src/Provers/Arith/fast_lin_arith.ML
author wenzelm
Tue Sep 18 16:08:00 2007 +0200 (2007-09-18)
changeset 24630 351a308ab58d
parent 24112 6c4e7d17f9b0
child 24920 2a45e400fdad
permissions -rw-r--r--
simplified type int (eliminated IntInf.int, integer);
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(*  Title:      Provers/Arith/fast_lin_arith.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow and Tjark Weber
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A generic linear arithmetic package.  It provides two tactics
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(cut_lin_arith_tac, lin_arith_tac) and a simplification procedure
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(lin_arith_simproc).
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Only take premises and conclusions into account that are already
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(negated) (in)equations. lin_arith_simproc tries to prove or disprove
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the term.
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*)
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(* Debugging: set Fast_Arith.trace *)
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(*** Data needed for setting up the linear arithmetic package ***)
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signature LIN_ARITH_LOGIC =
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sig
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  val conjI       : thm  (* P ==> Q ==> P & Q *)
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  val ccontr      : thm  (* (~ P ==> False) ==> P *)
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  val notI        : thm  (* (P ==> False) ==> ~ P *)
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  val not_lessD   : thm  (* ~(m < n) ==> n <= m *)
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  val not_leD     : thm  (* ~(m <= n) ==> n < m *)
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  val sym         : thm  (* x = y ==> y = x *)
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  val mk_Eq       : thm -> thm
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  val atomize     : thm -> thm list
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  val mk_Trueprop : term -> term
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  val neg_prop    : term -> term
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  val is_False    : thm -> bool
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  val is_nat      : typ list * term -> bool
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  val mk_nat_thm  : theory -> term -> thm
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end;
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(*
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mk_Eq(~in) = `in == False'
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mk_Eq(in) = `in == True'
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where `in' is an (in)equality.
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neg_prop(t) = neg  if t is wrapped up in Trueprop and neg is the
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  (logically) negated version of t (again wrapped up in Trueprop),
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  where the negation of a negative term is the term itself (no
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  double negation!); raises TERM ("neg_prop", [t]) if t is not of
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  the form 'Trueprop $ _'
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is_nat(parameter-types,t) =  t:nat
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mk_nat_thm(t) = "0 <= t"
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*)
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signature LIN_ARITH_DATA =
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sig
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  (*internal representation of linear (in-)equations:*)
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  type decompT = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool
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  val decomp: Proof.context -> term -> decompT option
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  val domain_is_nat: term -> bool
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  (*preprocessing, performed on a representation of subgoals as list of premises:*)
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  val pre_decomp: Proof.context -> typ list * term list -> (typ list * term list) list
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  (*preprocessing, performed on the goal -- must do the same as 'pre_decomp':*)
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  val pre_tac: Proof.context -> int -> tactic
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  val number_of: int * typ -> term
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  (*the limit on the number of ~= allowed; because each ~= is split
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    into two cases, this can lead to an explosion*)
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  val fast_arith_neq_limit: int Config.T
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end;
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(*
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decomp(`x Rel y') should yield (p,i,Rel,q,j,d)
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   where Rel is one of "<", "~<", "<=", "~<=" and "=" and
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         p (q, respectively) is the decomposition of the sum term x
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         (y, respectively) into a list of summand * multiplicity
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         pairs and a constant summand and d indicates if the domain
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         is discrete.
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domain_is_nat(`x Rel y') t should yield true iff x is of type "nat".
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The relationship between pre_decomp and pre_tac is somewhat tricky.  The
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internal representation of a subgoal and the corresponding theorem must
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be modified by pre_decomp (pre_tac, resp.) in a corresponding way.  See
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the comment for split_items below.  (This is even necessary for eta- and
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beta-equivalent modifications, as some of the lin. arith. code is not
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insensitive to them.)
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ss must reduce contradictory <= to False.
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   It should also cancel common summands to keep <= reduced;
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   otherwise <= can grow to massive proportions.
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*)
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signature FAST_LIN_ARITH =
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sig
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  val map_data: ({add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
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                 lessD: thm list, neqE: thm list, simpset: Simplifier.simpset}
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                 -> {add_mono_thms: thm list, mult_mono_thms: thm list, inj_thms: thm list,
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                     lessD: thm list, neqE: thm list, simpset: Simplifier.simpset})
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                -> Context.generic -> Context.generic
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  val trace: bool ref
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  val cut_lin_arith_tac: simpset -> int -> tactic
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  val lin_arith_tac: Proof.context -> bool -> int -> tactic
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  val lin_arith_simproc: simpset -> term -> thm option
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end;
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functor Fast_Lin_Arith
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  (structure LA_Logic: LIN_ARITH_LOGIC and LA_Data: LIN_ARITH_DATA): FAST_LIN_ARITH =
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struct
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(** theory data **)
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structure Data = GenericDataFun
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(
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  type T =
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   {add_mono_thms: thm list,
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    mult_mono_thms: thm list,
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    inj_thms: thm list,
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    lessD: thm list,
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    neqE: thm list,
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    simpset: Simplifier.simpset};
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  val empty = {add_mono_thms = [], mult_mono_thms = [], inj_thms = [],
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               lessD = [], neqE = [], simpset = Simplifier.empty_ss};
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  val extend = I;
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  fun merge _
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    ({add_mono_thms= add_mono_thms1, mult_mono_thms= mult_mono_thms1, inj_thms= inj_thms1,
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      lessD = lessD1, neqE=neqE1, simpset = simpset1},
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     {add_mono_thms= add_mono_thms2, mult_mono_thms= mult_mono_thms2, inj_thms= inj_thms2,
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      lessD = lessD2, neqE=neqE2, simpset = simpset2}) =
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    {add_mono_thms = Thm.merge_thms (add_mono_thms1, add_mono_thms2),
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     mult_mono_thms = Thm.merge_thms (mult_mono_thms1, mult_mono_thms2),
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     inj_thms = Thm.merge_thms (inj_thms1, inj_thms2),
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     lessD = Thm.merge_thms (lessD1, lessD2),
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     neqE = Thm.merge_thms (neqE1, neqE2),
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     simpset = Simplifier.merge_ss (simpset1, simpset2)};
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);
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val map_data = Data.map;
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val get_data = Data.get o Context.Proof;
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(*** A fast decision procedure ***)
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(*** Code ported from HOL Light ***)
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(* possible optimizations:
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   use (var,coeff) rep or vector rep  tp save space;
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   treat non-negative atoms separately rather than adding 0 <= atom
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*)
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val trace = ref false;
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datatype lineq_type = Eq | Le | Lt;
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datatype injust = Asm of int
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                | Nat of int (* index of atom *)
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                | LessD of injust
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                | NotLessD of injust
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                | NotLeD of injust
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                | NotLeDD of injust
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                | Multiplied of int * injust
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                | Multiplied2 of int * injust
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                | Added of injust * injust;
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datatype lineq = Lineq of int * lineq_type * int list * injust;
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(* ------------------------------------------------------------------------- *)
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(* Finding a (counter) example from the trace of a failed elimination        *)
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(* ------------------------------------------------------------------------- *)
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(* Examples are represented as rational numbers,                             *)
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(* Dont blame John Harrison for this code - it is entirely mine. TN          *)
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exception NoEx;
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(* Coding: (i,true,cs) means i <= cs and (i,false,cs) means i < cs.
