src/HOL/Tools/lin_arith.ML
author haftmann
Wed Feb 10 14:12:04 2010 +0100 (2010-02-10)
changeset 35092 cfe605c54e50
parent 35084 e25eedfc15ce
child 35230 be006fbcfb96
child 35267 8dfd816713c6
permissions -rw-r--r--
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.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: Proof.context -> 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 ->
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    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 warning_count: int Unsynchronized.ref
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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 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 (@{const_name "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 (@{const_name 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 = 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 instantiate ([], [(cn, ct)]) @{thm le0} end;
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end;  (* LA_Logic *)
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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, setup_split_limit) = Attrib.config_int "linarith_split_limit" 9;
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val (neq_limit, setup_neq_limit) = Attrib.config_int "linarith_neq_limit" 9;
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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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(* 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 Algebras.times}, _)) $ s $ t, m) =
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      (case s of Const (@{const_name Algebras.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 Rings.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 Algebras.uminus}, _) $ t, m) = demult (t, Rat.neg m)
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    | demult (Const (@{const_name Algebras.zero}, _), m) = (NONE, Rat.zero)
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    | demult (Const (@{const_name Algebras.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 Algebras.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 Algebras.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 Algebras.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 Algebras.zero}, _), _, pi) =
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        pi
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    | poly (Const (@{const_name Algebras.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 Algebras.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 Rings.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 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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  | "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, @{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 ("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 ("Trueprop", _)) $
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  (Const ("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 = 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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fun domain_is_nat (_ $ (Const (_, T) $ _ $ _))                      = nT T
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  | domain_is_nat (_ $ (Const ("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 thm =
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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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   294
                                    @{const_name Orderings.min},
haftmann@35092
   295
                                    @{const_name Groups.abs},
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   296
                                    @{const_name Algebras.minus},
haftmann@34974
   297
                                    "Int.nat" (*DYNAMIC BINDING!*),
haftmann@34974
   298
                                    "Divides.div_class.mod" (*DYNAMIC BINDING!*),
haftmann@34974
   299
                                    "Divides.div_class.div" (*DYNAMIC BINDING!*)] a
wenzelm@24092
   300
    | _            => (warning ("Lin. Arith.: wrong format for split rule " ^
wenzelm@32091
   301
                                 Display.string_of_thm_without_context thm);
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   302
                       false))
wenzelm@24092
   303
  | _ => (warning ("Lin. Arith.: wrong format for split rule " ^
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   304
                   Display.string_of_thm_without_context thm);
wenzelm@24092
   305
          false);
wenzelm@24092
   306
wenzelm@24092
   307
(* substitute new for occurrences of old in a term, incrementing bound       *)
wenzelm@24092
   308
(* variables as needed when substituting inside an abstraction               *)
wenzelm@24092
   309
wenzelm@24092
   310
fun subst_term ([] : (term * term) list) (t : term) = t
wenzelm@24092
   311
  | subst_term pairs                     t          =
nipkow@29528
   312
      (case AList.lookup Pattern.aeconv pairs t of
wenzelm@24092
   313
        SOME new =>
wenzelm@24092
   314
          new
wenzelm@24092
   315
      | NONE     =>
wenzelm@24092
   316
          (case t of Abs (a, T, body) =>
wenzelm@24092
   317
            let val pairs' = map (pairself (incr_boundvars 1)) pairs
wenzelm@24092
   318
            in  Abs (a, T, subst_term pairs' body)  end
wenzelm@24092
   319
          | t1 $ t2                   =>
wenzelm@24092
   320
            subst_term pairs t1 $ subst_term pairs t2
wenzelm@24092
   321
          | _ => t));
wenzelm@24092
   322
wenzelm@24092
   323
(* approximates the effect of one application of split_tac (followed by NNF  *)
wenzelm@24092
   324
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a     *)
wenzelm@24092
   325
(* list of new subgoals (each again represented by a typ list for bound      *)
wenzelm@24092
   326
(* variables and a term list for premises), or NONE if split_tac would fail  *)
wenzelm@24092
   327
(* on the subgoal                                                            *)
wenzelm@24092
   328
wenzelm@24092
   329
(* FIXME: currently only the effect of certain split theorems is reproduced  *)
wenzelm@24092
   330
(*        (which is why we need 'is_split_thm').  A more canonical           *)
wenzelm@24092
   331
(*        implementation should analyze the right-hand side of the split     *)
wenzelm@24092
   332
(*        theorem that can be applied, and modify the subgoal accordingly.   *)
wenzelm@24092
   333
(*        Or even better, the splitter should be extended to provide         *)
wenzelm@24092
   334
(*        splitting on terms as well as splitting on theorems (where the     *)
wenzelm@24092
   335
(*        former can have a faster implementation as it does not need to be  *)
wenzelm@24092
   336
(*        proof-producing).                                                  *)
wenzelm@24092
