1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/Provers/Arith/cancel_numeral_factor.ML	Wed Nov 29 10:22:38 2000 +0100
     1.3 @@ -0,0 +1,87 @@
     1.4 +(*  Title:      Provers/Arith/cancel_numeral_factor.ML
     1.5 +    ID:         $Id$
     1.6 +    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     1.7 +    Copyright   2000  University of Cambridge
     1.8 +
     1.9 +Cancel common coefficients in balanced expressions:
    1.10 +
    1.11 +     u*#m ~~ u'*#m'  ==  #n*u ~~ #n'*u'
    1.12 +
    1.13 +where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
    1.14 +and d = gcd(m,m') and n=m/d and n'=m'/d.
    1.15 +
    1.16 +It works by (a) massaging both sides to bring gcd(m,m') to the front:
    1.17 +
    1.18 +     u*#m ~~ u'*#m'  ==  #d*(#n*u) ~~ #d*(#n'*u')
    1.19 +
    1.20 +(b) then using the rule "cancel" to reach #n*u ~~ #n'*u'.
    1.21 +*)
    1.22 +
    1.23 +signature CANCEL_NUMERAL_FACTOR_DATA =
    1.24 +sig
    1.25 +  (*abstract syntax*)
    1.26 +  val mk_bal: term * term -> term
    1.27 +  val dest_bal: term -> term * term
    1.28 +  val mk_coeff: int * term -> term
    1.29 +  val dest_coeff: term -> int * term
    1.30 +  (*rules*)
    1.31 +  val cancel: thm
    1.32 +  val neg_exchanges: bool  (*true if a negative coeff swaps the two operands,
    1.33 +                             as with < and <= but not = and div*)
    1.34 +  (*proof tools*)
    1.35 +  val prove_conv: tactic list -> Sign.sg -> 
    1.36 +                  thm list -> term * term -> thm option
    1.37 +  val trans_tac: thm option -> tactic (*applies the initial lemma*)
    1.38 +  val norm_tac: tactic                (*proves the initial lemma*)
    1.39 +  val numeral_simp_tac: tactic        (*proves the final theorem*)
    1.40 +  val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
    1.41 +end;
    1.42 +
    1.43 +
    1.44 +functor CancelNumeralFactorFun(Data: CANCEL_NUMERAL_FACTOR_DATA):
    1.45 +  sig
    1.46 +  val proc: Sign.sg -> thm list -> term -> thm option
    1.47 +  end 
    1.48 +=
    1.49 +struct
    1.50 +
    1.51 +
    1.52 +(* greatest common divisor *)
    1.53 +fun gcd (0, n) = abs n
    1.54 +  | gcd (m, n) = gcd (n mod m, m);
    1.55 +
    1.56 +(*the simplification procedure*)
    1.57 +fun proc sg hyps t =
    1.58 +  let (*first freeze any Vars in the term to prevent flex-flex problems*)
    1.59 +      val rand_s = gensym"_"
    1.60 +      fun mk_inst (var as Var((a,i),T))  = 
    1.61 +	    (var,  Free((a ^ rand_s ^ string_of_int i), T))
    1.62 +      val t' = subst_atomic (map mk_inst (term_vars t)) t
    1.63 +      val (t1,t2) = Data.dest_bal t' 
    1.64 +      val (m1, t1') = Data.dest_coeff t1
    1.65 +      and (m2, t2') = Data.dest_coeff t2
    1.66 +      val d = (*if both are negative, also divide through by ~1*)
    1.67 +          if m1<0 andalso m2<0 then ~ (gcd(m1,m2)) else gcd(m1,m2)
    1.68 +      val _ = if d=1 then   (*trivial, so do nothing*)
    1.69 +		      raise TERM("cancel_numeral_factor", []) 
    1.70 +              else ()
    1.71 +      fun newshape (i,t) = Data.mk_coeff(d, Data.mk_coeff(i,t))
    1.72 +      val n1 = m1 div d and n2 = m2 div d
    1.73 +      val rhs = if d<0 andalso Data.neg_exchanges
    1.74 +                then Data.mk_bal (Data.mk_coeff(n2,t2'), Data.mk_coeff(n1,t1'))
    1.75 +                else Data.mk_bal (Data.mk_coeff(n1,t1'), Data.mk_coeff(n2,t2'))
    1.76 +      val reshape =  (*Move d to the front and put the rest into standard form
    1.77 +		       i * #m * j == #d * (#n * (j * k)) *)
    1.78 +	    Data.prove_conv [Data.norm_tac] sg hyps 
    1.79 +	      (t',   Data.mk_bal (newshape(n1,t1'), newshape(n2,t2')))
    1.80 +  in
    1.81 +      apsome Data.simplify_meta_eq
    1.82 +       (Data.prove_conv 
    1.83 +	       [Data.trans_tac reshape, rtac Data.cancel 1,
    1.84 +		Data.numeral_simp_tac] sg hyps (t', rhs))
    1.85 +  end
    1.86 +  handle TERM _ => None
    1.87 +       | TYPE _ => None;   (*Typically (if thy doesn't include Numeral)
    1.88 +			     Undeclared type constructor "Numeral.bin"*)
    1.89 +
    1.90 +end;