src/HOL/Tools/lin_arith.ML
author wenzelm
Thu Mar 27 14:41:09 2008 +0100 (2008-03-27)
changeset 26424 a6cad32a27b0
parent 26110 06eacfd8dd9f
child 26942 87e4208700d1
permissions -rw-r--r--
eliminated theory ProtoPure;
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(*  Title:      HOL/Tools/lin_arith.ML
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    ID:         $Id$
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    Author:     Tjark Weber and Tobias Nipkow
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HOL setup for linear arithmetic (see Provers/Arith/fast_lin_arith.ML).
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*)
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signature BASIC_LIN_ARITH =
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sig
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  type arith_tactic
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  val mk_arith_tactic: string -> (Proof.context -> int -> tactic) -> arith_tactic
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  val eq_arith_tactic: arith_tactic * arith_tactic -> bool
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  val arith_split_add: attribute
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  val arith_discrete: string -> Context.generic -> Context.generic
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  val arith_inj_const: string * typ -> Context.generic -> Context.generic
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  val arith_tactic_add: arith_tactic -> Context.generic -> Context.generic
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  val fast_arith_split_limit: int Config.T
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  val fast_arith_neq_limit: int Config.T
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  val lin_arith_pre_tac: Proof.context -> int -> tactic
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  val fast_arith_tac: Proof.context -> int -> tactic
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  val fast_ex_arith_tac: Proof.context -> bool -> int -> tactic
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  val trace_arith: bool ref
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  val lin_arith_simproc: simpset -> term -> thm option
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  val fast_nat_arith_simproc: simproc
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  val simple_arith_tac: Proof.context -> int -> tactic
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  val arith_tac: Proof.context -> int -> tactic
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  val silent_arith_tac: Proof.context -> int -> tactic
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end;
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signature LIN_ARITH =
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sig
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  include BASIC_LIN_ARITH
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  val map_data:
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    ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
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      lessD: thm list, neqE: thm list, simpset: Simplifier.simpset} ->
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     {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
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      lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}) ->
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    Context.generic -> Context.generic
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  val setup: Context.generic -> Context.generic
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end;
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structure LinArith: LIN_ARITH =
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struct
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(* Parameters data for general linear arithmetic functor *)
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structure LA_Logic: LIN_ARITH_LOGIC =
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struct
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val ccontr = ccontr;
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val conjI = conjI;
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val notI = notI;
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val sym = sym;
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val not_lessD = @{thm linorder_not_less} RS iffD1;
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val not_leD = @{thm linorder_not_le} RS iffD1;
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val le0 = thm "le0";
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fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
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val mk_Trueprop = HOLogic.mk_Trueprop;
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fun atomize thm = case Thm.prop_of thm of
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    Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) =>
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    atomize(thm RS conjunct1) @ atomize(thm RS conjunct2)
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  | _ => [thm];
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fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t
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  | neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t)
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  | neg_prop t = raise TERM ("neg_prop", [t]);
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fun is_False thm =
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  let val _ $ t = Thm.prop_of thm
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  in t = Const("False",HOLogic.boolT) end;
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fun is_nat(t) = fastype_of1 t = HOLogic.natT;
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fun mk_nat_thm sg t =
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  let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
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  in instantiate ([],[(cn,ct)]) le0 end;
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end;
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(* arith context data *)
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datatype arith_tactic =
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  ArithTactic of {name: string, tactic: Proof.context -> int -> tactic, id: stamp};
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fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()};
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fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2);
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structure ArithContextData = GenericDataFun
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(
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  type T = {splits: thm list,
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            inj_consts: (string * typ) list,
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            discrete: string list,
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            tactics: arith_tactic list};
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  val empty = {splits = [], inj_consts = [], discrete = [], tactics = []};
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  val extend = I;
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  fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1},
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             {splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) =
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   {splits = Library.merge Thm.eq_thm_prop (splits1, splits2),
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    inj_consts = Library.merge (op =) (inj_consts1, inj_consts2),
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    discrete = Library.merge (op =) (discrete1, discrete2),
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    tactics = Library.merge eq_arith_tactic (tactics1, tactics2)};
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);
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val get_arith_data = ArithContextData.get o Context.Proof;
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val arith_split_add = Thm.declaration_attribute (fn thm =>
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  ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
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    {splits = update Thm.eq_thm_prop thm splits,
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     inj_consts = inj_consts, discrete = discrete, tactics = tactics}));
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fun arith_discrete d = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
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  {splits = splits, inj_consts = inj_consts,
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   discrete = update (op =) d discrete, tactics = tactics});
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fun arith_inj_const c = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
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  {splits = splits, inj_consts = update (op =) c inj_consts,
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   discrete = discrete, tactics= tactics});
