src/HOL/Tools/lin_arith.ML
author huffman
Mon Aug 08 09:52:09 2011 -0700 (2011-08-08)
changeset 44064 5bce8ff0d9ae
parent 43609 20760e3608fa
child 44654 d80fe56788a5
permissions -rw-r--r--
moved division ring stuff from Rings.thy to Fields.thy
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(*  Title:      HOL/Tools/lin_arith.ML
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    Author:     Tjark Weber and Tobias Nipkow, TU Muenchen
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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 LIN_ARITH =
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sig
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  val pre_tac: simpset -> int -> tactic
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  val simple_tac: Proof.context -> int -> tactic
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  val tac: Proof.context -> int -> tactic
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  val simproc: simpset -> term -> thm option
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  val add_inj_thms: thm list -> Context.generic -> Context.generic
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  val add_lessD: thm -> Context.generic -> Context.generic
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  val add_simps: thm list -> Context.generic -> Context.generic
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  val add_simprocs: simproc list -> Context.generic -> Context.generic
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  val add_inj_const: string * typ -> Context.generic -> Context.generic
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  val add_discrete_type: string -> Context.generic -> Context.generic
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  val set_number_of: (theory -> typ -> int -> cterm) -> Context.generic -> Context.generic
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  val setup: Context.generic -> Context.generic
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  val global_setup: theory -> theory
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  val split_limit: int Config.T
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  val neq_limit: int Config.T
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  val verbose: bool Config.T
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  val trace: bool Unsynchronized.ref
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end;
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structure Lin_Arith: 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 trueI = TrueI;
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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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fun mk_Eq thm = thm RS @{thm Eq_FalseI} handle THM _ => thm RS @{thm 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 (@{const_name Trueprop}, _) $ (Const (@{const_name HOL.conj}, _) $ _ $ _) =>
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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(@{const_name Trueprop}, _)) $ (Const (@{const_name Not}, _) $ t)) = TP $ t
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  | neg_prop ((TP as Const(@{const_name 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 = HOLogic.false_const end;
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fun is_nat t = (fastype_of1 t = HOLogic.natT);
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fun mk_nat_thm thy t =
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  let
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    val cn = cterm_of thy (Var (("n", 0), HOLogic.natT))
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    and ct = cterm_of thy t
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  in Drule.instantiate_normalize ([], [(cn, ct)]) @{thm le0} end;
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end;
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(* arith context data *)
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structure Lin_Arith_Data = Generic_Data
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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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  val empty = {splits = [], inj_consts = [], discrete = []};
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  val extend = I;
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  fun merge
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   ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1},
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    {splits= splits2, inj_consts= inj_consts2, discrete= discrete2}) : T =
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   {splits = Thm.merge_thms (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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);
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val get_arith_data = Lin_Arith_Data.get o Context.Proof;
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fun add_split thm = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
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  {splits = update Thm.eq_thm_prop thm splits,
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   inj_consts = inj_consts, discrete = discrete});
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fun add_discrete_type d = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
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  {splits = splits, inj_consts = inj_consts,
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   discrete = update (op =) d discrete});
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fun add_inj_const c = Lin_Arith_Data.map (fn {splits, inj_consts, discrete} =>
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  {splits = splits, inj_consts = update (op =) c inj_consts,
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   discrete = discrete});
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val split_limit = Attrib.setup_config_int @{binding linarith_split_limit} (K 9);
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val neq_limit = Attrib.setup_config_int @{binding linarith_neq_limit} (K 9);
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val verbose  = Attrib.setup_config_bool @{binding linarith_verbose} (K true);
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structure LA_Data =
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struct
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val fast_arith_neq_limit = neq_limit;
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val fast_arith_verbose = verbose;
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(* Decomposition of terms *)
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(*internal representation of linear (in-)equations*)
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type decomp =
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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 Pattern.aeconv p t of
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      NONE   => ((t, m) :: p, i)
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    | SOME n => (AList.update Pattern.aeconv (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 Groups.times}, _)) $ s $ t, m) =
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      (case s of Const (@{const_name Groups.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 Fields.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 Groups.uminus}, _) $ t, m) = demult (t, Rat.neg m)
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    | demult (Const (@{const_name Groups.zero}, _), m) = (NONE, Rat.zero)
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    | demult (Const (@{const_name Groups.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 Groups.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 Groups.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 Groups.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 Groups.zero}, _), _, pi) =
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        pi
