src/Provers/Arith/cancel_numeral_factor.ML
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(*  Title:      Provers/Arith/cancel_numeral_factor.ML
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   2000  University of Cambridge
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Cancel common coefficients in balanced expressions:
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     u*#m ~~ u'*#m'  ==  #n*u ~~ #n'*u'
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where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
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and d = gcd(m,m') and n=m/d and n'=m'/d.
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It works by (a) massaging both sides to bring gcd(m,m') to the front:
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     u*#m ~~ u'*#m'  ==  #d*(#n*u) ~~ #d*(#n'*u')
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(b) then using the rule "cancel" to reach #n*u ~~ #n'*u'.
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*)
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signature CANCEL_NUMERAL_FACTOR_DATA =
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sig
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  (*abstract syntax*)
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  val mk_bal: term * term -> term
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  val dest_bal: term -> term * term
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  val mk_coeff: int * term -> term
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  val dest_coeff: term -> int * term
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  (*rules*)
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  val cancel: thm
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  val neg_exchanges: bool  (*true if a negative coeff swaps the two operands,
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                             as with < and <= but not = and div*)
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  (*proof tools*)
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  val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option
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  val trans_tac: thm option -> tactic                (*applies the initial lemma*)
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  val norm_tac: Proof.context -> tactic              (*proves the initial lemma*)
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  val numeral_simp_tac: Proof.context -> tactic      (*proves the final theorem*)
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  val simplify_meta_eq: Proof.context -> thm -> thm  (*simplifies the final theorem*)
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end;
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functor CancelNumeralFactorFun(Data: CANCEL_NUMERAL_FACTOR_DATA):
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  sig
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  val proc: Proof.context -> term -> thm option
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  end
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=
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struct
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(*the simplification procedure*)
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fun proc ctxt t =
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  let
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    val prems = Simplifier.prems_of ctxt;
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    val ([t'], ctxt') = Variable.import_terms true [t] ctxt
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    val export = singleton (Variable.export ctxt' ctxt)
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    (* FIXME ctxt vs. ctxt' *)
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    val (t1,t2) = Data.dest_bal t'
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    val (m1, t1') = Data.dest_coeff t1
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    and (m2, t2') = Data.dest_coeff t2
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    val d = (*if both are negative, also divide through by ~1*)
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      if (m1<0 andalso m2<=0) orelse
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         (m1<=0 andalso m2<0) then ~ (abs (Integer.gcd m1 m2)) else abs (Integer.gcd m1 m2)
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    val _ = if d=1 then   (*trivial, so do nothing*)
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              raise TERM("cancel_numeral_factor", [])
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            else ()
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    fun newshape (i,t) = Data.mk_coeff(d, Data.mk_coeff(i,t))
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    val n1 = m1 div d and n2 = m2 div d
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    val rhs = if d<0 andalso Data.neg_exchanges
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              then Data.mk_bal (Data.mk_coeff(n2,t2'), Data.mk_coeff(n1,t1'))
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              else Data.mk_bal (Data.mk_coeff(n1,t1'), Data.mk_coeff(n2,t2'))
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    val reshape =  (*Move d to the front and put the rest into standard form
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                       i * #m * j == #d * (#n * (j * k)) *)
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      Data.prove_conv [Data.norm_tac ctxt] ctxt prems
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        (t', Data.mk_bal (newshape(n1,t1'), newshape(n2,t2')))
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  in
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    Option.map (export o Data.simplify_meta_eq ctxt)
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      (Data.prove_conv
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         [Data.trans_tac reshape, rtac Data.cancel 1,
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          Data.numeral_simp_tac ctxt] ctxt prems (t', rhs))
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  end
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  (* FIXME avoid handling of generic exceptions *)
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  handle TERM _ => NONE
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       | TYPE _ => NONE;   (*Typically (if thy doesn't include Numeral)
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                             Undeclared type constructor "Numeral.bin"*)
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end;