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   In general, true means the bound is included, false means it is excluded.
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   Need to know if it is a lower or upper bound for unambiguous interpretation!
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*)
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fun elim_eqns (Lineq (i, Le, cs, _)) = [(i, true, cs)]
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  | elim_eqns (Lineq (i, Eq, cs, _)) = [(i, true, cs),(~i, true, map ~ cs)]
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  | elim_eqns (Lineq (i, Lt, cs, _)) = [(i, false, cs)];
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(* PRE: ex[v] must be 0! *)
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fun eval ex v (a, le, cs) =
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  let
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    val rs = map Rat.rat_of_int cs;
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    val rsum = fold2 (Rat.add oo Rat.mult) rs ex Rat.zero;
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  in (Rat.mult (Rat.add (Rat.rat_of_int a) (Rat.neg rsum)) (Rat.inv (nth rs v)), le) end;
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(* If nth rs v < 0, le should be negated.
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   Instead this swap is taken into account in ratrelmin2.
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*)
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fun ratrelmin2 (x as (r, ler), y as (s, les)) =
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  case Rat.ord (r, s)
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   of EQUAL => (r, (not ler) andalso (not les))
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    | LESS => x
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    | GREATER => y;
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fun ratrelmax2 (x as (r, ler), y as (s, les)) =
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  case Rat.ord (r, s)
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   of EQUAL => (r, ler andalso les)
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    | LESS => y
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    | GREATER => x;
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val ratrelmin = foldr1 ratrelmin2;
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val ratrelmax = foldr1 ratrelmax2;
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fun ratexact up (r, exact) =
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  if exact then r else
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  let
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    val (p, q) = Rat.quotient_of_rat r;
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    val nth = Rat.inv (Rat.rat_of_int q);
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  in Rat.add r (if up then nth else Rat.neg nth) end;
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fun ratmiddle (r, s) = Rat.mult (Rat.add r s) (Rat.inv Rat.two);
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fun choose2 d ((lb, exactl), (ub, exactu)) =
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  let val ord = Rat.sign lb in
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  if (ord = LESS orelse exactl) andalso (ord = GREATER orelse exactu)
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    then Rat.zero
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    else if not d then
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      if ord = GREATER
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        then if exactl then lb else ratmiddle (lb, ub)
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        else if exactu then ub else ratmiddle (lb, ub)
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      else (*discrete domain, both bounds must be exact*)
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      if ord = GREATER
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        then let val lb' = Rat.roundup lb in
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          if Rat.le lb' ub then lb' else raise NoEx end
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        else let val ub' = Rat.rounddown ub in
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          if Rat.le lb ub' then ub' else raise NoEx end
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  end;
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fun findex1 discr (v, lineqs) ex =
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  let
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    val nz = filter (fn (Lineq (_, _, cs, _)) => nth cs v <> 0) lineqs;
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    val ineqs = maps elim_eqns nz;
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    val (ge, le) = List.partition (fn (_,_,cs) => nth cs v > 0) ineqs
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    val lb = ratrelmax (map (eval ex v) ge)
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    val ub = ratrelmin (map (eval ex v) le)
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  in nth_map v (K (choose2 (nth discr v) (lb, ub))) ex end;
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fun elim1 v x =
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  map (fn (a,le,bs) => (Rat.add a (Rat.neg (Rat.mult (nth bs v) x)), le,
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                        nth_map v (K Rat.zero) bs));
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fun single_var v (_, _, cs) = case filter_out (curry (op =) EQUAL o Rat.sign) cs
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 of [x] => x =/ nth cs v
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  | _ => false;
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(* The base case:
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   all variables occur only with positive or only with negative coefficients *)
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fun pick_vars discr (ineqs,ex) =
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  let val nz = filter_out (fn (_,_,cs) => forall (curry (op =) EQUAL o Rat.sign) cs) ineqs
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  in case nz of [] => ex
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     | (_,_,cs) :: _ =>
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       let val v = find_index (not o curry (op =) EQUAL o Rat.sign) cs
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           val d = nth discr v;
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           val pos = not (Rat.sign (nth cs v) = LESS);
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           val sv = filter (single_var v) nz;
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           val minmax =
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             if pos then if d then Rat.roundup o fst o ratrelmax
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                         else ratexact true o ratrelmax
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                    else if d then Rat.rounddown o fst o ratrelmin
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                         else ratexact false o ratrelmin
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           val bnds = map (fn (a,le,bs) => (Rat.mult a (Rat.inv (nth bs v)), le)) sv
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           val x = minmax((Rat.zero,if pos then true else false)::bnds)
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           val ineqs' = elim1 v x nz
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           val ex' = nth_map v (K x) ex
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       in pick_vars discr (ineqs',ex') end
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  end;
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fun findex0 discr n lineqs =
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  let val ineqs = maps elim_eqns lineqs
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      val rineqs = map (fn (a,le,cs) => (Rat.rat_of_int a, le, map Rat.rat_of_int cs))
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                       ineqs
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  in pick_vars discr (rineqs,replicate n Rat.zero) end;
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(* ------------------------------------------------------------------------- *)
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(* End of counterexample finder. The actual decision procedure starts here.  *)
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(* ------------------------------------------------------------------------- *)
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(* ------------------------------------------------------------------------- *)
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(* Calculate new (in)equality type after addition.                           *)
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(* ------------------------------------------------------------------------- *)
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fun find_add_type(Eq,x) = x
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  | find_add_type(x,Eq) = x
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  | find_add_type(_,Lt) = Lt
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  | find_add_type(Lt,_) = Lt
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  | find_add_type(Le,Le) = Le;
