src/Provers/Arith/cancel_numerals.ML

tuned Isar interfaces;
tuned IsarThy.theorem_i;

(*  Title:      Provers/Arith/cancel_numerals.ML
    ID:         $Id$
    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.curried_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;