moved lin_arith stuff to Tools/lin_arith.ML;
authorwenzelm
Tue Jul 31 19:40:26 2007 +0200 (2007-07-31)
changeset 24095785c3cd7fcb5
parent 24094 6db35c14146d
child 24096 74926cdbf071
moved lin_arith stuff to Tools/lin_arith.ML;
src/HOL/arith_data.ML
     1.1 --- a/src/HOL/arith_data.ML	Tue Jul 31 19:40:25 2007 +0200
     1.2 +++ b/src/HOL/arith_data.ML	Tue Jul 31 19:40:26 2007 +0200
     1.3 @@ -1,13 +1,11 @@
     1.4  (*  Title:      HOL/arith_data.ML
     1.5      ID:         $Id$
     1.6 -    Author:     Markus Wenzel, Stefan Berghofer and Tobias Nipkow
     1.7 +    Author:     Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
     1.8  
     1.9 -Various arithmetic proof procedures.
    1.10 +Basic arithmetic proof tools.
    1.11  *)
    1.12  
    1.13 -(*---------------------------------------------------------------------------*)
    1.14 -(* 1. Cancellation of common terms                                           *)
    1.15 -(*---------------------------------------------------------------------------*)
    1.16 +(*** cancellation of common terms ***)
    1.17  
    1.18  structure NatArithUtils =
    1.19  struct
    1.20 @@ -28,6 +26,7 @@
    1.21      funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
    1.22    end;
    1.23  
    1.24 +
    1.25  (* dest_sum *)
    1.26  
    1.27  val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} HOLogic.natT;
    1.28 @@ -42,6 +41,8 @@
    1.29            SOME (t, u) => dest_sum t @ dest_sum u
    1.30          | NONE => [tm]));
    1.31  
    1.32 +
    1.33 +
    1.34  (** generic proof tools **)
    1.35  
    1.36  (* prove conversions *)
    1.37 @@ -61,21 +62,25 @@
    1.38  fun prep_simproc (name, pats, proc) =
    1.39    Simplifier.simproc (the_context ()) name pats proc;
    1.40  
    1.41 -end;  (* NatArithUtils *)
    1.42 +end;
    1.43 +
    1.44  
    1.45  
    1.46 +(** ArithData **)
    1.47 +
    1.48  signature ARITH_DATA =
    1.49  sig
    1.50    val nat_cancel_sums_add: simproc list
    1.51    val nat_cancel_sums: simproc list
    1.52 +  val arith_data_setup: Context.generic -> Context.generic
    1.53  end;
    1.54  
    1.55 -
    1.56  structure ArithData: ARITH_DATA =
    1.57  struct
    1.58  
    1.59  open NatArithUtils;
    1.60  
    1.61 +
    1.62  (** cancel common summands **)
    1.63  
    1.64  structure Sum =
    1.65 @@ -92,6 +97,7 @@
    1.66  fun gen_uncancel_tac rule ct =
    1.67    rtac (instantiate' [] [NONE, SOME ct] (rule RS @{thm subst_equals})) 1;
    1.68  
    1.69 +
    1.70  (* nat eq *)
    1.71  
    1.72  structure EqCancelSums = CancelSumsFun
    1.73 @@ -102,6 +108,7 @@
    1.74    val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel"};
    1.75  end);
    1.76  
    1.77 +
    1.78  (* nat less *)
    1.79  
    1.80  structure LessCancelSums = CancelSumsFun
    1.81 @@ -112,6 +119,7 @@
    1.82    val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_less"};
    1.83  end);
    1.84  
    1.85 +
    1.86  (* nat le *)
    1.87  
    1.88  structure LeCancelSums = CancelSumsFun
    1.89 @@ -122,6 +130,7 @@
    1.90    val uncancel_tac = gen_uncancel_tac @{thm "nat_add_left_cancel_le"};
    1.91  end);
    1.92  
    1.93 +
    1.94  (* nat diff *)
    1.95  
    1.96  structure DiffCancelSums = CancelSumsFun
    1.97 @@ -132,7 +141,8 @@
    1.98    val uncancel_tac = gen_uncancel_tac @{thm "diff_cancel"};
    1.99  end);
   1.100  
   1.101 -(** prepare nat_cancel simprocs **)
   1.102 +
   1.103 +(* prepare nat_cancel simprocs *)
   1.104  
   1.105  val nat_cancel_sums_add = map prep_simproc
   1.106    [("nateq_cancel_sums",
   1.107 @@ -150,848 +160,11 @@
   1.108      ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"],
   1.109      K DiffCancelSums.proc)];
   1.110  
   1.111 -end;  (* ArithData *)
   1.112 -
   1.113 -open ArithData;
   1.114 -
   1.115 -
   1.116 -(*---------------------------------------------------------------------------*)
   1.117 -(* 2. Linear arithmetic                                                      *)
   1.118 -(*---------------------------------------------------------------------------*)
   1.119 -
   1.120 -(* Parameters data for general linear arithmetic functor *)
   1.121 -
   1.122 -structure LA_Logic: LIN_ARITH_LOGIC =
   1.123 -struct
   1.124 -
   1.125 -val ccontr = ccontr;
   1.126 -val conjI = conjI;
   1.127 -val notI = notI;
   1.128 -val sym = sym;
   1.129 -val not_lessD = @{thm linorder_not_less} RS iffD1;
   1.130 -val not_leD = @{thm linorder_not_le} RS iffD1;
   1.131 -val le0 = thm "le0";
   1.132 -
   1.133 -fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
   1.134 -
   1.135 -val mk_Trueprop = HOLogic.mk_Trueprop;
   1.136 -
   1.137 -fun atomize thm = case Thm.prop_of thm of