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(* ------------------------------------------------------------------------- *)
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(* Multiply out an (in)equation.                                             *)
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(* ------------------------------------------------------------------------- *)
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fun multiply_ineq n (i as Lineq(k,ty,l,just)) =
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  if n = 1 then i
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  else if n = 0 andalso ty = Lt then sys_error "multiply_ineq"
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  else if n < 0 andalso (ty=Le orelse ty=Lt) then sys_error "multiply_ineq"
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  else Lineq (n * k, ty, map (curry op* n) l, Multiplied (n, just));
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(* ------------------------------------------------------------------------- *)
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(* Add together (in)equations.                                               *)
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(* ------------------------------------------------------------------------- *)
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fun add_ineq (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) =
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  let val l = map2 (curry (op +)) l1 l2
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  in Lineq(k1+k2,find_add_type(ty1,ty2),l,Added(just1,just2)) end;
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(* ------------------------------------------------------------------------- *)
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(* Elimination of variable between a single pair of (in)equations.           *)
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(* If they're both inequalities, 1st coefficient must be +ve, 2nd -ve.       *)
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(* ------------------------------------------------------------------------- *)
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fun elim_var v (i1 as Lineq(k1,ty1,l1,just1)) (i2 as Lineq(k2,ty2,l2,just2)) =
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  let val c1 = nth l1 v and c2 = nth l2 v
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      val m = Integer.lcm (abs c1) (abs c2)
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      val m1 = m div (abs c1) and m2 = m div (abs c2)
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      val (n1,n2) =
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        if (c1 >= 0) = (c2 >= 0)
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        then if ty1 = Eq then (~m1,m2)
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   319
             else if ty2 = Eq then (m1,~m2)
nipkow@5982
   320
                  else sys_error "elim_var"
nipkow@5982
   321
        else (m1,m2)
nipkow@5982
   322
      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
nipkow@5982
   323
                    then (~n1,~n2) else (n1,n2)
nipkow@5982
   324
  in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end;
nipkow@5982
   325
nipkow@5982
   326
(* ------------------------------------------------------------------------- *)
nipkow@5982
   327
(* The main refutation-finding code.                                         *)
nipkow@5982
   328
(* ------------------------------------------------------------------------- *)
nipkow@5982
   329
nipkow@5982
   330
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
nipkow@5982
   331
nipkow@5982
   332
fun is_answer (ans as Lineq(k,ty,l,_)) =
nipkow@5982
   333
  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
nipkow@5982
   334
wenzelm@24630
   335
fun calc_blowup l =
haftmann@17496
   336
  let val (p,n) = List.partition (curry (op <) 0) (List.filter (curry (op <>) 0) l)
wenzelm@24630
   337
  in length p * length n end;
nipkow@5982
   338
nipkow@5982
   339
(* ------------------------------------------------------------------------- *)
nipkow@5982
   340
(* Main elimination code:                                                    *)
nipkow@5982
   341
(*                                                                           *)
nipkow@5982
   342
(* (1) Looks for immediate solutions (false assertions with no variables).   *)
nipkow@5982
   343
(*                                                                           *)
nipkow@5982
   344
(* (2) If there are any equations, picks a variable with the lowest absolute *)
nipkow@5982
   345
(* coefficient in any of them, and uses it to eliminate.                     *)
nipkow@5982
   346
(*                                                                           *)
nipkow@5982
   347
(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
nipkow@5982
   348
(* blowup (number of consequences generated) and eliminates it.              *)
nipkow@5982
   349
(* ------------------------------------------------------------------------- *)
nipkow@5982
   350
nipkow@5982
   351
fun allpairs f xs ys =
webertj@20217
   352
  List.concat (map (fn x => map (fn y => f x y) ys) xs);
nipkow@5982
   353
nipkow@5982
   354
fun extract_first p =
skalberg@15531
   355
  let fun extract xs (y::ys) = if p y then (SOME y,xs@ys)
nipkow@5982
   356
                               else extract (y::xs) ys
skalberg@15531
   357
        | extract xs []      = (NONE,xs)
nipkow@5982
   358
  in extract [] end;
nipkow@5982
   359
nipkow@6056
   360
fun print_ineqs ineqs =
paulson@9073
   361
  if !trace then
wenzelm@12262
   362
     tracing(cat_lines(""::map (fn Lineq(c,t,l,_) =>
wenzelm@24630
   363
       string_of_int c ^
paulson@9073
   364
       (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
wenzelm@24630
   365
       commas(map string_of_int l)) ineqs))
paulson@9073
   366
  else ();
nipkow@6056
   367
nipkow@13498
   368
type history = (int * lineq list) list;
nipkow@13498
   369
datatype result = Success of injust | Failure of history;
nipkow@13498
   370
webertj@20217
   371
fun elim (ineqs, hist) =
webertj@20217
   372
  let val dummy = print_ineqs ineqs
webertj@20217
   373
      val (triv, nontriv) = List.partition is_trivial ineqs in
webertj@20217
   374
  if not (null triv)
nipkow@13186
   375
  then case Library.find_first is_answer triv of
webertj@20217
   376
         NONE => elim (nontriv, hist)
skalberg@15531
   377
       | SOME(Lineq(_,_,_,j)) => Success j
nipkow@5982
   378
  else
webertj@20217
   379
  if null nontriv then Failure hist
nipkow@13498
   380
  else
webertj@20217
   381
  let val (eqs, noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
webertj@20217
   382
  if not (null eqs) then
skalberg@15570
   383
     let val clist = Library.foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs)
wenzelm@24630
   384
         val sclist = sort (fn (x,y) => int_ord (abs x, abs y))
skalberg@15570
   385
                           (List.filter (fn i => i<>0) clist)
nipkow@5982
   386
         val c = hd sclist
skalberg@15531
   387
         val (SOME(eq as Lineq(_,_,ceq,_)),othereqs) =
nipkow@5982
   388
               extract_first (fn Lineq(_,_,l,_) => c mem l) eqs
webertj@20217
   389
         val v = find_index_eq c ceq
haftmann@23063
   390
         val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0)
nipkow@5982
   391
                                     (othereqs @ noneqs)
nipkow@5982
   392
         val others = map (elim_var v eq) roth @ ioth
nipkow@13498
   393
     in elim(others,(v,nontriv)::hist) end
nipkow@5982
   394
  else
nipkow@5982
   395
  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
haftmann@23063
   396
      val numlist = 0 upto (length (hd lists) - 1)
haftmann@23063
   397
      val coeffs = map (fn i => map (fn xs => nth xs i) lists) numlist
nipkow@5982
   398
      val blows = map calc_blowup coeffs
nipkow@5982
   399
      val iblows = blows ~~ numlist
haftmann@23063
   400
      val nziblows = filter_out (fn (i, _) => i = 0) iblows
nipkow@13498
   401
  in if null nziblows then Failure((~1,nontriv)::hist)
nipkow@13498
   402
     else
nipkow@5982
   403
     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
haftmann@23063
   404
         val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) ineqs
haftmann@23063
   405
         val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => nth l v > 0) yes
nipkow@13498
   406
     in elim(no @ allpairs (elim_var v) pos neg, (v,nontriv)::hist) end
nipkow@5982
   407
  end
nipkow@5982
   408
  end
nipkow@5982
   409
  end;
nipkow@5982
   410
nipkow@5982
   411
(* ------------------------------------------------------------------------- *)
nipkow@5982
   412
(* Translate back a proof.                                                   *)
nipkow@5982
   413
(* ------------------------------------------------------------------------- *)
nipkow@5982
   414
wenzelm@24076
   415
fun trace_thm msg th =
wenzelm@24076
   416
  (if !trace then (tracing msg; tracing (Display.string_of_thm th)) else (); th);
paulson@9073
   417
wenzelm@24076
   418
fun trace_term ctxt msg t =
wenzelm@24076
   419
  (if !trace then tracing (cat_lines [msg, ProofContext.string_of_term ctxt t]) else (); t)
wenzelm@24076
   420
wenzelm@24076
   421
fun trace_msg msg =
wenzelm@24076
   422
  if !trace then tracing msg else ();
paulson@9073
   423
nipkow@13498
   424
(* FIXME OPTIMIZE!!!! (partly done already)
nipkow@6056
   425
   Addition/Multiplication need i*t representation rather than t+t+...