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fun arith_tactic_add tac = ArithContextData.map (fn {splits, inj_consts, discrete, tactics} =>
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  {splits = splits, inj_consts = inj_consts, discrete = discrete,
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   tactics = update eq_arith_tactic tac tactics});
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val (fast_arith_split_limit, setup1) = Attrib.config_int "fast_arith_split_limit" 9;
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val (fast_arith_neq_limit, setup2) = Attrib.config_int "fast_arith_neq_limit" 9;
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val setup_options = setup1 #> setup2;
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structure LA_Data_Ref =
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struct
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val fast_arith_neq_limit = fast_arith_neq_limit;
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(* Decomposition of terms *)
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(*internal representation of linear (in-)equations*)
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type decompT =
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  ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool);
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fun nT (Type ("fun", [N, _])) = (N = HOLogic.natT)
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  | nT _                      = false;
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fun add_atom (t : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) :
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             (term * Rat.rat) list * Rat.rat =
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  case AList.lookup (op =) p t of
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      NONE   => ((t, m) :: p, i)
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    | SOME n => (AList.update (op =) (t, Rat.add n m) p, i);
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(* decompose nested multiplications, bracketing them to the right and combining
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   all their coefficients
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   inj_consts: list of constants to be ignored when encountered
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               (e.g. arithmetic type conversions that preserve value)
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   m: multiplicity associated with the entire product
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   returns either (SOME term, associated multiplicity) or (NONE, constant)
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*)
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fun demult (inj_consts : (string * typ) list) : term * Rat.rat -> term option * Rat.rat =
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let
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  fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) =
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      (case s of Const (@{const_name HOL.times}, _) $ s1 $ s2 =>
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        (* bracketing to the right: '(s1 * s2) * t' becomes 's1 * (s2 * t)' *)
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        demult (mC $ s1 $ (mC $ s2 $ t), m)
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      | _ =>
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        (* product 's * t', where either factor can be 'NONE' *)
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        (case demult (s, m) of
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          (SOME s', m') =>
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            (case demult (t, m') of
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              (SOME t', m'') => (SOME (mC $ s' $ t'), m'')
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            | (NONE,    m'') => (SOME s', m''))
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        | (NONE,    m') => demult (t, m')))
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    | demult ((mC as Const (@{const_name HOL.divide}, _)) $ s $ t, m) =
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      (* FIXME: Shouldn't we simplify nested quotients, e.g. '(s/t)/u' could
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         become 's/(t*u)', and '(s*t)/u' could become 's*(t/u)' ?   Note that
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         if we choose to do so here, the simpset used by arith must be able to
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         perform the same simplifications. *)
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      (* FIXME: Currently we treat the numerator as atomic unless the
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         denominator can be reduced to a numeric constant.  It might be better
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         to demult the numerator in any case, and invent a new term of the form
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         '1 / t' if the numerator can be reduced, but the denominator cannot. *)
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      (* FIXME: Currently we even treat the whole fraction as atomic unless the
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         denominator can be reduced to a numeric constant.  It might be better
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         to use the partially reduced denominator (i.e. 's / (2*t)' could be
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         demult'ed to 's / t' with multiplicity .5).   This would require a
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         very simple change only below, but it breaks existing proofs. *)
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      (* quotient 's / t', where the denominator t can be NONE *)
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      (* Note: will raise Rat.DIVZERO iff m' is Rat.zero *)
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      (case demult (t, Rat.one) of
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        (SOME _, _) => (SOME (mC $ s $ t), m)
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      | (NONE,  m') => apsnd (Rat.mult (Rat.inv m')) (demult (s, m)))
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    (* terms that evaluate to numeric constants *)
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    | demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m)
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    | demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero)
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    | demult (Const (@{const_name HOL.one}, _), m) = (NONE, m)
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    (*Warning: in rare cases number_of encloses a non-numeral,
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      in which case dest_numeral raises TERM; hence all the handles below.
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      Same for Suc-terms that turn out not to be numerals -
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      although the simplifier should eliminate those anyway ...*)
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    | demult (t as Const ("Int.number_class.number_of", _) $ n, m) =
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      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n)))
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        handle TERM _ => (SOME t, m))
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    | demult (t as Const (@{const_name Suc}, _) $ _, m) =
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      ((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat t)))
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        handle TERM _ => (SOME t, m))
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    (* injection constants are ignored *)
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    | demult (t as Const f $ x, m) =
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      if member (op =) inj_consts f then demult (x, m) else (SOME t, m)
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    (* everything else is considered atomic *)
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    | demult (atom, m) = (SOME atom, m)
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in demult end;
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fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) :
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            ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option =
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let
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  (* Turns a term 'all' and associated multiplicity 'm' into a list 'p' of
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     summands and associated multiplicities, plus a constant 'i' (with implicit
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     multiplicity 1) *)
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  fun poly (Const (@{const_name HOL.plus}, _) $ s $ t,
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        m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = poly (s, m, poly (t, m, pi))
