src/Provers/Arith/fast_lin_arith.ML
author wenzelm
Sun May 18 15:04:31 2008 +0200 (2008-05-18)
changeset 26945 9cd13e810998
parent 26835 404550067389
child 27020 b5b8afc9fdcd
permissions -rw-r--r--
renamed type decompT to decomp;
refute: proper context for trace_ex;
some attempts to improve readability;
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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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(*** 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 decomp = (term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool
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  val decomp: Proof.context -> term -> decomp 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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             else if ty2 = Eq then (m1,~m2)
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                  else sys_error "elim_var"
nipkow@5982
   319
        else (m1,m2)
nipkow@5982
   320
      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
nipkow@5982
   321
                    then (~n1,~n2) else (n1,n2)
nipkow@5982
   322
  in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end;
nipkow@5982
   323
nipkow@5982
   324
(* ------------------------------------------------------------------------- *)
nipkow@5982
   325
(* The main refutation-finding code.                                         *)
nipkow@5982
   326
(* ------------------------------------------------------------------------- *)
nipkow@5982
   327
nipkow@5982
   328
fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
nipkow@5982
   329
nipkow@5982
   330
fun is_answer (ans as Lineq(k,ty,l,_)) =
nipkow@5982
   331
  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
nipkow@5982
   332
wenzelm@24630
   333
fun calc_blowup l =
haftmann@17496
   334
  let val (p,n) = List.partition (curry (op <) 0) (List.filter (curry (op <>) 0) l)
wenzelm@24630
   335
  in length p * length n end;
nipkow@5982
   336
nipkow@5982
   337
(* ------------------------------------------------------------------------- *)
nipkow@5982
   338
(* Main elimination code:                                                    *)
nipkow@5982
   339
(*                                                                           *)
nipkow@5982
   340
(* (1) Looks for immediate solutions (false assertions with no variables).   *)
nipkow@5982
   341
(*                                                                           *)
nipkow@5982
   342
(* (2) If there are any equations, picks a variable with the lowest absolute *)
nipkow@5982
   343
(* coefficient in any of them, and uses it to eliminate.                     *)
nipkow@5982
   344
(*                                                                           *)
nipkow@5982
   345
(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
nipkow@5982
   346
(* blowup (number of consequences generated) and eliminates it.              *)
nipkow@5982
   347
(* ------------------------------------------------------------------------- *)
nipkow@5982
   348
nipkow@5982
   349
fun allpairs f xs ys =
wenzelm@26945
   350
  maps (fn x => map (fn y => f x y) ys) xs;
nipkow@5982
   351
nipkow@5982
   352
fun extract_first p =
skalberg@15531
   353
  let fun extract xs (y::ys) = if p y then (SOME y,xs@ys)
nipkow@5982
   354
                               else extract (y::xs) ys
skalberg@15531
   355
        | extract xs []      = (NONE,xs)
nipkow@5982
   356
  in extract [] end;
nipkow@5982
   357
nipkow@6056
   358
fun print_ineqs ineqs =
paulson@9073
   359
  if !trace then
wenzelm@12262
   360
     tracing(cat_lines(""::map (fn Lineq(c,t,l,_) =>
wenzelm@24630
   361
       string_of_int c ^
paulson@9073
   362
       (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
wenzelm@24630
   363
       commas(map string_of_int l)) ineqs))
paulson@9073
   364
  else ();
nipkow@6056
   365
nipkow@13498
   366
type history = (int * lineq list) list;
nipkow@13498
   367
datatype result = Success of injust | Failure of history;
nipkow@13498
   368
webertj@20217
   369
fun elim (ineqs, hist) =
webertj@20217
   370
  let val dummy = print_ineqs ineqs
webertj@20217
   371
      val (triv, nontriv) = List.partition is_trivial ineqs in
webertj@20217
   372
  if not (null triv)
nipkow@13186
   373
  then case Library.find_first is_answer triv of
webertj@20217
   374
         NONE => elim (nontriv, hist)
skalberg@15531
   375
       | SOME(Lineq(_,_,_,j)) => Success j
nipkow@5982
   376
  else
webertj@20217
   377
  if null nontriv then Failure hist
nipkow@13498
   378
  else
webertj@20217
   379
  let val (eqs, noneqs) = List.partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