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    | poly (Const (@{const_name Groups.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 Groups.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 Fields.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 member (op =) inj_consts f 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 Orderings.less}    => SOME (p, i, "<", q, j)
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  | @{const_name Orderings.less_eq} => SOME (p, i, "<=", q, j)
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  | @{const_name HOL.eq}            => 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, @{sort Rings.linordered_idom});
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fun allows_lin_arith thy (discrete : string list) (U as Type (D, [])) : bool * bool =
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      if of_lin_arith_sort thy U then (true, member (op =) discrete D)
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      else if member (op =) discrete D then (true, true) else (false, false)
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  | allows_lin_arith sg discrete U = (of_lin_arith_sort sg U, false);
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fun decomp_typecheck (thy, discrete, inj_consts) (T : typ, xxx) : decomp 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 (@{const_name Trueprop}, _)) $ (Const (rel, T) $ lhs $ rhs)) : decomp option =
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      decomp_typecheck data (T, (rel, lhs, rhs))
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  | decomp_negation data ((Const (@{const_name Trueprop}, _)) $
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  (Const (@{const_name Not}, _) $ (Const (rel, T) $ lhs $ rhs))) =
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      negate (decomp_typecheck data (T, (rel, lhs, rhs)))
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  | decomp_negation data _ =
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      NONE;
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fun decomp ctxt : term -> decomp option =
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  let
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    val thy = Proof_Context.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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fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T
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  | domain_is_nat (_ $ (Const (@{const_name Not}, _) $ (Const (_, T) $ _ $ _))) = nT T
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  | domain_is_nat _ = false;
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(*---------------------------------------------------------------------------*)
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(* the following code performs splitting of certain constants (e.g., min,    *)
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(* max) in a linear arithmetic problem; similar to what split_tac later does *)
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(* to the proof state                                                        *)
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(*---------------------------------------------------------------------------*)
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(* checks if splitting with 'thm' is implemented                             *)
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fun is_split_thm ctxt thm =
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  (case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) =>
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    (* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *)
wenzelm@42439
   293
    (case head_of lhs of
wenzelm@42439
   294
      Const (a, _) =>
wenzelm@42439
   295
        member (op =)
wenzelm@42439
   296
         [@{const_name Orderings.max},
wenzelm@42439
   297
          @{const_name Orderings.min},
wenzelm@42439
   298
          @{const_name Groups.abs},
wenzelm@42439
   299
          @{const_name Groups.minus},
wenzelm@42439
   300
          "Int.nat" (*DYNAMIC BINDING!*),
wenzelm@42439
   301
          "Divides.div_class.mod" (*DYNAMIC BINDING!*),
wenzelm@42439
   302
          "Divides.div_class.div" (*DYNAMIC BINDING!*)] a
wenzelm@42439
   303
    | _ =>
wenzelm@42439
   304
      (warning ("Lin. Arith.: wrong format for split rule " ^ Display.string_of_thm ctxt thm);
wenzelm@42439
   305
        false))
wenzelm@42439
   306
  | _ =>
wenzelm@42439
   307
    (warning ("Lin. Arith.: wrong format for split rule " ^ Display.string_of_thm ctxt thm);
wenzelm@42439
   308
      false));
wenzelm@24092
   309
wenzelm@24092
   310
(* substitute new for occurrences of old in a term, incrementing bound       *)
wenzelm@24092
   311
(* variables as needed when substituting inside an abstraction               *)
wenzelm@24092
   312
wenzelm@24092
   313
fun subst_term ([] : (term * term) list) (t : term) = t
wenzelm@24092
   314
  | subst_term pairs                     t          =
nipkow@29528
   315
      (case AList.lookup Pattern.aeconv pairs t of
wenzelm@24092
   316
        SOME new =>
wenzelm@24092
   317
          new
wenzelm@24092
   318
      | NONE     =>
wenzelm@24092
   319
          (case t of Abs (a, T, body) =>
wenzelm@24092
   320
            let val pairs' = map (pairself (incr_boundvars 1)) pairs
wenzelm@24092
   321
            in  Abs (a, T, subst_term pairs' body)  end
wenzelm@24092
   322
          | t1 $ t2                   =>
wenzelm@24092
   323
            subst_term pairs t1 $ subst_term pairs t2
wenzelm@24092
   324
          | _ => t));
wenzelm@24092
   325
wenzelm@24092
   326
(* approximates the effect of one application of split_tac (followed by NNF  *)
wenzelm@24092
   327
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)
wenzelm@24092
   328
(* list of new subgoals (each again represented by a typ list for bound      *)
wenzelm@24092
   329
(* variables and a term list for premises), or NONE if split_tac would fail  *)
wenzelm@24092
   330
(* on the subgoal                                                            *)
wenzelm@24092
   331
wenzelm@24092
   332
(* FIXME: currently only the effect of certain split theorems is reproduced  *)
wenzelm@24092
   333
(*        (which is why we need 'is_split_thm').  A more canonical           *)
wenzelm@24092
   334
(*        implementation should analyze the right-hand side of the split     *)
wenzelm@24092
   335
(*        theorem that can be applied, and modify the subgoal accordingly.   *)
wenzelm@24092
   336
(*        Or even better, the splitter should be extended to provide         *)
wenzelm@24092
   337
(*        splitting on terms as well as splitting on theorems (where the     *)
wenzelm@24092
   338
(*        former can have a faster implementation as it does not need to be  *)
wenzelm@24092
   339
(*        proof-producing).                                                  *)
wenzelm@24092
   340
wenzelm@24092
   341
fun split_once_items ctxt (Ts : typ list, terms : term list) :
wenzelm@24092
   342
                     (typ list * term list) list option =
wenzelm@24092
   343
let
wenzelm@42361
   344
  val thy = Proof_Context.theory_of ctxt
wenzelm@24092
   345
  (* takes a list  [t1, ..., tn]  to the term                                *)
wenzelm@24092
   346
  (*   tn' --> ... --> t1' --> False  ,                                      *)
wenzelm@24092
   347
  (* where ti' = HOLogic.dest_Trueprop ti                                    *)
webertj@32369
   348
  fun REPEAT_DETERM_etac_rev_mp tms =
webertj@32369
   349
    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop tms)
webertj@32369
   350
      HOLogic.false_const
wenzelm@42439
   351
  val split_thms  = filter (is_split_thm ctxt) (#splits (get_arith_data ctxt))
webertj@32369
   352
  val cmap        = Splitter.cmap_of_split_thms split_thms
webertj@32369
   353
  val goal_tm     = REPEAT_DETERM_etac_rev_mp terms
webertj@32369
   354