nipkow@10691
   426
   Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n
nipkow@10691
   427
   because Numerals are not known early enough.
nipkow@6056
   428
nipkow@6056
   429
Simplification may detect a contradiction 'prematurely' due to type
nipkow@6056
   430
information: n+1 <= 0 is simplified to False and does not need to be crossed
nipkow@6056
   431
with 0 <= n.
nipkow@6056
   432
*)
nipkow@6056
   433
local
wenzelm@24076
   434
  exception FalseE of thm
nipkow@6056
   435
in
wenzelm@24076
   436
fun mkthm ss asms (just: injust) =
wenzelm@24076
   437
  let
wenzelm@24076
   438
    val ctxt = Simplifier.the_context ss;
wenzelm@24076
   439
    val thy = ProofContext.theory_of ctxt;
wenzelm@24076
   440
    val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = get_data ctxt;
wenzelm@24076
   441
    val simpset' = Simplifier.inherit_context ss simpset;
wenzelm@24076
   442
    val atoms = Library.foldl (fn (ats, (lhs,_,_,rhs,_,_)) =>
nipkow@6056
   443
                            map fst lhs  union  (map fst rhs  union  ats))
webertj@20217
   444
                        ([], List.mapPartial (fn thm => if Thm.no_prems thm
wenzelm@24076
   445
                                              then LA_Data.decomp ctxt (Thm.concl_of thm)
webertj@20217
   446
                                              else NONE) asms)
nipkow@6056
   447
nipkow@10575
   448
      fun add2 thm1 thm2 =
nipkow@6102
   449
        let val conj = thm1 RS (thm2 RS LA_Logic.conjI)
skalberg@15531
   450
        in get_first (fn th => SOME(conj RS th) handle THM _ => NONE) add_mono_thms
nipkow@5982
   451
        end;
skalberg@15531
   452
      fun try_add [] _ = NONE
nipkow@10575
   453
        | try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of
skalberg@15531
   454
             NONE => try_add thm1s thm2 | some => some;
nipkow@10575
   455
nipkow@10575
   456
      fun addthms thm1 thm2 =
nipkow@10575
   457
        case add2 thm1 thm2 of
skalberg@15531
   458
          NONE => (case try_add ([thm1] RL inj_thms) thm2 of
webertj@20217
   459
                     NONE => ( the (try_add ([thm2] RL inj_thms) thm1)
wenzelm@15660
   460
                               handle Option =>
nipkow@14360
   461
                               (trace_thm "" thm1; trace_thm "" thm2;
webertj@20217
   462
                                sys_error "Lin.arith. failed to add thms")
webertj@20217
   463
                             )
skalberg@15531
   464
                   | SOME thm => thm)
skalberg@15531
   465
        | SOME thm => thm;
nipkow@10575
   466
nipkow@5982
   467
      fun multn(n,thm) =
nipkow@5982
   468
        let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th)
nipkow@6102
   469
        in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end;
webertj@20217
   470
nipkow@15184
   471
      fun multn2(n,thm) =
skalberg@15531
   472
        let val SOME(mth) =
skalberg@15531
   473
              get_first (fn th => SOME(thm RS th) handle THM _ => NONE) mult_mono_thms
wenzelm@22596
   474
            fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (Thm.theory_of_thm th) var;
nipkow@15184
   475
            val cv = cvar(mth, hd(prems_of mth));
wenzelm@24076
   476
            val ct = cterm_of thy (LA_Data.number_of(n,#T(rep_cterm cv)))
nipkow@15184
   477
        in instantiate ([],[(cv,ct)]) mth end
nipkow@10691
   478
nipkow@6056
   479
      fun simp thm =
wenzelm@17515
   480
        let val thm' = trace_thm "Simplified:" (full_simplify simpset' thm)
nipkow@6102
   481
        in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end
nipkow@6056
   482
wenzelm@24076
   483
      fun mk (Asm i) = trace_thm ("Asm " ^ string_of_int i) (nth asms i)
wenzelm@24076
   484
        | mk (Nat i) = trace_thm ("Nat " ^ string_of_int i) (LA_Logic.mk_nat_thm thy (nth atoms i))
webertj@20254
   485
        | mk (LessD j)            = trace_thm "L" (hd ([mk j] RL lessD))
webertj@20254
   486
        | mk (NotLeD j)           = trace_thm "NLe" (mk j RS LA_Logic.not_leD)
webertj@20254
   487
        | mk (NotLeDD j)          = trace_thm "NLeD" (hd ([mk j RS LA_Logic.not_leD] RL lessD))
webertj@20254
   488
        | mk (NotLessD j)         = trace_thm "NL" (mk j RS LA_Logic.not_lessD)
webertj@20254
   489
        | mk (Added (j1, j2))     = simp (trace_thm "+" (addthms (mk j1) (mk j2)))
wenzelm@24630
   490
        | mk (Multiplied (n, j))  = (trace_msg ("*" ^ string_of_int n); trace_thm "*" (multn (n, mk j)))
wenzelm@24630
   491
        | mk (Multiplied2 (n, j)) = simp (trace_msg ("**" ^ string_of_int n); trace_thm "**" (multn2 (n, mk j)))
nipkow@5982
   492
paulson@9073
   493
  in trace_msg "mkthm";
nipkow@12932
   494
     let val thm = trace_thm "Final thm:" (mk just)
wenzelm@17515
   495
     in let val fls = simplify simpset' thm
nipkow@13186
   496
        in trace_thm "After simplification:" fls;
nipkow@13186
   497
           if LA_Logic.is_False fls then fls
nipkow@13186
   498
           else
webertj@20217
   499
            (tracing "Assumptions:"; List.app (tracing o Display.string_of_thm) asms;
webertj@20217
   500
             tracing "Proved:"; tracing (Display.string_of_thm fls);
nipkow@13186
   501
             warning "Linear arithmetic should have refuted the assumptions.\n\
nipkow@13186
   502
                     \Please inform Tobias Nipkow (nipkow@in.tum.de).";
nipkow@13186
   503
             fls)