webertj@20217
   380
  if not (null eqs) then
skalberg@15570
   381
     let val clist = Library.foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs)
wenzelm@24630
   382
         val sclist = sort (fn (x,y) => int_ord (abs x, abs y))
skalberg@15570
   383
                           (List.filter (fn i => i<>0) clist)
nipkow@5982
   384
         val c = hd sclist
skalberg@15531
   385
         val (SOME(eq as Lineq(_,_,ceq,_)),othereqs) =
nipkow@5982
   386
               extract_first (fn Lineq(_,_,l,_) => c mem l) eqs
webertj@20217
   387
         val v = find_index_eq c ceq
haftmann@23063
   388
         val (ioth,roth) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0)
nipkow@5982
   389
                                     (othereqs @ noneqs)
nipkow@5982
   390
         val others = map (elim_var v eq) roth @ ioth
nipkow@13498
   391
     in elim(others,(v,nontriv)::hist) end
nipkow@5982
   392
  else
nipkow@5982
   393
  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
haftmann@23063
   394
      val numlist = 0 upto (length (hd lists) - 1)
haftmann@23063
   395
      val coeffs = map (fn i => map (fn xs => nth xs i) lists) numlist
nipkow@5982
   396
      val blows = map calc_blowup coeffs
nipkow@5982
   397
      val iblows = blows ~~ numlist
haftmann@23063
   398
      val nziblows = filter_out (fn (i, _) => i = 0) iblows
nipkow@13498
   399
  in if null nziblows then Failure((~1,nontriv)::hist)
nipkow@13498
   400
     else
nipkow@5982
   401
     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
haftmann@23063
   402
         val (no,yes) = List.partition (fn (Lineq(_,_,l,_)) => nth l v = 0) ineqs
haftmann@23063
   403
         val (pos,neg) = List.partition(fn (Lineq(_,_,l,_)) => nth l v > 0) yes
nipkow@13498
   404
     in elim(no @ allpairs (elim_var v) pos neg, (v,nontriv)::hist) end
nipkow@5982
   405
  end
nipkow@5982
   406
  end
nipkow@5982
   407
  end;
nipkow@5982
   408
nipkow@5982
   409
(* ------------------------------------------------------------------------- *)
nipkow@5982
   410
(* Translate back a proof.                                                   *)
nipkow@5982
   411
(* ------------------------------------------------------------------------- *)
nipkow@5982
   412
wenzelm@24076
   413
fun trace_thm msg th =
wenzelm@24076
   414
  (if !trace then (tracing msg; tracing (Display.string_of_thm th)) else (); th);
paulson@9073
   415
wenzelm@24076
   416
fun trace_term ctxt msg t =
wenzelm@24920
   417
  (if !trace then tracing (cat_lines [msg, Syntax.string_of_term ctxt t]) else (); t)
wenzelm@24076
   418
wenzelm@24076
   419
fun trace_msg msg =
wenzelm@24076
   420
  if !trace then tracing msg else ();
paulson@9073
   421
berghofe@26835
   422
val union_term = curry (gen_union Pattern.aeconv);
berghofe@26835
   423
val union_bterm = curry (gen_union
berghofe@26835
   424
  (fn ((b:bool, t), (b', t')) => b = b' andalso Pattern.aeconv (t, t')));
berghofe@26835
   425
nipkow@13498
   426
(* FIXME OPTIMIZE!!!! (partly done already)
nipkow@6056
   427
   Addition/Multiplication need i*t representation rather than t+t+...
nipkow@10691
   428
   Get rid of Mulitplied(2). For Nat LA_Data.number_of should return Suc^n
nipkow@10691
   429
   because Numerals are not known early enough.
nipkow@6056
   430
nipkow@6056
   431
Simplification may detect a contradiction 'prematurely' due to type
nipkow@6056
   432
information: n+1 <= 0 is simplified to False and does not need to be crossed
nipkow@6056
   433
with 0 <= n.
nipkow@6056
   434
*)
nipkow@6056
   435
local
wenzelm@24076
   436
  exception FalseE of thm
nipkow@6056
   437
in
wenzelm@24076
   438
fun mkthm ss asms (just: injust) =
wenzelm@24076
   439
  let
wenzelm@24076
   440
    val ctxt = Simplifier.the_context ss;
wenzelm@24076
   441
    val thy = ProofContext.theory_of ctxt;
wenzelm@24076
   442
    val {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset, ...} = get_data ctxt;
wenzelm@24076
   443
    val simpset' = Simplifier.inherit_context ss simpset;
wenzelm@24076
   444
    val atoms = Library.foldl (fn (ats, (lhs,_,_,rhs,_,_)) =>
berghofe@26835
   445
                            union_term (map fst lhs) (union_term (map fst rhs) ats))
webertj@20217
   446
                        ([], List.mapPartial (fn thm => if Thm.no_prems thm
wenzelm@24076
   447
                                              then LA_Data.decomp ctxt (Thm.concl_of thm)
webertj@20217
   448
                                              else NONE) asms)
nipkow@6056
   449
nipkow@10575
   450
      fun add2 thm1 thm2 =
nipkow@6102
   451
        let val conj = thm1 RS (thm2 RS LA_Logic.conjI)
skalberg@15531
   452
