src/HOL/Tools/Presburger/qelim.ML

A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
the tactic has also changed and allows the abstaction over fuction occurences whose type is nat or int.

(*  Title:      HOL/Integ/qelim.ML
    ID:         $Id$
    Author:     Amine Chaieb and Tobias Nipkow, TU Muenchen
    License:    GPL (GNU GENERAL PUBLIC LICENSE)

File containing the implementation of the proof protocoling
and proof generation for multiple quantified formulae.
*)

signature QELIM =
 sig
 val tproof_of_mlift_qelim: Sign.sg -> (term -> bool) ->
   (string list -> term -> thm) -> (term -> thm) ->
   (term -> thm) -> term -> thm

end;

structure Qelim : QELIM =
struct
open CooperDec;
open CooperProof;

val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;

(* List of the theorems to be used in the following*)

val qe_ex_conj = thm "qe_ex_conj";
val qe_ex_nconj = thm "qe_ex_nconj";
val qe_ALL = thm "qe_ALL";


(*============= Compact version =====*)


fun decomp_qe is_at afnp nfnp qfnp sg P = 
   if is_at P then ([], fn [] => afnp [] P) else 
   case P of
   (Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
   |(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
   |(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
   |(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
   |(Const("Not",_)$p) => ([p],fn [th] => th RS qe_Not)
   |(Const("Ex",_)$Abs(xn,xT,p)) => 
      let val (xn1,p1) = variant_abs(xn,xT,p) 
      in ([p1],
        fn [th1_1] => 
          let val th2 = [th1_1,nfnp (snd (qe_get_terms th1_1))] MRS trans
              val eth1 = (forall_intr (cterm_of sg (Free(xn1,xT))) th2) COMP  qe_exI
              val th3 = qfnp (snd(qe_get_terms eth1))
          in [eth1,th3] MRS trans
          end )
      end

   |(Const("All",_)$Abs(xn,xT,p)) => ([(HOLogic.exists_const xT)$Abs(xn,xT,HOLogic.Not $ p)], fn [th] => th RS qe_ALL)
   | _ => ([],fn [] => instantiate' [Some (ctyp_of sg (type_of P))] [Some (cterm_of sg P)] refl);
 

fun tproof_of_mlift_qelim sg isat afnp nfnp qfnp p = 
   let val th1 = thm_of sg (decomp_qe isat afnp nfnp qfnp sg) p
       val th2 = nfnp (snd (qe_get_terms th1))
    in [th1,th2] MRS trans
    end;
end;