  val splits      = Splitter.split_posns cmap thy Ts goal_tm
haftmann@31082
   355
  val split_limit = Config.get ctxt split_limit
wenzelm@24092
   356
in
webertj@32369
   357
  if length splits > split_limit then (
webertj@32369
   358
    tracing ("linarith_split_limit exceeded (current value is " ^
webertj@32369
   359
      string_of_int split_limit ^ ")");
webertj@32369
   360
    NONE
webertj@32369
   361
  ) else case splits of
webertj@32369
   362
    [] =>
wenzelm@24092
   363
    (* split_tac would fail: no possible split *)
wenzelm@24092
   364
    NONE
webertj@32369
   365
  | (_, _::_, _, _, _) :: _ =>
webertj@32369
   366
    (* disallow a split that involves non-locally bound variables (except    *)
webertj@32369
   367
    (* when bound by outermost meta-quantifiers)                             *)
webertj@32369
   368
    NONE
webertj@32369
   369
  | (_, [], _, split_type, split_term) :: _ =>
webertj@32369
   370
    (* ignore all but the first possible split                               *)
webertj@32369
   371
    (case strip_comb split_term of
wenzelm@24092
   372
    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
wenzelm@24092
   373
      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
wenzelm@24092
   374
      let
wenzelm@24092
   375
        val rev_terms     = rev terms
wenzelm@24092
   376
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   377
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
haftmann@35092
   378
        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   379
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   380
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   381
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   382
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   383
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   384
      in
wenzelm@24092
   385
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   386
      end
wenzelm@24092
   387
    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
wenzelm@24092
   388
    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
wenzelm@24092
   389
      let
wenzelm@24092
   390
        val rev_terms     = rev terms
wenzelm@24092
   391
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   392
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
haftmann@35092
   393
        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   394
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   395
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   396
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   397
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   398
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   399
      in
wenzelm@24092
   400
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   401
      end
wenzelm@24092
   402
    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
haftmann@35092
   403
    | (Const (@{const_name Groups.abs}, _), [t1]) =>
wenzelm@24092
   404
      let
wenzelm@24092
   405
        val rev_terms   = rev terms
wenzelm@24092
   406
        val terms1      = map (subst_term [(split_term, t1)]) rev_terms
haftmann@35267
   407
        val terms2      = map (subst_term [(split_term, Const (@{const_name Groups.uminus},
wenzelm@24092
   408
                            split_type --> split_type) $ t1)]) rev_terms
haftmann@35267
   409
        val zero        = Const (@{const_name Groups.zero}, split_type)
haftmann@35092
   410
        val zero_leq_t1 = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   411
                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1
haftmann@35092
   412
        val t1_lt_zero  = Const (@{const_name Orderings.less},
wenzelm@24092
   413
                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
wenzelm@24092
   414
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   415
        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
wenzelm@24092
   416
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   417
      in
wenzelm@24092
   418
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   419
      end
wenzelm@24092
   420
    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
haftmann@35267
   421
    | (Const (@{const_name Groups.minus}, _), [t1, t2]) =>
wenzelm@24092
   422
      let
wenzelm@24092
   423
        (* "d" in the above theorem becomes a new bound variable after NNF   *)
wenzelm@24092
   424
        (* transformation, therefore some adjustment of indices is necessary *)
wenzelm@24092
   425
        val rev_terms       = rev terms
haftmann@35267
   426
        val zero            = Const (@{const_name Groups.zero}, split_type)
wenzelm@24092
   427
        val d               = Bound 0
wenzelm@24092
   428
        val terms1          = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   429
        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])
wenzelm@24092
   430
                                (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   431
        val t1'             = incr_boundvars 1 t1
wenzelm@24092
   432
        val t2'             = incr_boundvars 1 t2
haftmann@35092
   433
        val t1_lt_t2        = Const (@{const_name Orderings.less},
wenzelm@24092
   434
                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
haftmann@38864
   435
        val t1_eq_t2_plus_d = Const (@{const_name HOL.eq}, split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@35267
   436
                                (Const (@{const_name Groups.plus},
wenzelm@24092
   437
                                  split_type --> split_type --> split_type) $ t2' $ d)
wenzelm@24092
   438
        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   439
        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
wenzelm@24092
   440
        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
wenzelm@24092
   441
      in
wenzelm@24092
   442
        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
wenzelm@24092
   443
      end
webertj@33728
   444
    (* ?P (nat ?i) = ((ALL n. ?i = of_nat n --> ?P n) & (?i < 0 --> ?P 0)) *)
haftmann@25919
   445
    | (Const ("Int.nat", _), [t1]) =>
wenzelm@24092
   446
      let
wenzelm@24092
   447
        val rev_terms   = rev terms
haftmann@35267
   448
        val zero_int    = Const (@{const_name Groups.zero}, HOLogic.intT)
haftmann@35267
   449
        val zero_nat    = Const (@{const_name Groups.zero}, HOLogic.natT)
wenzelm@24092
   450
        val n           = Bound 0
wenzelm@24092
   451
        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])
wenzelm@24092
   452
                            (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   453
        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms
wenzelm@24092
   454
        val t1'         = incr_boundvars 1 t1
haftmann@38864
   455
        val t1_eq_nat_n = Const (@{const_name HOL.eq}, HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
haftmann@24196
   456
                            (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
haftmann@35092
   457
        val t1_lt_zero  = Const (@{const_name Orderings.less},
wenzelm@24092
   458
                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
wenzelm@24092
   459
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   460
        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_nat_n) :: terms1 @ [not_false]
wenzelm@24092
   461
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   462
      in
wenzelm@24092
   463
        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   464
      end
webertj@33719
   465
    (* ?P ((?n::nat) mod (number_of ?k)) =
wenzelm@24092
   466
         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
wenzelm@24092
   467
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
haftmann@37388
   468
    | (Const ("Divides.div_class.mod", Type ("fun", [@{typ nat}, _])), [t1, t2]) =>
wenzelm@24092
   469
      let
wenzelm@24092
   470
        val rev_terms               = rev terms
haftmann@35267
   471
        val zero                    = Const (@{const_name Groups.zero}, split_type)
