src/Provers/Arith/fast_lin_arith.ML
author wenzelm
Mon Jul 24 23:48:29 2000 +0200 (2000-07-24)
changeset 9420 d4e9f60fe25a
parent 9073 40d8dfac96b8
child 10575 c78d26d5c3c1
permissions -rw-r--r--
avoid global references;
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(*  Title:      Provers/Arith/fast_lin_arith.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1998  TU Munich
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A generic linear arithmetic package.
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It provides two tactics
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    lin_arith_tac:         int -> tactic
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cut_lin_arith_tac: thms -> int -> tactic
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and a simplification procedure
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    lin_arith_prover: Sign.sg -> thm list -> term -> thm option
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Only take premises and conclusions into account that are already (negated)
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(in)equations. lin_arith_prover tries to prove or disprove 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
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  val ccontr:           thm (* (~ P ==> False) ==> P *)
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  val neqE:             thm (* [| m ~= n; m < n ==> P; n < m ==> P |] ==> 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 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: Sign.sg -> 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
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  nt is the (logically) negated version of t, where the negation
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  of a negative term is the term itself (no double negation!);
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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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  val decomp:
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    Sign.sg -> term -> ((term*int)list * int * string * (term*int)list * int * bool)option
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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 is the decomposition of the sum terms x/y into a list
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         of summand * multiplicity pairs and a constant summand and
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         d indicates if the domain is discrete.
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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 setup: (theory -> theory) list
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  val map_data: ({add_mono_thms: thm list, lessD: thm list, simpset: Simplifier.simpset}
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    -> {add_mono_thms: thm list, lessD: thm list, simpset: Simplifier.simpset})
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    -> theory -> theory
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  val trace           : bool ref
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  val lin_arith_prover: Sign.sg -> thm list -> term -> thm option
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  val     lin_arith_tac:             int -> tactic
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  val cut_lin_arith_tac: thm list -> int -> tactic
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end;
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functor Fast_Lin_Arith(structure LA_Logic:LIN_ARITH_LOGIC 
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                       and       LA_Data:LIN_ARITH_DATA) : FAST_LIN_ARITH =
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struct
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(** theory data **)
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(* data kind 'Provers/fast_lin_arith' *)
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structure DataArgs =
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struct
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  val name = "Provers/fast_lin_arith";
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  type T = {add_mono_thms: thm list, lessD: thm list, simpset: Simplifier.simpset};
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  val empty = {add_mono_thms = [], lessD = [], simpset = Simplifier.empty_ss};
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  val copy = I;
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  val prep_ext = I;
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  fun merge ({add_mono_thms = add_mono_thms1, lessD = lessD1, simpset = simpset1},
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      {add_mono_thms = add_mono_thms2, lessD = lessD2, simpset = simpset2}) =
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    {add_mono_thms = Drule.merge_rules (add_mono_thms1, add_mono_thms2),
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      lessD = Drule.merge_rules (lessD1, lessD2),
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      simpset = Simplifier.merge_ss (simpset1, simpset2)};
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  fun print _ _ = ();
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end;
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structure Data = TheoryDataFun(DataArgs);
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val map_data = Data.map;
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val setup = [Data.init];
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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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                | 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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(* 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 (apl(n,op * )) 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 (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 gcd x y =
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  let fun gxd x y =
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    if y = 0 then x else gxd y (x mod y)
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  in if x < y then gxd y x else gxd x y end;
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fun lcm x y = (x * y) div gcd x y;
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fun el 0 (h::_) = h
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  | el n (_::t) = el (n - 1) t
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  | el _ _  = sys_error "el";
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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 = el v l1 and c2 = el v l2
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      val m = 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"
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        else (m1,m2)
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      val (p1,p2) = if ty1=Eq andalso ty2=Eq andalso (n1 = ~1 orelse n2 = ~1)
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                    then (~n1,~n2) else (n1,n2)
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  in add_ineq (multiply_ineq n1 i1) (multiply_ineq n2 i2) end;
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(* ------------------------------------------------------------------------- *)
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(* The main refutation-finding code.                                         *)