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    | poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) =
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        if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi))
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    | poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) =
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        if nT T then add_atom all m pi else poly (t, Rat.neg m, pi)
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    | poly (Const (@{const_name HOL.zero}, _), _, pi) =
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        pi
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    | poly (Const (@{const_name HOL.one}, _), m, (p, i)) =
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        (p, Rat.add i m)
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    | poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) =
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        poly (t, m, (p, Rat.add i m))
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    | poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) =
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        (case demult inj_consts (all, m) of
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           (NONE,   m') => (p, Rat.add i m')
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         | (SOME u, m') => add_atom u m' pi)
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    | poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) =
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        (case demult inj_consts (all, m) of
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           (NONE,   m') => (p, Rat.add i m')
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         | (SOME u, m') => add_atom u m' pi)
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    | poly (all as Const ("Int.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) =
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        (let val k = HOLogic.dest_numeral t
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            val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k
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        in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end
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        handle TERM _ => add_atom all m pi)
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    | poly (all as Const f $ x, m, pi) =
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        if f mem inj_consts then poly (x, m, pi) else add_atom all m pi
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    | poly (all, m, pi) =
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        add_atom all m pi
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  val (p, i) = poly (lhs, Rat.one, ([], Rat.zero))
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  val (q, j) = poly (rhs, Rat.one, ([], Rat.zero))
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in
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  case rel of
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    @{const_name HOL.less}    => SOME (p, i, "<", q, j)
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  | @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j)
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  | "op ="              => SOME (p, i, "=", q, j)
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  | _                   => NONE
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end handle Rat.DIVZERO => NONE;
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fun of_lin_arith_sort thy U =
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  Sign.of_sort thy (U, ["Ring_and_Field.ordered_idom"]);
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fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool =
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  if of_lin_arith_sort sg U then
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    (true, D mem discrete)
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  else (* special cases *)
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    if D mem discrete then  (true, true)  else  (false, false)
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  | allows_lin_arith sg discrete U =
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  (of_lin_arith_sort sg U, false);
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fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decompT option =
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  case T of
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    Type ("fun", [U, _]) =>
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      (case allows_lin_arith thy discrete U of
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        (true, d) =>
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          (case decomp0 inj_consts xxx of
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            NONE                   => NONE
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          | SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d))
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      | (false, _) =>
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          NONE)
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  | _ => NONE;
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fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d)
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  | negate NONE                        = NONE;
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fun decomp_negation data
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  ((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option =
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      decomp_typecheck data (T, (rel, lhs, rhs))
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  | decomp_negation data ((Const ("Trueprop", _)) $
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  (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) =
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   295
      negate (decomp_typecheck data (T, (rel, lhs, rhs)))
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   296
  | decomp_negation data _ =
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      NONE;
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   298
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   299
fun decomp ctxt : term -> decompT option =
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  let
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    val thy = ProofContext.theory_of ctxt
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    val {discrete, inj_consts, ...} = get_arith_data ctxt
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  in decomp_negation (thy, discrete, inj_consts) end;
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   305
fun domain_is_nat (_ $ (Const (_, T) $ _ $ _))                      = nT T
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   306
  | domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T
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   307
  | domain_is_nat _                                                 = false;
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   308
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   309
fun number_of (n, T) = HOLogic.mk_number T n;
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   310
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   311
(*---------------------------------------------------------------------------*)
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   312
(* the following code performs splitting of certain constants (e.g. min,     *)
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   313
(* max) in a linear arithmetic problem; similar to what split_tac later does *)
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   314
(* to the proof state                                                        *)
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   315
(*---------------------------------------------------------------------------*)
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   316
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   317
(* checks if splitting with 'thm' is implemented                             *)
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   318
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   319
fun is_split_thm (thm : thm) : bool =
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  case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => (
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    (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
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    case head_of lhs of
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      Const (a, _) => member (op =) [@{const_name Orderings.max},
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   324
                                    @{const_name Orderings.min},
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   325
                                    @{const_name HOL.abs},
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                                    @{const_name HOL.minus},
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                                    "Int.nat",