wenzelm@24092
   472
        val i                       = Bound 1
wenzelm@24092
   473
        val j                       = Bound 0
wenzelm@24092
   474
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   475
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   476
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   477
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   478
        val t2'                     = incr_boundvars 2 t2
haftmann@38864
   479
        val t2_eq_zero              = Const (@{const_name HOL.eq},
wenzelm@24092
   480
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@38864
   481
        val t2_neq_zero             = HOLogic.mk_not (Const (@{const_name HOL.eq},
wenzelm@24092
   482
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
haftmann@35092
   483
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   484
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@38864
   485
        val t1_eq_t2_times_i_plus_j = Const (@{const_name HOL.eq}, split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@35267
   486
                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $
haftmann@35267
   487
                                         (Const (@{const_name Groups.times},
wenzelm@24092
   488
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   489
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   490
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   491
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   492
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   493
                                          @ terms2 @ [not_false]
wenzelm@24092
   494
      in
wenzelm@24092
   495
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   496
      end
webertj@33719
   497
    (* ?P ((?n::nat) div (number_of ?k)) =
wenzelm@24092
   498
         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
wenzelm@24092
   499
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
haftmann@37388
   500
    | (Const ("Divides.div_class.div", Type ("fun", [@{typ nat}, _])), [t1, t2]) =>
wenzelm@24092
   501
      let
wenzelm@24092
   502
        val rev_terms               = rev terms
haftmann@35267
   503
        val zero                    = Const (@{const_name Groups.zero}, split_type)
wenzelm@24092
   504
        val i                       = Bound 1
wenzelm@24092
   505
        val j                       = Bound 0
wenzelm@24092
   506
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   507
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   508
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   509
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   510
        val t2'                     = incr_boundvars 2 t2
haftmann@38864
   511
        val t2_eq_zero              = Const (@{const_name HOL.eq},
wenzelm@24092
   512
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@38864
   513
        val t2_neq_zero             = HOLogic.mk_not (Const (@{const_name HOL.eq},
wenzelm@24092
   514
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
haftmann@35092
   515
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   516
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@38864
   517
        val t1_eq_t2_times_i_plus_j = Const (@{const_name HOL.eq}, split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@35267
   518
                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $
haftmann@35267
   519
                                         (Const (@{const_name Groups.times},
wenzelm@24092
   520
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   521
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   522
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   523
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   524
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   525
                                          @ terms2 @ [not_false]
wenzelm@24092
   526
      in
wenzelm@24092
   527
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   528
      end
webertj@33719
   529
    (* ?P ((?n::int) mod (number_of ?k)) =
webertj@33728
   530
         ((number_of ?k = 0 --> ?P ?n) &
webertj@33728
   531
          (0 < number_of ?k -->
webertj@33728
   532
            (ALL i j.
webertj@33728
   533
              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
webertj@33728
   534
          (number_of ?k < 0 -->
webertj@33728
   535
            (ALL i j.
webertj@33728
   536
              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   537
    | (Const ("Divides.div_class.mod",
webertj@33728
   538
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>
wenzelm@24092
   539
      let
wenzelm@24092
   540
        val rev_terms               = rev terms
haftmann@35267
   541
        val zero                    = Const (@{const_name Groups.zero}, split_type)
wenzelm@24092
   542
        val i                       = Bound 1
wenzelm@24092
   543
        val j                       = Bound 0
wenzelm@24092
   544
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   545
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   546
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   547
        val t1'                     = incr_boundvars 2 t1
webertj@33728
   548
        val t2'                     = incr_boundvars 2 t2
haftmann@38864
   549
        val t2_eq_zero              = Const (@{const_name HOL.eq},
webertj@33728
   550
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@35092
   551
        val zero_lt_t2              = Const (@{const_name Orderings.less},
webertj@33728
   552
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'
haftmann@35092
   553
        val t2_lt_zero              = Const (@{const_name Orderings.less},
webertj@33728
   554
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero
haftmann@35092
   555
        val zero_leq_j              = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   556
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
haftmann@35092
   557
        val j_leq_zero              = Const (@{const_name Orderings.less_eq},
webertj@33728
   558
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
haftmann@35092
   559
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   560
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@35092
   561
        val t2_lt_j                 = Const (@{const_name Orderings.less},
webertj@33728
   562
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
haftmann@38864
   563
        val t1_eq_t2_times_i_plus_j = Const (@{const_name HOL.eq}, split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@35267
   564
                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $
haftmann@35267
   565
                                         (Const (@{const_name Groups.times},
wenzelm@24092
   566
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   567
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   568
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
webertj@33728
   569
        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])
wenzelm@24092
   570
                                        @ hd terms2_3
wenzelm@24092
   571
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   572
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   573
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
webertj@33728
   574
        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])
wenzelm@24092
   575
                                        @ hd terms2_3
wenzelm@24092
   576
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   577
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   578
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   579
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   580
      in
wenzelm@24092
   581
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   582
      end
webertj@33719
   583
    (* ?P ((?n::int) div (number_of ?k)) =
webertj@33728
   584
         ((number_of ?k = 0 --> ?P 0) &
webertj@33728
   585
          (0 < number_of ?k -->
webertj@33728
   586
            (ALL i j.