   337
wenzelm@24092
   338
fun split_once_items ctxt (Ts : typ list, terms : term list) :
wenzelm@24092
   339
                     (typ list * term list) list option =
wenzelm@24092
   340
let
wenzelm@24092
   341
  val thy = ProofContext.theory_of ctxt
wenzelm@24092
   342
  (* takes a list  [t1, ..., tn]  to the term                                *)
wenzelm@24092
   343
  (*   tn' --> ... --> t1' --> False  ,                                      *)
wenzelm@24092
   344
  (* where ti' = HOLogic.dest_Trueprop ti                                    *)
webertj@32369
   345
  fun REPEAT_DETERM_etac_rev_mp tms =
webertj@32369
   346
    fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop tms)
webertj@32369
   347
      HOLogic.false_const
webertj@32369
   348
  val split_thms  = filter is_split_thm (#splits (get_arith_data ctxt))
webertj@32369
   349
  val cmap        = Splitter.cmap_of_split_thms split_thms
webertj@32369
   350
  val goal_tm     = REPEAT_DETERM_etac_rev_mp terms
webertj@32369
   351
  val splits      = Splitter.split_posns cmap thy Ts goal_tm
haftmann@31082
   352
  val split_limit = Config.get ctxt split_limit
wenzelm@24092
   353
in
webertj@32369
   354
  if length splits > split_limit then (
webertj@32369
   355
    tracing ("linarith_split_limit exceeded (current value is " ^
webertj@32369
   356
      string_of_int split_limit ^ ")");
webertj@32369
   357
    NONE
webertj@32369
   358
  ) else case splits of
webertj@32369
   359
    [] =>
wenzelm@24092
   360
    (* split_tac would fail: no possible split *)
wenzelm@24092
   361
    NONE
webertj@32369
   362
  | (_, _::_, _, _, _) :: _ =>
webertj@32369
   363
    (* disallow a split that involves non-locally bound variables (except    *)
webertj@32369
   364
    (* when bound by outermost meta-quantifiers)                             *)
webertj@32369
   365
    NONE
webertj@32369
   366
  | (_, [], _, split_type, split_term) :: _ =>
webertj@32369
   367
    (* ignore all but the first possible split                               *)
webertj@32369
   368
    (case strip_comb split_term of
wenzelm@24092
   369
    (* ?P (max ?i ?j) = ((?i <= ?j --> ?P ?j) & (~ ?i <= ?j --> ?P ?i)) *)
wenzelm@24092
   370
      (Const (@{const_name Orderings.max}, _), [t1, t2]) =>
wenzelm@24092
   371
      let
wenzelm@24092
   372
        val rev_terms     = rev terms
wenzelm@24092
   373
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   374
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
haftmann@35092
   375
        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   376
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   377
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   378
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   379
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   380
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   381
      in
wenzelm@24092
   382
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   383
      end
wenzelm@24092
   384
    (* ?P (min ?i ?j) = ((?i <= ?j --> ?P ?i) & (~ ?i <= ?j --> ?P ?j)) *)
wenzelm@24092
   385
    | (Const (@{const_name Orderings.min}, _), [t1, t2]) =>
wenzelm@24092
   386
      let
wenzelm@24092
   387
        val rev_terms     = rev terms
wenzelm@24092
   388
        val terms1        = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   389
        val terms2        = map (subst_term [(split_term, t2)]) rev_terms
haftmann@35092
   390
        val t1_leq_t2     = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   391
                                    split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   392
        val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2
wenzelm@24092
   393
        val not_false     = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   394
        val subgoal1      = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false]
wenzelm@24092
   395
        val subgoal2      = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false]
wenzelm@24092
   396
      in
wenzelm@24092
   397
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   398
      end
wenzelm@24092
   399
    (* ?P (abs ?a) = ((0 <= ?a --> ?P ?a) & (?a < 0 --> ?P (- ?a))) *)
haftmann@35092
   400
    | (Const (@{const_name Groups.abs}, _), [t1]) =>
wenzelm@24092
   401
      let
wenzelm@24092
   402
        val rev_terms   = rev terms
wenzelm@24092
   403
        val terms1      = map (subst_term [(split_term, t1)]) rev_terms
haftmann@34974
   404
        val terms2      = map (subst_term [(split_term, Const (@{const_name Algebras.uminus},
wenzelm@24092
   405
                            split_type --> split_type) $ t1)]) rev_terms
haftmann@34974
   406
        val zero        = Const (@{const_name Algebras.zero}, split_type)
haftmann@35092
   407
        val zero_leq_t1 = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   408
                            split_type --> split_type --> HOLogic.boolT) $ zero $ t1
haftmann@35092
   409
        val t1_lt_zero  = Const (@{const_name Orderings.less},
wenzelm@24092
   410
                            split_type --> split_type --> HOLogic.boolT) $ t1 $ zero
wenzelm@24092
   411
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   412
        val subgoal1    = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false]
wenzelm@24092
   413
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   414
      in
wenzelm@24092
   415
        SOME [(Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   416
      end
wenzelm@24092
   417
    (* ?P (?a - ?b) = ((?a < ?b --> ?P 0) & (ALL d. ?a = ?b + d --> ?P d)) *)
haftmann@34974
   418
    | (Const (@{const_name Algebras.minus}, _), [t1, t2]) =>
wenzelm@24092
   419
      let
wenzelm@24092
   420
        (* "d" in the above theorem becomes a new bound variable after NNF   *)
wenzelm@24092
   421
        (* transformation, therefore some adjustment of indices is necessary *)
wenzelm@24092
   422
        val rev_terms       = rev terms
haftmann@34974
   423
        val zero            = Const (@{const_name Algebras.zero}, split_type)
wenzelm@24092
   424
        val d               = Bound 0
wenzelm@24092
   425
        val terms1          = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   426
        val terms2          = map (subst_term [(incr_boundvars 1 split_term, d)])
wenzelm@24092
   427
                                (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   428
        val t1'             = incr_boundvars 1 t1
wenzelm@24092
   429
        val t2'             = incr_boundvars 1 t2
haftmann@35092
   430
        val t1_lt_t2        = Const (@{const_name Orderings.less},
wenzelm@24092
   431
                                split_type --> split_type --> HOLogic.boolT) $ t1 $ t2
wenzelm@24092
   432
        val t1_eq_t2_plus_d = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@34974
   433
                                (Const (@{const_name Algebras.plus},
wenzelm@24092
   434
