(* Title: ZF/Tools/datatype_package.ML
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;
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 ->
(Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list *
(Facts.ref * Attrib.src list) list -> theory -> theory * inductive_result * datatype_result
end;
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
val coind : bool): DATATYPE_PACKAGE =
struct
(*con_ty_lists specifies the constructors in the form (name, prems, mixfix) *)
(*univ or quniv constitutes the sum domain for mutual recursion;
it is applied to the datatype parameters and to Consts occurring in the
definition other than Nat.nat and the datatype sets themselves.
FIXME: could insert all constant set expressions, e.g. nat->nat.*)
fun data_domain co (rec_tms, con_ty_lists) =
let val rec_hds = map head_of rec_tms
val dummy = assert_all is_Const rec_hds
(fn t => "Datatype set not previously declared as constant: " ^
Syntax.string_of_term_global @{theory IFOL} t);
val rec_names = (*nat doesn't have to be added*)
@{const_name nat} :: map (#1 o dest_Const) rec_hds
val u = if co then @{const QUniv.quniv} else @{const Univ.univ}
val cs = (fold o fold) (fn (_, _, _, prems) => prems |> (fold o fold_aterms)
(fn t as Const (a, _) => if member (op =) rec_names a then I else insert (op =) t
| _ => I)) con_ty_lists [];
in u $ Ind_Syntax.union_params (hd rec_tms, cs) end;
fun add_datatype_i (dom_sum, rec_tms) con_ty_lists (monos, type_intrs, type_elims) thy =
let
val dummy = (*has essential ancestors?*)
Theory.requires thy "Datatype_ZF" "(co)datatype definitions";
val rec_hds = map head_of rec_tms;
val dummy = assert_all is_Const rec_hds
(fn t => "Datatype set not previously declared as constant: " ^
Syntax.string_of_term_global thy t);
val rec_names = map (#1 o dest_Const) rec_hds
val rec_base_names = map Long_Name.base_name rec_names
val big_rec_base_name = space_implode "_" rec_base_names
val thy_path = thy |> Sign.add_path big_rec_base_name
val big_rec_name = Sign.intern_const thy_path big_rec_base_name;
val intr_tms = Ind_Syntax.mk_all_intr_tms thy_path (rec_tms, con_ty_lists);
val dummy =
writeln ((if coind then "Codatatype" else "Datatype") ^ " definition " ^ quote big_rec_name);
val case_varname = "f"; (*name for case variables*)
(** Define the constructors **)
(*The empty tuple is 0*)
fun mk_tuple [] = @{const zero}
| mk_tuple args = foldr1 (fn (t1, t2) => Pr.pair $ t1 $ t2) args;
fun mk_inject n k u = Balanced_Tree.access
{left = fn t => Su.inl $ t, right = fn t => Su.inr $ t, init = u} n k;
val npart = length rec_names; (*number of mutually recursive parts*)
val full_name = Sign.full_bname thy_path;
(*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*)
Misc_Legacy.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", @{typ i}, free)
| call_f (free,args) =
CP.ap_split (foldr1 CP.mk_prod (map (#2 o dest_Free) args))
@{typ i}
free
in Balanced_Tree.make (fn (t1, t2) => Su.elim $ t1 $ t2) (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) = fold_rev add_case con_ty_list (opno, [])
in (opno', case_list :: case_lists) end;
(*Treatment of all parts*)
val (_, case_lists) = fold_rev add_case_list con_ty_lists (1, []);
(*extract the types of all the variables*)
val case_typ = maps (map (#2 o #1)) con_ty_lists ---> @{typ "i => i"};
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 = maps (map #1) case_lists;
val case_const = Const (case_name, case_typ);
val case_tm = list_comb (case_const, case_args);
val case_def = Misc_Legacy.mk_defpair
(case_tm, Balanced_Tree.make (fn (t1, t2) => Su.elim $ t1 $ t2) (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 = @{const apply} $ Free ("rec", @{typ i});
(*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
List.foldr add_abs
(list_comb (recursor_var,
bound_args @ rec_args)) case_args
end
(*Find each recursive argument and add a recursive call for it*)
fun rec_args [] = []
| rec_args ((Const(@{const_name mem},_)$arg$X)::prems) =
(case head_of X of
Const(a,_) => (*recursive occurrence?*)
if member (op =) rec_names a
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 _ => @{typ i}) 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) = fold_rev add_case_list rec_ty_lists (1, []);
(*extract the types of all the variables*)
val recursor_typ = maps (map (#2 o #1)) rec_ty_lists ---> @{typ "i => i"};
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 = maps (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 =
Misc_Legacy.mk_defpair
(recursor_tm,
@{const Univ.Vrecursor} $
absfree ("rec", @{typ i}, list_comb (case_const, recursor_cases)));
(* Build the new theory *)
val need_recursor = (not coind andalso recursor_typ <> case_typ);
fun add_recursor thy =
if need_recursor then
thy
|> Sign.add_consts_i