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(* ------------------------------------------------------------------------- *)
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fun is_trivial (Lineq(_,_,l,_)) = forall (fn i => i=0) l;
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fun is_answer (ans as Lineq(k,ty,l,_)) =
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  case ty  of Eq => k <> 0 | Le => k > 0 | Lt => k >= 0;
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fun calc_blowup l =
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  let val (p,n) = partition (apl(0,op<)) (filter (apl(0,op<>)) l)
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  in (length p) * (length n) end;
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(* ------------------------------------------------------------------------- *)
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(* Main elimination code:                                                    *)
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(*                                                                           *)
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(* (1) Looks for immediate solutions (false assertions with no variables).   *)
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(*                                                                           *)
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(* (2) If there are any equations, picks a variable with the lowest absolute *)
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(* coefficient in any of them, and uses it to eliminate.                     *)
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(*                                                                           *)
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(* (3) Otherwise, chooses a variable in the inequality to minimize the       *)
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(* blowup (number of consequences generated) and eliminates it.              *)
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(* ------------------------------------------------------------------------- *)
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fun allpairs f xs ys =
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  flat(map (fn x => map (fn y => f x y) ys) xs);
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fun extract_first p =
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  let fun extract xs (y::ys) = if p y then (Some y,xs@ys)
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                               else extract (y::xs) ys
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        | extract xs []      = (None,xs)
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  in extract [] end;
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fun print_ineqs ineqs =
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  if !trace then
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     writeln(cat_lines(""::map (fn Lineq(c,t,l,_) =>
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       string_of_int c ^
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       (case t of Eq => " =  " | Lt=> " <  " | Le => " <= ") ^
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       commas(map string_of_int l)) ineqs))
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  else ();
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fun elim ineqs =
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  let val dummy = print_ineqs ineqs;
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      val (triv,nontriv) = partition is_trivial ineqs in
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  if not(null triv)
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  then case find_first is_answer triv of
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         None => elim nontriv | some => some
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  else
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  if null nontriv then None else
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  let val (eqs,noneqs) = partition (fn (Lineq(_,ty,_,_)) => ty=Eq) nontriv in
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  if not(null eqs) then
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     let val clist = foldl (fn (cs,Lineq(_,_,l,_)) => l union cs) ([],eqs)
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         val sclist = sort (fn (x,y) => int_ord(abs(x),abs(y)))
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                           (filter (fn i => i<>0) clist)
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         val c = hd sclist
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         val (Some(eq as Lineq(_,_,ceq,_)),othereqs) =
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               extract_first (fn Lineq(_,_,l,_) => c mem l) eqs
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         val v = find_index (fn k => k=c) ceq
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         val (ioth,roth) = partition (fn (Lineq(_,_,l,_)) => el v l = 0)
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                                     (othereqs @ noneqs)
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         val others = map (elim_var v eq) roth @ ioth
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     in elim others end
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  else
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  let val lists = map (fn (Lineq(_,_,l,_)) => l) noneqs
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      val numlist = 0 upto (length(hd lists) - 1)
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      val coeffs = map (fn i => map (el i) lists) numlist
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      val blows = map calc_blowup coeffs
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      val iblows = blows ~~ numlist
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      val nziblows = filter (fn (i,_) => i<>0) iblows
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  in if null nziblows then None else
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     let val (c,v) = hd(sort (fn (x,y) => int_ord(fst(x),fst(y))) nziblows)
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         val (no,yes) = partition (fn (Lineq(_,_,l,_)) => el v l = 0) ineqs
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         val (pos,neg) = partition(fn (Lineq(_,_,l,_)) => el v l > 0) yes
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     in elim (no @ allpairs (elim_var v) pos neg) end
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  end
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  end
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  end;
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(* ------------------------------------------------------------------------- *)
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(* Translate back a proof.                                                   *)
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(* ------------------------------------------------------------------------- *)
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fun trace_thm msg th = 
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    if !trace then (writeln msg; prth th) else th;
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fun trace_msg msg = 
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    if !trace then writeln msg else ();
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(* FIXME OPTIMIZE!!!!
