paulson@8760: (*  Title:      Provers/Arith/cancel_numerals.ML
paulson@8736:     ID:         $Id$
paulson@8736:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@8736:     Copyright   2000  University of Cambridge
paulson@8736: 
paulson@8760: Cancel common coefficients in balanced expressions:
paulson@8736: 
paulson@8760:      i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
paulson@8736: 
paulson@8736: where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
paulson@8760: 
paulson@8760: It works by (a) massaging both sides to bring the selected term to the front:
paulson@8760: 
paulson@8760:      #m*u + (i + j) ~~ #m'*u + (i' + j') 
paulson@8760: 
paulson@8760: (b) then using bal_add1 or bal_add2 to reach
paulson@8760: 
paulson@8760:      #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
paulson@8760: 
paulson@8760: or
paulson@8760: 
paulson@8760:      i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
paulson@8736: *)
paulson@8736: 
paulson@8736: signature CANCEL_NUMERALS_DATA =
paulson@8736: sig
paulson@8736:   (*abstract syntax*)
paulson@14387:   val mk_sum: typ -> term list -> term
paulson@8736:   val dest_sum: term -> term list
paulson@8736:   val mk_bal: term * term -> term
paulson@8736:   val dest_bal: term -> term * term
paulson@8760:   val mk_coeff: int * term -> term
paulson@8760:   val dest_coeff: term -> int * term
paulson@8760:   val find_first_coeff: term -> term list -> int * term list
paulson@8760:   (*rules*)
paulson@8760:   val bal_add1: thm
paulson@8760:   val bal_add2: thm
paulson@8736:   (*proof tools*)
paulson@9546:   val prove_conv: tactic list -> Sign.sg -> 
wenzelm@13484:                   thm list -> string list -> term * term -> thm option
paulson@8799:   val trans_tac: thm option -> tactic (*applies the initial lemma*)
paulson@8799:   val norm_tac: tactic                (*proves the initial lemma*)
paulson@8799:   val numeral_simp_tac: tactic        (*proves the final theorem*)
paulson@8799:   val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
paulson@8736: end;
paulson@8736: 
paulson@8736: 
paulson@8760: functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
paulson@8760:   sig
wenzelm@15027:   val proc: Sign.sg -> simpset -> term -> thm option
paulson@8760:   end 
paulson@8760: =
paulson@8736: struct
paulson@8736: 
paulson@8779: (*For t = #n*u then put u in the table*)
paulson@8760: fun update_by_coeff (tab, t) =
paulson@8779:   Termtab.update ((#2 (Data.dest_coeff t), ()), tab);
paulson@8760: 
paulson@8760: (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
paulson@8760:   #m*u is in terms2 for some m*)
paulson@8760: fun find_common (terms1,terms2) =
skalberg@15570:   let val tab2 = Library.foldl update_by_coeff (Termtab.empty, terms2)
paulson@8760:       fun seek [] = raise TERM("find_common", []) 
paulson@8760: 	| seek (t::terms) =
paulson@8760: 	      let val (_,u) = Data.dest_coeff t 
skalberg@15570: 	      in  if isSome (Termtab.lookup (tab2, u)) then u
paulson@8760: 		  else seek terms
paulson@8760: 	      end
paulson@8760:   in  seek terms1 end;
paulson@8736: 
paulson@8736: (*the simplification procedure*)
wenzelm@15027: fun proc sg ss t =
wenzelm@15027:   let
wenzelm@15027:       val hyps = prems_of_ss ss;
wenzelm@15027:       (*first freeze any Vars in the term to prevent flex-flex problems*)
wenzelm@13484:       val (t', xs) = Term.adhoc_freeze_vars t;
paulson@9191:       val (t1,t2) = Data.dest_bal t' 
paulson@8760:       val terms1 = Data.dest_sum t1
paulson@8760:       and terms2 = Data.dest_sum t2
paulson@8760:       val u = find_common (terms1,terms2)
paulson@8760:       val (n1, terms1') = Data.find_first_coeff u terms1
paulson@8760:       and (n2, terms2') = Data.find_first_coeff u terms2
paulson@14387:       and T = Term.fastype_of u
paulson@14387:       fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
paulson@8779:       val reshape =  (*Move i*u to the front and put j*u into standard form
paulson@8779: 		       i + #m + j + k == #m + i + (j + k) *)
paulson@8772: 	    if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
paulson@8772: 		raise TERM("cancel_numerals", []) 
wenzelm@13484: 	    else Data.prove_conv [Data.norm_tac] sg hyps xs
paulson@9191: 			(t', 
paulson@8772: 			 Data.mk_bal (newshape(n1,terms1'), 
paulson@8779: 				      newshape(n2,terms2')))
paulson@8736:   in
skalberg@15570:       Option.map Data.simplify_meta_eq
paulson@8799:        (if n2<=n1 then 
paulson@8799: 	    Data.prove_conv 
paulson@8799: 	       [Data.trans_tac reshape, rtac Data.bal_add1 1,
wenzelm@13484: 		Data.numeral_simp_tac] sg hyps xs
paulson@9191: 	       (t', Data.mk_bal (newshape(n1-n2,terms1'), 
paulson@14387: 				 Data.mk_sum T terms2'))
paulson@8799: 	else
paulson@8799: 	    Data.prove_conv 
paulson@8799: 	       [Data.trans_tac reshape, rtac Data.bal_add2 1,
wenzelm@13484: 		Data.numeral_simp_tac] sg hyps xs
paulson@14387: 	       (t', Data.mk_bal (Data.mk_sum T terms1', 
paulson@9191: 				 newshape(n2-n1,terms2'))))
paulson@8736:   end
skalberg@15531:   handle TERM _ => NONE
skalberg@15531:        | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
paulson@8779: 			     Undeclared type constructor "Numeral.bin"*)
paulson@8736: 
paulson@8736: end;