        in get_first (fn th => SOME(conj RS th) handle THM _ => NONE) add_mono_thms
nipkow@5982
   453
        end;
skalberg@15531
   454
      fun try_add [] _ = NONE
nipkow@10575
   455
        | try_add (thm1::thm1s) thm2 = case add2 thm1 thm2 of
skalberg@15531
   456
             NONE => try_add thm1s thm2 | some => some;
nipkow@10575
   457
nipkow@10575
   458
      fun addthms thm1 thm2 =
nipkow@10575
   459
        case add2 thm1 thm2 of
skalberg@15531
   460
          NONE => (case try_add ([thm1] RL inj_thms) thm2 of
webertj@20217
   461
                     NONE => ( the (try_add ([thm2] RL inj_thms) thm1)
wenzelm@15660
   462
                               handle Option =>
nipkow@14360
   463
                               (trace_thm "" thm1; trace_thm "" thm2;
webertj@20217
   464
                                sys_error "Lin.arith. failed to add thms")
webertj@20217
   465
                             )
skalberg@15531
   466
                   | SOME thm => thm)
skalberg@15531
   467
        | SOME thm => thm;
nipkow@10575
   468
nipkow@5982
   469
      fun multn(n,thm) =
nipkow@5982
   470
        let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th)
nipkow@6102
   471
        in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end;
webertj@20217
   472
nipkow@15184
   473
      fun multn2(n,thm) =
skalberg@15531
   474
        let val SOME(mth) =
skalberg@15531
   475
              get_first (fn th => SOME(thm RS th) handle THM _ => NONE) mult_mono_thms
wenzelm@22596
   476
            fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (Thm.theory_of_thm th) var;
nipkow@15184
   477
            val cv = cvar(mth, hd(prems_of mth));
wenzelm@24076
   478
            val ct = cterm_of thy (LA_Data.number_of(n,#T(rep_cterm cv)))
nipkow@15184
   479
        in instantiate ([],[(cv,ct)]) mth end
nipkow@10691
   480
nipkow@6056
   481
      fun simp thm =
wenzelm@17515
   482
        let val thm' = trace_thm "Simplified:" (full_simplify simpset' thm)
nipkow@6102
   483
        in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end
nipkow@6056
   484
wenzelm@24076
   485
      fun mk (Asm i) = trace_thm ("Asm " ^ string_of_int i) (nth asms i)
wenzelm@24076
   486
        | mk (Nat i) = trace_thm ("Nat " ^ string_of_int i) (LA_Logic.mk_nat_thm thy (nth atoms i))
webertj@20254
   487
        | mk (LessD j)            = trace_thm "L" (hd ([mk j] RL lessD))
webertj@20254
   488
        | mk (NotLeD j)           = trace_thm "NLe" (mk j RS LA_Logic.not_leD)
webertj@20254
   489
        | mk (NotLeDD j)          = trace_thm "NLeD" (hd ([mk j RS LA_Logic.not_leD] RL lessD))
webertj@20254
   490
        | mk (NotLessD j)         = trace_thm "NL" (mk j RS LA_Logic.not_lessD)
webertj@20254
   491
        | mk (Added (j1, j2))     = simp (trace_thm "+" (addthms (mk j1) (mk j2)))
wenzelm@24630
   492
        | mk (Multiplied (n, j))  = (trace_msg ("*" ^ string_of_int n); trace_thm "*" (multn (n, mk j)))
wenzelm@24630
   493
        | mk (Multiplied2 (n, j)) = simp (trace_msg ("**" ^ string_of_int n); trace_thm "**" (multn2 (n, mk j)))
nipkow@5982
   494
paulson@9073
   495
  in trace_msg "mkthm";
nipkow@12932
   496
     let val thm = trace_thm "Final thm:" (mk just)
wenzelm@17515
   497
     in let val fls = simplify simpset' thm
nipkow@13186
   498
        in trace_thm "After simplification:" fls;
nipkow@13186
   499
           if LA_Logic.is_False fls then fls
nipkow@13186
   500
           else
webertj@20217
   501
            (tracing "Assumptions:"; List.app (tracing o Display.string_of_thm) asms;
webertj@20217
   502
             tracing "Proved:"; tracing (Display.string_of_thm fls);
nipkow@13186
   503
             warning "Linear arithmetic should have refuted the assumptions.\n\
nipkow@13186
   504
                     \Please inform Tobias Nipkow (nipkow@in.tum.de).";
nipkow@13186
   505
             fls)
nipkow@12932
   506
        end
webertj@20217
   507
     end handle FalseE thm => trace_thm "False reached early:" thm
nipkow@12932
   508
  end
nipkow@6056
   509
end;
nipkow@5982
   510
haftmann@23261
   511
fun coeff poly atom =
berghofe@26835
   512
  AList.lookup Pattern.aeconv poly atom |> the_default 0;
nipkow@10691
   513
nipkow@10691
   514
fun integ(rlhs,r,rel,rrhs,s,d) =
haftmann@17951
   515
let val (rn,rd) = Rat.quotient_of_rat r and (sn,sd) = Rat.quotient_of_rat s
wenzelm@24630
   516
    val m = Integer.lcms(map (abs o snd o Rat.quotient_of_rat) (r :: s :: map snd rlhs @ map snd rrhs))
wenzelm@22846
   517
    fun mult(t,r) =
haftmann@17951
   518
        let val (i,j) = Rat.quotient_of_rat r
paulson@15965
   519
        in (t,i * (m div j)) end
nipkow@12932
   520
