(* Title: ZF/Tools/datatype_package.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Datatype/Codatatype Definitions
The functor will be instantiated for normal sums/products (datatype defs)
and non-standard sums/products (codatatype defs)
Sums are used only for mutual recursion;
Products are used only to derive "streamlined" induction rules for relations
*)
type datatype_result =
{con_defs : thm list, (*definitions made in thy*)
case_eqns : thm list, (*equations for case operator*)
recursor_eqns : thm list, (*equations for the recursor*)
free_iffs : thm list, (*freeness rewrite rules*)
free_SEs : thm list, (*freeness destruct rules*)
mk_free : string -> thm}; (*function to make freeness theorems*)
signature DATATYPE_ARG =
sig
val intrs : thm list
val elims : thm list
end;
(*Functor's result signature*)
signature DATATYPE_PACKAGE =
sig
(*Insert definitions for the recursive sets, which
must *already* be declared as constants in parent theory!*)
val add_datatype_i :
term * term list * Ind_Syntax.constructor_spec list list *
thm list * thm list * thm list
-> theory -> theory * inductive_result * datatype_result
val add_datatype :
string * string list *
(string * string list * mixfix) list list *
thm list * thm list * thm list
-> theory -> theory * inductive_result * datatype_result
end;
(*Declares functions to add fixedpoint/constructor defs to a theory.
Recursive sets must *already* be declared as constants.*)
functor Add_datatype_def_Fun
(structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU
and Ind_Package : INDUCTIVE_PACKAGE
and Datatype_Arg : DATATYPE_ARG)
: DATATYPE_PACKAGE =
struct
(*con_ty_lists specifies the constructors in the form (name,prems,mixfix) *)
fun add_datatype_i (dom_sum, rec_tms, con_ty_lists,
monos, type_intrs, type_elims) thy =
let
open BasisLibrary
val dummy = (*has essential ancestors?*)
Theory.requires thy "Datatype" "(co)datatype definitions";
val rec_names = map (#1 o dest_Const o head_of) rec_tms
val rec_base_names = map Sign.base_name rec_names
val big_rec_base_name = space_implode "_" rec_base_names
val thy_path = thy |> Theory.add_path big_rec_base_name
val sign = sign_of thy_path
val big_rec_name = Sign.intern_const sign big_rec_base_name;
val intr_tms = Ind_Syntax.mk_all_intr_tms sign (rec_tms, con_ty_lists)
val dummy =
writeln ((if (#1 (dest_Const Fp.oper) = "Fixedpt.lfp") then "Datatype"
else "Codatatype")
^ " definition " ^ big_rec_name)
val case_varname = "f"; (*name for case variables*)
(** Define the constructors **)
(*The empty tuple is 0*)
fun mk_tuple [] = Const("0",iT)
| mk_tuple args = foldr1 (app Pr.pair) args;
fun mk_inject n k u = access_bal (ap Su.inl, ap Su.inr, u) n k;
val npart = length rec_names; (*number of mutually recursive parts*)
val full_name = Sign.full_name sign;
(*Make constructor definition;
kpart is the number of this mutually recursive part*)
fun mk_con_defs (kpart, con_ty_list) =
let val ncon = length con_ty_list (*number of constructors*)
fun mk_def (((id,T,syn), name, args, prems), kcon) =
(*kcon is index of constructor*)
Logic.mk_defpair (list_comb (Const (full_name name, T), args),
mk_inject npart kpart
(mk_inject ncon kcon (mk_tuple args)))
in ListPair.map mk_def (con_ty_list, 1 upto ncon) end;
(*** Define the case operator ***)
(*Combine split terms using case; yields the case operator for one part*)
fun call_case case_list =
let fun call_f (free,[]) = Abs("null", iT, free)
| call_f (free,args) =
CP.ap_split (foldr1 CP.mk_prod (map (#2 o dest_Free) args))
Ind_Syntax.iT
free
in fold_bal (app Su.elim) (map call_f case_list) end;
(** Generating function variables for the case definition
Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)
(*The function variable for a single constructor*)
fun add_case (((_, T, _), name, args, _), (opno, cases)) =
if Syntax.is_identifier name then
(opno, (Free (case_varname ^ "_" ^ name, T), args) :: cases)
else
(opno + 1, (Free (case_varname ^ "_op_" ^ string_of_int opno, T), args)
:: cases);
(*Treatment of a list of constructors, for one part
Result adds a list of terms, each a function variable with arguments*)
fun add_case_list (con_ty_list, (opno, case_lists)) =
let val (opno', case_list) = foldr add_case (con_ty_list, (opno, []))
in (opno', case_list :: case_lists) end;
(*Treatment of all parts*)
val (_, case_lists) = foldr add_case_list (con_ty_lists, (1,[]));
(*extract the types of all the variables*)
val case_typ = flat (map (map (#2 o #1)) con_ty_lists) ---> (iT-->iT);
val case_base_name = big_rec_base_name ^ "_case";
val case_name = full_name case_base_name;
(*The list of all the function variables*)
val case_args = flat (map (map #1) case_lists);
val case_const = Const (case_name, case_typ);
val case_tm = list_comb (case_const, case_args);
val case_def = Logic.mk_defpair
(case_tm, fold_bal (app Su.elim) (map call_case case_lists));
(** Generating function variables for the recursor definition
Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)
(*a recursive call for x is the application rec`x *)
val rec_call = Ind_Syntax.apply_const $ Free ("rec", iT);
(*look back down the "case args" (which have been reversed) to
determine the de Bruijn index*)
fun make_rec_call ([], _) arg = error
"Internal error in datatype (variable name mismatch)"
| make_rec_call (a::args, i) arg =
if a = arg then rec_call $ Bound i
else make_rec_call (args, i+1) arg;
(*creates one case of the "X_case" definition of the recursor*)
fun call_recursor ((case_var, case_args), (recursor_var, recursor_args)) =
let fun add_abs (Free(a,T), u) = Abs(a,T,u)
val ncase_args = length case_args
val bound_args = map Bound ((ncase_args - 1) downto 0)
val rec_args = map (make_rec_call (rev case_args,0))
(List.drop(recursor_args, ncase_args))
in
foldr add_abs
(case_args, list_comb (recursor_var,
bound_args @ rec_args))
end
(*Find each recursive argument and add a recursive call for it*)
fun rec_args [] = []
| rec_args ((Const("op :",_)$arg$X)::prems) =
(case head_of X of
Const(a,_) => (*recursive occurrence?*)
if a mem_string rec_names
then arg :: rec_args prems
else rec_args prems
| _ => rec_args prems)
| rec_args (_::prems) = rec_args prems;
(*Add an argument position for each occurrence of a recursive set.
Strictly speaking, the recursive arguments are the LAST of the function
variable, but they all have type "i" anyway*)
fun add_rec_args args' T = (map (fn _ => iT) args') ---> T
(*Plug in the function variable type needed for the recursor
as well as the new arguments (recursive calls)*)
fun rec_ty_elem ((id, T, syn), name, args, prems) =
let val args' = rec_args prems
in ((id, add_rec_args args' T, syn),
name, args @ args', prems)
end;
val rec_ty_lists = (map (map rec_ty_elem) con_ty_lists);
(*Treatment of all parts*)
val (_, recursor_lists) = foldr add_case_list (rec_ty_lists, (1,[]));
(*extract the types of all the variables*)
val recursor_typ = flat (map (map (#2 o #1)) rec_ty_lists)
---> (iT-->iT);
val recursor_base_name = big_rec_base_name ^ "_rec";
val recursor_name = full_name recursor_base_name;
(*The list of all the function variables*)
val recursor_args = flat (map (map #1) recursor_lists);
val recursor_tm =
list_comb (Const (recursor_name, recursor_typ), recursor_args);
val recursor_cases = map call_recursor
(flat case_lists ~~ flat recursor_lists)
val recursor_def =
Logic.mk_defpair
(recursor_tm,
Ind_Syntax.Vrecursor_const $
absfree ("rec", iT, list_comb (case_const, recursor_cases)));
(* Build the new theory *)
val need_recursor =
(#1 (dest_Const Fp.oper) = "Fixedpt.lfp" andalso recursor_typ <> case_typ);
fun add_recursor thy =
if need_recursor then
thy |> Theory.add_consts_i
[(recursor_base_name, recursor_typ, NoSyn)]
|> PureThy.add_defs_i [Thm.no_attributes recursor_def]
else thy;
val thy0 = thy_path
|> Theory.add_consts_i
((case_base_name, case_typ, NoSyn) ::
map #1 (flat con_ty_lists))
|> PureThy.add_defs_i
(map Thm.no_attributes
(case_def ::
flat (ListPair.map mk_con_defs
(1 upto npart, con_ty_lists))))
|> add_recursor
|> Theory.parent_path
val con_defs = get_def thy0 case_name ::
map (get_def thy0 o #2) (flat con_ty_lists);
val (thy1, ind_result) =
thy0 |> Ind_Package.add_inductive_i
false (rec_tms, dom_sum, intr_tms,
monos, con_defs,
type_intrs @ Datatype_Arg.intrs,
type_elims @ Datatype_Arg.elims)
(**** Now prove the datatype theorems in this theory ****)
(*** Prove the case theorems ***)
(*Each equation has the form
case(f_con1,...,f_conn)(coni(args)) = f_coni(args) *)
fun mk_case_eqn (((_,T,_), name, args, _), case_free) =
FOLogic.mk_Trueprop
(FOLogic.mk_eq
(case_tm $
(list_comb (Const (Sign.intern_const (sign_of thy1) name,T),
args)),
list_comb (case_free, args)));
val case_trans = hd con_defs RS Ind_Syntax.def_trans
and split_trans = Pr.split_eq RS meta_eq_to_obj_eq RS trans;
(*Proves a single case equation. Could use simp_tac, but it's slower!*)
fun case_tacsf con_def _ =
[rewtac con_def,
rtac case_trans 1,
REPEAT (resolve_tac [refl, split_trans,
Su.case_inl RS trans,
Su.case_inr RS trans] 1)];
fun prove_case_eqn (arg,con_def) =
prove_goalw_cterm []
(Ind_Syntax.traceIt "next case equation = "
(cterm_of (sign_of thy1) (mk_case_eqn arg)))
(case_tacsf con_def);
val free_iffs = con_defs RL [Ind_Syntax.def_swap_iff];
val case_eqns =
map prove_case_eqn
(flat con_ty_lists ~~ case_args ~~ tl con_defs);
(*** Prove the recursor theorems ***)
val recursor_eqns = case try (get_def thy1) recursor_base_name of
None => (writeln " [ No recursion operator ]";
[])
| Some recursor_def =>
let
(*Replace subterms rec`x (where rec is a Free var) by recursor_tm(x) *)
fun subst_rec (Const("op `",_) $ Free _ $ arg) = recursor_tm $ arg
| subst_rec tm =
let val (head, args) = strip_comb tm
in list_comb (head, map subst_rec args) end;
(*Each equation has the form
REC(coni(args)) = f_coni(args, REC(rec_arg), ...)
