src/Provers/Arith/abel_cancel.ML

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(*  Title:      Provers/Arith/abel_cancel.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

Simplification procedures for abelian groups (e.g. integers, reals)

- Cancel complementary terms in sums 
- Cancel like terms on opposite sides of relations
*)


signature ABEL_CANCEL =
sig
  val ss		: simpset	(*basic simpset of object-logtic*)
  val eq_reflection	: thm		(*object-equality to meta-equality*)

  val sg_ref 		: Sign.sg_ref	(*signature of the theory of the group*)
  val T			: typ		(*the type of group elements*)

  val zero		: term
  val restrict_to_left  : thm
  val add_cancel_21	: thm
  val add_cancel_end	: thm
  val add_left_cancel	: thm
  val add_assoc		: thm
  val add_commute 	: thm
  val add_left_commute 	: thm
  val add_0 		: thm
  val add_0_right 	: thm

  val eq_diff_eq 	: thm
  val eqI_rules		: thm list
  val dest_eqI		: thm -> term

  val diff_def		: thm
  val minus_add_distrib	: thm
  val minus_minus	: thm
  val minus_0		: thm

  val add_inverses	: thm list
  val cancel_simps	: thm list
end;


functor Abel_Cancel (Data: ABEL_CANCEL) =
struct

open Data;

 val prepare_ss = Data.ss addsimps [add_assoc, diff_def, 
				    minus_add_distrib, minus_minus,
				    minus_0, add_0, add_0_right];

 (*prove while suppressing timing information*)
 fun prove ct = setmp Library.timing false (prove_goalw_cterm [] ct);

 val plus = Const ("op +", [Data.T,Data.T] ---> Data.T);
 val minus = Const ("uminus", Data.T --> Data.T);

 (*Cancel corresponding terms on the two sides of the equation, NOT on
   the same side!*)
 val cancel_ss = 
   Data.ss addsimps [add_cancel_21, add_cancel_end, minus_minus] @ 
                    (map (fn th => th RS restrict_to_left) Data.cancel_simps);

 val inverse_ss = Data.ss addsimps Data.add_inverses @ Data.cancel_simps;

 fun mk_sum []  = Data.zero
   | mk_sum tms = foldr1 (fn (x,y) => plus $ x $ y) tms;

 (*We map -t to t and (in other cases) t to -t.  No need to check the type of
   uminus, since the simproc is only called on sums of type T.*)
 fun negate (Const("uminus",_) $ t) = t
   | negate t                       = minus $ t;

 (*Flatten a formula built from +, binary - and unary -.
   No need to check types PROVIDED they are checked upon entry!*)
 fun add_terms neg (Const ("op +", _) $ x $ y, ts) =
	 add_terms neg (x, add_terms neg (y, ts))
   | add_terms neg (Const ("op -", _) $ x $ y, ts) =
	 add_terms neg (x, add_terms (not neg) (y, ts))
   | add_terms neg (Const ("uminus", _) $ x, ts) = 
	 add_terms (not neg) (x, ts)
   | add_terms neg (x, ts) = 
	 (if neg then negate x else x) :: ts;

 fun terms fml = add_terms false (fml, []);

 exception Cancel;

 (*Cancels just the first occurrence of u, leaving the rest unchanged*)
 fun cancelled (u, t::ts) = if u aconv t then ts else t :: cancelled(u,ts)
   | cancelled _          = raise Cancel;


 val trace = ref false;

 (*Make a simproc to cancel complementary terms in sums.  Examples:
    x-x = 0    x+(y-x) = y   -x+(y+(x+z)) = y+z
   It will unfold the definition of diff and associate to the right if 
   necessary.  Rewriting is faster if the formula is already
   in that form.
 *)

 fun sum_proc sg _ lhs =
   let val _ = if !trace then writeln ("cancel_sums: LHS = " ^ 
				       string_of_cterm (cterm_of sg lhs))
	       else ()
       val (head::tail) = terms lhs
       val head' = negate head
       val rhs = mk_sum (cancelled (head',tail))
       and chead' = Thm.cterm_of sg head'
       val _ = if !trace then 
		 writeln ("RHS = " ^ string_of_cterm (Thm.cterm_of sg rhs))
	       else ()
       val ct = Thm.cterm_of sg (Logic.mk_equals (lhs, rhs))
       val thm = prove ct 
		   (fn _ => [rtac eq_reflection 1,
			     simp_tac prepare_ss 1,
			     IF_UNSOLVED (simp_tac cancel_ss 1),
			     IF_UNSOLVED (simp_tac inverse_ss 1)])
	 handle ERROR =>
	 error("cancel_sums simproc:\nfailed to prove " ^
	       string_of_cterm ct)
   in Some thm end
   handle Cancel => None;


 val sum_conv = 
     Simplifier.mk_simproc "cancel_sums"
       (map (Thm.read_cterm (Sign.deref sg_ref))
	[("x + y", Data.T), ("x - y", Data.T)])
       sum_proc;


 (*A simproc to cancel like terms on the opposite sides of relations:
     (x + y - z < -z + x) = (y < 0)
   Works for (=) and (<=) as well as (<), if the necessary rules are supplied.
   Reduces the problem to subtraction and calls the previous simproc.
 *)

 (*Cancel the FIRST occurrence of a term.  If it's repeated, then repeated
   calls to the simproc will be needed.*)
 fun cancel1 ([], u)    = raise Match (*impossible: it's a common term*)
   | cancel1 (t::ts, u) = if t aconv u then ts
			  else t :: cancel1 (ts,u);


 val sum_cancel_ss = Data.ss addsimprocs [sum_conv]
			     addsimps    [add_0, add_0_right];

 val add_ac_ss = Data.ss addsimps [add_assoc,add_commute,add_left_commute];

 fun rel_proc sg _ (lhs as (rel$lt$rt)) =
   let val _ = if !trace then writeln ("cancel_relations: LHS = " ^ 
				       string_of_cterm (cterm_of sg lhs))
	       else ()
       val ltms = terms lt
       and rtms = terms rt
       val common = (*inter_term miscounts repetitions, so squash them*)
		    gen_distinct (op aconv) (inter_term (ltms, rtms))
       val _ = if null common then raise Cancel  (*nothing to do*)
				   else ()

       fun cancelled tms = mk_sum (foldl cancel1 (tms, common))

       val lt' = cancelled ltms
       and rt' = cancelled rtms

       val rhs = rel$lt'$rt'
       val _ = if !trace then 
		 writeln ("RHS = " ^ string_of_cterm (Thm.cterm_of sg rhs))
	       else ()
       val ct = Thm.cterm_of sg (Logic.mk_equals (lhs,rhs))

       val thm = prove ct 
		   (fn _ => [rtac eq_reflection 1,
			     resolve_tac eqI_rules 1,
			     simp_tac prepare_ss 1,
			     simp_tac sum_cancel_ss 1,
			     IF_UNSOLVED (simp_tac add_ac_ss 1)])
	 handle ERROR =>
	 error("cancel_relations simproc:\nfailed to prove " ^
	       string_of_cterm ct)
   in Some thm end
   handle Cancel => None;

 val rel_conv = 
     Simplifier.mk_simproc "cancel_relations"
       (map (Thm.cterm_of (Sign.deref sg_ref) o Data.dest_eqI) eqI_rules)
       rel_proc;

end;