in (m,(map mult rlhs, rn*(m div rd), rel, map mult rrhs, sn*(m div sd), d)) end
nipkow@10691
   521
nipkow@13498
   522
fun mklineq n atoms =
webertj@20217
   523
  fn (item, k) =>
webertj@20217
   524
  let val (m, (lhs,i,rel,rhs,j,discrete)) = integ item
nipkow@13498
   525
      val lhsa = map (coeff lhs) atoms
nipkow@13498
   526
      and rhsa = map (coeff rhs) atoms
haftmann@18330
   527
      val diff = map2 (curry (op -)) rhsa lhsa
nipkow@13498
   528
      val c = i-j
nipkow@13498
   529
      val just = Asm k
nipkow@13498
   530
      fun lineq(c,le,cs,j) = Lineq(c,le,cs, if m=1 then j else Multiplied2(m,j))
nipkow@13498
   531
  in case rel of
nipkow@13498
   532
      "<="   => lineq(c,Le,diff,just)
nipkow@13498
   533
     | "~<=" => if discrete
nipkow@13498
   534
                then lineq(1-c,Le,map (op ~) diff,NotLeDD(just))
nipkow@13498
   535
                else lineq(~c,Lt,map (op ~) diff,NotLeD(just))
nipkow@13498
   536
     | "<"   => if discrete
nipkow@13498
   537
                then lineq(c+1,Le,diff,LessD(just))
nipkow@13498
   538
                else lineq(c,Lt,diff,just)
nipkow@13498
   539
     | "~<"  => lineq(~c,Le,map (op~) diff,NotLessD(just))
nipkow@13498
   540
     | "="   => lineq(c,Eq,diff,just)
wenzelm@22846
   541
     | _     => sys_error("mklineq" ^ rel)
nipkow@5982
   542
  end;
nipkow@5982
   543
nipkow@13498
   544
(* ------------------------------------------------------------------------- *)
nipkow@13498
   545
(* Print (counter) example                                                   *)
nipkow@13498
   546
(* ------------------------------------------------------------------------- *)
nipkow@13498
   547
nipkow@13498
   548
fun print_atom((a,d),r) =
haftmann@17951
   549
  let val (p,q) = Rat.quotient_of_rat r
wenzelm@24630
   550
      val s = if d then string_of_int p else
nipkow@13498
   551
              if p = 0 then "0"
wenzelm@24630
   552
              else string_of_int p ^ "/" ^ string_of_int q
nipkow@13498
   553
  in a ^ " = " ^ s end;
nipkow@13498
   554
wenzelm@19049
   555
fun produce_ex sds =
haftmann@17496
   556
  curry (op ~~) sds
haftmann@17496
   557
  #> map print_atom
haftmann@17496
   558
  #> commas
webertj@23197
   559
  #> curry (op ^) "Counterexample (possibly spurious):\n";
nipkow@13498
   560
wenzelm@24076
   561
fun trace_ex ctxt params atoms discr n (hist: history) =
webertj@20217
   562
  case hist of
webertj@20217
   563
    [] => ()
webertj@20217
   564
  | (v, lineqs) :: hist' =>
wenzelm@24076
   565
      let
wenzelm@24076
   566
        val frees = map Free params
wenzelm@24920
   567
        fun show_term t = Syntax.string_of_term ctxt (subst_bounds (frees, t))
wenzelm@24076
   568
        val start =
wenzelm@24076
   569
          if v = ~1 then (hist', findex0 discr n lineqs)
haftmann@22950
   570
          else (hist, replicate n Rat.zero)
wenzelm@24076
   571
        val ex = SOME (produce_ex (map show_term atoms ~~ discr)
wenzelm@24076
   572
            (uncurry (fold (findex1 discr)) start))
webertj@20217
   573
          handle NoEx => NONE
wenzelm@24076
   574
      in
wenzelm@24076
   575
        case ex of
wenzelm@24076
   576
          SOME s => (warning "arith failed - see trace for a counterexample"; tracing s)
wenzelm@24076
   577
        | NONE => warning "arith failed"
wenzelm@24076
   578
      end;
nipkow@13498
   579
webertj@20217
   580
(* ------------------------------------------------------------------------- *)
webertj@20217
   581
webertj@20268
   582
fun mknat (pTs : typ list) (ixs : int list) (atom : term, i : int) : lineq option =
webertj@20217
   583
  if LA_Logic.is_nat (pTs, atom)
nipkow@6056
   584
  then let val l = map (fn j => if j=i then 1 else 0) ixs
webertj@20217
   585
       in SOME (Lineq (0, Le, l, Nat i)) end
webertj@20217
   586
  else NONE;
nipkow@6056
   587
nipkow@13186
   588
(* This code is tricky. It takes a list of premises in the order they occur
skalberg@15531
   589
in the subgoal. Numerical premises are coded as SOME(tuple), non-numerical
skalberg@15531
   590
ones as NONE. Going through the premises, each numeric one is converted into
nipkow@13186
   591
a Lineq. The tricky bit is to convert ~= which is split into two cases < and
nipkow@13498
   592
>. Thus split_items returns a list of equation systems. This may blow up if
nipkow@13186
   593
there are many ~=, but in practice it does not seem to happen. The really
nipkow@13186
   594
tricky bit is to arrange the order of the cases such that they coincide with
nipkow@13186
   595
the order in which the cases are in the end generated by the tactic that
nipkow@13186
   596
applies the generated refutation thms (see function 'refute_tac').