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   Addition/Multiplication need i*t representation rather than t+t+...
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Simplification may detect a contradiction 'prematurely' due to type
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information: n+1 <= 0 is simplified to False and does not need to be crossed
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with 0 <= n.
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*)
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local
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 exception FalseE of thm
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in
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fun mkthm sg asms just =
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  let val {add_mono_thms, lessD, simpset} = Data.get_sg sg;
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      val atoms = foldl (fn (ats,(lhs,_,_,rhs,_,_)) =>
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                            map fst lhs  union  (map fst rhs  union  ats))
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                        ([], mapfilter (LA_Data.decomp sg o concl_of) asms)
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      fun addthms thm1 thm2 =
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        let val conj = thm1 RS (thm2 RS LA_Logic.conjI)
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        in the(get_first (fn th => Some(conj RS th) handle _ => None) add_mono_thms)
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        end;
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      fun multn(n,thm) =
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        let fun mul(i,th) = if i=1 then th else mul(i-1, addthms thm th)
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        in if n < 0 then mul(~n,thm) RS LA_Logic.sym else mul(n,thm) end;
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      fun simp thm =
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        let val thm' = simplify simpset thm
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        in if LA_Logic.is_False thm' then raise FalseE thm' else thm' end
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      fun mk(Asm i) = trace_thm "Asm" (nth_elem(i,asms))
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        | mk(Nat(i)) = (trace_msg "Nat";
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			LA_Logic.mk_nat_thm sg (nth_elem(i,atoms)))
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        | mk(LessD(j)) = trace_thm "L" (hd([mk j] RL lessD))
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        | mk(NotLeD(j)) = trace_thm "NLe" (mk j RS LA_Logic.not_leD)
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        | mk(NotLeDD(j)) = trace_thm "NLeD" (hd([mk j RS LA_Logic.not_leD] RL lessD))
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        | mk(NotLessD(j)) = trace_thm "NL" (mk j RS LA_Logic.not_lessD)
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        | mk(Added(j1,j2)) = simp (trace_thm "+" (addthms (mk j1) (mk j2)))
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        | mk(Multiplied(n,j)) = (trace_msg "*"; multn(n,mk j))
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  in trace_msg "mkthm";
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     simplify simpset (mk just) handle FalseE thm => thm end
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end;
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fun coeff poly atom = case assoc(poly,atom) of None => 0 | Some i => i;
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fun mklineq atoms =
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  let val n = length atoms in
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    fn ((lhs,i,rel,rhs,j,discrete),k) =>
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   332
    let val lhsa = map (coeff lhs) atoms
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   333
        and rhsa = map (coeff rhs) atoms
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        val diff = map2 (op -) (rhsa,lhsa)
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   335
        val c = i-j
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        val just = Asm k
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   337
    in case rel of
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   338
        "<="   => Some(Lineq(c,Le,diff,just))
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   339
       | "~<=" => if discrete
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   340
                  then Some(Lineq(1-c,Le,map (op ~) diff,NotLeDD(just)))
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   341
                  else Some(Lineq(~c,Lt,map (op ~) diff,NotLeD(just)))
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   342
       | "<"   => if discrete
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   343
                  then Some(Lineq(c+1,Le,diff,LessD(just)))
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   344
                  else Some(Lineq(c,Lt,diff,just))
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   345
       | "~<"  => Some(Lineq(~c,Le,map (op~) diff,NotLessD(just)))
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   346
       | "="   => Some(Lineq(c,Eq,diff,just))
