src/HOL/arith_data.ML
author obua
Mon Apr 10 16:00:34 2006 +0200 (2006-04-10)
changeset 19404 9bf2cdc9e8e8
parent 19297 8f6e097d7b23
child 19481 a6205c6203ea
permissions -rw-r--r--
Moved stuff from Ring_and_Field to Matrix
     1 (*  Title:      HOL/arith_data.ML
     2     ID:         $Id$
     3     Author:     Markus Wenzel, Stefan Berghofer and Tobias Nipkow
     4 
     5 Various arithmetic proof procedures.
     6 *)
     7 
     8 (*---------------------------------------------------------------------------*)
     9 (* 1. Cancellation of common terms                                           *)
    10 (*---------------------------------------------------------------------------*)
    11 
    12 structure NatArithUtils =
    13 struct
    14 
    15 (** abstract syntax of structure nat: 0, Suc, + **)
    16 
    17 (* mk_sum, mk_norm_sum *)
    18 
    19 val one = HOLogic.mk_nat 1;
    20 val mk_plus = HOLogic.mk_binop "HOL.plus";
    21 
    22 fun mk_sum [] = HOLogic.zero
    23   | mk_sum [t] = t
    24   | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
    25 
    26 (*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
    27 fun mk_norm_sum ts =
    28   let val (ones, sums) = List.partition (equal one) ts in
    29     funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
    30   end;
    31 
    32 
    33 (* dest_sum *)
    34 
    35 val dest_plus = HOLogic.dest_bin "HOL.plus" HOLogic.natT;
    36 
    37 fun dest_sum tm =
    38   if HOLogic.is_zero tm then []
    39   else
    40     (case try HOLogic.dest_Suc tm of
    41       SOME t => one :: dest_sum t
    42     | NONE =>
    43         (case try dest_plus tm of
    44           SOME (t, u) => dest_sum t @ dest_sum u
    45         | NONE => [tm]));
    46 
    47 
    48 (** generic proof tools **)
    49 
    50 (* prove conversions *)
    51 
    52 fun prove_conv expand_tac norm_tac sg ss tu =  (* FIXME avoid standard *)
    53   mk_meta_eq (standard (Goal.prove sg [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq tu))
    54     (K (EVERY [expand_tac, norm_tac ss]))));
    55 
    56 val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s"
    57   (fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]);
    58 
    59 
    60 (* rewriting *)
    61 
    62 fun simp_all_tac rules =
    63   let val ss0 = HOL_ss addsimps rules
    64   in fn ss => ALLGOALS (simp_tac (Simplifier.inherit_context ss ss0)) end;
    65 
    66 val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right];
    67 val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right];
    68 
    69 fun prep_simproc (name, pats, proc) =
    70   Simplifier.simproc (the_context ()) name pats proc;
    71 
    72 end;
    73 
    74 signature ARITH_DATA =
    75 sig
    76   val nat_cancel_sums_add: simproc list
    77   val nat_cancel_sums: simproc list
    78 end;
    79 
    80 structure ArithData: ARITH_DATA =
    81 struct
    82 
    83 open NatArithUtils;
    84 
    85 
    86 (** cancel common summands **)
    87 
    88 structure Sum =
    89 struct
    90   val mk_sum = mk_norm_sum;
    91   val dest_sum = dest_sum;
    92   val prove_conv = prove_conv;
    93   val norm_tac1 = simp_all_tac add_rules;
    94   val norm_tac2 = simp_all_tac add_ac;
    95   fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss;
    96 end;
    97 
    98 fun gen_uncancel_tac rule ct =
    99   rtac (instantiate' [] [NONE, SOME ct] (rule RS subst_equals)) 1;
   100 
   101 
   102 (* nat eq *)
   103 
   104 structure EqCancelSums = CancelSumsFun
   105 (struct
   106   open Sum;
   107   val mk_bal = HOLogic.mk_eq;
   108   val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
   109   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel;
   110 end);
   111 
   112 
   113 (* nat less *)
   114 
   115 structure LessCancelSums = CancelSumsFun
   116 (struct
   117   open Sum;
   118   val mk_bal = HOLogic.mk_binrel "Orderings.less";
   119   val dest_bal = HOLogic.dest_bin "Orderings.less" HOLogic.natT;
   120   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel_less;
   121 end);
   122 
   123 
   124 (* nat le *)
   125 
   126 structure LeCancelSums = CancelSumsFun
   127 (struct
   128   open Sum;
   129   val mk_bal = HOLogic.mk_binrel "Orderings.less_eq";
   130   val dest_bal = HOLogic.dest_bin "Orderings.less_eq" HOLogic.natT;
   131   val uncancel_tac = gen_uncancel_tac nat_add_left_cancel_le;
   132 end);
   133 
   134 
   135 (* nat diff *)
   136 
   137 structure DiffCancelSums = CancelSumsFun
   138 (struct
   139   open Sum;
   140   val mk_bal = HOLogic.mk_binop "HOL.minus";