nipkow@13186
   597
nipkow@13186
   598
For variables n of type nat, a constraint 0 <= n is added.
nipkow@13186
   599
*)
webertj@20217
   600
webertj@20217
   601
(* FIXME: To optimize, the splitting of cases and the search for refutations *)
webertj@20276
   602
(*        could be intertwined: separate the first (fully split) case,       *)
webertj@20217
   603
(*        refute it, continue with splitting and refuting.  Terminate with   *)
webertj@20217
   604
(*        failure as soon as a case could not be refuted; i.e. delay further *)
webertj@20217
   605
(*        splitting until after a refutation for other cases has been found. *)
webertj@20217
   606
wenzelm@26945
   607
fun split_items ctxt do_pre (Ts, terms) : (typ list * (LA_Data.decomp * int) list) list =
webertj@20276
   608
let
webertj@20276
   609
  (* splits inequalities '~=' into '<' and '>'; this corresponds to *)
webertj@20276
   610
  (* 'REPEAT_DETERM (eresolve_tac neqE i)' at the theorem/tactic    *)
webertj@20276
   611
  (* level                                                          *)
webertj@20276
   612
  (* FIXME: this is currently sensitive to the order of theorems in *)
webertj@20276
   613
  (*        neqE:  The theorem for type "nat" must come first.  A   *)
webertj@20276
   614
  (*        better (i.e. less likely to break when neqE changes)    *)
webertj@20276
   615
  (*        implementation should *test* which theorem from neqE    *)
webertj@20276
   616
  (*        can be applied, and split the premise accordingly.      *)
wenzelm@26945
   617
  fun elim_neq (ineqs : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   618
               (LA_Data.decomp option * bool) list list =
webertj@20276
   619
  let
wenzelm@26945
   620
    fun elim_neq' nat_only ([] : (LA_Data.decomp option * bool) list) :
wenzelm@26945
   621
                  (LA_Data.decomp option * bool) list list =
webertj@20276
   622
          [[]]
webertj@20276
   623
      | elim_neq' nat_only ((NONE, is_nat) :: ineqs) =
webertj@20276
   624
          map (cons (NONE, is_nat)) (elim_neq' nat_only ineqs)
webertj@20276
   625
      | elim_neq' nat_only ((ineq as (SOME (l, i, rel, r, j, d), is_nat)) :: ineqs) =
webertj@20276
   626
          if rel = "~=" andalso (not nat_only orelse is_nat) then
webertj@20276
   627
            (* [| ?l ~= ?r; ?l < ?r ==> ?R; ?r < ?l ==> ?R |] ==> ?R *)
webertj@20276
   628
            elim_neq' nat_only (ineqs @ [(SOME (l, i, "<", r, j, d), is_nat)]) @
webertj@20276
   629
            elim_neq' nat_only (ineqs @ [(SOME (r, j, "<", l, i, d), is_nat)])
webertj@20276
   630
          else
webertj@20276
   631
            map (cons ineq) (elim_neq' nat_only ineqs)
webertj@20276
   632
  in
webertj@20276
   633
    ineqs |> elim_neq' true
wenzelm@26945
   634
          |> maps (elim_neq' false)
webertj@20276
   635
  end
nipkow@13464
   636
webertj@20276
   637
  fun number_hyps _ []             = []
webertj@20276
   638
    | number_hyps n (NONE::xs)     = number_hyps (n+1) xs
webertj@20276
   639
    | number_hyps n ((SOME x)::xs) = (x, n) :: number_hyps (n+1) xs
webertj@20276
   640
webertj@20276
   641
  val result = (Ts, terms)
webertj@20276
   642
    |> (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   643
       (if do_pre then LA_Data.pre_decomp ctxt else Library.single)
webertj@23195
   644
    |> tap (fn subgoals => trace_msg ("Preprocessing yields " ^
webertj@23195
   645
         string_of_int (length subgoals) ^ " subgoal(s) total."))
