(* Title: HOL/Tools/BNF/bnf_fp_def_sugar.ML
Author: Jasmin Blanchette, TU Muenchen
Author: Martin Desharnais, TU Muenchen
Copyright 2012, 2013, 2014
Sugared datatype and codatatype constructions.
*)
signature BNF_FP_DEF_SUGAR =
sig
val fp_sugar_of: Proof.context -> string -> BNF_FP_Util.fp_sugar option
val fp_sugars_of: Proof.context -> BNF_FP_Util.fp_sugar list
val fp_sugar_interpretation: (BNF_FP_Util.fp_sugar list -> theory -> theory) -> theory -> theory
val register_fp_sugars: BNF_FP_Util.fp_sugar list -> local_theory -> local_theory
val co_induct_of: 'a list -> 'a
val strong_co_induct_of: 'a list -> 'a
val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
'a list
val nesting_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
type lfp_sugar_thms =
(thm list * thm * Token.src list) * (thm list list * Token.src list)
val morph_lfp_sugar_thms: morphism -> lfp_sugar_thms -> lfp_sugar_thms
val transfer_lfp_sugar_thms: theory -> lfp_sugar_thms -> lfp_sugar_thms
type gfp_sugar_thms =
((thm list * thm) list * (Token.src list * Token.src list))
* thm list list
* thm list list
* (thm list list * Token.src list)
* (thm list list list * Token.src list)
val morph_gfp_sugar_thms: morphism -> gfp_sugar_thms -> gfp_sugar_thms
val transfer_gfp_sugar_thms: theory -> gfp_sugar_thms -> gfp_sugar_thms
val mk_co_recs_prelims: BNF_Util.fp_kind -> typ list list list -> typ list -> typ list ->
typ list -> typ list -> int list -> int list list -> term list -> Proof.context ->
(term list
* (typ list list * typ list list list list * term list list * term list list list list) option
* (string * term list * term list list
* ((term list list * term list list list) * typ list)) option)
* Proof.context
val repair_nullary_single_ctr: typ list list -> typ list list
val mk_corec_p_pred_types: typ list -> int list -> typ list list
val mk_corec_fun_arg_types: typ list list list -> typ list -> typ list -> typ list -> int list ->
int list list -> term ->
typ list list
* (typ list list list list * typ list list list * typ list list list list * typ list)
val define_rec:
typ list list * typ list list list list * term list list * term list list list list ->
(string -> binding) -> typ list -> typ list -> term list -> term -> Proof.context ->
(term * thm) * Proof.context
val define_corec: 'a * term list * term list list
* ((term list list * term list list list) * typ list) -> (string -> binding) -> 'b list ->
typ list -> term list -> term -> local_theory -> (term * thm) * local_theory
val derive_induct_recs_thms_for_types: BNF_Def.bnf list ->
('a * typ list list list list * term list list * 'b) option -> thm -> thm list ->
BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list -> typ list ->
typ list list list -> thm list -> thm list -> thm list -> term list list -> thm list list ->
term list -> thm list -> Proof.context -> lfp_sugar_thms
val derive_coinduct_corecs_thms_for_types: BNF_Def.bnf list ->
string * term list * term list list * ((term list list * term list list list) * typ list) ->
thm -> thm list -> thm list -> thm list -> BNF_Def.bnf list -> typ list -> typ list ->
typ list -> typ list list list -> int list list -> int list list -> int list -> thm list ->
thm list -> (thm -> thm) -> thm list list -> Ctr_Sugar.ctr_sugar list -> term list ->
thm list -> (thm list -> thm list) -> Proof.context -> gfp_sugar_thms
val co_datatypes: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->
BNF_FP_Util.fp_result * local_theory) ->
(bool * bool)
* ((((((binding option * (typ * sort)) list * binding) * mixfix)
* ((binding, binding * typ) Ctr_Sugar.ctr_spec * mixfix) list) * (binding * binding))
* term list) list ->
local_theory -> local_theory