                                  split_type --> split_type --> split_type) $ t2' $ d)
wenzelm@24092
   435
        val not_false       = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   436
        val subgoal1        = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false]
wenzelm@24092
   437
        val subgoal2        = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false]
wenzelm@24092
   438
      in
wenzelm@24092
   439
        SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)]
wenzelm@24092
   440
      end
webertj@33728
   441
    (* ?P (nat ?i) = ((ALL n. ?i = of_nat n --> ?P n) & (?i < 0 --> ?P 0)) *)
haftmann@25919
   442
    | (Const ("Int.nat", _), [t1]) =>
wenzelm@24092
   443
      let
wenzelm@24092
   444
        val rev_terms   = rev terms
haftmann@34974
   445
        val zero_int    = Const (@{const_name Algebras.zero}, HOLogic.intT)
haftmann@34974
   446
        val zero_nat    = Const (@{const_name Algebras.zero}, HOLogic.natT)
wenzelm@24092
   447
        val n           = Bound 0
wenzelm@24092
   448
        val terms1      = map (subst_term [(incr_boundvars 1 split_term, n)])
wenzelm@24092
   449
                            (map (incr_boundvars 1) rev_terms)
wenzelm@24092
   450
        val terms2      = map (subst_term [(split_term, zero_nat)]) rev_terms
wenzelm@24092
   451
        val t1'         = incr_boundvars 1 t1
webertj@33728
   452
        val t1_eq_nat_n = Const ("op =", HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1' $
haftmann@24196
   453
                            (Const (@{const_name of_nat}, HOLogic.natT --> HOLogic.intT) $ n)
haftmann@35092
   454
        val t1_lt_zero  = Const (@{const_name Orderings.less},
wenzelm@24092
   455
                            HOLogic.intT --> HOLogic.intT --> HOLogic.boolT) $ t1 $ zero_int
wenzelm@24092
   456
        val not_false   = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   457
        val subgoal1    = (HOLogic.mk_Trueprop t1_eq_nat_n) :: terms1 @ [not_false]
wenzelm@24092
   458
        val subgoal2    = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false]
wenzelm@24092
   459
      in
wenzelm@24092
   460
        SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)]
wenzelm@24092
   461
      end
webertj@33719
   462
    (* ?P ((?n::nat) mod (number_of ?k)) =
wenzelm@24092
   463
         ((number_of ?k = 0 --> ?P ?n) & (~ (number_of ?k = 0) -->
wenzelm@24092
   464
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   465
    | (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   466
      let
wenzelm@24092
   467
        val rev_terms               = rev terms
haftmann@34974
   468
        val zero                    = Const (@{const_name Algebras.zero}, split_type)
wenzelm@24092
   469
        val i                       = Bound 1
wenzelm@24092
   470
        val j                       = Bound 0
wenzelm@24092
   471
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   472
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   473
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   474
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   475
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   476
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   477
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   478
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   479
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
haftmann@35092
   480
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   481
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   482
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@34974
   483
                                       (Const (@{const_name Algebras.plus}, split_type --> split_type --> split_type) $
haftmann@34974
   484
                                         (Const (@{const_name Algebras.times},
wenzelm@24092
   485
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   486
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   487
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   488
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   489
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   490
                                          @ terms2 @ [not_false]
wenzelm@24092
   491
      in
wenzelm@24092
   492
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   493
      end
webertj@33719
   494
    (* ?P ((?n::nat) div (number_of ?k)) =
wenzelm@24092
   495
         ((number_of ?k = 0 --> ?P 0) & (~ (number_of ?k = 0) -->
wenzelm@24092
   496
           (ALL i j. j < number_of ?k --> ?n = number_of ?k * i + j --> ?P i))) *)
wenzelm@24092
   497
    | (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) =>
wenzelm@24092
   498
      let
wenzelm@24092
   499
        val rev_terms               = rev terms
haftmann@34974
   500
        val zero                    = Const (@{const_name Algebras.zero}, split_type)
wenzelm@24092
   501
        val i                       = Bound 1
wenzelm@24092
   502
        val j                       = Bound 0
wenzelm@24092
   503
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   504
        val terms2                  = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   505
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   506
        val t1'                     = incr_boundvars 2 t1
wenzelm@24092
   507
        val t2'                     = incr_boundvars 2 t2
wenzelm@24092
   508
        val t2_eq_zero              = Const ("op =",
wenzelm@24092
   509
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
wenzelm@24092
   510
        val t2_neq_zero             = HOLogic.mk_not (Const ("op =",
wenzelm@24092
   511
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero)
haftmann@35092
   512
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   513
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
wenzelm@24092
   514
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@34974
   515
                                       (Const (@{const_name Algebras.plus}, split_type --> split_type --> split_type) $
haftmann@34974
   516
                                         (Const (@{const_name Algebras.times},
wenzelm@24092
   517
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   518
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
wenzelm@24092
   519
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
wenzelm@24092
   520
        val subgoal2                = (map HOLogic.mk_Trueprop
wenzelm@24092
   521
                                        [t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   522
                                          @ terms2 @ [not_false]
wenzelm@24092
   523
      in
wenzelm@24092
   524
        SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)]
wenzelm@24092
   525
      end
webertj@33719
   526
    (* ?P ((?n::int) mod (number_of ?k)) =
webertj@33728
   527
         ((number_of ?k = 0 --> ?P ?n) &
webertj@33728
   528
          (0 < number_of ?k -->
webertj@33728
   529
            (ALL i j.