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   328
                                    "Divides.div_class.mod",
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   329
                                    "Divides.div_class.div"] a
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   330
    | _            => (warning ("Lin. Arith.: wrong format for split rule " ^
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   331
                                 Display.string_of_thm thm);
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   332
                       false))
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   333
  | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
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   334
                   Display.string_of_thm thm);
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   335
          false);
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   336
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   337
(* substitute new for occurrences of old in a term, incrementing bound       *)
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   338
(* variables as needed when substituting inside an abstraction               *)
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   339
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   340
fun subst_term ([] : (term * term) list) (t : term) = t
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   341
  | subst_term pairs                     t          =
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      (case AList.lookup (op aconv) pairs t of
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   343
        SOME new =>
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   344
          new
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   345
      | NONE     =>
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   346
          (case t of Abs (a, T, body) =>
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   347
            let val pairs' = map (pairself (incr_boundvars 1)) pairs
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   348
            in  Abs (a, T, subst_term pairs' body)  end
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   349
          | t1 $ t2                   =>
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   350
            subst_term pairs t1 $ subst_term pairs t2
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   351
          | _ => t));
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   352
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   353
(* approximates the effect of one application of split_tac (followed by NNF  *)
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   354
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)
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   355
(* list of new subgoals (each again represented by a typ list for bound      *)
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   356
(* variables and a term list for premises), or NONE if split_tac would fail  *)
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   357
(* on the subgoal                                                            *)
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   358
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   359
(* FIXME: currently only the effect of certain split theorems is reproduced  *)
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   360
(*        (which is why we need 'is_split_thm').  A more canonical           *)
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   361
(*        implementation should analyze the right-hand side of the split     *)
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   362
(*        theorem that can be applied, and modify the subgoal accordingly.   *)
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   363
(*        Or even better, the splitter should be extended to provide         *)
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   364
(*        splitting on terms as well as splitting on theorems (where the     *)
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   365
(*        former can have a faster implementation as it does not need to be  *)
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   366
(*        proof-producing).                                                  *)
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   367
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   368
fun split_once_items ctxt (Ts : typ list, terms : term list) :
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   369
                     (typ list * term list) list option =
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   370
let
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  val thy = ProofContext.theory_of ctxt
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   372
  (* takes a list  [t1, ..., tn]  to the term                                *)
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   373
  (*   tn' --> ... --> t1' --> False  ,                                      *)
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   374
  (* where ti' = HOLogic.dest_Trueprop ti                                    *)
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   375
  fun REPEAT_DETERM_etac_rev_mp terms' =
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   376
    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const
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   377
  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
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   378
  val cmap       = Splitter.cmap_of_split_thms split_thms
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   379
  val splits     = Splitter.split_posns cmap thy Ts (REPEAT_DETERM_etac_rev_mp terms)
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   380
  val split_limit = Config.get ctxt fast_arith_split_limit
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   381
in
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   382
  if length splits > split_limit then
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   383
   (tracing ("fast_arith_split_limit exceeded (current value is " ^
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   384
      string_of_int split_limit ^ ")"); NONE)
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   385
  else (
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  case splits of [] =>
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    (* split_tac would fail: no possible split *)
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    NONE
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   389
  | ((_, _, _, split_type, split_term) :: _) => (
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   390
    (* ignore all but the first possible split *)
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   391
    case strip_comb split_term of
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   392
    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
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   393
      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
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   394
      let
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   395
        val rev_terms     = rev terms
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   396
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
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   397
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
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   398
        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
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                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
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   400
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
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   401
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
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   402
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
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   403
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
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   404
      in
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   405
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
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   406
      end
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   407
    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
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   408
    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
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   409
      let
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   410
        val rev_terms     = rev terms
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   411
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
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   412
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
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   413
        val t1_leq_t2     = Const (@{const_name HOL.less_eq},
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   414
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