   141   val dest_bal = HOLogic.dest_bin "HOL.minus" HOLogic.natT;
   142   val uncancel_tac = gen_uncancel_tac diff_cancel;
   143 end);
   144 
   145 
   146 
   147 (** prepare nat_cancel simprocs **)
   148 
   149 val nat_cancel_sums_add = map prep_simproc
   150   [("nateq_cancel_sums",
   151      ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"], EqCancelSums.proc),
   152    ("natless_cancel_sums",
   153      ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"], LessCancelSums.proc),
   154    ("natle_cancel_sums",
   155      ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"], LeCancelSums.proc)];
   156 
   157 val nat_cancel_sums = nat_cancel_sums_add @
   158   [prep_simproc ("natdiff_cancel_sums",
   159     ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"], DiffCancelSums.proc)];
   160 
   161 end;
   162 
   163 open ArithData;
   164 
   165 
   166 (*---------------------------------------------------------------------------*)
   167 (* 2. Linear arithmetic                                                      *)
   168 (*---------------------------------------------------------------------------*)
   169 
   170 (* Parameters data for general linear arithmetic functor *)
   171 
   172 structure LA_Logic: LIN_ARITH_LOGIC =
   173 struct
   174 val ccontr = ccontr;
   175 val conjI = conjI;
   176 val notI = notI;
   177 val sym = sym;
   178 val not_lessD = linorder_not_less RS iffD1;
   179 val not_leD = linorder_not_le RS iffD1;
   180 
   181 
   182 fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
   183 
   184 val mk_Trueprop = HOLogic.mk_Trueprop;
   185 
   186 fun atomize thm = case #prop(rep_thm thm) of
   187     Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
   188     atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
   189   | _ => [thm];
   190 
   191 fun neg_prop(TP$(Const("Not",_)$t)) = TP$t
   192   | neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT-->HOLogic.boolT)$t);
   193 
   194 fun is_False thm =
   195   let val _ $ t = #prop(rep_thm thm)
   196   in t = Const("False",HOLogic.boolT) end;
   197 
   198 fun is_nat(t) = fastype_of1 t = HOLogic.natT;
   199 
   200 fun mk_nat_thm sg t =
   201   let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
   202   in instantiate ([],[(cn,ct)]) le0 end;
   203 
   204 end;
   205 
   206 
   207 (* arith theory data *)
   208 
   209 structure ArithTheoryData = TheoryDataFun
   210 (struct
   211   val name = "HOL/arith";
   212   type T = {splits: thm list, inj_consts: (string * typ)list, discrete: string  list, presburger: (int -> tactic) option};
   213 
   214   val empty = {splits = [], inj_consts = [], discrete = [], presburger = NONE};
   215   val copy = I;
   216   val extend = I;
   217   fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, presburger= presburger1},
   218              {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, presburger= presburger2}) =
   219    {splits = Drule.merge_rules (splits1, splits2),
   220     inj_consts = merge_lists inj_consts1 inj_consts2,
   221     discrete = merge_lists discrete1 discrete2,
   222     presburger = (case presburger1 of NONE => presburger2 | p => p)};
   223   fun print _ _ = ();
   224 end);
   225 
   226 val arith_split_add = Thm.declaration_attribute (fn thm =>
   227   Context.map_theory (ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>
   228     {splits= thm::splits, inj_consts= inj_consts, discrete= discrete, presburger= presburger})));
   229 
   230 fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>
   231   {splits = splits, inj_consts = inj_consts, discrete = d :: discrete, presburger= presburger});
   232 
   233 fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} =>
   234   {splits = splits, inj_consts = c :: inj_consts, discrete = discrete, presburger = presburger});
   235 
   236 
   237 structure LA_Data_Ref: LIN_ARITH_DATA =
   238 struct
   239 
   240 (* Decomposition of terms *)
   241 
   242 fun nT (Type("fun",[N,_])) = N = HOLogic.natT
   243   | nT _ = false;
   244 
   245 fun add_atom(t,m,(p,i)) = (case AList.lookup (op =) p t of NONE => ((t, m) :: p, i)
   246                            | SOME n => (AList.update (op =) (t, Rat.add (n, m)) p, i));
   247 
   248 exception Zero;
   249 
   250 fun rat_of_term (numt,dent) =
   251   let val num = HOLogic.dest_binum numt and den = HOLogic.dest_binum dent
   252   in if den = 0 then raise Zero else Rat.rat_of_quotient (num,den) end;
   253 
   254 (* Warning: in rare cases number_of encloses a non-numeral,
   255    in which case dest_binum raises TERM; hence all the handles below.