nipkow@12932
   504
        end
webertj@20217
   505
     end handle FalseE thm => trace_thm "False reached early:" thm
nipkow@12932
   506
  end
nipkow@6056
   507
end;
nipkow@5982
   508
haftmann@23261
   509
fun coeff poly atom =
wenzelm@24630
   510
  AList.lookup (op aconv) poly atom |> the_default 0;
nipkow@10691
   511
nipkow@10691
   512
fun integ(rlhs,r,rel,rrhs,s,d) =
haftmann@17951
   513
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s
wenzelm@24630
   514
    val m = Integer.lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs))
wenzelm@22846
   515
    fun mult(t,r) =
haftmann@17951
   516
        let val (i,j) = Rat.quotient_of_rat r
paulson@15965
   517
        in (t,i * (m div j)) end
nipkow@12932
   518
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end
nipkow@10691
   519
nipkow@13498
   520
fun mklineq n atoms =
webertj@20217
   521
  fn (item, k) =>
webertj@20217
   522
  let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item
nipkow@13498
   523
      val lhsa = map (coeff lhs) atoms
nipkow@13498
   524
      and rhsa = map (coeff rhs) atoms
haftmann@18330
   525
      val diff = map2 (curry (op -)) rhsa lhsa
nipkow@13498
   526
      val c = i-j
nipkow@13498
   527
      val just = Asm k
nipkow@13498
   528
      fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j))
nipkow@13498
   529
  in case rel of
nipkow@13498
   530
      "<="   => lineq(c,Le,diff,just)
nipkow@13498
   531
     | "~<=" => if discrete
nipkow@13498
   532
                then lineq(1-c,Le,map (op ~) diff,NotLeDD(just))
nipkow@13498
   533
                else lineq(~c,Lt,map (op ~) diff,NotLeD(just))
nipkow@13498
   534
     | "<"   => if discrete
nipkow@13498
   535
                then lineq(c+1,Le,diff,LessD(just))
nipkow@13498
   536
                else lineq(c,Lt,diff,just)
nipkow@13498
   537
     | "~<"  => lineq(~c,Le,map (op~) diff,NotLessD(just))
nipkow@13498
   538
     | "="   => lineq(c,Eq,diff,just)
wenzelm@22846
   539
     | _     => sys_error("mklineq" ^ rel)
nipkow@5982
   540
  end;
nipkow@5982
   541
nipkow@13498
   542
(* ------------------------------------------------------------------------- *)
nipkow@13498
   543
(* Print (counter) example                                                   *)
nipkow@13498
   544
(* ------------------------------------------------------------------------- *)
nipkow@13498
   545
nipkow@13498
   546
fun print_atom((a,d),r) =
haftmann@17951
   547
  let val (p,q) = Rat.quotient_of_rat r
wenzelm@24630
   548
      val s = if d then string_of_int p else
nipkow@13498
   549
              if p = 0 then "0"
wenzelm@24630
   550
              else string_of_int p ^ "/" ^ string_of_int q
nipkow@13498
   551
  in a ^ " = " ^ s end;
nipkow@13498
   552
wenzelm@19049
   553
fun produce_ex sds =
haftmann@17496
   554
  curry (op ~~) sds
haftmann@17496
   555
  #> map print_atom
haftmann@17496
   556
  #> commas
webertj@23197
   557
  #> curry (op ^) "Counterexample (possibly spurious):\n";
nipkow@13498
   558
wenzelm@24076
   559
fun trace_ex ctxt params atoms discr n (hist: history) =
webertj@20217
   560
  case hist of
webertj@20217
   561
    [] => ()
webertj@20217
   562
  | (v, lineqs) :: hist' =>
wenzelm@24076
   563
      let
wenzelm@24076
   564
        val frees = map Free params
wenzelm@24076
   565
        fun show_term t = ProofContext.string_of_term ctxt (subst_bounds (frees, t))
wenzelm@24076
   566
        val start =
wenzelm@24076
   567
          if v = ~1 then (hist', findex0 discr n lineqs)
haftmann@22950
   568
          else (hist, replicate n Rat.zero)
wenzelm@24076
   569
        val ex = SOME (produce_ex (map show_term atoms ~~ discr)
wenzelm@24076
   570
            (uncurry (fold (findex1 discr)) start))
webertj@20217
   571
          handle NoEx => NONE
wenzelm@24076
   572
      in
wenzelm@24076
   573
        case ex of
wenzelm@24076
   574
          SOME s => (warning "arith failed - see trace for a counterexample"; tracing s)
wenzelm@24076
   575
        | NONE => warning "arith failed"
wenzelm@24076
   576
      end;
nipkow@13498
   577
webertj@20217
   578
(* ------------------------------------------------------------------------- *)
webertj@20217
   579
webertj@20268
   580
fun mknat (pTs : typ list) (ixs : int list) (atom : term, i : int) : lineq option =
webertj@20217
   581
  if LA_Logic.is_nat (pTs, atom)
nipkow@6056
   582
  then let val l = map (fn j => if j=i then 1 else 0) ixs
webertj@20217
   583
       in SOME (Lineq (0, Le, l, Nat i)) end
webertj@20217
   584
  else NONE;
nipkow@6056
   585
nipkow@13186
   586
(* This code is tricky. It takes a list of premises in the order they occur
skalberg@15531
   587
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical
skalberg@15531
   588
ones as NONE. Going through the premises, each numeric one is converted into
nipkow@13186
   589
a Lineq. The tricky bit is to convert ~= which is split into two cases < and
nipkow@13498
   590
>. Thus split_items returns a list of equation systems. This may blow up if
nipkow@13186
   591
there are many ~=, but in practice it does not seem to happen. The really
nipkow@13186
   592
tricky bit is to arrange the order of the cases such that they coincide with
nipkow@13186
   593
the order in which the cases are in the end generated by the tactic that
nipkow@13186
   594
applies the generated refutation thms (see function 'refute_tac').