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   415
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
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   416
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
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   417
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
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   418
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
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   419
      in
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   420
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
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   421
      end
wenzelm@24092
   422
    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
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   423
    | (Const (@{const_name HOL.abs}, _), [t1]) =>
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   424
      let
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   425
        val rev_terms   = rev terms
wenzelm@24092
   426
        val terms1      = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   427
        val terms2      = map (subst_term [(split_term, Const (@{const_name HOL.uminus},
wenzelm@24092
   428
                            split_type --> split_type) $ t1)]) rev_terms
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   429
        val zero        = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   430
        val zero_leq_t1 = Const (@{const_name HOL.less_eq},
wenzelm@24092
   431
                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1
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   432
        val t1_lt_zero  = Const (@{const_name HOL.less},
wenzelm@24092
   433
                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
wenzelm@24092
   434
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
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   435
        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
wenzelm@24092
   436
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
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   437
      in
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   438
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   439
      end
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   440
    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
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   441
    | (Const (@{const_name HOL.minus}, _), [t1, t2]) =>
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   442
      let
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   443
        (* "d" in the above theorem becomes a new bound variable after NNF   *)
wenzelm@24092
   444
        (* transformation, therefore some adjustment of indices is necessary *)
wenzelm@24092
   445
        val rev_terms       = rev terms
wenzelm@24092
   446
        val zero            = Const (@{const_name HOL.zero}, split_type)
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   447
        val d               = Bound 0
wenzelm@24092
   448
        val terms1          = map (subst_term [(split_term, zero)]) rev_terms
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   449
        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])
wenzelm@24092
   450
                                (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   451
        val t1'             = incr_boundvars 1 t1
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   452
        val t2'             = incr_boundvars 1 t2
wenzelm@24092
   453
        val t1_lt_t2        = Const (@{const_name HOL.less},
wenzelm@24092
   454
                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   455
        val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   456
                                (Const (@{const_name HOL.plus},
wenzelm@24092
   457
                                  split_type --> split_type --> split_type) $ t2' $ d)
wenzelm@24092
   458
        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   459
        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
wenzelm@24092
   460
        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
wenzelm@24092
   461
      in
wenzelm@24092
   462
        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
wenzelm@24092
   463
      end
wenzelm@24092
   464
    (* ?P (nat ?i) = ((ALL n. ?i = int n --> ?P n) & (?i < 0 --> ?P 0)) *)
haftmann@25919
   465
    | (Const ("Int.nat", _), [t1]) =>
wenzelm@24092
   466
      let
wenzelm@24092
   467
        val rev_terms   = rev terms
wenzelm@24092
   468
        val zero_int    = Const (@{const_name HOL.zero}, HOLogic.intT)
wenzelm@24092
   469
        val zero_nat    = Const (@{const_name HOL.zero}, HOLogic.natT)
wenzelm@24092
   470
        val n           = Bound 0
wenzelm@24092
   471
        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])
wenzelm@24092
   472
                            (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   473
        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms
wenzelm@24092
   474
        val t1'         = incr_boundvars 1 t1
wenzelm@24092
   475
        val t1_eq_int_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
haftmann@24196
   476
                            (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
wenzelm@24092
   477
        val t1_lt_zero  = Const (@{const_name HOL.less},
wenzelm@24092
   478
                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
wenzelm@24092
   479
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   480
        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false]
wenzelm@24092
   481
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   482
      in
wenzelm@24092
   483
        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   484
      end
wenzelm@24092
   485
    (* "?P ((?n::nat) mod (number_of ?k)) =
wenzelm@24092
   486
         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
wenzelm@24092
   487
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   488
    | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   489
      let
wenzelm@24092
   490
        val rev_terms               = rev terms
wenzelm@24092
   491
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   492
        val i                       = Bound 1
wenzelm@24092
   493
        val j                       = Bound 0
wenzelm@24092
   494
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   495
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   496
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   497
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   498
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   499
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   500
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   501
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   502
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
wenzelm@24092
   503
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   504
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   505
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   506
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   507
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   508
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   509
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   510
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   511
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   512
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   513
                                          @ terms2 @ [not_false]
wenzelm@24092
   514
      in
wenzelm@24092
   515
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   516
      end
wenzelm@24092
   517
    (* "?P ((?n::nat) div (number_of ?k)) =
wenzelm@24092
   518
         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
wenzelm@24092
   519
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
wenzelm@24092
   520
    | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   521
      let
wenzelm@24092
   522
        val rev_terms               = rev terms
wenzelm@24092
   523
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   524
        val i                       = Bound 1
wenzelm@24092
   525
        val j                       = Bound 0
wenzelm@24092
   526
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   527
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   528
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   529
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   530
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   531
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   532
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   533
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   534
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
wenzelm@24092
   535
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   536
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   537
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   538
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   539
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   540
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   541
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   542
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   543
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   544