wenzelm@22846
   646
    |> (* produce the internal encoding of (in-)equalities *)
wenzelm@24076
   647
       map (apsnd (map (fn t => (LA_Data.decomp ctxt t, LA_Data.domain_is_nat t))))
webertj@20276
   648
    |> (* splitting of inequalities *)
webertj@20276
   649
       map (apsnd elim_neq)
wenzelm@22846
   650
    |> maps (fn (Ts, subgoals) => map (pair Ts o map fst) subgoals)
webertj@20276
   651
    |> (* numbering of hypotheses, ignoring irrelevant ones *)
webertj@20276
   652
       map (apsnd (number_hyps 0))
webertj@23195
   653
in
webertj@23195
   654
  trace_msg ("Splitting of inequalities yields " ^
webertj@23195
   655
    string_of_int (length result) ^ " subgoal(s) total.");
webertj@23195
   656
  result
webertj@23195
   657
end;
nipkow@13464
   658
wenzelm@26945
   659
fun add_atoms (ats : term list, ((lhs,_,_,rhs,_,_) : LA_Data.decomp, _)) : term list =
berghofe@26835
   660
    union_term (map fst lhs) (union_term (map fst rhs) ats);
webertj@20217
   661
wenzelm@26945
   662
fun add_datoms (dats : (bool * term) list, ((lhs,_,_,rhs,_,d) : LA_Data.decomp, _)) :
webertj@20268
   663
  (bool * term) list =
berghofe@26835
   664
  union_bterm (map (pair d o fst) lhs) (union_bterm (map (pair d o fst) rhs) dats);
nipkow@13498
   665
wenzelm@26945
   666
fun discr (initems : (LA_Data.decomp * int) list) : bool list =
webertj@20268
   667
  map fst (Library.foldl add_datoms ([],initems));
webertj@20217
   668
wenzelm@24076
   669
fun refutes ctxt params show_ex :
wenzelm@26945
   670
    (typ list * (LA_Data.decomp * int) list) list -> injust list -> injust list option =
wenzelm@26945
   671
  let
wenzelm@26945
   672
    fun refute ((Ts, initems : (LA_Data.decomp * int) list) :: initemss) (js: injust list) =
wenzelm@26945
   673
          let
wenzelm@26945
   674
            val atoms = Library.foldl add_atoms ([], initems)
wenzelm@26945
   675
            val n = length atoms
wenzelm@26945
   676
            val mkleq = mklineq n atoms
wenzelm@26945
   677
            val ixs = 0 upto (n - 1)
wenzelm@26945
   678
            val iatoms = atoms ~~ ixs
wenzelm@26945
   679
            val natlineqs = List.mapPartial (mknat Ts ixs) iatoms
wenzelm@26945
   680
            val ineqs = map mkleq initems @ natlineqs
wenzelm@26945
   681
          in case elim (ineqs, []) of
wenzelm@26945
   682
               Success j =>
wenzelm@26945
   683
                 (trace_msg ("Contradiction! (" ^ string_of_int (length js + 1) ^ ")");
wenzelm@26945
   684
                  refute initemss (js @ [j]))
wenzelm@26945
   685
             | Failure hist =>
wenzelm@26945
   686
                 (if not show_ex then ()
wenzelm@26945
   687
                  else
wenzelm@26945
   688
                    let
wenzelm@26945
   689
                      val (param_names, ctxt') = ctxt |> Variable.variant_fixes (map fst params)
wenzelm@26945
   690
                      val (more_names, ctxt'') = ctxt' |> Variable.variant_fixes
wenzelm@26945
   691
                        (Name.invents (Variable.names_of ctxt') Name.uu (length Ts - length params))
wenzelm@26945
   692
                      val params' = (more_names @ param_names) ~~ Ts
wenzelm@26945
   693
                    in
wenzelm@26945
   694
                      trace_ex ctxt'' params' atoms (discr initems) n hist
wenzelm@26945
   695
                    end; NONE)
wenzelm@26945
   696
          end
wenzelm@26945
   697
      | refute [] js = SOME js
wenzelm@26945
   698
  in refute end;
nipkow@5982
   699
wenzelm@24076
   700
fun refute ctxt params show_ex do_pre terms : injust list option =
wenzelm@24076
   701
  refutes ctxt params show_ex (split_items ctxt do_pre (map snd params, terms)) [];
webertj@20254
   702
haftmann@22950
   703
fun count P xs = length (filter P xs);
webertj@20254
   704
wenzelm@26945
   705
fun prove ctxt params show_ex do_pre Hs concl : injust list option =
webertj@20254
   706
  let
webertj@23190
   707
    val _ = trace_msg "prove:"
webertj@20254
   708
    (* append the negated conclusion to 'Hs' -- this corresponds to     *)
webertj@20254
   709
    (* 'DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i)' at the *)
webertj@20254
   710
    (* theorem/tactic level                                             *)
webertj@20254
   711
    val Hs' = Hs @ [LA_Logic.neg_prop concl]
webertj@20254
   712
    fun is_neq NONE                 = false
webertj@20254
   713
      | is_neq (SOME (_,_,r,_,_,_)) = (r = "~=")
wenzelm@24112