   1.138 -    Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
   1.139 -    atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
   1.140 -  | _ => [thm];
   1.141 -
   1.142 -fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
   1.143 -  | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
   1.144 -  | neg_prop t = raise TERM ("neg_prop", [t]);
   1.145 -
   1.146 -fun is_False thm =
   1.147 -  let val _ $ t = Thm.prop_of thm
   1.148 -  in t = Const("False",HOLogic.boolT) end;
   1.149 -
   1.150 -fun is_nat(t) = fastype_of1 t = HOLogic.natT;
   1.151 -
   1.152 -fun mk_nat_thm sg t =
   1.153 -  let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
   1.154 -  in instantiate ([],[(cn,ct)]) le0 end;
   1.155 -
   1.156 -end;  (* LA_Logic *)
   1.157 -
   1.158 -
   1.159 -(* arith theory data *)
   1.160 -
   1.161 -datatype arithtactic = ArithTactic of {name: string, tactic: int -> tactic, id: stamp};
   1.162 -
   1.163 -fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
   1.164 -
   1.165 -fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
   1.166 -
   1.167 -structure ArithContextData = GenericDataFun
   1.168 -(
   1.169 -  type T = {splits: thm list,
   1.170 -            inj_consts: (string * typ) list,
   1.171 -            discrete: string list,
   1.172 -            tactics: arithtactic list};
   1.173 -  val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
   1.174 -  val extend = I;
   1.175 -  fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
   1.176 -             {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
   1.177 -   {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
   1.178 -    inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
   1.179 -    discrete = Library.merge (op =) (discrete1, discrete2),
   1.180 -    tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
   1.181 -);
   1.182 -
   1.183 -val get_arith_data = ArithContextData.get o Context.Proof;
   1.184 -
   1.185 -val arith_split_add = Thm.declaration_attribute (fn thm =>
   1.186 -  ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
   1.187 -    {splits = insert Thm.eq_thm_prop thm splits,
   1.188 -     inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
   1.189 -
   1.190 -fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
   1.191 -  {splits = splits, inj_consts = inj_consts,
   1.192 -   discrete = insert (op =) d discrete, tactics = tactics});
   1.193 -
   1.194 -fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
   1.195 -  {splits = splits, inj_consts = insert (op =) c inj_consts,
   1.196 -   discrete = discrete, tactics= tactics});
   1.197 -
   1.198 -fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
   1.199 -  {splits = splits, inj_consts = inj_consts, discrete = discrete,
   1.200 -   tactics = insert eq_arith_tactic tac tactics});
   1.201 -
   1.202 -
   1.203 -signature HOL_LIN_ARITH_DATA =
   1.204 -sig
   1.205 -  include LIN_ARITH_DATA
   1.206 -  val fast_arith_split_limit: int ConfigOption.T
   1.207 -  val setup_options: theory -> theory
   1.208 -end;
   1.209 -
   1.210 -structure LA_Data_Ref: HOL_LIN_ARITH_DATA =
   1.211 -struct
   1.212 -
   1.213 -val (fast_arith_split_limit, setup1) = ConfigOption.int "fast_arith_split_limit" 9;
   1.214 -val (fast_arith_neq_limit, setup2) = ConfigOption.int "fast_arith_neq_limit" 9;
   1.215 -val setup_options = setup1 #> setup2;
   1.216 +val arith_data_setup =
   1.217 +  Simplifier.map_ss (fn ss => ss addsimprocs nat_cancel_sums);
   1.218  
   1.219  
   1.220 -(* internal representation of linear (in-)equations *)
   1.221 -type decompT = ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
   1.222 -
   1.223 -(* Decomposition of terms *)
   1.224 -
   1.225 -fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
   1.226 -  | nT _                      = false;
   1.227 -
   1.228 -fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
   1.229 -             (term * Rat.rat) list * Rat.rat =
   1.230 -  case AList.lookup (op =) p t of NONE   => ((t, m) :: p, i)
   1.231 -                                | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
   1.232 -
   1.233 -exception Zero;
   1.234 -
   1.235 -fun rat_of_term (numt, dent) =
   1.236 -  let
   1.237 -    val num = HOLogic.dest_numeral numt
   1.238 -    val den = HOLogic.dest_numeral dent
   1.239 -  in
   1.240 -    if den = 0 then raise Zero else Rat.rat_of_quotient (num, den)
   1.241 -  end;
   1.242 -
   1.243 -(* Warning: in rare cases number_of encloses a non-numeral,
   1.244 -   in which case dest_numeral raises TERM; hence all the handles below.
   1.245 -   Same for Suc-terms that turn out not to be numerals -
   1.246 -   although the simplifier should eliminate those anyway ...