   256    Same for Suc-terms that turn out not to be numerals -
   257    although the simplifier should eliminate those anyway...
   258 *)
   259 
   260 fun number_of_Sucs (Const("Suc",_) $ n) = number_of_Sucs n + 1
   261   | number_of_Sucs t = if HOLogic.is_zero t then 0
   262                        else raise TERM("number_of_Sucs",[])
   263 
   264 (* decompose nested multiplications, bracketing them to the right and combining all
   265    their coefficients
   266 *)
   267 
   268 fun demult inj_consts =
   269 let
   270 fun demult((mC as Const("HOL.times",_)) $ s $ t,m) = ((case s of
   271         Const("Numeral.number_of",_)$n
   272         => demult(t,Rat.mult(m,Rat.rat_of_intinf(HOLogic.dest_binum n)))
   273       | Const("HOL.uminus",_)$(Const("Numeral.number_of",_)$n)
   274         => demult(t,Rat.mult(m,Rat.rat_of_intinf(~(HOLogic.dest_binum n))))
   275       | Const("Suc",_) $ _
   276         => demult(t,Rat.mult(m,Rat.rat_of_int(number_of_Sucs s)))
   277       | Const("HOL.times",_) $ s1 $ s2 => demult(mC $ s1 $ (mC $ s2 $ t),m)
   278       | Const("HOL.divide",_) $ numt $ (Const("Numeral.number_of",_)$dent) =>
   279           let val den = HOLogic.dest_binum dent
   280           in if den = 0 then raise Zero
   281              else demult(mC $ numt $ t,Rat.mult(m, Rat.inv(Rat.rat_of_intinf den)))
   282           end
   283       | _ => atomult(mC,s,t,m)
   284       ) handle TERM _ => atomult(mC,s,t,m))
   285   | demult(atom as Const("HOL.divide",_) $ t $ (Const("Numeral.number_of",_)$dent), m) =
   286       (let val den = HOLogic.dest_binum dent
   287        in if den = 0 then raise Zero else demult(t,Rat.mult(m, Rat.inv(Rat.rat_of_intinf den))) end
   288        handle TERM _ => (SOME atom,m))
   289   | demult(Const("0",_),m) = (NONE, Rat.rat_of_int 0)
   290   | demult(Const("1",_),m) = (NONE, m)
   291   | demult(t as Const("Numeral.number_of",_)$n,m) =
   292       ((NONE,Rat.mult(m,Rat.rat_of_intinf(HOLogic.dest_binum n)))
   293        handle TERM _ => (SOME t,m))
   294   | demult(Const("HOL.uminus",_)$t, m) = demult(t,Rat.mult(m,Rat.rat_of_int(~1)))
   295   | demult(t as Const f $ x, m) =
   296       (if f mem inj_consts then SOME x else SOME t,m)
   297   | demult(atom,m) = (SOME atom,m)
   298 
   299 and atomult(mC,atom,t,m) = (case demult(t,m) of (NONE,m') => (SOME atom,m')
   300                             | (SOME t',m') => (SOME(mC $ atom $ t'),m'))
   301 in demult end;
   302 
   303 fun decomp2 inj_consts (rel,lhs,rhs) =
   304 let
   305 (* Turn term into list of summand * multiplicity plus a constant *)
   306 fun poly(Const("HOL.plus",_) $ s $ t, m, pi) = poly(s,m,poly(t,m,pi))
   307   | poly(all as Const("HOL.minus",T) $ s $ t, m, pi) =
   308       if nT T then add_atom(all,m,pi) else poly(s,m,poly(t,Rat.neg m,pi))
   309   | poly(all as Const("HOL.uminus",T) $ t, m, pi) =
   310       if nT T then add_atom(all,m,pi) else poly(t,Rat.neg m,pi)
   311   | poly(Const("0",_), _, pi) = pi
   312   | poly(Const("1",_), m, (p,i)) = (p,Rat.add(i,m))
