(* Title: HOL/Codatatype/Tools/bnf_fp_sugar.ML
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Sugar for constructing LFPs and GFPs.
*)
signature BNF_FP_SUGAR =
sig
end;
structure BNF_FP_Sugar : BNF_FP_SUGAR =
struct
open BNF_Util
open BNF_Wrap
open BNF_FP_Util
open BNF_LFP
open BNF_GFP
open BNF_FP_Sugar_Tactics
val caseN = "case";
fun retype_free (Free (s, _)) T = Free (s, T);
fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
fun mk_doubly_uncurried_fun f xss =
mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);
fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
fun merge_type_arg_constrained ctxt (T, c) (T', c') =
if T = T' then
(case (c, c') of
(_, NONE) => (T, c)
| (NONE, _) => (T, c')
| _ =>
if c = c' then
(T, c)
else
error ("Inconsistent sort constraints for type variable " ^
quote (Syntax.string_of_typ ctxt T)))
else
cannot_merge_types ();
fun merge_type_args_constrained ctxt (cAs, cAs') =
if length cAs = length cAs' then map2 (merge_type_arg_constrained ctxt) cAs cAs'
else cannot_merge_types ();
fun type_args_constrained_of (((cAs, _), _), _) = cAs;
val type_args_of = map fst o type_args_constrained_of;
fun type_binder_of (((_, b), _), _) = b;
fun mixfix_of ((_, mx), _) = mx;
fun ctr_specs_of (_, ctr_specs) = ctr_specs;
fun disc_of (((disc, _), _), _) = disc;
fun ctr_of (((_, ctr), _), _) = ctr;
fun args_of ((_, args), _) = args;
fun ctr_mixfix_of (_, mx) = mx;
fun prepare_datatype prepare_typ gfp specs fake_lthy lthy =
let
val constrained_As =
map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs
|> Library.foldr1 (merge_type_args_constrained lthy);
val As = map fst constrained_As;
val As' = map dest_TFree As;
val _ = (case duplicates (op =) As of [] => ()
| A :: _ => error ("Duplicate type parameter " ^ quote (Syntax.string_of_typ lthy A)));
(* TODO: use sort constraints on type args *)
val N = length specs;
fun mk_fake_T b =
Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
As);
val bs = map type_binder_of specs;
val fake_Ts = map mk_fake_T bs;
val mixfixes = map mixfix_of specs;
val _ = (case duplicates Binding.eq_name bs of [] => ()
| b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
val ctr_specss = map ctr_specs_of specs;
val disc_binderss = map (map disc_of) ctr_specss;
val ctr_binderss = map (map ctr_of) ctr_specss;
val ctr_argsss = map (map args_of) ctr_specss;
val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
val sel_bindersss = map (map (map fst)) ctr_argsss;
val fake_ctr_Tsss = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
val _ = (case subtract (op =) As' rhs_As' of
[] => ()
| A' :: _ => error ("Extra type variables on rhs: " ^
quote (Syntax.string_of_typ lthy (TFree A'))));
val ((Cs, Xs), _) =
lthy
|> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs
|> mk_TFrees N
||>> mk_TFrees N;
fun is_same_recT (T as Type (s, Us)) (Type (s', Us')) =
s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
quote (Syntax.string_of_typ fake_lthy T)))
| is_same_recT _ _ = false;
fun freeze_recXs (T as Type (s, Us)) =
(case find_index (is_same_recT T) fake_Ts of
~1 => Type (s, map freeze_recXs Us)
| i => nth Xs i)
| freeze_recXs T = T;
val ctr_TsssXs = map (map (map freeze_recXs)) fake_ctr_Tsss;
val sum_prod_TsXs = map (mk_sumTN o map HOLogic.mk_tupleT) ctr_TsssXs;
val eqs = map dest_TFree Xs ~~ sum_prod_TsXs;
val ((raw_unfs, raw_flds, raw_fp_iters, raw_fp_recs, unf_flds, fld_unfs, fld_injects), lthy') =
fp_bnf (if gfp then bnf_gfp else bnf_lfp) bs mixfixes As' eqs lthy;