nipkow@13186
   595
nipkow@13186
   596
For variables n of type nat, a constraint 0 <= n is added.
nipkow@13186
   597
*)
webertj@20217
   598
webertj@20217
   599
(* FIXME: To optimize, the splitting of cases and the search for refutations *)
webertj@20276
   600
(*        could be intertwined: separate the first (fully split) case,       *)
webertj@20217
   601
(*        refute it, continue with splitting and refuting.  Terminate with   *)
webertj@20217
   602
(*        failure as soon as a case could not be refuted; i.e. delay further *)
webertj@20217
   603
(*        splitting until after a refutation for other cases has been found. *)
webertj@20217
   604
wenzelm@24076
   605
fun split_items ctxt do_pre (Ts, terms) : (typ list * (LA_Data.decompT * int) list) list =
webertj@20276
   606
let
webertj@20276
   607
  (* splits inequalities '~=' into '<' and '>'; this corresponds to *)
webertj@20276
   608
  (* 'REPEAT_DETERM (eresolve_tac neqE i)' at the theorem/tactic    *)
webertj@20276
   609
  (* level                                                          *)
webertj@20276
   610
  (* FIXME: this is currently sensitive to the order of theorems in *)
webertj@20276
   611
  (*        neqE:  The theorem for type "nat" must come first.  A   *)
webertj@20276
   612
  (*        better (i.e. less likely to break when neqE changes)    *)
webertj@20276
   613
  (*        implementation should *test* which theorem from neqE    *)
webertj@20276
   614
  (*        can be applied, and split the premise accordingly.      *)
webertj@20276
   615
  fun elim_neq (ineqs : (LA_Data.decompT option * bool) list) :
webertj@20276
   616
               (LA_Data.decompT option * bool) list list =
webertj@20276
   617
  let
webertj@20276
   618
    fun elim_neq' nat_only ([] : (LA_Data.decompT option * bool) list) :
webertj@20276
   619
                  (LA_Data.decompT option * bool) list list =
webertj@20276
   620
          [[]]
webertj@20276
   621
      | elim_neq' nat_only ((NONE, is_nat) :: ineqs) =
webertj@20276
   622
          map (cons (NONE, is_nat)) (elim_neq' nat_only ineqs)
webertj@20276
   623
      | elim_neq' nat_only ((ineq as (SOME (l, i, rel, r, j, d), is_nat)) :: ineqs) =
webertj@20276
   624
          if rel = "~=" andalso (not nat_only orelse is_nat) then
webertj@20276
   625
            (* [| ?l ~= ?r; ?l < ?r ==> ?R; ?r < ?l ==> ?R |] ==> ?R *)
webertj@20276
   626
            elim_neq' nat_only (ineqs @ [(SOME (l, i, "<", r, j, d), is_nat)]) @
webertj@20276
   627
            elim_neq' nat_only (ineqs @ [(SOME (r, j, "<", l, i, d), is_nat)])
webertj@20276
   628
          else
webertj@20276
   629
            map (cons ineq) (elim_neq' nat_only ineqs)
webertj@20276
   630
  in
webertj@20276
   631
    ineqs |> elim_neq' true
webertj@20276
   632
          |> map (elim_neq' false)
webertj@20276
   633
          |> List.concat
webertj@20276
   634
  end
nipkow@13464
   635
webertj@20276
   636
  fun number_hyps _ []             = []
webertj@20276
   637
    | number_hyps n (NONE::xs)     = number_hyps (n+1) xs
webertj@20276
   638
    | number_hyps n ((SOME x)::xs) = (x, n) :: number_hyps (n+1) xs
webertj@20276
   639
webertj@20276
   640
  val result = (Ts, terms)
webertj@20276
   641
    |> (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   642
       (if do_pre then LA_Data.pre_decomp ctxt else Library.single)
webertj@23195
   643
    |> tap (fn subgoals => trace_msg ("Preprocessing yields " ^
webertj@23195
   644
         string_of_int (length subgoals) ^ " subgoal(s) total."))
wenzelm@22846
   645
    |> (* produce the internal encoding of (in-)equalities *)
wenzelm@24076
   646
       map (apsnd (map (fn t => (LA_Data.decomp ctxt t, LA_Data.domain_is_nat t))))
webertj@20276
   647
    |> (* splitting of inequalities *)
webertj@20276
   648
       map (apsnd elim_neq)
wenzelm@22846
   649
    |> maps (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals)
webertj@20276
   650
    |> (* numbering of hypotheses, ignoring irrelevant ones *)
webertj@20276
   651
       map (apsnd (number_hyps 0))
webertj@23195
   652
in
webertj@23195
   653
  trace_msg ("Splitting of inequalities yields " ^
webertj@23195
   654
    string_of_int (length result) ^ " subgoal(s) total.");
webertj@23195
   655
  result
webertj@23195
   656
end;
nipkow@13464
   657
webertj@20268
   658
fun add_atoms (ats : term list, ((lhs,_,_,rhs,_,_) : LA_Data.decompT, _)) : term list =
webertj@20217
   659
    (map fst lhs) union ((map fst rhs) union ats);
webertj@20217
   660
webertj@20268
   661
fun add_datoms (dats : (bool * term) list, ((lhs,_,_,rhs,_,d) : LA_Data.decompT, _)) :
webertj@20268
   662
  (bool * term) list =
webertj@20268
   663
  (map (pair d o fst) lhs) union ((map (pair d o fst) rhs) union dats);
nipkow@13498
   664
webertj@20268
   665
fun discr (initems : (LA_Data.decompT * int) list) : bool list =
webertj@20268
   666
  map fst (Library.foldl add_datoms ([],initems));
webertj@20217
   667
wenzelm@24076
   668
fun refutes ctxt params show_ex :
webertj@20268
   669
  (typ list * (LA_Data.decompT * int) list) list -> injust list -> injust list option =
nipkow@13498
   670
let
webertj@20268
   671
  fun refute ((Ts : typ list, initems : (LA_Data.decompT * int) list)::initemss)
webertj@20268
   672
             (js : injust list) : injust list option =
wenzelm@24076
   673
    let
wenzelm@24076