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   545
                                          @ terms2 @ [not_false]
wenzelm@24092
   546
      in
wenzelm@24092
   547
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   548
      end
wenzelm@24092
   549
    (* "?P ((?n::int) mod (number_of ?k)) =
wenzelm@24092
   550
         ((iszero (number_of ?k) --> ?P ?n) &
wenzelm@24092
   551
          (neg (number_of (uminus ?k)) -->
wenzelm@24092
   552
            (ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
wenzelm@24092
   553
          (neg (number_of ?k) -->
wenzelm@24092
   554
            (ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   555
    | (Const ("Divides.div_class.mod",
haftmann@25919
   556
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
wenzelm@24092
   557
      let
wenzelm@24092
   558
        val rev_terms               = rev terms
wenzelm@24092
   559
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   560
        val i                       = Bound 1
wenzelm@24092
   561
        val j                       = Bound 0
wenzelm@24092
   562
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   563
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   564
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   565
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   566
        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
haftmann@25919
   567
        val iszero_t2               = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
haftmann@25919
   568
        val neg_minus_k             = Const ("Int.neg", split_type --> HOLogic.boolT) $
wenzelm@24092
   569
                                        (number_of $
wenzelm@24092
   570
                                          (Const (@{const_name HOL.uminus},
wenzelm@24092
   571
                                            HOLogic.intT --> HOLogic.intT) $ k'))
wenzelm@24092
   572
        val zero_leq_j              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   573
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
wenzelm@24092
   574
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   575
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   576
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   577
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   578
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   579
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
haftmann@25919
   580
        val neg_t2                  = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
wenzelm@24092
   581
        val t2_lt_j                 = Const (@{const_name HOL.less},
wenzelm@24092
   582
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
wenzelm@24092
   583
        val j_leq_zero              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   584
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
wenzelm@24092
   585
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   586
        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
wenzelm@24092
   587
        val subgoal2                = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j])
wenzelm@24092
   588
                                        @ hd terms2_3
wenzelm@24092
   589
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   590
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   591
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   592
        val subgoal3                = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j])
wenzelm@24092
   593
                                        @ hd terms2_3
wenzelm@24092
   594
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   595
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   596
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   597
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   598
      in
wenzelm@24092
   599
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   600
      end
wenzelm@24092
   601
    (* "?P ((?n::int) div (number_of ?k)) =
wenzelm@24092
   602
         ((iszero (number_of ?k) --> ?P 0) &
wenzelm@24092
   603
          (neg (number_of (uminus ?k)) -->
wenzelm@24092
   604
            (ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) --> ?P i)) &
wenzelm@24092
   605
          (neg (number_of ?k) -->
wenzelm@24092
   606
            (ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) --> ?P i))) *)
wenzelm@24092
   607
    | (Const ("Divides.div_class.div",
haftmann@25919
   608
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2 as (number_of $ k)]) =>
wenzelm@24092
   609
      let
wenzelm@24092
   610
        val rev_terms               = rev terms
wenzelm@24092
   611
        val zero                    = Const (@{const_name HOL.zero}, split_type)
wenzelm@24092
   612
        val i                       = Bound 1
wenzelm@24092
   613
        val j                       = Bound 0
wenzelm@24092
   614
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   615
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   616
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   617
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   618
        val (t2' as (_ $ k'))       = incr_boundvars 2 t2
haftmann@25919
   619
        val iszero_t2               = Const ("Int.iszero", split_type --> HOLogic.boolT) $ t2
haftmann@25919
   620
        val neg_minus_k             = Const ("Int.neg", split_type --> HOLogic.boolT) $
wenzelm@24092
   621
                                        (number_of $
wenzelm@24092
   622
                                          (Const (@{const_name HOL.uminus},
wenzelm@24092
   623
                                            HOLogic.intT --> HOLogic.intT) $ k'))
wenzelm@24092
   624
        val zero_leq_j              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   625
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
wenzelm@24092
   626
        val j_lt_t2                 = Const (@{const_name HOL.less},
wenzelm@24092
   627
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   628
        val t1_eq_t2_times_i_plus_j = Const ("op =",
wenzelm@24092
   629
                                        split_type --> split_type --> HOLogic.boolT) $ t1' $
wenzelm@24092
   630
                                       (Const (@{const_name HOL.plus}, split_type --> split_type --> split_type) $
wenzelm@24092
   631
                                         (Const (@{const_name HOL.times},
wenzelm@24092
   632
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
haftmann@25919
   633
        val neg_t2                  = Const ("Int.neg", split_type --> HOLogic.boolT) $ t2'
wenzelm@24092
   634
        val t2_lt_j                 = Const (@{const_name HOL.less},
wenzelm@24092
   635
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
wenzelm@24092
   636
        val j_leq_zero              = Const (@{const_name HOL.less_eq},
wenzelm@24092
   637
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
wenzelm@24092
   638
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   639
        val subgoal1                = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false]
wenzelm@24092
   640
        val subgoal2                = (HOLogic.mk_Trueprop neg_minus_k)
wenzelm@24092
   641
                                        :: terms2_3
wenzelm@24092
   642
                                        @ not_false
wenzelm@24092
   643
                                        :: (map HOLogic.mk_Trueprop
wenzelm@24092
   644
                                             [zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   645
        val subgoal3                = (HOLogic.mk_Trueprop neg_t2)
wenzelm@24092
   646
                                        :: terms2_3
wenzelm@24092
   647
                                        @ not_false
wenzelm@24092
   648
                                        :: (map HOLogic.mk_Trueprop
wenzelm@24092
   649
                                             [t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   650
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   651
      in
wenzelm@24092
   652
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   653
      end
wenzelm@24092
   654
    (* this will only happen if a split theorem can be applied for which no  *)
wenzelm@24092
   655
    (* code exists above -- in which case either the split theorem should be *)
wenzelm@24092
   656
    (* implemented above, or 'is_split_thm' should be modified to filter it  *)
wenzelm@24092
   657
    (* out                                                                   *)
wenzelm@24092
   658
    | (t, ts) => (
wenzelm@24920
   659
      warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
wenzelm@24092
   660
               " (with " ^ string_of_int (length ts) ^
wenzelm@24092
   661
               " argument(s)) not implemented; proof reconstruction is likely to fail");
wenzelm@24092
   662
      NONE
wenzelm@24092
   663
    ))
wenzelm@24092
   664
  )
wenzelm@24092
   665
end;
wenzelm@24092
   666
wenzelm@24092
   667
(* remove terms that do not satisfy 'p'; change the order of the remaining   *)
wenzelm@24092
   668
(* terms in the same way as filter_prems_tac does                            *)
wenzelm@24092
   669
wenzelm@24092
   670
fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
wenzelm@24092
   671
let
wenzelm@24092
   672
  fun filter_prems (t, (left, right)) =
wenzelm@24092
   673
    if  p t  then  (left, right @ [t])  else  (left @ right, [])
wenzelm@24092