   1.247 -*)
   1.248 -fun number_of_Sucs (Const ("Suc", _) $ n) : int =
   1.249 -      number_of_Sucs n + 1
   1.250 -  | number_of_Sucs t =
   1.251 -      if HOLogic.is_zero t then 0 else raise TERM ("number_of_Sucs", []);
   1.252 -
   1.253 -(* decompose nested multiplications, bracketing them to the right and combining
   1.254 -   all their coefficients
   1.255 -*)
   1.256 -fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
   1.257 -let
   1.258 -  fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) = (
   1.259 -    (case s of
   1.260 -      Const ("Numeral.number_class.number_of", _) $ n =>
   1.261 -        demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
   1.262 -    | Const (@{const_name HOL.uminus}, _) $ (Const ("Numeral.number_class.number_of", _) $ n) =>
   1.263 -        demult (t, Rat.mult m (Rat.rat_of_int (~(HOLogic.dest_numeral n))))
   1.264 -    | Const (@{const_name Suc}, _) $ _ =>
   1.265 -        demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat s)))
   1.266 -    | Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
   1.267 -        demult (mC $ s1 $ (mC $ s2 $ t), m)
   1.268 -    | Const (@{const_name HOL.divide}, _) $ numt $ (Const ("Numeral.number_class.number_of", _) $ dent) =>
   1.269 -        let
   1.270 -          val den = HOLogic.dest_numeral dent
   1.271 -        in
   1.272 -          if den = 0 then
   1.273 -            raise Zero
   1.274 -          else
   1.275 -            demult (mC $ numt $ t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
   1.276 -        end
   1.277 -    | _ =>
   1.278 -        atomult (mC, s, t, m)
   1.279 -    ) handle TERM _ => atomult (mC, s, t, m)
   1.280 -  )
   1.281 -    | demult (atom as Const(@{const_name HOL.divide}, _) $ t $ (Const ("Numeral.number_class.number_of", _) $ dent), m) =
   1.282 -      (let
   1.283 -        val den = HOLogic.dest_numeral dent
   1.284 -      in
   1.285 -        if den = 0 then
   1.286 -          raise Zero
   1.287 -        else
   1.288 -          demult (t, Rat.mult m (Rat.inv (Rat.rat_of_int den)))
   1.289 -      end
   1.290 -        handle TERM _ => (SOME atom, m))
   1.291 -    | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
   1.292 -    | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
   1.293 -    | demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) =
   1.294 -        ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
   1.295 -          handle TERM _ => (SOME t, m))
   1.296 -    | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
   1.297 -    | demult (t as Const f $ x, m) =
   1.298 -        (if member (op =) inj_consts f then SOME x else SOME t, m)
   1.299 -    | demult (atom, m) = (SOME atom, m)
   1.300 -and
   1.301 -  atomult (mC, atom, t, m) = (
   1.302 -    case demult (t, m) of (NONE, m')    => (SOME atom, m')
   1.303 -                        | (SOME t', m') => (SOME (mC $ atom $ t'), m')
   1.304 -  )
   1.305 -in demult end;
   1.306 -
   1.307 -fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
   1.308 -            ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
   1.309 -let
   1.310 -  (* Turn term into list of summand * multiplicity plus a constant *)
   1.311 -  fun poly (Const (@{const_name HOL.plus}, _) $ s $ t, m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) =
   1.312 -        poly (s, m, poly (t, m, pi))
   1.313 -    | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
   1.314 -        if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
   1.315 -    | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
   1.316 -        if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
   1.317 -    | poly (Const (@{const_name HOL.zero}, _), _, pi) =
   1.318 -        pi
   1.319 -    | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
   1.320 -        (p, Rat.add i m)
   1.321 -    | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
   1.322 -        poly (t, m, (p, Rat.add i m))
   1.323 -    | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
   1.324 -        (case demult inj_consts (all, m) of
   1.325 -           (NONE,   m') => (p, Rat.add i m')
   1.326 -         | (SOME u, m') => add_atom u m' pi)
   1.327 -    | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
   1.328 -        (case demult inj_consts (all, m) of
   1.329 -           (NONE,   m') => (p, Rat.add i m')
   1.330 -         | (SOME u, m') => add_atom u m' pi)
   1.331 -    | poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
   1.332 -        (let val k = HOLogic.dest_numeral t
   1.333 -            val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
   1.334 -        in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
   1.335 -        handle TERM _ => add_atom all m pi)
   1.336 -    | poly (all as Const f $ x, m, pi) =
   1.337 -        if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
   1.338 -    | poly (all, m, pi) =
   1.339 -        add_atom all m pi
   1.340 -  val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
   1.341 -  val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
   1.342 -in
   1.343 -  case rel of
   1.344 -    @{const_name HOL.less}    => SOME (p, i, "<", q, j)
   1.345 -  | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
   1.346 -  | "op ="              => SOME (p, i, "=", q, j)
   1.347 -  | _                   => NONE
   1.348 -end handle Zero => NONE;
   1.349 -
   1.350 -fun of_lin_arith_sort sg (U : typ) : bool =
   1.351 -  Type.of_sort (Sign.tsig_of sg) (U, ["Ring_and_Field.ordered_idom"])
   1.352 -
   1.353 -fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
   1.354 -  if of_lin_arith_sort sg U then
   1.355 -    (true, D mem discrete)
   1.356 -  else (* special cases *)
   1.357 -    if D mem discrete then  (true, true)  else  (false, false)
   1.358 -  | allows_lin_arith sg discrete U =
   1.359 -  (of_lin_arith_sort sg U, false);
   1.360 -
   1.361 -fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decompT option =
   1.362 -  case T of
   1.363 -    Type ("fun", [U, _]) =>
   1.364 -      (case allows_lin_arith thy discrete U of
   1.365 -        (true, d) =>
   1.366 -          (case decomp0 inj_consts xxx of
   1.367 -            NONE                   => NONE
   1.368 -          | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
   1.369 -      | (false, _) =>
   1.370 -          NONE)
   1.371 -  | _ => NONE;
   1.372 -
   1.373 -fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
   1.374 -  | negate NONE                        = NONE;
   1.375 -
   1.376 -fun decomp_negation data
   1.377 -  ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option =
   1.378 -      decomp_typecheck data (T, (rel, lhs, rhs))
   1.379 -  | decomp_negation data ((Const ("Trueprop", _)) $
   1.380 -  (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
   1.381 -      negate (decomp_typecheck data (T, (rel, lhs, rhs)))
   1.382 -  | decomp_negation data _ =
   1.383 -      NONE;
   1.384 -
   1.385 -fun decomp ctxt : term -> decompT option =
   1.386 -  let
   1.387 -    val thy = ProofContext.theory_of ctxt
   1.388 -    val {discrete, inj_consts, ...} = get_arith_data ctxt
   1.389 -  in decomp_negation (thy, discrete, inj_consts) end;
   1.390 -
   1.391 -fun domain_is_nat (_ $ (Const (_, T) $ _ $ _))                      = nT T