   313   | poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,Rat.add(i,m)))
   314   | poly(t as Const("HOL.times",_) $ _ $ _, m, pi as (p,i)) =
   315       (case demult inj_consts (t,m) of
   316          (NONE,m') => (p,Rat.add(i,m))
   317        | (SOME u,m') => add_atom(u,m',pi))
   318   | poly(t as Const("HOL.divide",_) $ _ $ _, m, pi as (p,i)) =
   319       (case demult inj_consts (t,m) of
   320          (NONE,m') => (p,Rat.add(i,m'))
   321        | (SOME u,m') => add_atom(u,m',pi))
   322   | poly(all as (Const("Numeral.number_of",_)$t,m,(p,i))) =
   323       ((p,Rat.add(i,Rat.mult(m,Rat.rat_of_intinf(HOLogic.dest_binum t))))
   324        handle TERM _ => add_atom all)
   325   | poly(all as Const f $ x, m, pi) =
   326       if f mem inj_consts then poly(x,m,pi) else add_atom(all,m,pi)
   327   | poly x  = add_atom x;
   328 
   329 val (p,i) = poly(lhs,Rat.rat_of_int 1,([],Rat.rat_of_int 0))
   330 and (q,j) = poly(rhs,Rat.rat_of_int 1,([],Rat.rat_of_int 0))
   331 
   332   in case rel of
   333        "Orderings.less"  => SOME(p,i,"<",q,j)
   334      | "Orderings.less_eq" => SOME(p,i,"<=",q,j)
   335      | "op ="  => SOME(p,i,"=",q,j)
   336      | _       => NONE
   337   end handle Zero => NONE;
   338 
   339 fun negate(SOME(x,i,rel,y,j,d)) = SOME(x,i,"~"^rel,y,j,d)
   340   | negate NONE = NONE;
   341 
   342 fun of_lin_arith_sort sg U =
   343   Type.of_sort (Sign.tsig_of sg) (U,["Ring_and_Field.ordered_idom"])
   344 
   345 fun allows_lin_arith sg discrete (U as Type(D,[])) =
   346       if of_lin_arith_sort sg U
   347       then (true, D mem discrete)
   348       else (* special cases *)
   349            if D mem discrete then (true,true) else (false,false)
   350   | allows_lin_arith sg discrete U = (of_lin_arith_sort sg U, false);
   351 
   352 fun decomp1 (sg,discrete,inj_consts) (T,xxx) =
   353   (case T of
   354      Type("fun",[U,_]) =>
   355        (case allows_lin_arith sg discrete U of
   356           (true,d) => (case decomp2 inj_consts xxx of NONE => NONE
   357                        | SOME(p,i,rel,q,j) => SOME(p,i,rel,q,j,d))
   358         | (false,_) => NONE)
   359    | _ => NONE);
   360 
   361 fun decomp2 data (_$(Const(rel,T)$lhs$rhs)) = decomp1 data (T,(rel,lhs,rhs))
   362   | decomp2 data (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
   363       negate(decomp1 data (T,(rel,lhs,rhs)))
   364   | decomp2 data _ = NONE
   365 
   366 fun decomp sg =
   367   let val {discrete, inj_consts, ...} = ArithTheoryData.get sg
   368   in decomp2 (sg,discrete,inj_consts) end
   369 
   370 fun number_of(n,T) = HOLogic.number_of_const T $ (HOLogic.mk_bin n)
   371 
   372 end;
   373 
   374 
   375 structure Fast_Arith =
   376   Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
   377 
   378 val fast_arith_tac    = Fast_Arith.lin_arith_tac false
   379 and fast_ex_arith_tac = Fast_Arith.lin_arith_tac
   380 and trace_arith    = Fast_Arith.trace
   381 and fast_arith_neq_limit = Fast_Arith.fast_arith_neq_limit;
   382 
   383 local
   384 
   385 (* reduce contradictory <= to False.
