src/Provers/Arith/cancel_numerals.ML
author paulson
Fri Apr 21 11:31:03 2000 +0200 (2000-04-21)
changeset 8760 9139453d7033
parent 8736 0bfd6678a5fa
child 8772 ebb07113c4f7
permissions -rw-r--r--
now works for coefficients, not just for numerals
no longer works by subtraction, so no need for inverse_fold
     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: 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: int * term -> term
    33   val dest_coeff: term -> int * term
    34   val find_first_coeff: term -> term list -> int * term list
    35   (*rules*)
    36   val bal_add1: thm
    37   val bal_add2: thm
    38   (*proof tools*)
    39   val prove_conv: tactic list -> Sign.sg -> term * term -> thm option
    40   val norm_tac: tactic
    41   val numeral_simp_tac: tactic
    42 end;
    43 
    44 
    45 functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
    46   sig
    47   val proc: Sign.sg -> thm list -> term -> thm option
    48   end 
    49 =
    50 struct
    51 
    52 fun listof None = []
    53   | listof (Some x) = [x];
    54 
    55 (*If 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   handle TERM _ => tab;
    59 
    60 (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
    61   #m*u is in terms2 for some m*)
    62 fun find_common (terms1,terms2) =
    63   let val tab2 = foldl update_by_coeff (Termtab.empty, terms2)
    64       fun seek [] = raise TERM("find_common", []) 
    65 	| seek (t::terms) =
    66 	      let val (_,u) = Data.dest_coeff t 
    67 	      in  if is_some (Termtab.lookup (tab2, u)) then u
    68 		  else seek terms
    69 	      end
    70 	      handle TERM _ => seek terms
    71   in  seek terms1 end;
    72 
    73 (*the simplification procedure*)
    74 fun proc sg _ t =
    75   let val (t1,t2) = Data.dest_bal t 
    76       val terms1 = Data.dest_sum t1
    77       and terms2 = Data.dest_sum t2
    78       val u = find_common (terms1,terms2)
    79       val (n1, terms1') = Data.find_first_coeff u terms1
    80       and (n2, terms2') = Data.find_first_coeff u terms2
    81       fun newshape (i,terms) = Data.mk_sum (Data.mk_coeff(i,u)::terms)
    82       val reshapes =  (*Move i*u to the front and put j*u into standard form
    83 		      i + #m + j + k == #m + i + (j + k) *)
    84 	    listof (Data.prove_conv [Data.norm_tac] sg 
    85 		    (t, 
    86 		     Data.mk_bal (newshape(n1,terms1'), 
    87 				  newshape(n2,terms2'))))
    88   in
    89 
    90       if n2<=n1 then 
    91 	  Data.prove_conv 
    92 	     [rewrite_goals_tac reshapes, rtac Data.bal_add1 1,
    93 	      Data.numeral_simp_tac] sg
    94 	     (t, Data.mk_bal (newshape(n1-n2,terms1'), Data.mk_sum terms2'))
    95       else
    96 	  Data.prove_conv 
    97 	     [rewrite_goals_tac reshapes, rtac Data.bal_add2 1,
    98 	      Data.numeral_simp_tac] sg
    99 	     (t, Data.mk_bal (Data.mk_sum terms1', newshape(n2-n1,terms2')))
   100   end
   101   handle TERM _ => None;
   102 
   103 end;