src/Provers/Arith/cancel_numerals.ML
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(*  Title:      Provers/Arith/cancel_numerals.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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     i + #m*u + j ~~ i' + #m'*u + j'  ==  #(m-m')*u + i + j ~~ i' + j'
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where ~~ is an appropriate balancing operation (e.g. =, <=, <, -).
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It works by (a) massaging both sides to bring the selected term to the front:
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     #m*u + (i + j) ~~ #m'*u + (i' + j')
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(b) then using bal_add1 or bal_add2 to reach
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     #(m-m')*u + i + j ~~ i' + j'       (if m'<=m)
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or
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     i + j ~~ #(m'-m)*u + i' + j'       (otherwise)
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*)
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signature CANCEL_NUMERALS_DATA =
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sig
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  (*abstract syntax*)
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  val mk_sum: typ -> term list -> term
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  val dest_sum: term -> term list
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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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  val find_first_coeff: term -> term list -> int * term list
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  (*rules*)
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  val bal_add1: thm
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  val bal_add2: thm
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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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signature CANCEL_NUMERALS =
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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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functor CancelNumeralsFun(Data: CANCEL_NUMERALS_DATA): CANCEL_NUMERALS =
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struct
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(*For t = #n*u then put u in the table*)
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fun update_by_coeff t =
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  Termtab.update (#2 (Data.dest_coeff t), ());
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(*a left-to-right scan of terms1, seeking a term of the form #n*u, where
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  #m*u is in terms2 for some m*)
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fun find_common (terms1,terms2) =
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  let val tab2 = fold update_by_coeff terms2 Termtab.empty
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      fun seek [] = raise TERM("find_common", [])
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        | seek (t::terms) =
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              let val (_,u) = Data.dest_coeff t
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              in if Termtab.defined tab2 u then u else seek terms end
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  in  seek terms1 end;
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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 terms1 = Data.dest_sum t1
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    and terms2 = Data.dest_sum t2
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    val u = find_common (terms1, terms2)
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    val (n1, terms1') = Data.find_first_coeff u terms1
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    and (n2, terms2') = Data.find_first_coeff u terms2
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    and T = Term.fastype_of u
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    fun newshape (i,terms) = Data.mk_sum T (Data.mk_coeff(i,u)::terms)
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    val reshape =  (*Move i*u to the front and put j*u into standard form
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                       i + #m + j + k == #m + i + (j + k) *)
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        if n1=0 orelse n2=0 then   (*trivial, so do nothing*)
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          raise TERM("cancel_numerals", [])
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        else Data.prove_conv [Data.norm_tac ctxt] ctxt prems
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          (t', Data.mk_bal (newshape(n1,terms1'), newshape(n2,terms2')))
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  in
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    Option.map (export o Data.simplify_meta_eq ctxt)
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      (if n2 <= n1 then
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         Data.prove_conv
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           [Data.trans_tac reshape, rtac Data.bal_add1 1,
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            Data.numeral_simp_tac ctxt] ctxt prems
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           (t', Data.mk_bal (newshape(n1-n2,terms1'), Data.mk_sum T terms2'))
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       else
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         Data.prove_conv
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           [Data.trans_tac reshape, rtac Data.bal_add2 1,
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            Data.numeral_simp_tac ctxt] ctxt prems
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           (t', Data.mk_bal (Data.mk_sum T terms1', newshape(n2-n1,terms2'))))
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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;