webertj@33728
   587
              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P i)) &
webertj@33728
   588
          (number_of ?k < 0 -->
webertj@33728
   589
            (ALL i j.
webertj@33728
   590
              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P i))) *)
wenzelm@24092
   591
    | (Const ("Divides.div_class.div",
webertj@33728
   592
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>
wenzelm@24092
   593
      let
wenzelm@24092
   594
        val rev_terms               = rev terms
haftmann@35267
   595
        val zero                    = Const (@{const_name Groups.zero}, split_type)
wenzelm@24092
   596
        val i                       = Bound 1
wenzelm@24092
   597
        val j                       = Bound 0
wenzelm@24092
   598
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   599
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   600
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   601
        val t1'                     = incr_boundvars 2 t1
webertj@33728
   602
        val t2'                     = incr_boundvars 2 t2
haftmann@38864
   603
        val t2_eq_zero              = Const (@{const_name HOL.eq},
webertj@33728
   604
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@35092
   605
        val zero_lt_t2              = Const (@{const_name Orderings.less},
webertj@33728
   606
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'
haftmann@35092
   607
        val t2_lt_zero              = Const (@{const_name Orderings.less},
webertj@33728
   608
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero
haftmann@35092
   609
        val zero_leq_j              = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   610
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
haftmann@35092
   611
        val j_leq_zero              = Const (@{const_name Orderings.less_eq},
webertj@33728
   612
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
haftmann@35092
   613
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   614
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@35092
   615
        val t2_lt_j                 = Const (@{const_name Orderings.less},
webertj@33728
   616
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
haftmann@38864
   617
        val t1_eq_t2_times_i_plus_j = Const (@{const_name HOL.eq}, split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@35267
   618
                                       (Const (@{const_name Groups.plus}, split_type --> split_type --> split_type) $
haftmann@35267
   619
                                         (Const (@{const_name Groups.times},
wenzelm@24092
   620
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   621
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   622
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
webertj@33728
   623
        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])
webertj@33728
   624
                                        @ hd terms2_3
webertj@33728
   625
                                        :: (if tl terms2_3 = [] then [not_false] else [])
webertj@33728
   626
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
webertj@33728
   627
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
webertj@33728
   628
        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])
webertj@33728
   629
                                        @ hd terms2_3
webertj@33728
   630
                                        :: (if tl terms2_3 = [] then [not_false] else [])
webertj@33728
   631
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
webertj@33728
   632
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   633
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   634
      in
wenzelm@24092
   635
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   636
      end
wenzelm@24092
   637
    (* this will only happen if a split theorem can be applied for which no  *)
wenzelm@24092
   638
    (* code exists above -- in which case either the split theorem should be *)
wenzelm@24092
   639
    (* implemented above, or 'is_split_thm' should be modified to filter it  *)
wenzelm@24092
   640
    (* out                                                                   *)
wenzelm@24092
   641
    | (t, ts) => (
wenzelm@24920
   642
      warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
webertj@32369
   643
        " (with " ^ string_of_int (length ts) ^
webertj@32369
   644
        " argument(s)) not implemented; proof reconstruction is likely to fail");
wenzelm@24092
   645
      NONE
wenzelm@24092
   646
    ))
webertj@32369
   647
end;  (* split_once_items *)
wenzelm@24092
   648
wenzelm@24092
   649
(* remove terms that do not satisfy 'p'; change the order of the remaining   *)
wenzelm@24092
   650
(* terms in the same way as filter_prems_tac does                            *)
wenzelm@24092
   651
wenzelm@24092
   652
fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
wenzelm@42439
   653
  let
wenzelm@42439
   654
    fun filter_prems t (left, right) =