   674
  val (left, right) = foldl filter_prems ([], []) terms
wenzelm@24092
   675
in
wenzelm@24092
   676
  right @ left
wenzelm@24092
   677
end;
wenzelm@24092
   678
wenzelm@24092
   679
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)
wenzelm@24092
   680
(* subgoal that has 'terms' as premises                                      *)
wenzelm@24092
   681
wenzelm@24092
   682
fun negated_term_occurs_positively (terms : term list) : bool =
wenzelm@24092
   683
  List.exists
wenzelm@24092
   684
    (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
wenzelm@24092
   685
      | _                                   => false)
wenzelm@24092
   686
    terms;
wenzelm@24092
   687
wenzelm@24092
   688
fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
wenzelm@24092
   689
let
wenzelm@24092
   690
  (* repeatedly split (including newly emerging subgoals) until no further   *)
wenzelm@24092
   691
  (* splitting is possible                                                   *)
wenzelm@24092
   692
  fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
wenzelm@24092
   693
    | split_loop (subgoal::subgoals)                = (
wenzelm@24092
   694
        case split_once_items ctxt subgoal of
wenzelm@24092
   695
          SOME new_subgoals => split_loop (new_subgoals @ subgoals)
wenzelm@24092
   696
        | NONE              => subgoal :: split_loop subgoals
wenzelm@24092
   697
      )
wenzelm@24092
   698
  fun is_relevant t  = isSome (decomp ctxt t)
wenzelm@24092
   699
  (* filter_prems_tac is_relevant: *)
wenzelm@24092
   700
  val relevant_terms = filter_prems_tac_items is_relevant terms
wenzelm@24092
   701
  (* split_tac, NNF normalization: *)
wenzelm@24092
   702
  val split_goals    = split_loop [(Ts, relevant_terms)]
wenzelm@24092
   703
  (* necessary because split_once_tac may normalize terms: *)
wenzelm@24092
   704
  val beta_eta_norm  = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals
wenzelm@24092
   705
  (* TRY (etac notE) THEN eq_assume_tac: *)
wenzelm@24092
   706
  val result         = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm
wenzelm@24092
   707
in
wenzelm@24092
   708
  result
wenzelm@24092
   709
end;
wenzelm@24092
   710
wenzelm@24092
   711
(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)
wenzelm@24092
   712
(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)
wenzelm@24092
   713
(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)
wenzelm@24092
   714
(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)
wenzelm@24092
   715
(* disjunctions and existential quantifiers from the premises, possibly (in  *)
wenzelm@24092
   716
(* the case of disjunctions) resulting in several new subgoals, each of the  *)
wenzelm@24092
   717
(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)
wenzelm@24092
   718
(* !fast_arith_split_limit splits are possible.                              *)
wenzelm@24092
   719
wenzelm@24092
   720
local
wenzelm@24092
   721
  val nnf_simpset =
wenzelm@24092
   722
    empty_ss setmkeqTrue mk_eq_True
wenzelm@24092
   723
    setmksimps (mksimps mksimps_pairs)
wenzelm@24092
   724
    addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
wenzelm@24092
   725
      not_all, not_ex, not_not]
wenzelm@24092
   726
  fun prem_nnf_tac i st =
wenzelm@24092
   727
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st
wenzelm@24092
   728
in
wenzelm@24092
   729
wenzelm@24092
   730
fun split_once_tac ctxt split_thms =
wenzelm@24092
   731
  let
wenzelm@24092
   732
    val thy = ProofContext.theory_of ctxt
wenzelm@24092
   733
    val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
wenzelm@24092
   734
      let
wenzelm@24092
   735
        val Ts = rev (map snd (Logic.strip_params subgoal))
wenzelm@24092
   736
        val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
wenzelm@24092
   737
        val cmap = Splitter.cmap_of_split_thms split_thms
wenzelm@24092
   738
        val splits = Splitter.split_posns cmap thy Ts concl
wenzelm@24112
   739
        val split_limit = Config.get ctxt fast_arith_split_limit
wenzelm@24092
   740
      in
wenzelm@24092
   741
        if length splits > split_limit then no_tac
wenzelm@24092
   742
        else split_tac split_thms i
wenzelm@24092
   743
      end)
wenzelm@24092
   744
  in
wenzelm@24092
   745
    EVERY' [
wenzelm@24092
   746
      REPEAT_DETERM o etac rev_mp,
wenzelm@24092
   747
      cond_split_tac,
wenzelm@24092
   748
      rtac ccontr,
wenzelm@24092
   749
      prem_nnf_tac,
wenzelm@24092
   750
      TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
wenzelm@24092
   751
    ]
wenzelm@24092
   752
  end;
wenzelm@24092
   753
wenzelm@24092
   754
end;  (* local *)
wenzelm@24092
   755
wenzelm@24092
   756
(* remove irrelevant premises, then split the i-th subgoal (and all new      *)
wenzelm@24092
   757
(* subgoals) by using 'split_once_tac' repeatedly.  Beta-eta-normalize new   *)
wenzelm@24092
   758
(* subgoals and finally attempt to solve them by finding an immediate        *)
wenzelm@24092
   759
(* contradiction (i.e. a term and its negation) in their premises.           *)
wenzelm@24092
   760
wenzelm@24092
   761
fun pre_tac ctxt i =
wenzelm@24092
   762
let
wenzelm@24092
   763
  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
wenzelm@24092
   764
  fun is_relevant t = isSome (decomp ctxt t)
wenzelm@24092
   765
in
wenzelm@24092
   766
  DETERM (
wenzelm@24092
   767
    TRY (filter_prems_tac is_relevant i)
wenzelm@24092
   768
      THEN (
wenzelm@24092
   769
        (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
wenzelm@24092
   770
          THEN_ALL_NEW
wenzelm@24092
   771
            (CONVERSION Drule.beta_eta_conversion
wenzelm@24092
   772
              THEN'
wenzelm@24092
   773
            (TRY o (etac notE THEN' eq_assume_tac)))
wenzelm@24092
   774
      ) i
wenzelm@24092
   775
  )
wenzelm@24092
   776
end;
wenzelm@24092
   777
wenzelm@24092
   778
end;  (* LA_Data_Ref *)
wenzelm@24092
   779
wenzelm@24092
   780
wenzelm@24092
   781
val lin_arith_pre_tac = LA_Data_Ref.pre_tac;
wenzelm@24092
   782
wenzelm@24092
   783
structure Fast_Arith =
wenzelm@24092
   784
  Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
wenzelm@24092
   785
wenzelm@24092
   786
val map_data = Fast_Arith.map_data;
wenzelm@24092
   787
wenzelm@24092
   788
fun fast_arith_tac ctxt    = Fast_Arith.lin_arith_tac ctxt false;
wenzelm@24092
   789
val fast_ex_arith_tac      = Fast_Arith.lin_arith_tac;
wenzelm@24092
   790
val trace_arith            = Fast_Arith.trace;
wenzelm@24092
   791
wenzelm@24092
   792
(* reduce contradictory <= to False.
wenzelm@24092
   793
   Most of the work is done by the cancel tactics. *)
wenzelm@24092
   794
wenzelm@24092
   795
val init_arith_data =
wenzelm@24092
   796
 Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} =>
wenzelm@24092
   797
   {add_mono_thms = add_mono_thms @
wenzelm@24092
   798
    @{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field},
wenzelm@24092
   799
    mult_mono_thms = mult_mono_thms,
wenzelm@24092
   800
    inj_thms = inj_thms,
wenzelm@24092
   801
    lessD = lessD @ [thm "Suc_leI"],
wenzelm@24092
   802
    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}],
wenzelm@24092
   803
    simpset = HOL_basic_ss
wenzelm@24092
   804
      addsimps
wenzelm@24092
   805
       [@{thm "monoid_add_class.zero_plus.add_0_left"},
wenzelm@24092
   806
        @{thm "monoid_add_class.zero_plus.add_0_right"},
wenzelm@24092
   807
        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
wenzelm@24092
   808
        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
wenzelm@24092
   809
        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
wenzelm@24092
   810
        @{thm "not_one_less_zero"}]
wenzelm@24092
   811
      addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
wenzelm@24092
   812
       (*abel_cancel helps it work in abstract algebraic domains*)
haftmann@26101
   813
      addsimprocs ArithData.nat_cancel_sums_add}) #>
wenzelm@24092
   814
  arith_discrete "nat";
wenzelm@24092
   815
wenzelm@24092
   816
val lin_arith_simproc = Fast_Arith.lin_arith_simproc;
wenzelm@24092
   817
wenzelm@24092
   818
val fast_nat_arith_simproc =
wenzelm@24092
   819
  Simplifier.simproc (the_context ()) "fast_nat_arith"
wenzelm@24092
   820
    ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K Fast_Arith.lin_arith_simproc);
wenzelm@24092
   821
wenzelm@24092
   822
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
wenzelm@24092
   823
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@24092
   824
*not* themselves (in)equalities, because the latter activate
wenzelm@24092
   825
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@24092
   826
solver all the time rather than add the additional check. *)
wenzelm@24092
   827
wenzelm@24092
   828
haftmann@26110
   829
(* generic refutation procedure *)
haftmann@26110
   830
haftmann@26110
   831
(* parameters:
haftmann@26110
   832
haftmann@26110
   833
   test: term -> bool
haftmann@26110
   834
   tests if a term is at all relevant to the refutation proof;
haftmann@26110
   835
   if not, then it can be discarded. Can improve performance,
haftmann@26110
   836
   esp. if disjunctions can be discarded (no case distinction needed!).
haftmann@26110
   837
haftmann@26110
   838
   prep_tac: int -> tactic
haftmann@26110
   839
   A preparation tactic to be applied to the goal once all relevant premises
haftmann@26110
   840
   have been moved to the conclusion.