   674
      val atoms = Library.foldl add_atoms ([], initems)
wenzelm@24076
   675
      val n = length atoms
wenzelm@24076
   676
      val mkleq = mklineq n atoms
wenzelm@24076
   677
      val ixs = 0 upto (n - 1)
wenzelm@24076
   678
      val iatoms = atoms ~~ ixs
wenzelm@24076
   679
      val natlineqs = List.mapPartial (mknat Ts ixs) iatoms
wenzelm@24076
   680
      val ineqs = map mkleq initems @ natlineqs
webertj@20217
   681
    in case elim (ineqs, []) of
webertj@20217
   682
         Success j =>
wenzelm@24076
   683
           (trace_msg ("Contradiction! (" ^ string_of_int (length js + 1) ^ ")");
webertj@20268
   684
            refute initemss (js@[j]))
webertj@20217
   685
       | Failure hist =>
webertj@20217
   686
           (if not show_ex then
webertj@20217
   687
              ()
webertj@20217
   688
            else let
webertj@20276
   689
              (* invent names for bound variables that are new, i.e. in Ts,  *)
webertj@20276
   690
              (* but not in params; we assume that Ts still contains (map    *)
webertj@20276
   691
              (* snd params) as a suffix                                     *)
webertj@20217
   692
              val new_count = length Ts - length params - 1
webertj@20217
   693
              val new_names = map Name.bound (0 upto new_count)
webertj@20217
   694
              val params'   = (new_names @ map fst params) ~~ Ts
webertj@20217
   695
            in
wenzelm@24076
   696
              trace_ex ctxt params' atoms (discr initems) n hist
webertj@20217
   697
            end; NONE)
nipkow@13498
   698
    end
skalberg@15531
   699
    | refute [] js = SOME js
nipkow@13498
   700
in refute end;
nipkow@5982
   701
wenzelm@24076
   702
fun refute ctxt params show_ex do_pre terms : injust list option =
wenzelm@24076
   703
  refutes ctxt params show_ex (split_items ctxt do_pre (map snd params, terms)) [];
webertj@20254
   704
haftmann@22950
   705
fun count P xs = length (filter P xs);
webertj@20254
   706
wenzelm@24076
   707
fun prove ctxt (params : (string * Term.typ) list) show_ex do_pre Hs concl : injust list option =
webertj@20254
   708
  let
webertj@23190
   709
    val _ = trace_msg "prove:"
webertj@20254
   710
    (* append the negated conclusion to 'Hs' -- this corresponds to     *)
webertj@20254
   711
    (* 'DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i)' at the *)
webertj@20254
   712
    (* theorem/tactic level                                             *)
webertj@20254
   713
    val Hs' = Hs @ [LA_Logic.neg_prop concl]
webertj@20254
   714
    fun is_neq NONE                 = false
webertj@20254
   715
      | is_neq (SOME (_,_,r,_,_,_)) = (r = "~=")
wenzelm@24112
   716
    val neq_limit = Config.get ctxt LA_Data.fast_arith_neq_limit;
webertj@20254
   717
  in
wenzelm@24076
   718
    if count is_neq (map (LA_Data.decomp ctxt) Hs') > neq_limit then
wenzelm@24076
   719
     (trace_msg ("fast_arith_neq_limit exceeded (current value is " ^
wenzelm@24076
   720
        string_of_int neq_limit ^ ")"); NONE)
wenzelm@24076
   721
    else
wenzelm@24076
   722
      refute ctxt params show_ex do_pre Hs'
webertj@23190
   723
  end handle TERM ("neg_prop", _) =>
webertj@23190
   724
    (* since no meta-logic negation is available, we can only fail if   *)
webertj@23190
   725
    (* the conclusion is not of the form 'Trueprop $ _' (simply         *)
webertj@23190
   726
    (* dropping the conclusion doesn't work either, because even        *)
webertj@23190
   727
    (* 'False' does not imply arbitrary 'concl::prop')                  *)
webertj@23190
   728
    (trace_msg "prove failed (cannot negate conclusion)."; NONE);
webertj@20217
   729
wenzelm@22846
   730
fun refute_tac ss (i, justs) =
nipkow@6074
   731
  fn state =>
wenzelm@24076
   732
    let
wenzelm@24076
   733
      val ctxt = Simplifier.the_context ss;
wenzelm@24076
   734
      val _ = trace_thm ("refute_tac (on subgoal " ^ string_of_int i ^ ", with " ^
wenzelm@24076
   735
                             string_of_int (length justs) ^ " justification(s)):") state
wenzelm@24076
   736
      val {neqE, ...} = get_data ctxt;
wenzelm@24076
   737
      fun just1 j =
wenzelm@24076
   738
        (* eliminate inequalities *)
wenzelm@24076
   739
        REPEAT_DETERM (eresolve_tac neqE i) THEN
wenzelm@24076
   740
          PRIMITIVE (trace_thm "State after neqE:") THEN
wenzelm@24076
   741
          (* use theorems generated from the actual justifications *)
wenzelm@24076
   742
          METAHYPS (fn asms => rtac (mkthm ss asms j) 1) i
wenzelm@24076
   743
    in
wenzelm@24076
   744
      (* rewrite "[| A1; ...; An |] ==> B" to "[| A1; ...; An; ~B |] ==> False" *)
wenzelm@24076
   745
      DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i) THEN
wenzelm@24076
   746
      (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   747
      DETERM (LA_Data.pre_tac ctxt i) THEN
wenzelm@24076
   748
      PRIMITIVE (trace_thm "State after pre_tac:") THEN
wenzelm@24076
   749
      (* prove every resulting subgoal, using its justification *)
wenzelm@24076
   750
      EVERY (map just1 justs)
webertj@20217
   751
    end  state;
nipkow@6074
   752
nipkow@5982
   753
(*
nipkow@5982
   754
Fast but very incomplete decider. Only premises and conclusions
nipkow@5982
   755
that are already (negated) (in)equations are taken into account.
