src/Provers/Arith/cancel_numerals.ML
author wenzelm
Mon Aug 01 19:20:26 2005 +0200 (2005-08-01)
changeset 16973 b2a894562b8f
parent 15965 f422f8283491
child 17223 430edc6b7826
permissions -rw-r--r--
simprocs: Simplifier.inherit_bounds;
     1 (*  Title:      Provers/Arith/cancel_numerals.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   2000  University of Cambridge
     5 
     6 Cancel common coefficients in balanced expressions:
     7 
     8      i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
     9 
    10 where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
    11 
    12 It works by (a) massaging both sides to bring the selected term to the front:
    13 
    14      #m*u + (i + j) ~~ #m'*u + (i' + j') 
    15 
    16 (b) then using bal_add1 or bal_add2 to reach
    17 
    18      #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
    19 
    20 or
    21 
    22      i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
    23 *)
    24 
    25 signature CANCEL_NUMERALS_DATA =
    26 sig
    27   (*abstract syntax*)
    28   val mk_sum: typ -> term list -> term
    29   val dest_sum: term -> term list
    30   val mk_bal: term * term -> term
    31   val dest_bal: term -> term * term
    32   val mk_coeff: IntInf.int * term -> term
    33   val dest_coeff: term -> IntInf.int * term
    34   val find_first_coeff: term -> term list -> IntInf.int * term list
    35   (*rules*)
    36   val bal_add1: thm
    37   val bal_add2: thm
    38   (*proof tools*)
    39   val prove_conv: tactic list -> theory -> 
    40                   thm list -> string list -> term * term -> thm option
    41   val trans_tac: simpset -> thm option -> tactic (*applies the initial lemma*)
    42   val norm_tac: simpset -> tactic                (*proves the initial lemma*)
    43   val numeral_simp_tac: simpset -> tactic        (*proves the final theorem*)
    44   val simplify_meta_eq: simpset -> thm -> thm    (*simplifies the final theorem*)
    45 end;
    46 
    47 
    48 functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
    49   sig
    50   val proc: theory -> simpset -> term -> thm option
    51   end 
    52 =
    53 struct
    54 
    55 (*For t = #n*u then put u in the table*)
    56 fun update_by_coeff (tab, t) =
    57   Termtab.update ((#2 (Data.dest_coeff t), ()), tab);
    58 
    59 (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
    60   #m*u is in terms2 for some m*)
    61 fun find_common (terms1,terms2) =
    62   let val tab2 = Library.foldl update_by_coeff (Termtab.empty, terms2)
    63       fun seek [] = raise TERM("find_common", []) 
    64 	| seek (t::terms) =
    65 	      let val (_,u) = Data.dest_coeff t 
    66 	      in  if isSome (Termtab.lookup (tab2, u)) then u
    67 		  else seek terms
    68 	      end
    69   in  seek terms1 end;
    70 
    71 (*the simplification procedure*)
    72 fun proc thy ss t =
    73   let
    74       val hyps = prems_of_ss ss;
    75       (*first freeze any Vars in the term to prevent flex-flex problems*)
    76       val (t', xs) = Term.adhoc_freeze_vars t;
    77       val (t1,t2) = Data.dest_bal t' 
    78       val terms1 = Data.dest_sum t1
    79       and terms2 = Data.dest_sum t2
    80       val u = find_common (terms1,terms2)
    81       val (n1, terms1') = Data.find_first_coeff u terms1
    82       and (n2, terms2') = Data.find_first_coeff u terms2
    83       and T = Term.fastype_of u
    84       fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
    85       val reshape =  (*Move i*u to the front and put j*u into standard form
    86 		       i + #m + j + k == #m + i + (j + k) *)
    87 	    if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
    88 		raise TERM("cancel_numerals", []) 
    89 	    else Data.prove_conv [Data.norm_tac ss] thy hyps xs
    90 			(t', 
    91 			 Data.mk_bal (newshape(n1,terms1'), 
    92 				      newshape(n2,terms2')))
    93   in
    94       Option.map (Data.simplify_meta_eq ss)
    95        (if n2<=n1 then 
    96 	    Data.prove_conv 
    97 	       [Data.trans_tac ss reshape, rtac Data.bal_add1 1,
    98 		Data.numeral_simp_tac ss] thy hyps xs
    99 	       (t', Data.mk_bal (newshape(n1-n2,terms1'), 
   100 				 Data.mk_sum T terms2'))
   101 	else
   102 	    Data.prove_conv 
   103 	       [Data.trans_tac ss reshape, rtac Data.bal_add2 1,
   104 		Data.numeral_simp_tac ss] thy hyps xs
   105 	       (t', Data.mk_bal (Data.mk_sum T terms1', 
   106 				 newshape(n2-n1,terms2'))))
   107   end
   108   handle TERM _ => NONE
   109        | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
   110 			     Undeclared type constructor "Numeral.bin"*)
   111 
   112 end;