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(* Title: Provers/Arith/cancel_numeral_factor.ML
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 -> Proof.context -> thm list -> term * term -> thm option
val trans_tac: Proof.context -> thm option -> tactic (*applies the initial lemma*)
val norm_tac: Proof.context -> tactic (*proves the initial lemma*)
val numeral_simp_tac: Proof.context -> tactic (*proves the final theorem*)
val simplify_meta_eq: Proof.context -> thm -> thm (*simplifies the final theorem*)
end;
functor CancelNumeralFactorFun(Data: CANCEL_NUMERAL_FACTOR_DATA):
sig val proc: Proof.context -> cterm -> thm option end =
struct
(*the simplification procedure*)
fun proc ctxt ct =
let
val prems = Simplifier.prems_of ctxt;
val t = Thm.term_of ct;
val (t', ctxt') = yield_singleton (Variable.import_terms true) t ctxt
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 ~ (abs (Integer.gcd m1 m2)) else abs (Integer.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 ctxt'] ctxt' prems
(t', Data.mk_bal (newshape(n1,t1'), newshape(n2,t2')))
in
Data.prove_conv
[Data.trans_tac ctxt' reshape, resolve_tac ctxt' [Data.cancel] 1,
Data.numeral_simp_tac ctxt'] ctxt' prems (t', rhs)
|> Option.map
(Data.simplify_meta_eq ctxt' #>
singleton (Variable.export ctxt' ctxt))
end
(* FIXME avoid handling of generic exceptions *)
handle TERM _ => NONE
| TYPE _ => NONE; (*Typically (if thy doesn't include Numeral)
Undeclared type constructor "Numeral.bin"*)
end;
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