webertj@33728
   530
              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P j)) &
webertj@33728
   531
          (number_of ?k < 0 -->
webertj@33728
   532
            (ALL i j.
webertj@33728
   533
              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P j))) *)
wenzelm@24092
   534
    | (Const ("Divides.div_class.mod",
webertj@33728
   535
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>
wenzelm@24092
   536
      let
wenzelm@24092
   537
        val rev_terms               = rev terms
haftmann@34974
   538
        val zero                    = Const (@{const_name Algebras.zero}, split_type)
wenzelm@24092
   539
        val i                       = Bound 1
wenzelm@24092
   540
        val j                       = Bound 0
wenzelm@24092
   541
        val terms1                  = map (subst_term [(split_term, t1)]) rev_terms
wenzelm@24092
   542
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, j)])
wenzelm@24092
   543
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   544
        val t1'                     = incr_boundvars 2 t1
webertj@33728
   545
        val t2'                     = incr_boundvars 2 t2
webertj@33728
   546
        val t2_eq_zero              = Const ("op =",
webertj@33728
   547
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@35092
   548
        val zero_lt_t2              = Const (@{const_name Orderings.less},
webertj@33728
   549
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'
haftmann@35092
   550
        val t2_lt_zero              = Const (@{const_name Orderings.less},
webertj@33728
   551
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero
haftmann@35092
   552
        val zero_leq_j              = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   553
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
haftmann@35092
   554
        val j_leq_zero              = Const (@{const_name Orderings.less_eq},
webertj@33728
   555
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
haftmann@35092
   556
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   557
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@35092
   558
        val t2_lt_j                 = Const (@{const_name Orderings.less},
webertj@33728
   559
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
wenzelm@24092
   560
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@34974
   561
                                       (Const (@{const_name Algebras.plus}, split_type --> split_type --> split_type) $
haftmann@34974
   562
                                         (Const (@{const_name Algebras.times},
wenzelm@24092
   563
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   564
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   565
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
webertj@33728
   566
        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])
wenzelm@24092
   567
                                        @ hd terms2_3
wenzelm@24092
   568
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   569
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   570
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
webertj@33728
   571
        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])
wenzelm@24092
   572
                                        @ hd terms2_3
wenzelm@24092
   573
                                        :: (if tl terms2_3 = [] then [not_false] else [])
wenzelm@24092
   574
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
wenzelm@24092
   575
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   576
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   577
      in
wenzelm@24092
   578
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   579
      end
webertj@33719
   580
    (* ?P ((?n::int) div (number_of ?k)) =
webertj@33728
   581
         ((number_of ?k = 0 --> ?P 0) &
webertj@33728
   582
          (0 < number_of ?k -->
webertj@33728
   583
            (ALL i j.
webertj@33728
   584
              0 <= j & j < number_of ?k & ?n = number_of ?k * i + j --> ?P i)) &
webertj@33728
   585
          (number_of ?k < 0 -->
webertj@33728
   586
            (ALL i j.
