiring_normalize_wrapper ctxt res)) form));
+
+fun presimplify ctxt add_thms del_thms = asm_full_simp_tac (Simplifier.context ctxt
+ (HOL_basic_ss addsimps (Algebra_Simplification.get ctxt) delsimps del_thms addsimps add_thms));
+
+fun ring_tac add_ths del_ths ctxt =
+ Object_Logic.full_atomize_tac
+ THEN' presimplify ctxt add_ths del_ths
+ THEN' CSUBGOAL (fn (p, i) =>
+ rtac (let val form = Object_Logic.dest_judgment p
+ in case get_ring_ideal_convs ctxt form of
+ NONE => reflexive form
+ | SOME thy => #ring_conv thy form
+ end) i
+ handle TERM _ => no_tac
+ | CTERM _ => no_tac
+ | THM _ => no_tac);
+
+local
+ fun lhs t = case term_of t of
+ Const("op =",_)$_$_ => Thm.dest_arg1 t
+ | _=> raise CTERM ("ideal_tac - lhs",[t])
+ fun exitac NONE = no_tac
+ | exitac (SOME y) = rtac (instantiate' [SOME (ctyp_of_term y)] [NONE,SOME y] exI) 1
+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 = params, context = ctxt, schematics = scs} =
+ 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 o AList.lookup op aconvc cfs) evs
+ in EVERY (rev ws) THEN Method.insert_tac prems 1
+ THEN ring_tac add_ths del_ths ctxt 1
+ end
+ in
+ clarify_tac @{claset} i
+ THEN Object_Logic.full_atomize_tac i
+ THEN asm_full_simp_tac (Simplifier.context ctxt (#poly_eq_ss thy)) i
+ THEN clarify_tac @{claset} i
+ THEN (REPEAT (CONVERSION (#unwind_conv thy) 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
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+val addN = "add"
+val delN = "del"
+val any_keyword = keyword addN || keyword delN
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+
+in
+
+val algebra_method = ((Scan.optional (keyword addN |-- thms) []) --
+ (Scan.optional (keyword delN |-- thms) [])) >>
+ (fn (add_ths, del_ths) => fn ctxt =>
+ SIMPLE_METHOD' (algebra_tac add_ths del_ths ctxt))
+
+end;
+
+end;