nipkow@5982
   756
*)
wenzelm@24076
   757
fun simpset_lin_arith_tac ss show_ex = SUBGOAL (fn (A, i) =>
wenzelm@24076
   758
  let
wenzelm@24076
   759
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   760
    val params = rev (Logic.strip_params A)
wenzelm@24076
   761
    val Hs = Logic.strip_assums_hyp A
wenzelm@24076
   762
    val concl = Logic.strip_assums_concl A
wenzelm@24076
   763
    val _ = trace_term ctxt ("Trying to refute subgoal " ^ string_of_int i) A
wenzelm@24076
   764
  in
wenzelm@24076
   765
    case prove ctxt params show_ex true Hs concl of
wenzelm@24076
   766
      NONE => (trace_msg "Refutation failed."; no_tac)
wenzelm@24076
   767
    | SOME js => (trace_msg "Refutation succeeded."; refute_tac ss (i, js))
wenzelm@24076
   768
  end);
nipkow@5982
   769
wenzelm@24076
   770
fun cut_lin_arith_tac ss =
wenzelm@24076
   771
  cut_facts_tac (Simplifier.prems_of_ss ss) THEN'
wenzelm@24076
   772
  simpset_lin_arith_tac ss false;
wenzelm@17613
   773
wenzelm@24076
   774
fun lin_arith_tac ctxt =
wenzelm@24076
   775
  simpset_lin_arith_tac (Simplifier.context ctxt Simplifier.empty_ss);
wenzelm@24076
   776
wenzelm@24076
   777
nipkow@5982
   778
nipkow@13186
   779
(** Forward proof from theorems **)
nipkow@13186
   780
webertj@20433
   781
(* More tricky code. Needs to arrange the proofs of the multiple cases (due
webertj@20433
   782
to splits of ~= premises) such that it coincides with the order of the cases
webertj@20433
   783
generated by function split_items. *)
webertj@20433
   784
webertj@20433
   785
datatype splittree = Tip of thm list
webertj@20433
   786
                   | Spl of thm * cterm * splittree * cterm * splittree;
webertj@20433
   787
webertj@20433
   788
(* "(ct1 ==> ?R) ==> (ct2 ==> ?R) ==> ?R" is taken to (ct1, ct2) *)
webertj@20433
   789
webertj@20433
   790
fun extract (imp : cterm) : cterm * cterm =
webertj@20433
   791
let val (Il, r)    = Thm.dest_comb imp
webertj@20433
   792
    val (_, imp1)  = Thm.dest_comb Il
webertj@20433
   793
    val (Ict1, _)  = Thm.dest_comb imp1
webertj@20433
   794
    val (_, ct1)   = Thm.dest_comb Ict1
webertj@20433
   795
    val (Ir, _)    = Thm.dest_comb r
webertj@20433
   796
    val (_, Ict2r) = Thm.dest_comb Ir
webertj@20433
   797
    val (Ict2, _)  = Thm.dest_comb Ict2r
webertj@20433
   798
    val (_, ct2)   = Thm.dest_comb Ict2
webertj@20433
   799
in (ct1, ct2) end;
webertj@20433
   800
wenzelm@24076
   801
fun splitasms ctxt (asms : thm list) : splittree =
wenzelm@24076
   802
let val {neqE, ...} = get_data ctxt
webertj@20433
   803
    fun elim_neq (asms', []) = Tip (rev asms')
webertj@20433
   804
      | elim_neq (asms', asm::asms) =
webertj@20433
   805
      (case get_first (fn th => SOME (asm COMP th) handle THM _ => NONE) neqE of
webertj@20433
   806
        SOME spl =>
webertj@20433
   807
          let val (ct1, ct2) = extract (cprop_of spl)
webertj@20433
   808
              val thm1 = assume ct1
webertj@20433
   809
              val thm2 = assume ct2
webertj@20433
   810
          in Spl (spl, ct1, elim_neq (asms', asms@[thm1]), ct2, elim_neq (asms', asms@[thm2]))
webertj@20433
   811
          end
webertj@20433
   812
      | NONE => elim_neq (asm::asms', asms))
webertj@20433
   813
in elim_neq ([], asms) end;
webertj@20433
   814
wenzelm@24076
   815
fun fwdproof ss (Tip asms : splittree) (j::js : injust list) = (mkthm ss asms j, js)
wenzelm@24076
   816
  | fwdproof ss (Spl (thm, ct1, tree1, ct2, tree2)) js =
wenzelm@24076
   817
      let
wenzelm@24076
   818
        val (thm1, js1) = fwdproof ss tree1 js
wenzelm@24076
   819
        val (thm2, js2) = fwdproof ss tree2 js1
webertj@20433
   820
        val thm1' = implies_intr ct1 thm1
webertj@20433
   821
        val thm2' = implies_intr ct2 thm2
wenzelm@24076
   822
      in (thm2' COMP (thm1' COMP thm), js2) end;
wenzelm@24076
   823
      (* FIXME needs handle THM _ => NONE ? *)
webertj@20433
   824
wenzelm@24076
   825
fun prover ss thms Tconcl (js : injust list) pos : thm option =
wenzelm@24076
   826
  let
wenzelm@24076
   827
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   828
    val thy = ProofContext.theory_of ctxt
wenzelm@24076
   829
    val nTconcl = LA_Logic.neg_prop Tconcl
wenzelm@24076
   830
    val cnTconcl = cterm_of thy nTconcl
wenzelm@24076
   831
    val nTconclthm = assume cnTconcl
wenzelm@24076
   832
    val tree = splitasms ctxt (thms @ [nTconclthm])
wenzelm@24076
   833
    val (Falsethm, _) = fwdproof ss tree js
wenzelm@24076
   834
    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
wenzelm@24076
   835
    val concl = implies_intr cnTconcl Falsethm COMP contr
wenzelm@24076
   836
  in SOME (trace_thm "Proved by lin. arith. prover:" (LA_Logic.mk_Eq concl)) end
wenzelm@24076
   837
  (*in case concl contains ?-var, which makes assume fail:*)   (* FIXME Variable.import_terms *)
wenzelm@24076
   838
  handle THM _ => NONE;
nipkow@13186
   839
nipkow@13186
   840
(* PRE: concl is not negated!
nipkow@13186
   841
   This assumption is OK because
wenzelm@24076
   842
   1. lin_arith_simproc tries both to prove and disprove concl and
wenzelm@24076
   843
   2. lin_arith_simproc is applied by the Simplifier which
nipkow@13186
   844
      dives into terms and will thus try the non-negated concl anyway.
nipkow@13186
   845
*)
wenzelm@24076
   846
fun lin_arith_simproc ss concl =
wenzelm@24076
   847
  let
wenzelm@24076
   848
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   849
    val thms = List.concat (map LA_Logic.atomize (Simplifier.prems_of_ss ss))
wenzelm@24076
   850
    val Hs = map Thm.prop_of thms
nipkow@6102
   851
    val Tconcl = LA_Logic.mk_Trueprop concl
wenzelm@24076
   852
  in
wenzelm@24076
   853
    case prove ctxt [] false false Hs Tconcl of (* concl provable? *)
wenzelm@24076
   854
      SOME js => prover ss thms Tconcl js true
wenzelm@24076
   855
    | NONE =>
wenzelm@24076
   856
        let val nTconcl = LA_Logic.neg_prop Tconcl in
wenzelm@24076
   857
          case prove ctxt [] false false Hs nTconcl of (* ~concl provable? *)
wenzelm@24076
   858
            SOME js => prover ss thms nTconcl js false
wenzelm@24076
   859
          | NONE => NONE
wenzelm@24076
   860
        end
wenzelm@24076
   861
  end;
nipkow@6074
   862
nipkow@6074
   863
end;