webertj@33728
   587
              number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j --> ?P i))) *)
wenzelm@24092
   588
    | (Const ("Divides.div_class.div",
webertj@33728
   589
        Type ("fun", [Type ("Int.int", []), _])), [t1, t2]) =>
wenzelm@24092
   590
      let
wenzelm@24092
   591
        val rev_terms               = rev terms
haftmann@34974
   592
        val zero                    = Const (@{const_name Algebras.zero}, split_type)
wenzelm@24092
   593
        val i                       = Bound 1
wenzelm@24092
   594
        val j                       = Bound 0
wenzelm@24092
   595
        val terms1                  = map (subst_term [(split_term, zero)]) rev_terms
wenzelm@24092
   596
        val terms2_3                = map (subst_term [(incr_boundvars 2 split_term, i)])
wenzelm@24092
   597
                                        (map (incr_boundvars 2) rev_terms)
wenzelm@24092
   598
        val t1'                     = incr_boundvars 2 t1
webertj@33728
   599
        val t2'                     = incr_boundvars 2 t2
webertj@33728
   600
        val t2_eq_zero              = Const ("op =",
webertj@33728
   601
                                        split_type --> split_type --> HOLogic.boolT) $ t2 $ zero
haftmann@35092
   602
        val zero_lt_t2              = Const (@{const_name Orderings.less},
webertj@33728
   603
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ t2'
haftmann@35092
   604
        val t2_lt_zero              = Const (@{const_name Orderings.less},
webertj@33728
   605
                                        split_type --> split_type --> HOLogic.boolT) $ t2' $ zero
haftmann@35092
   606
        val zero_leq_j              = Const (@{const_name Orderings.less_eq},
wenzelm@24092
   607
                                        split_type --> split_type --> HOLogic.boolT) $ zero $ j
haftmann@35092
   608
        val j_leq_zero              = Const (@{const_name Orderings.less_eq},
webertj@33728
   609
                                        split_type --> split_type --> HOLogic.boolT) $ j $ zero
haftmann@35092
   610
        val j_lt_t2                 = Const (@{const_name Orderings.less},
wenzelm@24092
   611
                                        split_type --> split_type--> HOLogic.boolT) $ j $ t2'
haftmann@35092
   612
        val t2_lt_j                 = Const (@{const_name Orderings.less},
webertj@33728
   613
                                        split_type --> split_type--> HOLogic.boolT) $ t2' $ j
webertj@33728
   614
        val t1_eq_t2_times_i_plus_j = Const ("op =", split_type --> split_type --> HOLogic.boolT) $ t1' $
haftmann@34974
   615
                                       (Const (@{const_name Algebras.plus}, split_type --> split_type --> split_type) $
haftmann@34974
   616
                                         (Const (@{const_name Algebras.times},
wenzelm@24092
   617
                                           split_type --> split_type --> split_type) $ t2' $ i) $ j)
wenzelm@24092
   618
        val not_false               = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const)
webertj@33728
   619
        val subgoal1                = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false]
webertj@33728
   620
        val subgoal2                = (map HOLogic.mk_Trueprop [zero_lt_t2, zero_leq_j])
webertj@33728
   621
                                        @ hd terms2_3
webertj@33728
   622
                                        :: (if tl terms2_3 = [] then [not_false] else [])
webertj@33728
   623
                                        @ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j])
webertj@33728
   624
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
webertj@33728
   625
        val subgoal3                = (map HOLogic.mk_Trueprop [t2_lt_zero, t2_lt_j])
webertj@33728
   626
                                        @ hd terms2_3
webertj@33728
   627
                                        :: (if tl terms2_3 = [] then [not_false] else [])
webertj@33728
   628
                                        @ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j])
webertj@33728
   629
                                        @ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false])
wenzelm@24092
   630
        val Ts'                     = split_type :: split_type :: Ts
wenzelm@24092
   631
      in
wenzelm@24092
   632
        SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)]
wenzelm@24092
   633
      end
wenzelm@24092
   634
    (* this will only happen if a split theorem can be applied for which no  *)
wenzelm@24092
   635
    (* code exists above -- in which case either the split theorem should be *)
wenzelm@24092
   636
    (* implemented above, or 'is_split_thm' should be modified to filter it  *)
wenzelm@24092
   637
    (* out                                                                   *)
wenzelm@24092
   638
    | (t, ts) => (
wenzelm@24920
   639
      warning ("Lin. Arith.: split rule for " ^ Syntax.string_of_term ctxt t ^
webertj@32369
   640
        " (with " ^ string_of_int (length ts) ^
webertj@32369
   641
        " argument(s)) not implemented; proof reconstruction is likely to fail");
wenzelm@24092
   642
      NONE
wenzelm@24092
   643
    ))
webertj@32369