haftmann@26110
   841
haftmann@26110
   842
   ref_tac: int -> tactic
haftmann@26110
   843
   the actual refutation tactic. Should be able to deal with goals
haftmann@26110
   844
   [| A1; ...; An |] ==> False
haftmann@26110
   845
   where the Ai are atomic, i.e. no top-level &, | or EX
haftmann@26110
   846
*)
haftmann@26110
   847
haftmann@26110
   848
local
haftmann@26110
   849
  val nnf_simpset =
haftmann@26110
   850
    empty_ss setmkeqTrue mk_eq_True
haftmann@26110
   851
    setmksimps (mksimps mksimps_pairs)
haftmann@26110
   852
    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
haftmann@26110
   853
      @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];
haftmann@26110
   854
  fun prem_nnf_tac i st =
haftmann@26110
   855
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
haftmann@26110
   856
in
haftmann@26110
   857
fun refute_tac test prep_tac ref_tac =
haftmann@26110
   858
  let val refute_prems_tac =
haftmann@26110
   859
        REPEAT_DETERM
haftmann@26110
   860
              (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
haftmann@26110
   861
               filter_prems_tac test 1 ORELSE
haftmann@26110
   862
               etac @{thm disjE} 1) THEN
haftmann@26110
   863
        (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
haftmann@26110
   864
         ref_tac 1);
haftmann@26110
   865
  in EVERY'[TRY o filter_prems_tac test,
haftmann@26110
   866
            REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
haftmann@26110
   867
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
haftmann@26110
   868
  end;
haftmann@26110
   869
end;
haftmann@26110
   870
haftmann@26110
   871
wenzelm@24092
   872
(* arith proof method *)
wenzelm@24092
   873
wenzelm@24092
   874
local
wenzelm@24092
   875
wenzelm@24092
   876
fun raw_arith_tac ctxt ex =
wenzelm@24092
   877
  (* FIXME: K true should be replaced by a sensible test (perhaps "isSome o
wenzelm@24092
   878
     decomp sg"? -- but note that the test is applied to terms already before
wenzelm@24092
   879
     they are split/normalized) to speed things up in case there are lots of
wenzelm@24092
   880
     irrelevant terms involved; elimination of min/max can be optimized:
wenzelm@24092
   881
     (max m n + k <= r) = (m+k <= r & n+k <= r)
wenzelm@24092
   882
     (l <= min m n + k) = (l <= m+k & l <= n+k)
wenzelm@24092
   883
  *)
wenzelm@24092
   884
  refute_tac (K true)
wenzelm@24092
   885
    (* Splitting is also done inside fast_arith_tac, but not completely --   *)
wenzelm@24092
   886
    (* split_tac may use split theorems that have not been implemented in    *)
wenzelm@24092
   887
    (* fast_arith_tac (cf. pre_decomp and split_once_items above), and       *)
wenzelm@24092
   888
    (* fast_arith_split_limit may trigger.                                   *)
wenzelm@24092
   889
    (* Therefore splitting outside of fast_arith_tac may allow us to prove   *)
wenzelm@24092
   890
    (* some goals that fast_arith_tac alone would fail on.                   *)
wenzelm@24092
   891
    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
wenzelm@24092
   892
    (fast_ex_arith_tac ctxt ex);
wenzelm@24092
   893
wenzelm@24092
   894
fun more_arith_tacs ctxt =
wenzelm@24092
   895
  let val tactics = #tactics (get_arith_data ctxt)
wenzelm@24092
   896
  in FIRST' (map (fn ArithTactic {tactic, ...} => tactic ctxt) tactics) end;
wenzelm@24092
   897
wenzelm@24092
   898
in
wenzelm@24092
   899
wenzelm@24092
   900
fun simple_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   901
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true];
wenzelm@24092
   902
wenzelm@24092
   903
fun arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   904
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt true,
wenzelm@24092
   905
  more_arith_tacs ctxt];
wenzelm@24092
   906
wenzelm@24092
   907
fun silent_arith_tac ctxt = FIRST' [fast_arith_tac ctxt,
wenzelm@24092
   908
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac ctxt false,
wenzelm@24092
   909
  more_arith_tacs ctxt];
wenzelm@24092
   910
wenzelm@24092
   911
fun arith_method src =
wenzelm@24092
   912
  Method.syntax Args.bang_facts src
wenzelm@24092
   913
  #> (fn (prems, ctxt) => Method.METHOD (fn facts =>
wenzelm@24092
   914
      HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac ctxt)));
wenzelm@24092
   915
wenzelm@24092
   916
end;
wenzelm@24092
   917
wenzelm@24092
   918
wenzelm@24092
   919
(* context setup *)
wenzelm@24092
   920
wenzelm@24092
   921
val setup =
wenzelm@24092
   922
  init_arith_data #>
wenzelm@24092
   923
  Simplifier.map_ss (fn ss => ss addsimprocs [fast_nat_arith_simproc]
wenzelm@24092
   924
    addSolver (mk_solver' "lin_arith" Fast_Arith.cut_lin_arith_tac)) #>
wenzelm@24092
   925
  Context.mapping
wenzelm@24092
   926
   (setup_options #>
wenzelm@24092
   927
    Method.add_methods
huffman@26061
   928
      [("arith", arith_method, "decide linear arithmetic")] #>
wenzelm@24092
   929
    Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add,
wenzelm@24092
   930
      "declaration of split rules for arithmetic procedure")]) I;
wenzelm@24092
   931
wenzelm@24092
   932
end;
wenzelm@24092
   933
wenzelm@24092
   934
structure BasicLinArith: BASIC_LIN_ARITH = LinArith;
wenzelm@24092
   935
open BasicLinArith;