   1.392 -  | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
   1.393 -  | domain_is_nat _                                                 = false;
   1.394 -
   1.395 -fun number_of (n, T) = HOLogic.mk_number T n;
   1.396 -
   1.397 -(*---------------------------------------------------------------------------*)
   1.398 -(* the following code performs splitting of certain constants (e.g. min,     *)
   1.399 -(* max) in a linear arithmetic problem; similar to what split_tac later does *)
   1.400 -(* to the proof state                                                        *)
   1.401 -(*---------------------------------------------------------------------------*)
   1.402 -
   1.403 -(* checks if splitting with 'thm' is implemented                             *)
   1.404 -
   1.405 -fun is_split_thm (thm : thm) : bool =
   1.406 -  case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
   1.407 -    (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
   1.408 -    case head_of lhs of
   1.409 -      Const (a, _) => member (op =) [@{const_name Orderings.max},
   1.410 -                                    @{const_name Orderings.min},
   1.411 -                                    @{const_name HOL.abs},
   1.412 -                                    @{const_name HOL.minus},
   1.413 -                                    "IntDef.nat",
   1.414 -                                    "Divides.div_class.mod",
   1.415 -                                    "Divides.div_class.div"] a
   1.416 -    | _            => (warning ("Lin. Arith.: wrong format for split rule " ^
   1.417 -                                 Display.string_of_thm thm);
   1.418 -                       false))
   1.419 -  | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
   1.420 -                   Display.string_of_thm thm);
   1.421 -          false);
   1.422 -
   1.423 -(* substitute new for occurrences of old in a term, incrementing bound       *)
   1.424 -(* variables as needed when substituting inside an abstraction               *)
   1.425 -
   1.426 -fun subst_term ([] : (term * term) list) (t : term) = t
   1.427 -  | subst_term pairs                     t          =
   1.428 -      (case AList.lookup (op aconv) pairs t of
   1.429 -        SOME new =>
   1.430 -          new
   1.431 -      | NONE     =>
   1.432 -          (case t of Abs (a, T, body) =>
   1.433 -            let val pairs' = map (pairself (incr_boundvars 1)) pairs
   1.434 -            in  Abs (a, T, subst_term pairs' body)  end
   1.435 -          | t1 $ t2                   =>
   1.436 -            subst_term pairs t1 $ subst_term pairs t2
   1.437 -          | _ => t));
   1.438 -
   1.439 -(* approximates the effect of one application of split_tac (followed by NNF  *)
   1.440 -(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)
   1.441 -(* list of new subgoals (each again represented by a typ list for bound      *)
   1.442 -(* variables and a term list for premises), or NONE if split_tac would fail  *)
   1.443 -(* on the subgoal                                                            *)
   1.444 -
   1.445 -(* FIXME: currently only the effect of certain split theorems is reproduced  *)
   1.446 -(*        (which is why we need 'is_split_thm').  A more canonical           *)
   1.447 -(*        implementation should analyze the right-hand side of the split     *)
   1.448 -(*        theorem that can be applied, and modify the subgoal accordingly.   *)
   1.449 -(*        Or even better, the splitter should be extended to provide         *)
   1.450 -(*        splitting on terms as well as splitting on theorems (where the     *)
   1.451 -(*        former can have a faster implementation as it does not need to be  *)
   1.452 -(*        proof-producing).                                                  *)
   1.453 -
   1.454 -fun split_once_items ctxt (Ts : typ list, terms : term list) :
   1.455 -                     (typ list * term list) list option =
   1.456 -let
   1.457 -  val thy = ProofContext.theory_of ctxt
   1.458 -  (* takes a list  [t1, ..., tn]  to the term                                *)
   1.459 -  (*   tn' --> ... --> t1' --> False  ,                                      *)
   1.460 -  (* where ti' = HOLogic.dest_Trueprop ti                                    *)
   1.461 -  fun REPEAT_DETERM_etac_rev_mp terms' =
   1.462 -    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
   1.463 -  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
   1.464 -  val cmap       = Splitter.cmap_of_split_thms split_thms
   1.465 -  val splits     = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
   1.466 -  val split_limit = ConfigOption.get ctxt fast_arith_split_limit
   1.467 -in
   1.468 -  if length splits > split_limit then
   1.469 -   (tracing ("fast_arith_split_limit exceeded (current value is " ^
   1.470 -      string_of_int split_limit ^ ")"); NONE)
   1.471 -  else (
   1.472 -  case splits of [] =>
   1.473 -    (* split_tac would fail: no possible split *)
   1.474 -    NONE
   1.475 -  | ((_, _, _, split_type, split_term) :: _) => (
   1.476 -    (* ignore all but the first possible split *)
   1.477 -    case strip_comb split_term of
   1.478 -    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
   1.479 -      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
   1.480 -      let
   1.481 -        val rev_terms     = rev terms
   1.482 -        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
   1.483 -        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
   1.484 -        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
   1.485 -                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
   1.486 -        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
   1.487 -        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.488 -        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
   1.489 -        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
   1.490 -      in
   1.491 -        SOME [(Ts, subgoal1), (Ts, subgoal2)]
   1.492 -      end
   1.493 -    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
   1.494 -    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
   1.495 -      let
   1.496 -        val rev_terms     = rev terms
   1.497 -        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
   1.498 -        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
   1.499 -        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
   1.500 -                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
   1.501 -        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
   1.502 -        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.503 -        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
   1.504 -        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
   1.505 -      in
   1.506 -        SOME [(Ts, subgoal1), (Ts, subgoal2)]
   1.507 -      end
   1.508 -    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
   1.509 -    | (Const (@{const_name HOL.abs}, _), [t1]) =>
   1.510 -      let
   1.511 -        val rev_terms   = rev terms
   1.512 -        val terms1      = map (subst_term [(split_term, t1)]) rev_terms
   1.513 -        val terms2      = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