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   347
       | "~="  => None
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   348
       | _     => sys_error("mklineq" ^ rel)   
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   349
    end
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   350
  end;
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   351
nipkow@6056
   352
fun mknat pTs ixs (atom,i) =
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   353
  if LA_Logic.is_nat(pTs,atom)
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   354
  then let val l = map (fn j => if j=i then 1 else 0) ixs
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   355
       in Some(Lineq(0,Le,l,Nat(i))) end
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   356
  else None
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   357
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   358
fun abstract pTs items =
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   359
  let val atoms = foldl (fn (ats,((lhs,_,_,rhs,_,_),_)) =>
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   360
                            (map fst lhs) union ((map fst rhs) union ats))
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   361
                        ([],items)
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   362
      val ixs = 0 upto (length(atoms)-1)
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   363
      val iatoms = atoms ~~ ixs
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   364
  in mapfilter (mklineq atoms) items @ mapfilter (mknat pTs ixs) iatoms end;
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   365
nipkow@5982
   366
(* Ordinary refutation *)
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   367
fun refute1(pTs,items) =
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   368
  (case elim (abstract pTs items) of
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   369
       None => []
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   370
     | Some(Lineq(_,_,_,j)) => [j]);
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   371
nipkow@6074
   372
fun refute1_tac(i,just) =
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   373
  fn state =>
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   374
    let val sg = #sign(rep_thm state)
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   375
    in resolve_tac [LA_Logic.notI,LA_Logic.ccontr] i THEN
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   376
       METAHYPS (fn asms => rtac (mkthm sg asms just) 1) i
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   377
    end
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   378
    state;
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   379
nipkow@5982
   380
(* Double refutation caused by equality in conclusion *)
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   381
fun refute2(pTs,items, (rhs,i,_,lhs,j,d), nHs) =
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   382
  (case elim (abstract pTs (items@[((rhs,i,"<",lhs,j,d),nHs)])) of
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   383
    None => []
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   384
  | Some(Lineq(_,_,_,j1)) =>
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   385
      (case elim (abstract pTs (items@[((lhs,j,"<",rhs,i,d),nHs)])) of
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   386
        None => []
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   387
      | Some(Lineq(_,_,_,j2)) => [j1,j2]));
nipkow@6074
   388
nipkow@6074
   389
fun refute2_tac(i,just1,just2) =
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   390
  fn state => 
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   391
    let val sg = #sign(rep_thm state)
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   392
    in rtac LA_Logic.ccontr i THEN rotate_tac ~1 i THEN etac LA_Logic.neqE i THEN
nipkow@6074
   393
       METAHYPS (fn asms => rtac (mkthm sg asms just1) 1) i THEN
nipkow@6074
   394
       METAHYPS (fn asms => rtac (mkthm sg asms just2) 1) i
nipkow@6074
   395
    end
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   396
    state;
nipkow@6074
   397
wenzelm@9420
   398
fun prove sg (pTs,Hs,concl) =
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   399
let val nHs = length Hs
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   400
    val ixHs = Hs ~~ (0 upto (nHs-1))
wenzelm@9420
   401
    val Hitems = mapfilter (fn (h,i) => case LA_Data.decomp sg h of
nipkow@6074
   402
                                 None => None | Some(it) => Some(it,i)) ixHs
wenzelm@9420
   403
in case LA_Data.decomp sg concl of
nipkow@6074
   404
     None => if null Hitems then [] else refute1(pTs,Hitems)
nipkow@7551
   405
   | Some(citem as (r,i,rel,l,j,d)) =>
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   406
       if rel = "="
nipkow@6074
   407
       then refute2(pTs,Hitems,citem,nHs)
nipkow@6074
   408
       else let val neg::rel0 = explode rel
nipkow@6074
   409
                val nrel = if neg = "~" then implode rel0 else "~"^rel
nipkow@7551
   410
            in refute1(pTs, Hitems@[((r,i,nrel,l,j,d),nHs)]) end
nipkow@6074
   411
end;
nipkow@5982
   412
nipkow@5982
   413
(*
nipkow@5982
   414
Fast but very incomplete decider. Only premises and conclusions
nipkow@5982
   415
that are already (negated) (in)equations are taken into account.