wenzelm@42439
   655
      if p t then (left, right @ [t]) else (left @ right, [])
wenzelm@42439
   656
    val (left, right) = fold filter_prems terms ([], [])
wenzelm@42439
   657
  in
wenzelm@42439
   658
    right @ left
wenzelm@42439
   659
  end;
wenzelm@24092
   660
wenzelm@24092
   661
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)
wenzelm@24092
   662
(* subgoal that has 'terms' as premises                                      *)
wenzelm@24092
   663
wenzelm@24092
   664
fun negated_term_occurs_positively (terms : term list) : bool =
wenzelm@24092
   665
  List.exists
haftmann@38558
   666
    (fn (Trueprop $ (Const (@{const_name Not}, _) $ t)) =>
webertj@32369
   667
      member Pattern.aeconv terms (Trueprop $ t)
webertj@32369
   668
      | _ => false)
wenzelm@24092
   669
    terms;
wenzelm@24092
   670
wenzelm@24092
   671
fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
wenzelm@42439
   672
  let
wenzelm@42439
   673
    (* repeatedly split (including newly emerging subgoals) until no further   *)
wenzelm@42439
   674
    (* splitting is possible                                                   *)
wenzelm@42439
   675
    fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list)
wenzelm@42439
   676
      | split_loop (subgoal::subgoals) =
wenzelm@42439
   677
          (case split_once_items ctxt subgoal of
wenzelm@42439
   678
            SOME new_subgoals => split_loop (new_subgoals @ subgoals)
wenzelm@42439
   679
          | NONE => subgoal :: split_loop subgoals)
wenzelm@42439
   680
    fun is_relevant t  = is_some (decomp ctxt t)
wenzelm@42439
   681
    (* filter_prems_tac is_relevant: *)
wenzelm@42439
   682
    val relevant_terms = filter_prems_tac_items is_relevant terms
wenzelm@42439
   683
    (* split_tac, NNF normalization: *)
wenzelm@42439
   684
    val split_goals = split_loop [(Ts, relevant_terms)]
wenzelm@42439
   685
    (* necessary because split_once_tac may normalize terms: *)
wenzelm@42439
   686
    val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm)))
wenzelm@42439
   687
      split_goals
wenzelm@42439
   688
    (* TRY (etac notE) THEN eq_assume_tac: *)
wenzelm@42439
   689
    val result = filter_out (negated_term_occurs_positively o snd) beta_eta_norm
wenzelm@42439
   690
  in
wenzelm@42439
   691
    result
wenzelm@42439
   692
  end;
wenzelm@24092
   693
wenzelm@24092
   694
(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)
wenzelm@24092
   695
(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)
wenzelm@24092
   696
(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)
wenzelm@24092
   697
(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)
wenzelm@24092
   698
(* disjunctions and existential quantifiers from the premises, possibly (in  *)
wenzelm@24092
   699
(* the case of disjunctions) resulting in several new subgoals, each of the  *)
wenzelm@24092
   700
(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)
haftmann@31082
   701
(* !split_limit splits are possible.                              *)
wenzelm@24092
   702
wenzelm@24092
   703
local
wenzelm@24092
   704
  val nnf_simpset =
wenzelm@24092
   705
    empty_ss setmkeqTrue mk_eq_True
wenzelm@24092
   706
    setmksimps (mksimps mksimps_pairs)
wenzelm@35410
   707
    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
wenzelm@35410
   708
      @{thm de_Morgan_conj}, not_all, not_ex, not_not]
wenzelm@35230
   709
  fun prem_nnf_tac ss = full_simp_tac (Simplifier.inherit_context ss nnf_simpset)
wenzelm@24092
   710
in
wenzelm@24092
   711
wenzelm@35230
   712
fun split_once_tac ss split_thms =
wenzelm@24092
   713
  let
wenzelm@35230
   714
    val ctxt = Simplifier.the_context ss
wenzelm@42361
   715
    val thy = Proof_Context.theory_of ctxt
wenzelm@24092
   716
    val cond_split_tac = SUBGOAL (fn (subgoal, i) =>
wenzelm@24092
   717
      let
wenzelm@24092
   718
        val Ts = rev (map snd (Logic.strip_params subgoal))
wenzelm@24092
   719
        val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal)
wenzelm@24092
   720
        val cmap = Splitter.cmap_of_split_thms split_thms
wenzelm@24092
   721
        val splits = Splitter.split_posns cmap thy Ts concl
wenzelm@24092
   722
      in
webertj@32369
   723
        if null splits orelse length splits > Config.get ctxt split_limit then
webertj@32369
   724
          no_tac
webertj@32369
   725
        else if null (#2 (hd splits)) then
webertj@32369
   726
          split_tac split_thms i
webertj@32369
   727
        else
webertj@32369
   728
          (* disallow a split that involves non-locally bound variables      *)
webertj@32369
   729
          (* (except when bound by outermost meta-quantifiers)               *)
webertj@32369
   730
          no_tac
wenzelm@24092
   731
      end)
wenzelm@24092
   732
  in
wenzelm@24092
   733
    EVERY' [
wenzelm@24092
   734
      REPEAT_DETERM o etac rev_mp,
wenzelm@24092
   735
      cond_split_tac,
wenzelm@24092
   736
      rtac ccontr,
wenzelm@35230
   737
      prem_nnf_tac ss,
wenzelm@24092