   386    Most of the work is done by the cancel tactics.
   387 *)
   388 val add_rules =
   389  [add_zero_left,add_zero_right,Zero_not_Suc,Suc_not_Zero,le_0_eq,
   390   One_nat_def,
   391   order_less_irrefl, zero_neq_one, zero_less_one, zero_le_one,
   392   zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];
   393 
   394 val add_mono_thms_ordered_semiring = map (fn s => prove_goal (the_context ()) s
   395  (fn prems => [cut_facts_tac prems 1,
   396                blast_tac (claset() addIs [add_mono]) 1]))
   397 ["(i <= j) & (k <= l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)",
   398  "(i  = j) & (k <= l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)",
   399  "(i <= j) & (k  = l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)",
   400  "(i  = j) & (k  = l) ==> i + k  = j + (l::'a::pordered_ab_semigroup_add)"
   401 ];
   402 
   403 val mono_ss = simpset() addsimps
   404                 [add_mono,add_strict_mono,add_less_le_mono,add_le_less_mono];
   405 
   406 val add_mono_thms_ordered_field =
   407   map (fn s => prove_goal (the_context ()) s
   408                  (fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1]))
   409     ["(i<j) & (k=l)   ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)",
   410      "(i=j) & (k<l)   ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)",
   411      "(i<j) & (k<=l)  ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)",
   412      "(i<=j) & (k<l)  ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)",
   413      "(i<j) & (k<l)   ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)"];
   414 
   415 in
   416 
   417 val init_lin_arith_data =
   418  Fast_Arith.setup #>
   419  Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
   420    {add_mono_thms = add_mono_thms @
   421     add_mono_thms_ordered_semiring @ add_mono_thms_ordered_field,
   422     mult_mono_thms = mult_mono_thms,
   423     inj_thms = inj_thms,
   424     lessD = lessD @ [Suc_leI],
   425     neqE = [linorder_neqE_nat,
   426       get_thm (theory "Ring_and_Field") (Name "linorder_neqE_ordered_idom")],
   427     simpset = HOL_basic_ss addsimps add_rules
   428                    addsimprocs [ab_group_add_cancel.sum_conv,
   429                                 ab_group_add_cancel.rel_conv]
   430                    (*abel_cancel helps it work in abstract algebraic domains*)
   431                    addsimprocs nat_cancel_sums_add}) #>
   432   ArithTheoryData.init #>
   433   arith_discrete "nat";
   434 
   435 end;
   436 
   437 val fast_nat_arith_simproc =
   438   Simplifier.simproc (the_context ()) "fast_nat_arith"
   439     ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] Fast_Arith.lin_arith_prover;
   440 
   441 
   442 (* Because of fast_nat_arith_simproc, the arithmetic solver is really only
   443 useful to detect inconsistencies among the premises for subgoals which are
   444 *not* themselves (in)equalities, because the latter activate
   445 fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
   446 solver all the time rather than add the additional check. *)
   447 
   448 
   449 (* arith proof method *)
   450 
   451 (* FIXME: K true should be replaced by a sensible test to speed things up
   452    in case there are lots of irrelevant terms involved;
   453    elimination of min/max can be optimized:
   454    (max m n + k <= r) = (m+k <= r & n+k <= r)
   455    (l <= min m n + k) = (l <= m+k & l <= n+k)
   456 *)
   457 local
   458 (* a simpset for computations subject to optimization !!! *)
   459 (*
   460 val binarith = map thm
   461   ["Pls_0_eq", "Min_1_eq",
   462  "bin_pred_Pls","bin_pred_Min","bin_pred_1","bin_pred_0",
   463   "bin_succ_Pls", "bin_succ_Min", "bin_succ_1", "bin_succ_0",
   464   "bin_add_Pls", "bin_add_Min", "bin_add_BIT_0", "bin_add_BIT_10",
   465   "bin_add_BIT_11", "bin_minus_Pls", "bin_minus_Min", "bin_minus_1", 
   466   "bin_minus_0", "bin_mult_Pls", "bin_mult_Min", "bin_mult_1", "bin_mult_0", 
   467   "bin_add_Pls_right", "bin_add_Min_right"];
   468  val intarithrel = 
   469      (map thm ["int_eq_number_of_eq","int_neg_number_of_BIT", 
   470 		"int_le_number_of_eq","int_iszero_number_of_0",
   471 		"int_less_number_of_eq_neg"]) @
   472      (map (fn s => thm s RS thm "lift_bool") 
   473 	  ["int_iszero_number_of_Pls","int_iszero_number_of_1",
   474 	   "int_neg_number_of_Min"])@
   475      (map (fn s => thm s RS thm "nlift_bool") 