val timer = time (Timer.startRealTimer ());
fun mk_unf_or_fld get_T Ts t =
let val Type (_, Ts0) = get_T (fastype_of t) in
Term.subst_atomic_types (Ts0 ~~ Ts) t
end;
val mk_unf = mk_unf_or_fld domain_type;
val mk_fld = mk_unf_or_fld range_type;
val unfs = map (mk_unf As) raw_unfs;
val flds = map (mk_fld As) raw_flds;
val fp_Ts = map (domain_type o fastype_of) unfs;
val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fp_Ts)))) ctr_TsssXs;
fun mk_fp_iter_or_rec Ts Us t =
let
val (binders, body) = strip_type (fastype_of t);
val Type (_, Ts0) = if gfp then body else List.last binders;
val Us0 = map (if gfp then domain_type else body_type) (fst (split_last binders));
in
Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
end;
val fp_iters = map (mk_fp_iter_or_rec As Cs) raw_fp_iters;
val fp_recs = map (mk_fp_iter_or_rec As Cs) raw_fp_recs;
fun pour_sugar_on_type ((((((((((((((b, fp_T), C), fld), unf), fp_iter), fp_rec), fld_unf),
unf_fld), fld_inject), ctr_binders), ctr_mixfixes), ctr_Tss), disc_binders), sel_binderss)
no_defs_lthy =
let
val n = length ctr_Tss;
val ks = 1 upto n;
val ms = map length ctr_Tss;
val unf_T = domain_type (fastype_of fld);
val prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
val ((((u, v), fs), xss), _) =
lthy
|> yield_singleton (mk_Frees "u") unf_T
||>> yield_singleton (mk_Frees "v") fp_T
||>> mk_Frees "f" case_Ts
||>> mk_Freess "x" ctr_Tss;
val ctr_rhss =
map2 (fn k => fn xs =>
fold_rev Term.lambda xs (fld $ mk_InN prod_Ts (HOLogic.mk_tuple xs) k)) ks xss;
val case_binder = Binding.suffix_name ("_" ^ caseN) b;
val case_rhs =
fold_rev Term.lambda (fs @ [v]) (mk_sum_caseN (map2 mk_uncurried_fun fs xss) $ (unf $ v));
val (((raw_ctrs, raw_ctr_defs), (raw_case, raw_case_def)), (lthy', lthy)) = no_defs_lthy
|> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
ctr_binders ctr_mixfixes ctr_rhss
||>> (Local_Theory.define ((case_binder, NoSyn), ((Thm.def_binding case_binder, []),
case_rhs)) #>> apsnd snd)
||> `Local_Theory.restore;
(*transforms defined frees into consts (and more)*)
val phi = Proof_Context.export_morphism lthy lthy';
val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
val case_def = Morphism.thm phi raw_case_def;
val ctrs = map (Morphism.term phi) raw_ctrs;
val casex = Morphism.term phi raw_case;
fun exhaust_tac {context = ctxt, ...} =
let
val fld_iff_unf_thm =
let
val goal =
fold_rev Logic.all [u, v]
(mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));
in
Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unf_T, fp_T])
(certify lthy fld) (certify lthy unf) fld_unf unf_fld)
|> Thm.close_derivation
|> Morphism.thm phi
end;
val sumEN_thm' =
Local_Defs.unfold lthy @{thms all_unit_eq}
(Drule.instantiate' (map (SOME o certifyT lthy) prod_Ts) [] (mk_sumEN n))
|> Morphism.thm phi;
in
mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'
end;
val inject_tacss =
map2 (fn 0 => K []
| _ => fn ctr_def => [fn {context = ctxt, ...} =>
mk_inject_tac ctxt ctr_def fld_inject])
ms ctr_defs;
val half_distinct_tacss =
map (map (fn (def, def') => fn {context = ctxt, ...} =>
mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);
val case_tacs =
map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;
val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
(* (co)iterators, (co)recursors, (co)induction *)
val is_recT = member (op =) fp_Ts;
val ns = map length ctr_Tsss;
val mss = map (map length) ctr_Tsss;
val Css = map2 replicate ns Cs;