   644
end;  (* split_once_items *)
wenzelm@24092
   645
wenzelm@24092
   646
(* remove terms that do not satisfy 'p'; change the order of the remaining   *)
wenzelm@24092
   647
(* terms in the same way as filter_prems_tac does                            *)
wenzelm@24092
   648
wenzelm@24092
   649
fun filter_prems_tac_items (p : term -> bool) (terms : term list) : term list =
wenzelm@24092
   650
let
wenzelm@33339
   651
  fun filter_prems t (left, right) =
wenzelm@33339
   652
    if p t then (left, right @ [t]) else (left @ right, [])
wenzelm@33339
   653
  val (left, right) = fold filter_prems terms ([], [])
wenzelm@24092
   654
in
wenzelm@24092
   655
  right @ left
wenzelm@24092
   656
end;
wenzelm@24092
   657
wenzelm@24092
   658
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a     *)
wenzelm@24092
   659
(* subgoal that has 'terms' as premises                                      *)
wenzelm@24092
   660
wenzelm@24092
   661
fun negated_term_occurs_positively (terms : term list) : bool =
wenzelm@24092
   662
  List.exists
webertj@32369
   663
    (fn (Trueprop $ (Const ("Not", _) $ t)) =>
webertj@32369
   664
      member Pattern.aeconv terms (Trueprop $ t)
webertj@32369
   665
      | _ => false)
wenzelm@24092
   666
    terms;
wenzelm@24092
   667
wenzelm@24092
   668
fun pre_decomp ctxt (Ts : typ list, terms : term list) : (typ list * term list) list =
wenzelm@24092
   669
let
wenzelm@24092
   670
  (* repeatedly split (including newly emerging subgoals) until no further   *)
wenzelm@24092
   671
  (* splitting is possible                                                   *)
webertj@32369
   672
  fun split_loop ([] : (typ list * term list) list) =
webertj@32369
   673
      ([] : (typ list * term list) list)
webertj@32369
   674
    | split_loop (subgoal::subgoals) =
webertj@32369
   675
      (case split_once_items ctxt subgoal of
webertj@32369
   676
        SOME new_subgoals => split_loop (new_subgoals @ subgoals)
webertj@32369
   677
      | NONE              => subgoal :: split_loop subgoals)
wenzelm@33035
   678
  fun is_relevant t  = is_some (decomp ctxt t)
wenzelm@24092
   679
  (* filter_prems_tac is_relevant: *)
wenzelm@24092
   680
  val relevant_terms = filter_prems_tac_items is_relevant terms
wenzelm@24092
   681
  (* split_tac, NNF normalization: *)
wenzelm@24092
   682
  val split_goals    = split_loop [(Ts, relevant_terms)]
wenzelm@24092
   683
  (* necessary because split_once_tac may normalize terms: *)
webertj@32369
   684
  val beta_eta_norm  = map (apsnd (map (Envir.eta_contract o Envir.beta_norm)))
webertj@32369
   685
    split_goals
wenzelm@24092
   686
  (* TRY (etac notE) THEN eq_assume_tac: *)
wenzelm@33317
   687
  val result         = filter_out (negated_term_occurs_positively o snd)
webertj@32369
   688
    beta_eta_norm
wenzelm@24092
   689
in
wenzelm@24092
   690
  result
wenzelm@24092
   691
end;
wenzelm@24092
   692
wenzelm@24092
   693
(* takes the i-th subgoal  [| A1; ...; An |] ==> B  to                       *)
wenzelm@24092
   694
(* An --> ... --> A1 --> B,  performs splitting with the given 'split_thms'  *)
wenzelm@24092
   695
(* (resulting in a different subgoal P), takes  P  to  ~P ==> False,         *)
wenzelm@24092
   696
(* performs NNF-normalization of ~P, and eliminates conjunctions,            *)
wenzelm@24092
   697
(* disjunctions and existential quantifiers from the premises, possibly (in  *)
wenzelm@24092
   698
(* the case of disjunctions) resulting in several new subgoals, each of the  *)
wenzelm@24092
   699
(* general form  [| Q1; ...; Qm |] ==> False.  Fails if more than            *)
haftmann@31082
   700
(* !split_limit splits are possible.                              *)
wenzelm@24092
   701
wenzelm@24092
   702
local
wenzelm@24092
   703
  val nnf_simpset =
wenzelm@24092
   704
    empty_ss setmkeqTrue mk_eq_True
wenzelm@24092
   705
    setmksimps (mksimps mksimps_pairs)
wenzelm@24092
   706
    addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj,
wenzelm@24092
   707
      not_all, not_ex, not_not]
wenzelm@24092
   708
  fun prem_nnf_tac i st =
webertj@32369
   709
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset)
webertj@32369
   710
      i st
wenzelm@24092
   711
in
wenzelm@24092
   712
wenzelm@24092
   713
fun split_once_tac ctxt split_thms =
wenzelm@24092
   714
  let
wenzelm@24092
   715
    val thy = ProofContext.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@24092
   737
      prem_nnf_tac,
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@24092
   749
fun pre_tac ctxt i =
wenzelm@24092
   750
let
wenzelm@24092
   751
  val split_thms = filter is_split_thm (#splits (get_arith_data ctxt))
wenzelm@33035
   752
  fun is_relevant t = is_some (decomp ctxt t)
wenzelm@24092
   753
in
wenzelm@24092
   754
  DETERM (
wenzelm@24092
   755
    TRY (filter_prems_tac is_relevant i)
wenzelm@24092
   756
      THEN (
wenzelm@24092
   757
        (TRY o REPEAT_ALL_NEW (split_once_tac ctxt split_thms))
wenzelm@24092
   758
          THEN_ALL_NEW
wenzelm@24092
   759
            (CONVERSION Drule.beta_eta_conversion
wenzelm@24092
   760
              THEN'
wenzelm@24092
   761
            (TRY o (etac notE THEN' eq_assume_tac)))
wenzelm@24092
   762
      ) i
wenzelm@24092
   763
  )
wenzelm@24092
