tries harder to remove negative literals, e.g.
-2*x = 0 goes to x=0 rather than -x=0.
(* Title: Provers/Arith/cancel_numeral_factor.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2000 University of Cambridge
Cancel common coefficients in balanced expressions:
u*#m ~~ u'*#m' == #n*u ~~ #n'*u'
where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
and d = gcd(m,m') and n=m/d and n'=m'/d.
It works by (a) massaging both sides to bring gcd(m,m') to the front:
u*#m ~~ u'*#m' == #d*(#n*u) ~~ #d*(#n'*u')
(b) then using the rule "cancel" to reach #n*u ~~ #n'*u'.
*)
signature CANCEL_NUMERAL_FACTOR_DATA =
sig
(*abstract syntax*)
val mk_bal: term * term -> term
val dest_bal: term -> term * term
val mk_coeff: int * term -> term
val dest_coeff: term -> int * term
(*rules*)
val cancel: thm
val neg_exchanges: bool (*true if a negative coeff swaps the two operands,
as with < and <= but not = and div*)
(*proof tools*)
val prove_conv: tactic list -> Sign.sg ->
thm list -> term * term -> thm option
val trans_tac: thm option -> tactic (*applies the initial lemma*)
val norm_tac: tactic (*proves the initial lemma*)
val numeral_simp_tac: tactic (*proves the final theorem*)
val simplify_meta_eq: thm -> thm (*simplifies the final theorem*)
end;
functor CancelNumeralFactorFun(Data: CANCEL_NUMERAL_FACTOR_DATA):
sig
val proc: Sign.sg -> thm list -> term -> thm option
end
=
struct
(* greatest common divisor *)
fun gcd (0, n) = abs n
| gcd (m, n) = gcd (n mod m, m);
(*the simplification procedure*)
fun proc sg hyps t =
let (*first freeze any Vars in the term to prevent flex-flex problems*)
val rand_s = gensym"_"
fun mk_inst (var as Var((a,i),T)) =
(var, Free((a ^ rand_s ^ string_of_int i), T))
val t' = subst_atomic (map mk_inst (term_vars t)) t
val (t1,t2) = Data.dest_bal t'
val (m1, t1') = Data.dest_coeff t1
and (m2, t2') = Data.dest_coeff t2
val d = (*if both are negative, also divide through by ~1*)
if (m1<0 andalso m2<=0) orelse
(m1<=0 andalso m2<0) then ~ (gcd(m1,m2)) else gcd(m1,m2)
val _ = if d=1 then (*trivial, so do nothing*)
raise TERM("cancel_numeral_factor", [])
else ()
fun newshape (i,t) = Data.mk_coeff(d, Data.mk_coeff(i,t))
val n1 = m1 div d and n2 = m2 div d
val rhs = if d<0 andalso Data.neg_exchanges
then Data.mk_bal (Data.mk_coeff(n2,t2'), Data.mk_coeff(n1,t1'))
else Data.mk_bal (Data.mk_coeff(n1,t1'), Data.mk_coeff(n2,t2'))
val reshape = (*Move d to the front and put the rest into standard form
i * #m * j == #d * (#n * (j * k)) *)
Data.prove_conv [Data.norm_tac] sg hyps
(t', Data.mk_bal (newshape(n1,t1'), newshape(n2,t2')))
in
apsome Data.simplify_meta_eq
(Data.prove_conv
[Data.trans_tac reshape, rtac Data.cancel 1,
Data.numeral_simp_tac] sg hyps (t', rhs))
end
handle TERM _ => None
| TYPE _ => None; (*Typically (if thy doesn't include Numeral)
Undeclared type constructor "Numeral.bin"*)
end;