src/ZF/inductive.ML

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(*  Title: 	ZF/inductive.ML
    ID:         $Id$
    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

(Co)Inductive Definitions for Zermelo-Fraenkel Set Theory

Inductive definitions use least fixedpoints with standard products and sums
Coinductive definitions use greatest fixedpoints with Quine products and sums

Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)

local open Ind_Syntax
in
structure Lfp =
  struct
  val oper	= Const("lfp",      [iT,iT-->iT]--->iT)
  val bnd_mono	= Const("bnd_mono", [iT,iT-->iT]--->oT)
  val bnd_monoI	= bnd_monoI
  val subs	= def_lfp_subset
  val Tarski	= def_lfp_Tarski
  val induct	= def_induct
  end;

structure Standard_Prod =
  struct
  val sigma	= Const("Sigma", [iT, iT-->iT]--->iT)
  val pair	= Const("Pair", [iT,iT]--->iT)
  val split_const	= Const("split", [[iT,iT]--->iT, iT]--->iT)
  val fsplit_const	= Const("fsplit", [[iT,iT]--->oT, iT]--->oT)
  val pair_iff	= Pair_iff
  val split_eq	= split
  val fsplitI	= fsplitI
  val fsplitD	= fsplitD
  val fsplitE	= fsplitE
  end;

structure Standard_Sum =
  struct
  val sum	= Const("op +", [iT,iT]--->iT)
  val inl	= Const("Inl", iT-->iT)
  val inr	= Const("Inr", iT-->iT)
  val elim	= Const("case", [iT-->iT, iT-->iT, iT]--->iT)
  val case_inl	= case_Inl
  val case_inr	= case_Inr
  val inl_iff	= Inl_iff
  val inr_iff	= Inr_iff
  val distinct	= Inl_Inr_iff
  val distinct' = Inr_Inl_iff
  end;
end;

functor Ind_section_Fun (Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end) 
  : sig include INTR_ELIM INDRULE end =
struct
structure Intr_elim = 
    Intr_elim_Fun(structure Inductive=Inductive and Fp=Lfp and 
		  Pr=Standard_Prod and Su=Standard_Sum);

structure Indrule = Indrule_Fun (structure Inductive=Inductive and 
		                 Pr=Standard_Prod and Intr_elim=Intr_elim);

open Intr_elim Indrule
end;


structure Ind = Add_inductive_def_Fun
    (structure Fp=Lfp and Pr=Standard_Prod and Su=Standard_Sum);


signature INDUCTIVE_STRING =
  sig
  val thy_name   : string 		(*name of the new theory*)
  val rec_doms   : (string*string) list	(*recursion terms and their domains*)
  val sintrs     : string list		(*desired introduction rules*)
  end;


(*For upwards compatibility: can be called directly from ML*)
functor Inductive_Fun
 (Inductive: sig include INDUCTIVE_STRING INDUCTIVE_ARG end)
   : sig include INTR_ELIM INDRULE end =
Ind_section_Fun
   (open Inductive Ind_Syntax
    val sign = sign_of thy;
    val rec_tms = map (readtm sign iT o #1) rec_doms
    and domts   = map (readtm sign iT o #2) rec_doms
    and intr_tms = map (readtm sign propT) sintrs;
    val thy = thy |> Ind.add_fp_def_i(rec_tms, domts, intr_tms) 
                  |> add_thyname thy_name);



local open Ind_Syntax
in
structure Gfp =
  struct
  val oper	= Const("gfp",      [iT,iT-->iT]--->iT)
  val bnd_mono	= Const("bnd_mono", [iT,iT-->iT]--->oT)
  val bnd_monoI	= bnd_monoI
  val subs	= def_gfp_subset
  val Tarski	= def_gfp_Tarski
  val induct	= def_Collect_coinduct
  end;

structure Quine_Prod =
  struct
  val sigma	= Const("QSigma", [iT, iT-->iT]--->iT)
  val pair	= Const("QPair", [iT,iT]--->iT)
  val split_const	= Const("qsplit", [[iT,iT]--->iT, iT]--->iT)
  val fsplit_const	= Const("qfsplit", [[iT,iT]--->oT, iT]--->oT)
  val pair_iff	= QPair_iff
  val split_eq	= qsplit
  val fsplitI	= qfsplitI
  val fsplitD	= qfsplitD
  val fsplitE	= qfsplitE
  end;