fun dest_rec_pair (T as Type (@{type_name prod}, Us as [_, U])) =
if member (op =) Cs U then Us else [T]
| dest_rec_pair T = [T];
fun sugar_datatype no_defs_lthy =
let
val fp_y_Ts = map domain_type (fst (split_last (binder_types (fastype_of fp_iter))));
val y_prod_Tss = map2 dest_sumTN ns fp_y_Ts;
val y_Tsss = map2 (map2 dest_tupleT) mss y_prod_Tss;
val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
val iter_T = flat g_Tss ---> fp_T --> C;
val fp_z_Ts = map domain_type (fst (split_last (binder_types (fastype_of fp_rec))));
val z_prod_Tss = map2 dest_sumTN ns fp_z_Ts;
val z_Tsss = map2 (map2 dest_tupleT) mss z_prod_Tss;
val z_Tssss = map (map (map dest_rec_pair)) z_Tsss;
val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
val rec_T = flat h_Tss ---> fp_T --> C;
val ((gss, ysss), _) =
no_defs_lthy
|> mk_Freess "f" g_Tss
||>> mk_Freesss "x" y_Tsss;
val hss = map2 (map2 retype_free) gss h_Tss;
val (zssss, _) =
no_defs_lthy
|> mk_Freessss "x" z_Tssss;
val iter_binder = Binding.suffix_name ("_" ^ iterN) b;
val rec_binder = Binding.suffix_name ("_" ^ recN) b;
val iter_free = Free (Binding.name_of iter_binder, iter_T);
val rec_free = Free (Binding.name_of rec_binder, rec_T);
val iter_spec =
mk_Trueprop_eq (fold (fn gs => fn t => Term.list_comb (t, gs)) gss iter_free,
Term.list_comb (fp_iter, map2 (mk_sum_caseN oo map2 mk_uncurried_fun) gss ysss));
val rec_spec =
mk_Trueprop_eq (fold (fn hs => fn t => Term.list_comb (t, hs)) hss rec_free,
Term.list_comb (fp_rec,
map2 (mk_sum_caseN oo map2 mk_doubly_uncurried_fun) hss zssss));
val (([raw_iter, raw_rec], [raw_iter_def, raw_rec_def]), (lthy', lthy)) = no_defs_lthy
|> apfst split_list o fold_map (fn (b, spec) =>
Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
#>> apsnd snd) [(iter_binder, iter_spec), (rec_binder, rec_spec)]
||> `Local_Theory.restore;
in
lthy
end;
fun sugar_codatatype no_defs_lthy = no_defs_lthy;
in
wrap_datatype tacss ((ctrs, casex), (disc_binders, sel_binderss)) lthy'
|> (if gfp then sugar_codatatype else sugar_datatype)
end;
val lthy'' =
fold pour_sugar_on_type (bs ~~ fp_Ts ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~ fp_recs ~~
fld_unfs ~~ unf_flds ~~ fld_injects ~~ ctr_binderss ~~ ctr_mixfixess ~~ ctr_Tsss ~~
disc_binderss ~~ sel_bindersss) lthy';
val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
(if gfp then "co" else "") ^ "datatype"));
in
(timer; lthy'')
end;
fun datatype_cmd info specs lthy =
let
(*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
locale and shadows an existing global type*)
val fake_thy = Theory.copy
#> fold (fn spec => perhaps (try (Sign.add_type lthy
(type_binder_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
val fake_lthy = Proof_Context.background_theory fake_thy lthy;
in
prepare_datatype Syntax.read_typ info specs fake_lthy lthy
end;
val parse_opt_binding_colon = Scan.optional (Parse.binding --| Parse.$$$ ":") no_binder
val parse_ctr_arg =
Parse.$$$ "(" |-- parse_opt_binding_colon -- Parse.typ --| Parse.$$$ ")" ||
(Parse.typ >> pair no_binder);
val parse_single_spec =
Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
(@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
Scan.repeat parse_ctr_arg -- Parse.opt_mixfix));
val _ =
Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"
(Parse.and_list1 parse_single_spec >> datatype_cmd false);
val _ =
Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"
(Parse.and_list1 parse_single_spec >> datatype_cmd true);
end;