   738
      TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE))
wenzelm@24092
   739
    ]
wenzelm@24092
   740
  end;
wenzelm@24092
   741
wenzelm@24092
   742
end;  (* local *)
wenzelm@24092
   743
wenzelm@24092
   744
(* remove irrelevant premises, then split the i-th subgoal (and all new      *)
wenzelm@24092
   745
(* subgoals) by using 'split_once_tac' repeatedly.  Beta-eta-normalize new   *)
wenzelm@24092
   746
(* subgoals and finally attempt to solve them by finding an immediate        *)
webertj@32369
   747
(* contradiction (i.e., a term and its negation) in their premises.          *)
wenzelm@24092
   748
wenzelm@35230
   749
fun pre_tac ss i =
wenzelm@42439
   750
  let
wenzelm@42439
   751
    val ctxt = Simplifier.the_context ss;
wenzelm@42439
   752
    val split_thms = filter (is_split_thm ctxt) (#splits (get_arith_data ctxt))
wenzelm@42439
   753
    fun is_relevant t = is_some (decomp ctxt t)
wenzelm@42439
   754
  in
wenzelm@42439
   755
    DETERM (
wenzelm@42439
   756
      TRY (filter_prems_tac is_relevant i)
wenzelm@42439
   757
        THEN (
wenzelm@42439
   758
          (TRY o REPEAT_ALL_NEW (split_once_tac ss split_thms))
wenzelm@42439
   759
            THEN_ALL_NEW
wenzelm@42439
   760
              (CONVERSION Drule.beta_eta_conversion
wenzelm@42439
   761
                THEN'
wenzelm@42439
   762
              (TRY o (etac notE THEN' eq_assume_tac)))
wenzelm@42439
   763
        ) i
wenzelm@42439
   764
    )
wenzelm@42439
   765
  end;
wenzelm@24092
   766
haftmann@31100
   767
end;  (* LA_Data *)
wenzelm@24092
   768
wenzelm@24092
   769
haftmann@31100
   770
val pre_tac = LA_Data.pre_tac;
wenzelm@24092
   771
haftmann@31100
   772
structure Fast_Arith = Fast_Lin_Arith(structure LA_Logic = LA_Logic and LA_Data = LA_Data);
wenzelm@24092
   773
wenzelm@38762
   774
val add_inj_thms = Fast_Arith.add_inj_thms;
wenzelm@38762
   775
val add_lessD = Fast_Arith.add_lessD;
wenzelm@38762
   776
val add_simps = Fast_Arith.add_simps;
wenzelm@38762
   777
val add_simprocs = Fast_Arith.add_simprocs;
wenzelm@38762
   778
val set_number_of = Fast_Arith.set_number_of;
boehmes@31510
   779
haftmann@31101
   780
fun simple_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
haftmann@31101
   781
val lin_arith_tac = Fast_Arith.lin_arith_tac;
haftmann@31082
   782
val trace = Fast_Arith.trace;
wenzelm@24092
   783
wenzelm@24092
   784
(* reduce contradictory <= to False.
wenzelm@24092
   785
   Most of the work is done by the cancel tactics. *)
wenzelm@24092
   786
wenzelm@24092
   787
val init_arith_data =
boehmes@31510
   788
  Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, number_of, ...} =>
wenzelm@42439
   789
   {add_mono_thms = @{thms add_mono_thms_linordered_semiring} @
wenzelm@42439
   790
      @{thms add_mono_thms_linordered_field} @ add_mono_thms,
boehmes@31510
   791
    mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} ::
boehmes@31510
   792
      @{lemma "a = b ==> c*a = c*b" by (rule arg_cong)} :: mult_mono_thms,
wenzelm@24092
   793
    inj_thms = inj_thms,
haftmann@31082
   794
    lessD = lessD @ [@{thm "Suc_leI"}],
haftmann@35028
   795
    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_linordered_idom}],
wenzelm@24092
   796
    simpset = HOL_basic_ss
boehmes@31510
   797
      addsimps @{thms ring_distribs}
boehmes@31510
   798
      addsimps [@{thm if_True}, @{thm if_False}]
wenzelm@24092
   799
      addsimps
haftmann@35050
   800
       [@{thm add_0_left},
haftmann@35050
   801
        @{thm add_0_right},
wenzelm@24092
   802
        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
wenzelm@24092
   803
        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
wenzelm@24092
   804
        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
wenzelm@24092
   805
        @{thm "not_one_less_zero"}]
haftmann@37890
   806
      addsimprocs [@{simproc abel_cancel_sum}, @{simproc abel_cancel_relation}]
wenzelm@24092
   807
       (*abel_cancel helps it work in abstract algebraic domains*)
haftmann@31082
   808
      addsimprocs Nat_Arith.nat_cancel_sums_add
wenzelm@35410
   809
      addcongs [@{thm if_weak_cong}],
boehmes@31510
   810
    number_of = number_of}) #>
haftmann@31082
   811
  add_discrete_type @{type_name nat};
wenzelm@24092
   812
nipkow@29849
   813
fun add_arith_facts ss =
wenzelm@41225
   814
  Simplifier.add_prems (Arith_Data.get_arith_facts (Simplifier.the_context ss)) ss;
nipkow@29849
   815
haftmann@31101
   816
val simproc = add_arith_facts #> Fast_Arith.lin_arith_simproc;
wenzelm@24092
   817
wenzelm@24092
   818
haftmann@26110
   819
(* generic refutation procedure *)
haftmann@26110
   820
haftmann@26110
   821
(* parameters:
haftmann@26110
   822
haftmann@26110
   823
   test: term -> bool
haftmann@26110
   824
   tests if a term is at all relevant to the refutation proof;
haftmann@26110
   825
   if not, then it can be discarded. Can improve performance,
haftmann@26110
   826
   esp. if disjunctions can be discarded (no case distinction needed!).
