# File cancel_numerals.ML

    (*  Title:      Provers/Arith/cancel_numerals.ML
        ID:         $Id: cancel_numerals.ML,v 1.16 2005/09/15 15:16:58 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   2000  University of Cambridge

    Cancel common coefficients in balanced expressions:

         i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'

    where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).

    It works by (a) massaging both sides to bring the selected term to the front:

         #m*u + (i + j) ~~ #m'*u + (i' + j')

    (b) then using bal_add1 or bal_add2 to reach

         #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)

    or

         i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
    *)

    signature CANCEL_NUMERALS_DATA =
    sig
      (*abstract syntax*)
      val mk_sum: typ -> term list -> term
      val dest_sum: term -> term list
      val mk_bal: term * term -> term
      val dest_bal: term -> term * term
      val mk_coeff: IntInf.int * term -> term
      val dest_coeff: term -> IntInf.int * term
      val find_first_coeff: term -> term list -> IntInf.int * term list
      (*rules*)
      val bal_add1: thm
      val bal_add2: thm
      (*proof tools*)
      val prove_conv: tactic list -> theory ->
                      thm list -> string list -> term * term -> thm option
      val trans_tac: simpset -> thm option -> tactic (*applies the initial lemma*)
      val norm_tac: simpset -> tactic                (*proves the initial lemma*)
      val numeral_simp_tac: simpset -> tactic        (*proves the final theorem*)
      val simplify_meta_eq: simpset -> thm -> thm    (*simplifies the final theorem*)
    end;

    functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
      sig
      val proc: theory -> simpset -> term -> thm option
      end
    =
    struct

    (*For t = #n*u then put u in the table*)
    fun update_by_coeff t =
      Termtab.update (#2 (Data.dest_coeff t), ());

    (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
      #m*u is in terms2 for some m*)
    fun find_common (terms1,terms2) =
      let val tab2 = fold update_by_coeff terms2 Termtab.empty
          fun seek [] = raise TERM("find_common", [])
            | seek (t::terms) =
                  let val (_,u) = Data.dest_coeff t
                  in if Termtab.defined tab2 u then u else seek terms end
      in  seek terms1 end;

    (*the simplification procedure*)
    fun proc thy ss t =
      let
          val hyps = prems_of_ss ss;
          (*first freeze any Vars in the term to prevent flex-flex problems*)
          val (t', xs) = Term.adhoc_freeze_vars t;
          val (t1,t2) = Data.dest_bal t'
          val terms1 = Data.dest_sum t1
          and terms2 = Data.dest_sum t2
          val u = find_common (terms1,terms2)
          val (n1, terms1') = Data.find_first_coeff u terms1
          and (n2, terms2') = Data.find_first_coeff u terms2
          and T = Term.fastype_of u
          fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
          val reshape =  (*Move i*u to the front and put j*u into standard form
                           i + #m + j + k == #m + i + (j + k) *)
                if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
                    raise TERM("cancel_numerals", [])
                else Data.prove_conv [Data.norm_tac ss] thy hyps xs
                            (t',
                             Data.mk_bal (newshape(n1,terms1'),
                                          newshape(n2,terms2')))
      in
          Option.map (Data.simplify_meta_eq ss)
           (if n2<=n1 then
                Data.prove_conv
                   [Data.trans_tac ss reshape, rtac Data.bal_add1 1,
                    Data.numeral_simp_tac ss] thy hyps xs
                   (t', Data.mk_bal (newshape(n1-n2,terms1'),
                                     Data.mk_sum T terms2'))
            else
                Data.prove_conv
                   [Data.trans_tac ss reshape, rtac Data.bal_add2 1,
                    Data.numeral_simp_tac ss] thy hyps xs
                   (t', Data.mk_bal (Data.mk_sum T terms1',
                                     newshape(n2-n1,terms2'))))
      end
      handle TERM _ => NONE
           | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
                                 Undeclared type constructor "Numeral.bin"*)

    end;