(* 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 end structure Groebner : GROEBNER = struct open Conv Normalizer 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 = map2 (fn p => fn n => (p,Start n)) pols (0 upto (length pols - 1)) 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 =) ((map fst lis1) union (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_rev (fold_rev (curry Int.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",_)$Abs(_,_,_)) => true | _ => false; val mk_object_eq = fn th => th COMP meta_eq_to_obj_eq; val bool_simps = @{thms "bool_simps"}; val nnf_simps = @{thms "nnf_simps"}; val nnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps bool_simps addsimps nnf_simps) val weak_dnf_conv = Simplifier.rewrite (HOL_basic_ss addsimps @{thms "weak_dnf_simps"}); val initial_conv = Simplifier.rewrite (HOL_basic_ss addsimps nnf_simps addsimps [not_all, not_ex] addsimps map (fn th => th RS sym) (ex_simps @ all_simps)); val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec)); val cTrp = @{cterm "Trueprop"}; val cConj = @{cterm "op &"}; val (cNot,false_tm) = (@{cterm "Not"}, @{cterm "False"}); val assume_Trueprop = mk_comb cTrp #> assume; val list_mk_conj = list_mk_binop cConj; val conjs = list_dest_binop cConj; val mk_neg = mk_comb cNot; fun striplist dest = let fun h acc x = case try dest x of SOME (a,b) => h (h acc b) a | NONE => x::acc in h [] end; fun list_mk_binop b = foldr1 (fn (s,t) => Thm.capply (Thm.capply b s) t); val eq_commute = mk_meta_eq @{thm eq_commute}; fun sym_conv eq = let val (l,r) = Thm.dest_binop eq in instantiate' [SOME (ctyp_of_term l)] [SOME l, SOME r] eq_commute end; (* FIXME : copied from cqe.ML -- complex QE*) fun conjuncts ct = case term_of ct of @{term "op &"}$_$_ => (Thm.dest_arg1 ct)::(conjuncts (Thm.dest_arg ct)) | _ => [ct]; fun fold1 f = foldr1 (uncurry f); val list_conj = fold1 (fn c => fn c' => Thm.capply (Thm.capply @{cterm "op &"} c) c') ; fun mk_conj_tab th = let fun h acc th = case prop_of th of @{term "Trueprop"}$(@{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 = valOf (Termtab.lookup ctabl (term_of c)) fun tabr c = valOf (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) => TermOrd.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 (fn n => these (AList.lookup (op =) cert n) |> holify_polynomial vars) (0 upto (length pols - 1)) 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 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 => (valOf (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 = (valOf (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 (gen_inter op aconvc (frees m, vars))) (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) => TermOrd.term_ord (term_of s, term_of t)) in (eq,Library.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 |> valOf 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}$_ => find_term bounds (dest_arg tm) | _ => raise TERM ("find_term", [])) and find_args bounds tm = let val (t, u) = Thm.dest_binop tm in (find_term bounds t handle TERM _ => find_term bounds u) end and find_body bounds b = let val (_, b') = dest_abs (SOME (Name.bound bounds)) b in find_term (bounds + 1) b' end; fun get_ring_ideal_convs ctxt form = case try (find_term 0) form of NONE => NONE | SOME tm => (case NormalizerData.match ctxt tm of NONE => NONE | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) => SOME (ring_and_ideal_conv theory dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt) (semiring_normalize_wrapper ctxt res))) fun ring_solve ctxt form = (case try (find_term 0 (* FIXME !? *)) form of NONE => reflexive form | SOME tm => (case NormalizerData.match ctxt tm of NONE => reflexive form | SOME (res as (theory, {is_const, dest_const, mk_const, conv = ring_eq_conv})) => #ring_conv (ring_and_ideal_conv theory dest_const (mk_const (ctyp_of_term tm)) (ring_eq_conv ctxt) (semiring_normalize_wrapper ctxt res)) form)); fun ring_tac add_ths del_ths ctxt = ObjectLogic.full_atomize_tac THEN' asm_full_simp_tac (Simplifier.context ctxt (fst (NormalizerData.get ctxt)) delsimps del_ths addsimps add_ths) THEN' CSUBGOAL (fn (p, i) => rtac (let val form = (ObjectLogic.dest_judgment p) in case get_ring_ideal_convs ctxt form of NONE => reflexive form | SOME thy => #ring_conv thy form end) i handle TERM _ => no_tac | CTERM _ => no_tac | THM _ => no_tac); local fun lhs t = case term_of t of Const("op =",_)$_$_ => 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 = asm_full_simp_tac (Simplifier.context ctxt (fst (NormalizerData.get ctxt)) delsimps del_ths addsimps add_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 ObjectLogic.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 end;