src/Provers/Arith/cancel_numerals.ML
author paulson
Fri May 05 17:49:54 2000 +0200 (2000-05-05)
changeset 8799 89e9deef4bcb
parent 8779 2d4afbc46801
child 9191 300bd596d696
permissions -rw-r--r--
simprocs now simplify the RHS of their result
     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 trans_tac: thm option -> tactic (*applies the initial lemma*)
    41   val norm_tac: tactic                (*proves the initial lemma*)
    42   val numeral_simp_tac: tactic        (*proves the final theorem*)
    43   val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
    44 end;
    45 
    46 
    47 functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA):
    48   sig
    49   val proc: Sign.sg -> thm list -> term -> thm option
    50   end 
    51 =
    52 struct
    53 
    54 (*For t = #n*u then put u in the table*)
    55 fun update_by_coeff (tab, t) =
    56   Termtab.update ((#2 (Data.dest_coeff t), ()), tab);
    57 
    58 (*a left-to-right scan of terms1, seeking a term of the form #n*u, where
    59   #m*u is in terms2 for some m*)
    60 fun find_common (terms1,terms2) =
    61   let val tab2 = foldl update_by_coeff (Termtab.empty, terms2)
    62       fun seek [] = raise TERM("find_common", []) 
    63 	| seek (t::terms) =
    64 	      let val (_,u) = Data.dest_coeff t 
    65 	      in  if is_some (Termtab.lookup (tab2, u)) then u
    66 		  else seek terms
    67 	      end
    68   in  seek terms1 end;
    69 
    70 (*the simplification procedure*)
    71 fun proc sg _ t =
    72   let val (t1,t2) = Data.dest_bal t 
    73       val terms1 = Data.dest_sum t1
    74       and terms2 = Data.dest_sum t2
    75       val u = find_common (terms1,terms2)
    76       val (n1, terms1') = Data.find_first_coeff u terms1
    77       and (n2, terms2') = Data.find_first_coeff u terms2
    78       fun newshape (i,terms) = Data.mk_sum (Data.mk_coeff(i,u)::terms)
    79       val reshape =  (*Move i*u to the front and put j*u into standard form
    80 		       i + #m + j + k == #m + i + (j + k) *)
    81 	    if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
    82 		raise TERM("cancel_numerals", []) 
    83 	    else Data.prove_conv [Data.norm_tac] sg 
    84 			(t, 
    85 			 Data.mk_bal (newshape(n1,terms1'), 
    86 				      newshape(n2,terms2')))
    87   in
    88 
    89       apsome Data.simplify_meta_eq
    90        (if n2<=n1 then 
    91 	    Data.prove_conv 
    92 	       [Data.trans_tac reshape, 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 	       [Data.trans_tac reshape, 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        | TYPE _ => None;   (*Typically (if thy doesn't include Numeral)
   103 			     Undeclared type constructor "Numeral.bin"*)
   104 
   105 end;