nipkow@5982
   416
*)
wenzelm@9420
   417
fun lin_arith_tac i st = SUBGOAL (fn (A,n) =>
nipkow@6056
   418
  let val pTs = rev(map snd (Logic.strip_params A))
nipkow@6056
   419
      val Hs = Logic.strip_assums_hyp A
nipkow@6074
   420
      val concl = Logic.strip_assums_concl A
wenzelm@9420
   421
  in case prove (Thm.sign_of_thm st) (pTs,Hs,concl) of
nipkow@6074
   422
       [j] => refute1_tac(n,j)
nipkow@6074
   423
     | [j1,j2] => refute2_tac(n,j1,j2)
nipkow@6074
   424
     | _ => no_tac
wenzelm@9420
   425
  end) i st;
nipkow@5982
   426
nipkow@5982
   427
fun cut_lin_arith_tac thms i = cut_facts_tac thms i THEN lin_arith_tac i;
nipkow@5982
   428
nipkow@6079
   429
fun prover1(just,sg,thms,concl,pos) =
nipkow@6102
   430
let val nconcl = LA_Logic.neg_prop concl
nipkow@6074
   431
    val cnconcl = cterm_of sg nconcl
nipkow@6074
   432
    val Fthm = mkthm sg (thms @ [assume cnconcl]) just
nipkow@6102
   433
    val contr = if pos then LA_Logic.ccontr else LA_Logic.notI
nipkow@6102
   434
in Some(LA_Logic.mk_Eq ((implies_intr cnconcl Fthm) COMP contr)) end
nipkow@6074
   435
handle _ => None;
nipkow@6074
   436
nipkow@6074
   437
(* handle thm with equality conclusion *)
nipkow@6074
   438
fun prover2(just1,just2,sg,thms,concl) =
nipkow@6102
   439
let val nconcl = LA_Logic.neg_prop concl (* m ~= n *)
nipkow@6074
   440
    val cnconcl = cterm_of sg nconcl
nipkow@6074
   441
    val neqthm = assume cnconcl
nipkow@6102
   442
    val casethm = neqthm COMP LA_Logic.neqE (* [|m<n ==> R; n<m ==> R|] ==> R *)
nipkow@6074
   443
    val [lessimp1,lessimp2] = prems_of casethm
nipkow@6074
   444
    val less1 = fst(Logic.dest_implies lessimp1) (* m<n *)
nipkow@6074
   445
    and less2 = fst(Logic.dest_implies lessimp2) (* n<m *)
nipkow@6074
   446
    val cless1 = cterm_of sg less1 and cless2 = cterm_of sg less2
nipkow@6074
   447
    val thm1 = mkthm sg (thms @ [assume cless1]) just1
nipkow@6074
   448
    and thm2 = mkthm sg (thms @ [assume cless2]) just2
nipkow@6074
   449
    val dthm1 = implies_intr cless1 thm1 and dthm2 = implies_intr cless2 thm2
nipkow@6074
   450
    val thm = dthm2 COMP (dthm1 COMP casethm)
nipkow@6102
   451
in Some(LA_Logic.mk_Eq ((implies_intr cnconcl thm) COMP LA_Logic.ccontr)) end
nipkow@6074
   452
handle _ => None;
nipkow@6074
   453
nipkow@6079
   454
(* PRE: concl is not negated! *)
nipkow@6074
   455
fun lin_arith_prover sg thms concl =
nipkow@6074
   456
let val Hs = map (#prop o rep_thm) thms
nipkow@6102
   457
    val Tconcl = LA_Logic.mk_Trueprop concl
wenzelm@9420
   458
in case prove sg ([],Hs,Tconcl) of
nipkow@6079
   459
     [j] => prover1(j,sg,thms,Tconcl,true)
nipkow@6074
   460
   | [j1,j2] => prover2(j1,j2,sg,thms,Tconcl)
nipkow@6102
   461
   | _ => let val nTconcl = LA_Logic.neg_prop Tconcl
wenzelm@9420
   462
          in case prove sg ([],Hs,nTconcl) of
nipkow@6079
   463
               [j] => prover1(j,sg,thms,nTconcl,false)
nipkow@6079
   464
               (* [_,_] impossible because of negation *)
nipkow@6079
   465
             | _ => None
nipkow@6079
   466
          end
nipkow@5982
   467
end;
nipkow@6074
   468
nipkow@6074
   469
end;