   1.514 -                            split_type --> split_type) $ t1)]) rev_terms
   1.515 -        val zero        = Const (@{const_name HOL.zero}, split_type)
   1.516 -        val zero_leq_t1 = Const (@{const_name HOL.less_eq},
   1.517 -                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1
   1.518 -        val t1_lt_zero  = Const (@{const_name HOL.less},
   1.519 -                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
   1.520 -        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.521 -        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
   1.522 -        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
   1.523 -      in
   1.524 -        SOME [(Ts, subgoal1), (Ts, subgoal2)]
   1.525 -      end
   1.526 -    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
   1.527 -    | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
   1.528 -      let
   1.529 -        (* "d" in the above theorem becomes a new bound variable after NNF   *)
   1.530 -        (* transformation, therefore some adjustment of indices is necessary *)
   1.531 -        val rev_terms       = rev terms
   1.532 -        val zero            = Const (@{const_name HOL.zero}, split_type)
   1.533 -        val d               = Bound 0
   1.534 -        val terms1          = map (subst_term [(split_term, zero)]) rev_terms
   1.535 -        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])
   1.536 -                                (map (incr_boundvars 1) rev_terms)
   1.537 -        val t1'             = incr_boundvars 1 t1
   1.538 -        val t2'             = incr_boundvars 1 t2
   1.539 -        val t1_lt_t2        = Const (@{const_name HOL.less},
   1.540 -                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
   1.541 -        val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
   1.542 -                                (Const (@{const_name HOL.plus},
   1.543 -                                  split_type --> split_type --> split_type) $ t2' $ d)
   1.544 -        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.545 -        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
   1.546 -        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
   1.547 -      in
   1.548 -        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
   1.549 -      end
   1.550 -    (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
   1.551 -    | (Const ("IntDef.nat", _), [t1]) =>
   1.552 -      let
   1.553 -        val rev_terms   = rev terms
   1.554 -        val zero_int    = Const (@{const_name HOL.zero}, HOLogic.intT)
   1.555 -        val zero_nat    = Const (@{const_name HOL.zero}, HOLogic.natT)
   1.556 -        val n           = Bound 0
   1.557 -        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])
   1.558 -                            (map (incr_boundvars 1) rev_terms)
   1.559 -        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms
   1.560 -        val t1'         = incr_boundvars 1 t1
   1.561 -        val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
   1.562 -                            (Const ("Nat.of_nat", HOLogic.natT --> HOLogic.intT) $ n)
   1.563 -        val t1_lt_zero  = Const (@{const_name HOL.less},
   1.564 -                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
   1.565 -        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.566 -        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
   1.567 -        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
   1.568 -      in
   1.569 -        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
   1.570 -      end
   1.571 -    (* "?P ((?n::nat) mod (number_of ?k)) =
   1.572 -         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
   1.573 -           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
   1.574 -    | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
   1.575 -      let
   1.576 -        val rev_terms               = rev terms
   1.577 -        val zero                    = Const (@{const_name HOL.zero}, split_type)
   1.578 -        val i                       = Bound 1
   1.579 -        val j                       = Bound 0
   1.580 -        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
   1.581 -        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])
   1.582 -                                        (map (incr_boundvars 2) rev_terms)
   1.583 -        val t1'                     = incr_boundvars 2 t1
   1.584 -        val t2'                     = incr_boundvars 2 t2
   1.585 -        val t2_eq_zero              = Const ("op =",
   1.586 -                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
   1.587 -        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
   1.588 -                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
   1.589 -        val j_lt_t2                 = Const (@{const_name HOL.less},
   1.590 -                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
   1.591 -        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
   1.592 -                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
   1.593 -                                         (Const (@{const_name HOL.times},
   1.594 -                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
   1.595 -        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.596 -        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
   1.597 -        val subgoal2                = (map HOLogic.mk_Trueprop
   1.598 -                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
   1.599 -                                          @ terms2 @ [not_false]
   1.600 -      in
   1.601 -        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
   1.602 -      end
   1.603 -    (* "?P ((?n::nat) div (number_of ?k)) =
   1.604 -         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
   1.605 -           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
   1.606 -    | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
   1.607 -      let
   1.608 -        val rev_terms               = rev terms
   1.609 -        val zero                    = Const (@{const_name HOL.zero}, split_type)
   1.610 -        val i                       = Bound 1
   1.611 -        val j                       = Bound 0
   1.612 -        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
   1.613 -        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])
   1.614 -                                        (map (incr_boundvars 2) rev_terms)
   1.615 -        val t1'                     = incr_boundvars 2 t1
   1.616 -        val t2'                     = incr_boundvars 2 t2
   1.617 -        val t2_eq_zero              = Const ("op =",
   1.618 -                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
   1.619 -        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
   1.620 -                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
   1.621 -        val j_lt_t2                 = Const (@{const_name HOL.less},
   1.622 -                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
   1.623 -        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
   1.624 -                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