where REC = recursor(f_con1,...,f_conn) and rec_arg is a recursive
constructor argument.*)
fun mk_recursor_eqn (((_,T,_), name, args, _), recursor_case) =
FOLogic.mk_Trueprop
(FOLogic.mk_eq
(recursor_tm $
(list_comb (Const (Sign.intern_const (sign_of thy1) name,T),
args)),
subst_rec (foldl betapply (recursor_case, args))));
val recursor_trans = recursor_def RS def_Vrecursor RS trans;
(*Proves a single recursor equation.*)
fun recursor_tacsf _ =
[rtac recursor_trans 1,
simp_tac (rank_ss addsimps case_eqns) 1,
IF_UNSOLVED (simp_tac (rank_ss addsimps tl con_defs) 1)];
fun prove_recursor_eqn arg =
prove_goalw_cterm []
(Ind_Syntax.traceIt "next recursor equation = "
(cterm_of (sign_of thy1) (mk_recursor_eqn arg)))
recursor_tacsf
in
map prove_recursor_eqn (flat con_ty_lists ~~ recursor_cases)
end
val constructors =
map (head_of o #1 o Logic.dest_equals o #prop o rep_thm) (tl con_defs);
val free_SEs = Ind_Syntax.mk_free_SEs free_iffs;
val {elim, induct, mutual_induct, ...} = ind_result
(*Typical theorems have the form ~con1=con2, con1=con2==>False,
con1(x)=con1(y) ==> x=y, con1(x)=con1(y) <-> x=y, etc. *)
fun mk_free s =
prove_goalw (theory_of_thm elim) (*Don't use thy1: it will be stale*)
con_defs s
(fn prems => [cut_facts_tac prems 1,
fast_tac (ZF_cs addSEs free_SEs @ Su.free_SEs) 1]);
val simps = case_eqns @ recursor_eqns;
val dt_info =
{inductive = true,
constructors = constructors,
rec_rewrites = recursor_eqns,
case_rewrites = case_eqns,
induct = induct,
mutual_induct = mutual_induct,
exhaustion = elim};
val con_info =
{big_rec_name = big_rec_name,
constructors = constructors,
(*let primrec handle definition by cases*)
rec_rewrites = (case recursor_eqns of
[] => case_eqns | _ => recursor_eqns)};
(*associate with each constructor the datatype name and rewrites*)
val con_pairs = map (fn c => (#1 (dest_Const c), con_info)) constructors
in
(*Updating theory components: simprules and datatype info*)
(thy1 |> Theory.add_path big_rec_base_name
|> PureThy.add_thmss [(("simps", simps), [Simplifier.simp_add_global])]
|> DatatypesData.put
(Symtab.update
((big_rec_name, dt_info), DatatypesData.get thy1))
|> ConstructorsData.put
(foldr Symtab.update (con_pairs, ConstructorsData.get thy1))
|> Theory.parent_path,
ind_result,
{con_defs = con_defs,
case_eqns = case_eqns,
recursor_eqns = recursor_eqns,
free_iffs = free_iffs,
free_SEs = free_SEs,
mk_free = mk_free})
end;
fun add_datatype (sdom, srec_tms, scon_ty_lists,
monos, type_intrs, type_elims) thy =
let val sign = sign_of thy
val rec_tms = map (readtm sign Ind_Syntax.iT) srec_tms
val con_ty_lists = Ind_Syntax.read_constructs sign scon_ty_lists
val dom_sum =
if sdom = "" then
Ind_Syntax.data_domain (#1(dest_Const Fp.oper) <> "Fixedpt.lfp")
(rec_tms, con_ty_lists)
else readtm sign Ind_Syntax.iT sdom
in
add_datatype_i (dom_sum, rec_tms, con_ty_lists,
monos, type_intrs, type_elims) thy
end
end;