structure Quine_Sum =
  struct
  val sum	= Const("op <+>", [iT,iT]--->iT)
  val inl	= Const("QInl", iT-->iT)
  val inr	= Const("QInr", iT-->iT)
  val elim	= Const("qcase", [iT-->iT, iT-->iT, iT]--->iT)
  val case_inl	= qcase_QInl
  val case_inr	= qcase_QInr
  val inl_iff	= QInl_iff
  val inr_iff	= QInr_iff
  val distinct	= QInl_QInr_iff
  val distinct' = QInr_QInl_iff
  end;
end;


signature COINDRULE =
  sig
  val coinduct : thm
  end;


functor CoInd_section_Fun
 (Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end) 
    : sig include INTR_ELIM COINDRULE end =
struct
structure Intr_elim = 
    Intr_elim_Fun(structure Inductive=Inductive and Fp=Gfp and 
		  Pr=Quine_Prod and Su=Quine_Sum);

open Intr_elim 
val coinduct = raw_induct
end;


structure CoInd = 
    Add_inductive_def_Fun(structure Fp=Gfp and Pr=Quine_Prod and Su=Quine_Sum);


(*For upwards compatibility: can be called directly from ML*)
functor CoInductive_Fun
 (Inductive: sig include INDUCTIVE_STRING INDUCTIVE_ARG end)
   : sig include INTR_ELIM COINDRULE end =
CoInd_section_Fun
   (open Inductive Ind_Syntax
    val sign = sign_of thy;
    val rec_tms = map (readtm sign iT o #1) rec_doms
    and domts   = map (readtm sign iT o #2) rec_doms
    and intr_tms = map (readtm sign propT) sintrs;
    val thy = thy |> CoInd.add_fp_def_i(rec_tms, domts, intr_tms) 
                  |> add_thyname thy_name);



(*For installing the theory section.   co is either "" or "Co"*)
fun inductive_decl co =
  let open ThyParse Ind_Syntax
      fun mk_intr_name (s,_) =  (*the "op" cancels any infix status*)
	  if Syntax.is_identifier s then "op " ^ s  else "_"
      fun mk_params (((((domains: (string*string) list, ipairs), 
			monos), con_defs), type_intrs), type_elims) =
        let val big_rec_name = space_implode "_" 
		             (map (scan_to_id o trim o #1) domains)
	    and srec_tms = mk_list (map #1 domains)
            and sdoms    = mk_list (map #2 domains)
	    and sintrs   = mk_big_list (map snd ipairs)
            val stri_name = big_rec_name ^ "_Intrnl"
        in
	   (";\n\n\
            \structure " ^ stri_name ^ " =\n\
            \ let open Ind_Syntax in\n\
            \  struct\n\
            \  val rec_tms\t= map (readtm (sign_of thy) iT) "
	                     ^ srec_tms ^ "\n\
            \  and domts\t= map (readtm (sign_of thy) iT) "
	                     ^ sdoms ^ "\n\
            \  and intr_tms\t= map (readtm (sign_of thy) propT)\n"
	                     ^ sintrs ^ "\n\
            \  end\n\
            \ end;\n\n\
            \val thy = thy |> " ^ co ^ "Ind.add_fp_def_i \n    (" ^ 
	       stri_name ^ ".rec_tms, " ^
               stri_name ^ ".domts, " ^
               stri_name ^ ".intr_tms)"
           ,
	    "structure " ^ big_rec_name ^ " =\n\
            \  struct\n\
            \  val _ = writeln \"" ^ co ^ 
	               "Inductive definition " ^ big_rec_name ^ "\"\n\
            \  structure Result = " ^ co ^ "Ind_section_Fun\n\
            \  (open " ^ stri_name ^ "\n\
            \   val thy\t\t= thy\n\
            \   val monos\t\t= " ^ monos ^ "\n\
            \   val con_defs\t\t= " ^ con_defs ^ "\n\
            \   val type_intrs\t= " ^ type_intrs ^ "\n\
            \   val type_elims\t= " ^ type_elims ^ ");\n\n\
            \  val " ^ mk_list (map mk_intr_name ipairs) ^ " = Result.intrs;\n\
            \  open Result\n\
            \  end\n"
	   )
	end
      val domains = "domains" $$-- repeat1 (string --$$ "<=" -- !! string)
      val ipairs  = "intrs"   $$-- repeat1 (ident -- !! string)
      fun optstring s = optional (s $$-- string) "\"[]\"" >> trim
  in domains -- ipairs -- optstring "monos" -- optstring "con_defs"
             -- optstring "type_intrs" -- optstring "type_elims"
     >> mk_params
  end;