   714
    val neq_limit = Config.get ctxt LA_Data.fast_arith_neq_limit;
webertj@20254
   715
  in
wenzelm@24076
   716
    if count is_neq (map (LA_Data.decomp ctxt) Hs') > neq_limit then
wenzelm@24076
   717
     (trace_msg ("fast_arith_neq_limit exceeded (current value is " ^
wenzelm@24076
   718
        string_of_int neq_limit ^ ")"); NONE)
wenzelm@24076
   719
    else
wenzelm@24076
   720
      refute ctxt params show_ex do_pre Hs'
webertj@23190
   721
  end handle TERM ("neg_prop", _) =>
webertj@23190
   722
    (* since no meta-logic negation is available, we can only fail if   *)
webertj@23190
   723
    (* the conclusion is not of the form 'Trueprop $ _' (simply         *)
webertj@23190
   724
    (* dropping the conclusion doesn't work either, because even        *)
webertj@23190
   725
    (* 'False' does not imply arbitrary 'concl::prop')                  *)
webertj@23190
   726
    (trace_msg "prove failed (cannot negate conclusion)."; NONE);
webertj@20217
   727
wenzelm@22846
   728
fun refute_tac ss (i, justs) =
nipkow@6074
   729
  fn state =>
wenzelm@24076
   730
    let
wenzelm@24076
   731
      val ctxt = Simplifier.the_context ss;
wenzelm@24076
   732
      val _ = trace_thm ("refute_tac (on subgoal " ^ string_of_int i ^ ", with " ^
wenzelm@24076
   733
                             string_of_int (length justs) ^ " justification(s)):") state
wenzelm@24076
   734
      val {neqE, ...} = get_data ctxt;
wenzelm@24076
   735
      fun just1 j =
wenzelm@24076
   736
        (* eliminate inequalities *)
wenzelm@24076
   737
        REPEAT_DETERM (eresolve_tac neqE i) THEN
wenzelm@24076
   738
          PRIMITIVE (trace_thm "State after neqE:") THEN
wenzelm@24076
   739
          (* use theorems generated from the actual justifications *)
wenzelm@24076
   740
          METAHYPS (fn asms => rtac (mkthm ss asms j) 1) i
wenzelm@24076
   741
    in
wenzelm@24076
   742
      (* rewrite "[| A1; ...; An |] ==> B" to "[| A1; ...; An; ~B |] ==> False" *)
wenzelm@24076
   743
      DETERM (resolve_tac [LA_Logic.notI, LA_Logic.ccontr] i) THEN
wenzelm@24076
   744
      (* user-defined preprocessing of the subgoal *)
wenzelm@24076
   745
      DETERM (LA_Data.pre_tac ctxt i) THEN
wenzelm@24076
   746
      PRIMITIVE (trace_thm "State after pre_tac:") THEN
wenzelm@24076
   747
      (* prove every resulting subgoal, using its justification *)
wenzelm@24076
   748
      EVERY (map just1 justs)
webertj@20217
   749
    end  state;
nipkow@6074
   750
nipkow@5982
   751
(*
nipkow@5982
   752
Fast but very incomplete decider. Only premises and conclusions
nipkow@5982
   753
that are already (negated) (in)equations are taken into account.
nipkow@5982
   754
*)
wenzelm@24076
   755
fun simpset_lin_arith_tac ss show_ex = SUBGOAL (fn (A, i) =>
wenzelm@24076
   756
  let
wenzelm@24076
   757
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   758
    val params = rev (Logic.strip_params A)
wenzelm@24076
   759
    val Hs = Logic.strip_assums_hyp A
wenzelm@24076
   760
    val concl = Logic.strip_assums_concl A
wenzelm@24076
   761
    val _ = trace_term ctxt ("Trying to refute subgoal " ^ string_of_int i) A
wenzelm@24076
   762
  in
wenzelm@24076
   763
    case prove ctxt params show_ex true Hs concl of
wenzelm@24076
   764
      NONE => (trace_msg "Refutation failed."; no_tac)
wenzelm@24076
   765
    | SOME js => (trace_msg "Refutation succeeded."; refute_tac ss (i, js))
wenzelm@24076
   766
  end);
nipkow@5982
   767
wenzelm@24076
   768
fun cut_lin_arith_tac ss =
wenzelm@24076
   769
  cut_facts_tac (Simplifier.prems_of_ss ss) THEN'
wenzelm@24076
   770
  simpset_lin_arith_tac ss false;
wenzelm@17613
   771
wenzelm@24076
   772
fun lin_arith_tac ctxt =
wenzelm@24076
   773
  simpset_lin_arith_tac (Simplifier.context ctxt Simplifier.empty_ss);
wenzelm@24076
   774
wenzelm@24076
   775
nipkow@5982
   776
nipkow@13186
   777
(** Forward proof from theorems **)
nipkow@13186
   778
webertj@20433
   779
(* More tricky code. Needs to arrange the proofs of the multiple cases (due
webertj@20433
   780
to splits of ~= premises) such that it coincides with the order of the cases
webertj@20433
   781
generated by function split_items. *)
webertj@20433
   782
webertj@20433
   783
datatype splittree = Tip of thm list
webertj@20433
   784
                   | Spl of thm * cterm * splittree * cterm * splittree;
webertj@20433
   785
webertj@20433
   786
(* "(ct1 ==> ?R) ==> (ct2 ==> ?R) ==> ?R" is taken to (ct1, ct2) *)