[(Binding.name recursor_base_name, recursor_typ, NoSyn)]
|> (snd o Global_Theory.add_defs false [(Thm.no_attributes o apfst Binding.name) recursor_def])
else thy;
val (con_defs, thy0) = thy_path
|> Sign.add_consts_i
(map (fn (c, T, mx) => (Binding.name c, T, mx))
((case_base_name, case_typ, NoSyn) :: map #1 (flat con_ty_lists)))
|> Global_Theory.add_defs false
(map (Thm.no_attributes o apfst Binding.name)
(case_def ::
flat (ListPair.map mk_con_defs (1 upto npart, con_ty_lists))))
||> add_recursor
||> Sign.parent_path
val intr_names = map (Binding.name o #2) (flat con_ty_lists);
val (thy1, ind_result) =
thy0 |> Ind_Package.add_inductive_i
false (rec_tms, dom_sum) (map Thm.no_attributes (intr_names ~~ 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 thy1 name,T),
args)),
list_comb (case_free, args)));
val case_trans = hd con_defs RS @{thm def_trans}
and split_trans = Pr.split_eq RS @{thm meta_eq_to_obj_eq} RS @{thm trans};
fun prove_case_eqn (arg, con_def) =
Goal.prove_global thy1 [] []
(Ind_Syntax.traceIt "next case equation = " thy1 (mk_case_eqn arg))
(*Proves a single case equation. Could use simp_tac, but it's slower!*)
(fn _ => EVERY
[rewrite_goals_tac [con_def],
rtac case_trans 1,
REPEAT
(resolve_tac [@{thm refl}, split_trans,
Su.case_inl RS @{thm trans}, Su.case_inr RS @{thm trans}] 1)]);
val free_iffs = map Drule.export_without_context (con_defs RL [@{thm 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 (Misc_Legacy.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(@{const_name apply},_) $ 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 thy1 name,T),
args)),
subst_rec (Term.betapplys (recursor_case, args))));
val recursor_trans = recursor_def RS @{thm def_Vrecursor} RS @{thm trans};
fun prove_recursor_eqn arg =
Goal.prove_global thy1 [] []
(Ind_Syntax.traceIt "next recursor equation = " thy1 (mk_recursor_eqn arg))
(*Proves a single recursor equation.*)
(fn _ => EVERY
[rtac recursor_trans 1,
simp_tac (rank_ss addsimps case_eqns) 1,
IF_UNSOLVED (simp_tac (rank_ss addsimps tl con_defs) 1)]);
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 = map Drule.export_without_context (Ind_Syntax.mk_free_SEs free_iffs);
val {intrs, 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 =
let val thy = theory_of_thm elim in (*Don't use thy1: it will be stale*)
Goal.prove_global thy [] [] (Syntax.read_prop_global thy s)
(fn _ => EVERY
[rewrite_goals_tac con_defs,
fast_tac (ZF_cs addSEs free_SEs @ Su.free_SEs) 1])
end;
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*)
free_iffs = free_iffs,
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 |> Sign.add_path big_rec_base_name
|> Global_Theory.add_thmss
[((Binding.name "simps", simps), [Simplifier.simp_add]),
((Binding.empty , intrs), [Classical.safe_intro NONE]),
((Binding.name "con_defs", con_defs), []),
((Binding.name "case_eqns", case_eqns), []),
((Binding.name "recursor_eqns", recursor_eqns), []),
((Binding.name "free_iffs", free_iffs), []),
((Binding.name "free_elims", free_SEs), [])] |> snd
|> DatatypesData.map (Symtab.update (big_rec_name, dt_info))
|> ConstructorsData.map (fold Symtab.update con_pairs)
|> Sign.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 (raw_monos, raw_type_intrs, raw_type_elims) thy =
let
val ctxt = ProofContext.init_global thy;
fun read_is strs =
map (Syntax.parse_term ctxt #> Type.constraint @{typ i}) strs
|> Syntax.check_terms ctxt;
val rec_tms = read_is srec_tms;
val con_ty_lists = Ind_Syntax.read_constructs ctxt scon_ty_lists;
val dom_sum =
if sdom = "" then data_domain coind (rec_tms, con_ty_lists)
else singleton read_is sdom;
val monos = Attrib.eval_thms ctxt raw_monos;
val type_intrs = Attrib.eval_thms ctxt raw_type_intrs;
val type_elims = Attrib.eval_thms ctxt raw_type_elims;
in add_datatype_i (dom_sum, rec_tms) con_ty_lists (monos, type_intrs, type_elims) thy end;
(* outer syntax *)
fun mk_datatype ((((dom, dts), monos), type_intrs), type_elims) =
#1 o add_datatype (dom, map fst dts) (map snd dts) (monos, type_intrs, type_elims);
val con_decl =
Parse.name -- Scan.optional (Parse.$$$ "(" |-- Parse.list1 Parse.term --| Parse.$$$ ")") [] --
Parse.opt_mixfix >> Parse.triple1;
val datatype_decl =
(Scan.optional ((Parse.$$$ "⊆" || Parse.$$$ "<=") |-- Parse.!!! Parse.term) "") --
Parse.and_list1 (Parse.term -- (Parse.$$$ "=" |-- Parse.enum1 "|" con_decl)) --
Scan.optional (Parse.$$$ "monos" |-- Parse.!!! Parse_Spec.xthms1) [] --
Scan.optional (Parse.$$$ "type_intros" |-- Parse.!!! Parse_Spec.xthms1) [] --
Scan.optional (Parse.$$$ "type_elims" |-- Parse.!!! Parse_Spec.xthms1) []
>> (Toplevel.theory o mk_datatype);
val coind_prefix = if coind then "co" else "";
val _ =
Outer_Syntax.command (coind_prefix ^ "datatype")
("define " ^ coind_prefix ^ "datatype") Keyword.thy_decl datatype_decl;
end;