   764
end;
wenzelm@24092
   765
haftmann@31100
   766
end;  (* LA_Data *)
wenzelm@24092
   767
wenzelm@24092
   768
haftmann@31100
   769
val pre_tac = LA_Data.pre_tac;
wenzelm@24092
   770
haftmann@31100
   771
structure Fast_Arith = Fast_Lin_Arith(structure LA_Logic = LA_Logic and LA_Data = LA_Data);
wenzelm@24092
   772
boehmes@31510
   773
fun map_inj_thms f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
haftmann@31100
   774
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = f inj_thms,
boehmes@31510
   775
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = number_of};
haftmann@31100
   776
boehmes@31510
   777
fun map_lessD f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
haftmann@31100
   778
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
boehmes@31510
   779
    lessD = f lessD, neqE = neqE, simpset = simpset, number_of = number_of};
haftmann@31100
   780
boehmes@31510
   781
fun map_simpset f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
haftmann@31100
   782
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
boehmes@31510
   783
    lessD = lessD, neqE = neqE, simpset = f simpset, number_of = number_of};
boehmes@31510
   784
boehmes@31510
   785
fun map_number_of f {add_mono_thms, mult_mono_thms, inj_thms, lessD, neqE, simpset, number_of} =
boehmes@31510
   786
  {add_mono_thms = add_mono_thms, mult_mono_thms = mult_mono_thms, inj_thms = inj_thms,
boehmes@31510
   787
    lessD = lessD, neqE = neqE, simpset = simpset, number_of = f number_of};
haftmann@31100
   788
haftmann@31100
   789
fun add_inj_thms thms = Fast_Arith.map_data (map_inj_thms (append thms));
haftmann@31100
   790
fun add_lessD thm = Fast_Arith.map_data (map_lessD (fn thms => thms @ [thm]));
haftmann@31100
   791
fun add_simps thms = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimps thms));
haftmann@31100
   792
fun add_simprocs procs = Fast_Arith.map_data (map_simpset (fn simpset => simpset addsimprocs procs));
haftmann@31100
   793
boehmes@31510
   794
fun set_number_of f = Fast_Arith.map_data (map_number_of (K (serial (), f)))
boehmes@31510
   795
boehmes@31510
   796
haftmann@31101
   797
fun simple_tac ctxt = Fast_Arith.lin_arith_tac ctxt false;
haftmann@31101
   798
val lin_arith_tac = Fast_Arith.lin_arith_tac;
haftmann@31082
   799
val trace = Fast_Arith.trace;
wenzelm@27017
   800
val warning_count = Fast_Arith.warning_count;
wenzelm@24092
   801
wenzelm@24092
   802
(* reduce contradictory <= to False.
wenzelm@24092
   803
   Most of the work is done by the cancel tactics. *)
wenzelm@24092
   804
wenzelm@24092
   805
val init_arith_data =
boehmes@31510
   806
  Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, number_of, ...} =>
haftmann@35028
   807
   {add_mono_thms = @{thms add_mono_thms_linordered_semiring} @ @{thms add_mono_thms_linordered_field} @ add_mono_thms,
boehmes@31510
   808
    mult_mono_thms = @{thm mult_strict_left_mono} :: @{thm mult_left_mono} ::
boehmes@31510
   809
      @{lemma "a = b ==> c*a = c*b" by (rule arg_cong)} :: mult_mono_thms,
wenzelm@24092
   810
    inj_thms = inj_thms,
haftmann@31082
   811
    lessD = lessD @ [@{thm "Suc_leI"}],
haftmann@35028
   812
    neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_linordered_idom}],
wenzelm@24092
   813
    simpset = HOL_basic_ss
boehmes@31510
   814
      addsimps @{thms ring_distribs}
boehmes@31510
   815
      addsimps [@{thm if_True}, @{thm if_False}]
wenzelm@24092
   816
      addsimps
haftmann@35050
   817
       [@{thm add_0_left},
haftmann@35050
   818
        @{thm add_0_right},
wenzelm@24092
   819
        @{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"},
wenzelm@24092
   820
        @{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"},
wenzelm@24092
   821
        @{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"},
wenzelm@24092
   822
        @{thm "not_one_less_zero"}]
wenzelm@24092
   823
      addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv]
wenzelm@24092
   824
       (*abel_cancel helps it work in abstract algebraic domains*)
haftmann@31082
   825
      addsimprocs Nat_Arith.nat_cancel_sums_add
boehmes@31510
   826
      addcongs [if_weak_cong],
boehmes@31510
   827
    number_of = number_of}) #>
haftmann@31082
   828
  add_discrete_type @{type_name nat};
wenzelm@24092
   829
nipkow@29849
   830
fun add_arith_facts ss =
haftmann@30686
   831
  add_prems (Arith_Data.get_arith_facts (MetaSimplifier.the_context ss)) ss;
nipkow@29849
   832
haftmann@31101
   833
val simproc = add_arith_facts #> Fast_Arith.lin_arith_simproc;
wenzelm@24092
   834
wenzelm@24092
   835
haftmann@26110
   836
(* generic refutation procedure *)
haftmann@26110
   837
haftmann@26110
   838
(* parameters:
haftmann@26110
   839
haftmann@26110
   840
   test: term -> bool
haftmann@26110
   841
   tests if a term is at all relevant to the refutation proof;
haftmann@26110
   842
   if not, then it can be discarded. Can improve performance,
haftmann@26110
   843
   esp. if disjunctions can be discarded (no case distinction needed!).
haftmann@26110
   844
haftmann@26110
   845
   prep_tac: int -> tactic
haftmann@26110
   846
   A preparation tactic to be applied to the goal once all relevant premises
haftmann@26110
   847
   have been moved to the conclusion.