webertj@20433
   787
webertj@20433
   788
fun extract (imp : cterm) : cterm * cterm =
webertj@20433
   789
let val (Il, r)    = Thm.dest_comb imp
webertj@20433
   790
    val (_, imp1)  = Thm.dest_comb Il
webertj@20433
   791
    val (Ict1, _)  = Thm.dest_comb imp1
webertj@20433
   792
    val (_, ct1)   = Thm.dest_comb Ict1
webertj@20433
   793
    val (Ir, _)    = Thm.dest_comb r
webertj@20433
   794
    val (_, Ict2r) = Thm.dest_comb Ir
webertj@20433
   795
    val (Ict2, _)  = Thm.dest_comb Ict2r
webertj@20433
   796
    val (_, ct2)   = Thm.dest_comb Ict2
webertj@20433
   797
in (ct1, ct2) end;
webertj@20433
   798
wenzelm@24076
   799
fun splitasms ctxt (asms : thm list) : splittree =
wenzelm@24076
   800
let val {neqE, ...} = get_data ctxt
webertj@20433
   801
    fun elim_neq (asms', []) = Tip (rev asms')
webertj@20433
   802
      | elim_neq (asms', asm::asms) =
webertj@20433
   803
      (case get_first (fn th => SOME (asm COMP th) handle THM _ => NONE) neqE of
webertj@20433
   804
        SOME spl =>
webertj@20433
   805
          let val (ct1, ct2) = extract (cprop_of spl)
webertj@20433
   806
              val thm1 = assume ct1
webertj@20433
   807
              val thm2 = assume ct2
webertj@20433
   808
          in Spl (spl, ct1, elim_neq (asms', asms@[thm1]), ct2, elim_neq (asms', asms@[thm2]))
webertj@20433
   809
          end
webertj@20433
   810
      | NONE => elim_neq (asm::asms', asms))
webertj@20433
   811
in elim_neq ([], asms) end;
webertj@20433
   812
wenzelm@24076
   813
fun fwdproof ss (Tip asms : splittree) (j::js : injust list) = (mkthm ss asms j, js)
wenzelm@24076
   814
  | fwdproof ss (Spl (thm, ct1, tree1, ct2, tree2)) js =
wenzelm@24076
   815
      let
wenzelm@24076
   816
        val (thm1, js1) = fwdproof ss tree1 js
wenzelm@24076
   817
        val (thm2, js2) = fwdproof ss tree2 js1
webertj@20433
   818
        val thm1' = implies_intr ct1 thm1
webertj@20433
   819
        val thm2' = implies_intr ct2 thm2
wenzelm@24076
   820
      in (thm2' COMP (thm1' COMP thm), js2) end;
wenzelm@24076
   821
      (* FIXME needs handle THM _ => NONE ? *)
webertj@20433
   822
wenzelm@24076
   823
fun prover ss thms Tconcl (js : injust list) pos : thm option =
wenzelm@24076
   824
  let
wenzelm@24076
   825
    val ctxt = Simplifier.the_context ss
wenzelm@24076
   826
    val thy = ProofContext.theory_of ctxt
wenzelm@24076
   827
    val nTconcl = LA_Logic.neg_prop Tconcl
wenzelm@24076
   828
    val cnTconcl = cterm_of thy nTconcl
wenzelm@24076
   829
    val nTconclthm = assume cnTconcl
wenzelm@24076
   830
    val tree = splitasms ctxt (thms @ [nTconclthm])
wenzelm@24076
   831
    val (Falsethm, _) = fwdproof ss tree js
wenzelm@24076
   832
    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
wenzelm@24076
   833
    val concl = implies_intr cnTconcl Falsethm COMP contr
wenzelm@24076
   834
  in SOME (trace_thm "Proved by lin. arith. prover:" (LA_Logic.mk_Eq concl)) end
wenzelm@24076
   835
  (*in case concl contains ?-var, which makes assume fail:*)   (* FIXME Variable.import_terms *)
wenzelm@24076
   836
  handle THM _ => NONE;
nipkow@13186
   837
nipkow@13186
   838
(* PRE: concl is not negated!
nipkow@13186
   839
   This assumption is OK because
wenzelm@24076
   840
   1. lin_arith_simproc tries both to prove and disprove concl and
wenzelm@24076
   841
   2. lin_arith_simproc is applied by the Simplifier which
nipkow@13186
   842
      dives into terms and will thus try the non-negated concl anyway.
nipkow@13186
   843
*)
wenzelm@24076
   844
fun lin_arith_simproc ss concl =
wenzelm@24076
   845
  let
wenzelm@24076
   846
    val ctxt = Simplifier.the_context ss
wenzelm@26945
   847
    val thms = maps LA_Logic.atomize (Simplifier.prems_of_ss ss)
wenzelm@24076
   848
    val Hs = map Thm.prop_of thms
nipkow@6102
   849
    val Tconcl = LA_Logic.mk_Trueprop concl
wenzelm@24076
   850
  in
wenzelm@24076
   851
    case prove ctxt [] false false Hs Tconcl of (* concl provable? *)
wenzelm@24076
   852
      SOME js => prover ss thms Tconcl js true
wenzelm@24076
   853
    | NONE =>
wenzelm@24076
   854
        let val nTconcl = LA_Logic.neg_prop Tconcl in
wenzelm@24076
   855
          case prove ctxt [] false false Hs nTconcl of (* ~concl provable? *)
wenzelm@24076
   856
            SOME js => prover ss thms nTconcl js false
wenzelm@24076
   857
          | NONE => NONE
wenzelm@24076
   858
        end
wenzelm@24076
   859
  end;
nipkow@6074
   860
nipkow@6074
   861
end;