   1.625 -                                         (Const (@{const_name HOL.times},
   1.626 -                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
   1.627 -        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.628 -        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
   1.629 -        val subgoal2                = (map HOLogic.mk_Trueprop
   1.630 -                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
   1.631 -                                          @ terms2 @ [not_false]
   1.632 -      in
   1.633 -        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
   1.634 -      end
   1.635 -    (* "?P ((?n::int) mod (number_of ?k)) =
   1.636 -         ((iszero (number_of ?k) --> ?P ?n) &
   1.637 -          (neg (number_of (uminus ?k)) -->
   1.638 -            (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
   1.639 -          (neg (number_of ?k) -->
   1.640 -            (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
   1.641 -    | (Const ("Divides.div_class.mod",
   1.642 -        Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
   1.643 -      let
   1.644 -        val rev_terms               = rev terms
   1.645 -        val zero                    = Const (@{const_name HOL.zero}, split_type)
   1.646 -        val i                       = Bound 1
   1.647 -        val j                       = Bound 0
   1.648 -        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
   1.649 -        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])
   1.650 -                                        (map (incr_boundvars 2) rev_terms)
   1.651 -        val t1'                     = incr_boundvars 2 t1
   1.652 -        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
   1.653 -        val iszero_t2               = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
   1.654 -        val neg_minus_k             = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
   1.655 -                                        (number_of $
   1.656 -                                          (Const (@{const_name HOL.uminus},
   1.657 -                                            HOLogic.intT --> HOLogic.intT) $ k'))
   1.658 -        val zero_leq_j              = Const (@{const_name HOL.less_eq},
   1.659 -                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
   1.660 -        val j_lt_t2                 = Const (@{const_name HOL.less},
   1.661 -                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
   1.662 -        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
   1.663 -                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
   1.664 -                                         (Const (@{const_name HOL.times},
   1.665 -                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
   1.666 -        val neg_t2                  = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
   1.667 -        val t2_lt_j                 = Const (@{const_name HOL.less},
   1.668 -                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
   1.669 -        val j_leq_zero              = Const (@{const_name HOL.less_eq},
   1.670 -                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
   1.671 -        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.672 -        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
   1.673 -        val subgoal2                = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
   1.674 -                                        @ hd terms2_3
   1.675 -                                        :: (if tl terms2_3 = [] then [not_false] else [])
   1.676 -                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
   1.677 -                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
   1.678 -        val subgoal3                = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
   1.679 -                                        @ hd terms2_3
   1.680 -                                        :: (if tl terms2_3 = [] then [not_false] else [])
   1.681 -                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
   1.682 -                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
   1.683 -        val Ts'                     = split_type :: split_type :: Ts
   1.684 -      in
   1.685 -        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
   1.686 -      end
   1.687 -    (* "?P ((?n::int) div (number_of ?k)) =
   1.688 -         ((iszero (number_of ?k) --> ?P 0) &
   1.689 -          (neg (number_of (uminus ?k)) -->
   1.690 -            (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
   1.691 -          (neg (number_of ?k) -->
   1.692 -            (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
   1.693 -    | (Const ("Divides.div_class.div",
   1.694 -        Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) =>
   1.695 -      let
   1.696 -        val rev_terms               = rev terms
   1.697 -        val zero                    = Const (@{const_name HOL.zero}, split_type)
   1.698 -        val i                       = Bound 1
   1.699 -        val j                       = Bound 0
   1.700 -        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
   1.701 -        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])
   1.702 -                                        (map (incr_boundvars 2) rev_terms)
   1.703 -        val t1'                     = incr_boundvars 2 t1
   1.704 -        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
   1.705 -        val iszero_t2               = Const ("IntDef.iszero", split_type --> HOLogic.boolT) $ t2
   1.706 -        val neg_minus_k             = Const ("IntDef.neg", split_type --> HOLogic.boolT) $
   1.707 -                                        (number_of $
   1.708 -                                          (Const (@{const_name HOL.uminus},
   1.709 -                                            HOLogic.intT --> HOLogic.intT) $ k'))
   1.710 -        val zero_leq_j              = Const (@{const_name HOL.less_eq},
   1.711 -                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
   1.712 -        val j_lt_t2                 = Const (@{const_name HOL.less},
   1.713 -                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
   1.714 -        val t1_eq_t2_times_i_plus_j = Const ("op =",
   1.715 -                                        split_type --> split_type --> HOLogic.boolT) $ t1' $
   1.716 -                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
   1.717 -                                         (Const (@{const_name HOL.times},
   1.718 -                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
   1.719 -        val neg_t2                  = Const ("IntDef.neg", split_type --> HOLogic.boolT) $ t2'
   1.720 -        val t2_lt_j                 = Const (@{const_name HOL.less},
   1.721 -                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
   1.722 -        val j_leq_zero              = Const (@{const_name HOL.less_eq},
   1.723 -                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
   1.724 -        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
   1.725 -        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