haftmann@26110
   848
haftmann@26110
   849
   ref_tac: int -> tactic
haftmann@26110
   850
   the actual refutation tactic. Should be able to deal with goals
haftmann@26110
   851
   [| A1; ...; An |] ==> False
haftmann@26110
   852
   where the Ai are atomic, i.e. no top-level &, | or EX
haftmann@26110
   853
*)
haftmann@26110
   854
haftmann@26110
   855
local
haftmann@26110
   856
  val nnf_simpset =
haftmann@26110
   857
    empty_ss setmkeqTrue mk_eq_True
haftmann@26110
   858
    setmksimps (mksimps mksimps_pairs)
haftmann@26110
   859
    addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj},
haftmann@26110
   860
      @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}];
haftmann@26110
   861
  fun prem_nnf_tac i st =
haftmann@26110
   862
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
haftmann@26110
   863
in
haftmann@26110
   864
fun refute_tac test prep_tac ref_tac =
haftmann@26110
   865
  let val refute_prems_tac =
haftmann@26110
   866
        REPEAT_DETERM
haftmann@26110
   867
              (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE
haftmann@26110
   868
               filter_prems_tac test 1 ORELSE
haftmann@26110
   869
               etac @{thm disjE} 1) THEN
haftmann@26110
   870
        (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE
haftmann@26110
   871
         ref_tac 1);
haftmann@26110
   872
  in EVERY'[TRY o filter_prems_tac test,
haftmann@26110
   873
            REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac,
haftmann@26110
   874
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
haftmann@26110
   875
  end;
haftmann@26110
   876
end;
haftmann@26110
   877
haftmann@26110
   878
wenzelm@24092
   879
(* arith proof method *)
wenzelm@24092
   880
wenzelm@24092
   881
local
wenzelm@24092
   882
haftmann@31101
   883
fun raw_tac ctxt ex =
wenzelm@33035
   884
  (* FIXME: K true should be replaced by a sensible test (perhaps "is_some o
wenzelm@24092
   885
     decomp sg"? -- but note that the test is applied to terms already before
wenzelm@24092
   886
     they are split/normalized) to speed things up in case there are lots of
wenzelm@24092
   887
     irrelevant terms involved; elimination of min/max can be optimized:
wenzelm@24092
   888
     (max m n + k <= r) = (m+k <= r & n+k <= r)
wenzelm@24092
   889
     (l <= min m n + k) = (l <= m+k & l <= n+k)
wenzelm@24092
   890
  *)
wenzelm@24092
   891
  refute_tac (K true)
webertj@33728
   892
    (* Splitting is also done inside simple_tac, but not completely --    *)
webertj@33728
   893
    (* split_tac may use split theorems that have not been implemented in *)
webertj@33728
   894
    (* simple_tac (cf. pre_decomp and split_once_items above), and        *)
webertj@33728
   895
    (* split_limit may trigger.                                           *)
webertj@33728
   896
    (* Therefore splitting outside of simple_tac may allow us to prove    *)
webertj@33728
   897
    (* some goals that simple_tac alone would fail on.                    *)
wenzelm@24092
   898
    (REPEAT_DETERM o split_tac (#splits (get_arith_data ctxt)))
haftmann@31101
   899
    (lin_arith_tac ctxt ex);
wenzelm@24092
   900
wenzelm@24092
   901
in
wenzelm@24092
   902
haftmann@31101
   903
fun gen_tac ex ctxt = FIRST' [simple_tac ctxt,
haftmann@31101
   904
  ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_tac ctxt ex];
wenzelm@24092
   905
haftmann@31101
   906
val tac = gen_tac true;
wenzelm@24092
   907
wenzelm@24092
   908
end;
wenzelm@24092
   909
wenzelm@24092
   910
wenzelm@24092
   911
(* context setup *)
wenzelm@24092
   912
wenzelm@24092
   913
val setup =
wenzelm@24092
   914
  init_arith_data #>
haftmann@31101
   915
  Simplifier.map_ss (fn ss => ss addsimprocs [Simplifier.simproc (@{theory}) "fast_nat_arith"
haftmann@31101
   916
    ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] (K simproc)]
haftmann@31101
   917
    (* Because of fast_nat_arith_simproc, the arithmetic solver is really only
haftmann@31101
   918
    useful to detect inconsistencies among the premises for subgoals which are
haftmann@31101
   919
    *not* themselves (in)equalities, because the latter activate
haftmann@31101
   920
    fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
haftmann@31101
   921
    solver all the time rather than add the additional check. *)
nipkow@29850
   922
    addSolver (mk_solver' "lin_arith"
haftmann@31100
   923
      (add_arith_facts #> Fast_Arith.cut_lin_arith_tac)))
haftmann@31100
   924
haftmann@31100
   925
val global_setup =
haftmann@31100
   926
  setup_split_limit #> setup_neq_limit #>
haftmann@31100
   927
  Attrib.setup @{binding arith_split} (Scan.succeed (Thm.declaration_attribute add_split))
haftmann@31100
   928
    "declaration of split rules for arithmetic procedure" #>
haftmann@31100
   929
  Method.setup @{binding linarith}
wenzelm@33554
   930
    (Scan.succeed (fn ctxt =>
haftmann@31100
   931
      METHOD (fn facts =>
wenzelm@33554
   932
        HEADGOAL (Method.insert_tac (Arith_Data.get_arith_facts ctxt @ facts)
haftmann@31101
   933
          THEN' tac ctxt)))) "linear arithmetic" #>
haftmann@31101
   934
  Arith_Data.add_tactic "linear arithmetic" gen_tac;
wenzelm@24092
   935
wenzelm@24092
   936
end;