src/Provers/Arith/cancel_numerals.ML

prepared for Isabelle2001;

(*  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: term list -> term
  val dest_sum: term -> term list
  val mk_bal: term * term -> term
  val dest_bal: term -> term * term
  val mk_coeff: int * term -> term
  val dest_coeff: term -> int * term
  val find_first_coeff: term -> term list -> int * term list
  (*rules*)
  val bal_add1: thm
  val bal_add2: thm
  (*proof tools*)
  val prove_conv: tactic list -> Sign.sg -> 
                  thm list -> term * term -> thm option
  val trans_tac: thm option -> tactic (*applies the initial lemma*)
  val norm_tac: tactic                (*proves the initial lemma*)
  val numeral_simp_tac: tactic        (*proves the final theorem*)
  val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
end;


functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
  sig
  val proc: Sign.sg -> thm list -> term -> thm option
  end 
=
struct

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

(*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 = foldl update_by_coeff (Termtab.empty, terms2)
      fun seek [] = raise TERM("find_common", []) 
	| seek (t::terms) =
	      let val (_,u) = Data.dest_coeff t 
	      in  if is_some (Termtab.lookup (tab2, u)) then u
		  else seek terms
	      end
  in  seek terms1 end;

(*the simplification procedure*)
fun proc sg hyps t =
  let (*first freeze any Vars in the term to prevent flex-flex problems*)
      val rand_s = gensym"_"
      fun mk_inst (var as Var((a,i),T))  = 
	    (var,  Free((a ^ rand_s ^ string_of_int i), T))
      val t' = subst_atomic (map mk_inst (term_vars t)) 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
      fun newshape (i,terms) = Data.mk_sum (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] sg hyps
			(t', 
			 Data.mk_bal (newshape(n1,terms1'), 
				      newshape(n2,terms2')))
  in
      apsome Data.simplify_meta_eq
       (if n2<=n1 then 
	    Data.prove_conv 
	       [Data.trans_tac reshape, rtac Data.bal_add1 1,
		Data.numeral_simp_tac] sg hyps
	       (t', Data.mk_bal (newshape(n1-n2,terms1'), 
				 Data.mk_sum terms2'))
	else
	    Data.prove_conv 
	       [Data.trans_tac reshape, rtac Data.bal_add2 1,
		Data.numeral_simp_tac] sg hyps
	       (t', Data.mk_bal (Data.mk_sum terms1', 
				 newshape(n2-n1,terms2'))))
  end
  handle TERM _ => None
       | TYPE _ => None;   (*Typically (if thy doesn't include Numeral)
			     Undeclared type constructor "Numeral.bin"*)

end;