   1.726 -        val subgoal2                = (HOLogic.mk_Trueprop neg_minus_k)
   1.727 -                                        :: terms2_3
   1.728 -                                        @ not_false
   1.729 -                                        :: (map HOLogic.mk_Trueprop
   1.730 -                                             [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
   1.731 -        val subgoal3                = (HOLogic.mk_Trueprop neg_t2)
   1.732 -                                        :: terms2_3
   1.733 -                                        @ not_false
   1.734 -                                        :: (map HOLogic.mk_Trueprop
   1.735 -                                             [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
   1.736 -        val Ts'                     = split_type :: split_type :: Ts
   1.737 -      in
   1.738 -        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
   1.739 -      end
   1.740 -    (* this will only happen if a split theorem can be applied for which no  *)
   1.741 -    (* code exists above -- in which case either the split theorem should be *)
   1.742 -    (* implemented above, or 'is_split_thm' should be modified to filter it  *)
   1.743 -    (* out                                                                   *)
   1.744 -    | (t, ts) => (
   1.745 -      warning ("Lin. Arith.: split rule for " ^ ProofContext.string_of_term ctxt t ^
   1.746 -               " (with " ^ string_of_int (length ts) ^
   1.747 -               " argument(s)) not implemented; proof reconstruction is likely to fail");
   1.748 -      NONE
   1.749 -    ))
   1.750 -  )
   1.751 -end;
   1.752 -
   1.753 -(* remove terms that do not satisfy 'p'; change the order of the remaining   *)
   1.754 -(* terms in the same way as filter_prems_tac does                            *)
   1.755 -
   1.756 -fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
   1.757 -let
   1.758 -  fun filter_prems (t, (left, right)) =
   1.759 -    if  p t  then  (left, right @ [t])  else  (left @ right, [])
   1.760 -  val (left, right) = foldl filter_prems ([], []) terms
   1.761 -in
   1.762 -  right @ left
   1.763 -end;
   1.764 -
   1.765 -(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)
   1.766 -(* subgoal that has 'terms' as premises                                      *)
   1.767 -
   1.768 -fun negated_term_occurs_positively (terms : term list) : bool =
   1.769 -  List.exists
   1.770 -    (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
   1.771 -      | _                                   => false)
   1.772 -    terms;
   1.773 -
   1.774 -fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
   1.775 -let
   1.776 -  (* repeatedly split (including newly emerging subgoals) until no further   *)
   1.777 -  (* splitting is possible                                                   *)
   1.778 -  fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
   1.779 -    | split_loop (subgoal::subgoals)                = (
   1.780 -        case split_once_items ctxt subgoal of
   1.781 -          SOME new_subgoals => split_loop (new_subgoals @ subgoals)
   1.782 -        | NONE              => subgoal :: split_loop subgoals
   1.783 -      )
   1.784 -  fun is_relevant t  = isSome (decomp ctxt t)
   1.785 -  (* filter_prems_tac is_relevant: *)
   1.786 -  val relevant_terms = filter_prems_tac_items is_relevant terms
   1.787 -  (* split_tac, NNF normalization: *)
   1.788 -  val split_goals    = split_loop [(Ts, relevant_terms)]
   1.789 -  (* necessary because split_once_tac may normalize terms: *)
   1.790 -  val beta_eta_norm  = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
   1.791 -  (* TRY (etac notE) THEN eq_assume_tac: *)
   1.792 -  val result         = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
   1.793 -in
   1.794 -  result
   1.795 -end;
   1.796 -
   1.797 -(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)
   1.798 -(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)
   1.799 -(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)
   1.800 -(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)
   1.801 -(* disjunctions and existential quantifiers from the premises, possibly (in  *)
   1.802 -(* the case of disjunctions) resulting in several new subgoals, each of the  *)
   1.803 -(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)
   1.804 -(* !fast_arith_split_limit splits are possible.                              *)
   1.805 -
   1.806 -local
   1.807 -  val nnf_simpset =
   1.808 -    empty_ss setmkeqTrue mk_eq_True
   1.809 -    setmksimps (mksimps mksimps_pairs)
   1.810 -    addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
   1.811 -      not_all, not_ex, not_not]
   1.812 -  fun prem_nnf_tac i st =
   1.813 -    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
   1.814 -in
   1.815 -
   1.816 -fun split_once_tac ctxt split_thms =
   1.817 -  let
   1.818 -    val thy = ProofContext.theory_of ctxt
   1.819 -    val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
   1.820 -      let
   1.821 -        val Ts = rev (map snd (Logic.strip_params subgoal))
   1.822 -        val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
   1.823 -        val cmap = Splitter.cmap_of_split_thms split_thms
   1.824 -        val splits = Splitter.split_posns cmap thy Ts concl
   1.825 -        val split_limit = ConfigOption.get ctxt fast_arith_split_limit
   1.826 -      in
   1.827 -        if length splits > split_limit then no_tac
   1.828 -        else split_tac split_thms i
   1.829 -      end)
   1.830 -  in
   1.831 -    EVERY' [
   1.832 -      REPEAT_DETERM o etac rev_mp,
   1.833 -      cond_split_tac,
   1.834 -      rtac ccontr,
   1.835 -      prem_nnf_tac,
   1.836 -      TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
   1.837 -    ]
   1.838 -  end;
   1.839 -
   1.840 -end;  (* local *)
   1.841 -
   1.842 -(* remove irrelevant premises, then split the i-th subgoal (and all new      *)
   1.843 -(* subgoals) by using 'split_once_tac' repeatedly.  Beta-eta-normalize new   *)
   1.844 -(* subgoals and finally attempt to solve them by finding an immediate        *)
   1.845 -(* contradiction (i.e. a term and its negation) in their premises.           *)
   1.846 -
   1.847 -fun pre_tac ctxt i =
   1.848 -let
   1.849 -  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
   1.850 -  fun is_relevant t = isSome (decomp ctxt t)
   1.851 -in
   1.852 -  DETERM (
   1.853 -    TRY (filter_prems_tac is_relevant i)
   1.854 -      THEN (
   1.855 -        (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
   1.856 -          THEN_ALL_NEW
   1.857 -            (CONVERSION Drule.beta_eta_conversion
   1.858 -              THEN'
   1.859 -            (TRY o (etac notE THEN' eq_assume_tac)))
   1.860 -      ) i
   1.861 -  )
   1.862 -end;
   1.863 -
   1.864 -end;  (* LA_Data_Ref *)
   1.865 -
   1.866 -
   1.867 -structure Fast_Arith =
   1.868 -  Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
   1.869 -
   1.870 -fun fast_arith_tac ctxt    = Fast_Arith.lin_arith_tac ctxt false;
   1.871 -val fast_ex_arith_tac      = Fast_Arith.lin_arith_tac;
   1.872 -val trace_arith            = Fast_Arith.trace;
   1.873 -
   1.874 -(* reduce contradictory <= to False.