haftmann@26110
   827
haftmann@26110
   828
   prep_tac: int -> tactic
haftmann@26110
   829
   A preparation tactic to be applied to the goal once all relevant premises
haftmann@26110
   830
   have been moved to the conclusion.
haftmann@26110
   831
haftmann@26110
   832
   ref_tac: int -> tactic
haftmann@26110
   833
   the actual refutation tactic. Should be able to deal with goals
haftmann@26110
   834
   [| A1; ...; An |] ==> False
haftmann@26110
   835
   where the Ai are atomic, i.e. no top-level &, | or EX
haftmann@26110
   836
*)
haftmann@26110
   837
haftmann@26110
   838
local
haftmann@26110
   839
  val nnf_simpset =
haftmann@26110
   840
    empty_ss setmkeqTrue mk_eq_True
haftmann@26110
   841
    setmksimps (mksimps mksimps_pairs)
haftmann@26110
   842
    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
haftmann@26110
   843
      @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];
haftmann@26110
   844
  fun prem_nnf_tac i st =
wenzelm@35232
   845
    full_simp_tac (Simplifier.global_context (Thm.theory_of_thm st) nnf_simpset) i st;
haftmann@26110
   846
in
wenzelm@42439
   847
haftmann@26110
   848
fun refute_tac test prep_tac ref_tac =
haftmann@26110
   849
  let val refute_prems_tac =
haftmann@26110
   850
        REPEAT_DETERM
haftmann@26110
   851
              (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
haftmann@26110
   852
               filter_prems_tac test 1 ORELSE
haftmann@26110
   853
               etac @{thm disjE} 1) THEN
haftmann@26110
   854
        (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
haftmann@26110
   855
         ref_tac 1);
haftmann@26110
   856
  in EVERY'[TRY o filter_prems_tac test,
haftmann@26110
   857
            REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
haftmann@26110
   858
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
haftmann@26110
   859
  end;
wenzelm@42439
   860
haftmann@26110
   861
end;
haftmann@26110
   862
haftmann@26110
   863
wenzelm@24092
   864
(* arith proof method *)
wenzelm@24092
   865
wenzelm@24092
   866
local
wenzelm@24092
   867
haftmann@31101
   868
fun raw_tac ctxt ex =
wenzelm@33035
   869
  (* FIXME: K true should be replaced by a sensible test (perhaps "is_some o
wenzelm@24092
   870
     decomp sg"? -- but note that the test is applied to terms already before
wenzelm@24092
   871
     they are split/normalized) to speed things up in case there are lots of
wenzelm@24092
   872
     irrelevant terms involved; elimination of min/max can be optimized:
wenzelm@24092
   873
     (max m n + k <= r) = (m+k <= r & n+k <= r)
wenzelm@24092
   874
     (l <= min m n + k) = (l <= m+k & l <= n+k)
wenzelm@24092
   875
  *)
wenzelm@24092
   876
  refute_tac (K true)
webertj@33728
   877
    (* Splitting is also done inside simple_tac, but not completely --    *)
webertj@33728
   878
    (* split_tac may use split theorems that have not been implemented in *)
webertj@33728
   879
    (* simple_tac (cf. pre_decomp and split_once_items above), and        *)
webertj@33728
   880
    (* split_limit may trigger.                                           *)
webertj@33728
   881
    (* Therefore splitting outside of simple_tac may allow us to prove    *)
webertj@33728
   882
    (* some goals that simple_tac alone would fail on.                    *)
wenzelm@24092
   883
    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
haftmann@31101
   884
    (lin_arith_tac ctxt ex);
wenzelm@24092
   885
wenzelm@24092
   886
in
wenzelm@24092
   887
haftmann@31101
   888
fun gen_tac ex ctxt = FIRST' [simple_tac ctxt,
wenzelm@35625
   889
  Object_Logic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_tac ctxt ex];
wenzelm@24092
   890
haftmann@31101
   891
val tac = gen_tac true;
wenzelm@24092
   892
wenzelm@24092
   893
end;
wenzelm@24092
   894
wenzelm@24092
   895
wenzelm@24092
   896
(* context setup *)
wenzelm@24092
   897
wenzelm@24092
   898
val setup =
wenzelm@24092
   899
  init_arith_data #>
wenzelm@43595
   900
  Simplifier.map_ss (fn ss => ss
wenzelm@43596
   901
    addSolver (mk_solver "lin_arith" (add_arith_facts #> Fast_Arith.cut_lin_arith_tac)));
haftmann@31100
   902
haftmann@31100
   903
val global_setup =
haftmann@31100
   904
  Attrib.setup @{binding arith_split} (Scan.succeed (Thm.declaration_attribute add_split))
haftmann@31100
   905
    "declaration of split rules for arithmetic procedure" #>
haftmann@31100
   906
  Method.setup @{binding linarith}
wenzelm@33554
   907
    (Scan.succeed (fn ctxt =>
haftmann@31100
   908
      METHOD (fn facts =>
wenzelm@33554
   909
        HEADGOAL (Method.insert_tac (Arith_Data.get_arith_facts ctxt @ facts)
haftmann@31101
   910
          THEN' tac ctxt)))) "linear arithmetic" #>
haftmann@31101
   911
  Arith_Data.add_tactic "linear arithmetic" gen_tac;
wenzelm@24092
   912
wenzelm@24092
   913
end;