   476 	  ["int_nonzero_number_of_Min","int_not_neg_number_of_Pls"]);
   477      
   478 val intarith = map thm ["int_number_of_add_sym", "int_number_of_minus_sym",
   479 			"int_number_of_diff_sym", "int_number_of_mult_sym"];
   480 val natarith = map thm ["add_nat_number_of", "diff_nat_number_of",
   481 			"mult_nat_number_of", "eq_nat_number_of",
   482 			"less_nat_number_of"]
   483 val powerarith = 
   484     (map thm ["nat_number_of", "zpower_number_of_even", 
   485 	      "zpower_Pls", "zpower_Min"]) @ 
   486     [(Tactic.simplify true [thm "zero_eq_Numeral0_nring", 
   487 			   thm "one_eq_Numeral1_nring"] 
   488   (thm "zpower_number_of_odd"))]
   489 
   490 val comp_arith = binarith @ intarith @ intarithrel @ natarith 
   491 	    @ powerarith @[thm"not_false_eq_true", thm "not_true_eq_false"];
   492 
   493 val comp_ss = HOL_basic_ss addsimps comp_arith addsimps simp_thms;
   494 *)
   495 fun raw_arith_tac ex i st =
   496   refute_tac (K true)
   497    (REPEAT o split_tac (#splits (ArithTheoryData.get (Thm.theory_of_thm st))))
   498 (*   (REPEAT o 
   499     (fn i =>(split_tac (#splits (ArithTheoryData.get(Thm.theory_of_thm st))) i)
   500 		THEN (simp_tac comp_ss i))) *)
   501    ((REPEAT_DETERM o etac linorder_neqE) THEN' fast_ex_arith_tac ex)
   502    i st;
   503 
   504 fun presburger_tac i st =
   505   (case ArithTheoryData.get (Thm.theory_of_thm st) of
   506      {presburger = SOME tac, ...} =>
   507        (warning "Trying full Presburger arithmetic ..."; tac i st)
   508    | _ => no_tac st);
   509 
   510 in
   511 
   512 val simple_arith_tac = FIRST' [fast_arith_tac,
   513   ObjectLogic.atomize_tac THEN' raw_arith_tac true];
   514 
   515 val arith_tac = FIRST' [fast_arith_tac,
   516   ObjectLogic.atomize_tac THEN' raw_arith_tac true,
   517   presburger_tac];
   518 
   519 val silent_arith_tac = FIRST' [fast_arith_tac,
   520   ObjectLogic.atomize_tac THEN' raw_arith_tac false,
   521   presburger_tac];
   522 
   523 fun arith_method prems =
   524   Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac));
   525 
   526 end;
   527 
   528 (* antisymmetry:
   529    combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y
   530 
   531 local
   532 val antisym = mk_meta_eq order_antisym
   533 val not_lessD = linorder_not_less RS iffD1
   534 fun prp t thm = (#prop(rep_thm thm) = t)
   535 in
   536 fun antisym_eq prems thm =
   537   let
   538     val r = #prop(rep_thm thm);
   539   in
   540     case r of
   541       Tr $ ((c as Const("Orderings.less_eq",T)) $ s $ t) =>
   542         let val r' = Tr $ (c $ t $ s)
   543         in
   544           case Library.find_first (prp r') prems of
   545             NONE =>
   546               let val r' = Tr $ (HOLogic.Not $ (Const("Orderings.less",T) $ s $ t))
   547               in case Library.find_first (prp r') prems of
   548                    NONE => []
   549                  | SOME thm' => [(thm' RS not_lessD) RS (thm RS antisym)]
   550               end
   551           | SOME thm' => [thm' RS (thm RS antisym)]
   552         end
   553     | Tr $ (Const("Not",_) $ (Const("Orderings.less",T) $ s $ t)) =>
   554         let val r' = Tr $ (Const("Orderings.less_eq",T) $ s $ t)
   555         in
   556           case Library.find_first (prp r') prems of
   557             NONE =>
   558               let val r' = Tr $ (HOLogic.Not $ (Const("Orderings.less",T) $ t $ s))
   559               in case Library.find_first (prp r') prems of
   560                    NONE => []
   561                  | SOME thm' =>
   562                      [(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)]
   563               end
   564           | SOME thm' => [thm' RS ((thm RS not_lessD) RS antisym)]
   565         end
   566     | _ => []
   567   end
   568   handle THM _ => []
   569 end;
   570 *)
   571 
   572 (* theory setup *)
   573 
   574 val arith_setup =
   575   init_lin_arith_data #>
   576   (fn thy => (Simplifier.change_simpset_of thy (fn ss => ss
   577     addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc])
   578     addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)); thy)) #>
   579   Method.add_methods
   580     [("arith", (arith_method o #2) oo Method.syntax Args.bang_facts,
   581       "decide linear arithmethic")] #>
   582   Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
   583     "declaration of split rules for arithmetic procedure")];