   1.875 -   Most of the work is done by the cancel tactics. *)
   1.876 -
   1.877 -val init_arith_data =
   1.878 - Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
   1.879 -   {add_mono_thms = add_mono_thms @
   1.880 -    @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
   1.881 -    mult_mono_thms = mult_mono_thms,
   1.882 -    inj_thms = inj_thms,
   1.883 -    lessD = lessD @ [thm "Suc_leI"],
   1.884 -    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
   1.885 -    simpset = HOL_basic_ss
   1.886 -      addsimps
   1.887 -       [@{thm "monoid_add_class.zero_plus.add_0_left"},
   1.888 -        @{thm "monoid_add_class.zero_plus.add_0_right"},
   1.889 -        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
   1.890 -        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
   1.891 -        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
   1.892 -        @{thm "not_one_less_zero"}]
   1.893 -      addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
   1.894 -       (*abel_cancel helps it work in abstract algebraic domains*)
   1.895 -      addsimprocs nat_cancel_sums_add}) #>
   1.896 -  arith_discrete "nat";
   1.897 -
   1.898 -val fast_nat_arith_simproc =
   1.899 -  Simplifier.simproc (the_context ()) "fast_nat_arith"
   1.900 -    ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
   1.901 -
   1.902 -(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
   1.903 -useful to detect inconsistencies among the premises for subgoals which are
   1.904 -*not* themselves (in)equalities, because the latter activate
   1.905 -fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
   1.906 -solver all the time rather than add the additional check. *)
   1.907 -
   1.908 -
   1.909 -(* arith proof method *)
   1.910 -
   1.911 -local
   1.912 -
   1.913 -fun raw_arith_tac ctxt ex =
   1.914 -  (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
   1.915 -     decomp sg"? -- but note that the test is applied to terms already before
   1.916 -     they are split/normalized) to speed things up in case there are lots of
   1.917 -     irrelevant terms involved; elimination of min/max can be optimized:
   1.918 -     (max m n + k <= r) = (m+k <= r & n+k <= r)
   1.919 -     (l <= min m n + k) = (l <= m+k & l <= n+k)
   1.920 -  *)
   1.921 -  refute_tac (K true)
   1.922 -    (* Splitting is also done inside fast_arith_tac, but not completely --   *)
   1.923 -    (* split_tac may use split theorems that have not been implemented in    *)
   1.924 -    (* fast_arith_tac (cf. pre_decomp and split_once_items above), and       *)
   1.925 -    (* fast_arith_split_limit may trigger.                                   *)
   1.926 -    (* Therefore splitting outside of fast_arith_tac may allow us to prove   *)
   1.927 -    (* some goals that fast_arith_tac alone would fail on.                   *)
   1.928 -    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
   1.929 -    (fast_ex_arith_tac ctxt ex);
   1.930 -
   1.931 -fun more_arith_tacs ctxt =
   1.932 -  let val tactics = #tactics (get_arith_data ctxt)
   1.933 -  in FIRST' (map (fn ArithTactic {tactic, ...} => tactic) tactics) end;
   1.934 -
   1.935 -in
   1.936 -
   1.937 -fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
   1.938 -  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
   1.939 -
   1.940 -fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
   1.941 -  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
   1.942 -  more_arith_tacs ctxt];
   1.943 -
   1.944 -fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
   1.945 -  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
   1.946 -  more_arith_tacs ctxt];
   1.947 -
   1.948 -fun arith_method src =
   1.949 -  Method.syntax Args.bang_facts src
   1.950 -  #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
   1.951 -      HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
   1.952 -
   1.953 -end;
   1.954 -
   1.955 +(* FIXME dead code *)
   1.956  (* antisymmetry:
   1.957     combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y
   1.958  
   1.959 @@ -1036,17 +209,6 @@
   1.960  end;
   1.961  *)
   1.962  
   1.963 -(* theory setup *)
   1.964 +end;
   1.965  
   1.966 -val arith_setup =
   1.967 -  init_arith_data #>
   1.968 -  Simplifier.map_ss (fn ss => ss
   1.969 -    addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc])
   1.970 -    addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)) #>
   1.971 -  Context.mapping
   1.972 -   (LA_Data_Ref.setup_options #>
   1.973 -    Method.add_methods
   1.974 -      [("arith", arith_method,
   1.975 -        "decide linear arithmethic")] #>
   1.976 -    Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
   1.977 -      "declaration of split rules for arithmetic procedure")]) I;
   1.978 +open ArithData;