val parse_co_datatype_cmd: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->
BNF_FP_Util.fp_result * local_theory) ->
(local_theory -> local_theory) parser
end;
structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
struct
open Ctr_Sugar
open BNF_FP_Rec_Sugar_Util
open BNF_Util
open BNF_Comp
open BNF_Def
open BNF_FP_Util
open BNF_FP_Def_Sugar_Tactics
open BNF_LFP_Size
val EqN = "Eq_";
val case_transferN = "case_transfer";
val ctr_transferN = "ctr_transfer";
val disc_transferN = "disc_transfer";
val corec_codeN = "corec_code";
val map_disc_iffN = "map_disc_iff";
val map_selN = "map_sel";
val set_casesN = "set_cases";
val set_introsN = "set_intros";
val set_inductN = "set_induct";
val set_selN = "set_sel";
structure Data = Generic_Data
(
type T = fp_sugar Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data : T = Symtab.merge (K true) data;
);
fun zipping_map f =
let
fun zmap _ [] = []
| zmap xs (y :: ys) = f (xs, y, ys) :: zmap (xs @ [y]) ys
in zmap [] end;
fun choose_binary_fun fs AB =
find_first (fastype_of #> binder_types #> (fn [A, B] => AB = (A, B))) fs;
fun build_binary_fun_app fs t u =
Option.map (rapp u o rapp t) (choose_binary_fun fs (fastype_of t, fastype_of u));
fun build_the_rel rel_table ctxt Rs Ts A B =
build_rel rel_table ctxt Ts (the o choose_binary_fun Rs) (A, B);
fun build_rel_app ctxt Rs Ts t u =
build_the_rel [] ctxt Rs Ts (fastype_of t) (fastype_of u) $ t $ u;
fun mk_parametricity_goal ctxt Rs t u =
let val prem = build_the_rel [] ctxt Rs [] (fastype_of t) (fastype_of u) in
HOLogic.mk_Trueprop (prem $ t $ u)
end;
val name_of_set = name_of_const "set";
fun fp_sugar_of ctxt =
Symtab.lookup (Data.get (Context.Proof ctxt))
#> Option.map (transfer_fp_sugar (Proof_Context.theory_of ctxt));
fun fp_sugars_of ctxt =
Symtab.fold (cons o transfer_fp_sugar (Proof_Context.theory_of ctxt) o snd)
(Data.get (Context.Proof ctxt)) [];
fun co_induct_of (i :: _) = i;
fun strong_co_induct_of [_, s] = s;
structure FP_Sugar_Interpretation = Interpretation
(
type T = fp_sugar list;
val eq: T * T -> bool = op = o pairself (map #T);
);
fun with_repaired_path f (fp_sugars as ({T = Type (s, _), ...} : fp_sugar) :: _) thy =
thy
|> Sign.root_path
|> Sign.add_path (Long_Name.qualifier s)
|> f fp_sugars
|> Sign.restore_naming thy;
fun fp_sugar_interpretation f = FP_Sugar_Interpretation.interpretation (with_repaired_path f);
fun register_fp_sugars (fp_sugars as {fp, ...} :: _) =
fold (fn fp_sugar as {T = Type (s, _), ...} =>
Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi => Data.map (Symtab.update (s, morph_fp_sugar phi fp_sugar))))
fp_sugars
#> fp = Least_FP ? generate_lfp_size fp_sugars
#> Local_Theory.background_theory (FP_Sugar_Interpretation.data fp_sugars);
fun register_as_fp_sugars Ts BTs Xs fp pre_bnfs absT_infos fp_nesting_bnfs live_nesting_bnfs fp_res
ctrXs_Tsss ctr_defss ctr_sugars co_recs co_rec_defs mapss common_co_inducts co_inductss
co_rec_thmss disc_co_recss sel_co_recsss rel_injectss rel_distinctss noted =
let
val fp_sugars =
map_index (fn (kk, T) =>
{T = T, BT = nth BTs kk, X = nth Xs kk, fp = fp, fp_res = fp_res, fp_res_index = kk,
pre_bnf = nth pre_bnfs kk, absT_info = nth absT_infos kk,
fp_nesting_bnfs = fp_nesting_bnfs, live_nesting_bnfs = live_nesting_bnfs,
ctrXs_Tss = nth ctrXs_Tsss kk, ctr_defs = nth ctr_defss kk, ctr_sugar = nth ctr_sugars kk,
co_rec = nth co_recs kk, co_rec_def = nth co_rec_defs kk, maps = nth mapss kk,
common_co_inducts = common_co_inducts, co_inducts = nth co_inductss kk,
co_rec_thms = nth co_rec_thmss kk, disc_co_recs = nth disc_co_recss kk,
sel_co_recss = nth sel_co_recsss kk, rel_injects = nth rel_injectss kk,
rel_distincts = nth rel_distinctss kk}
|> morph_fp_sugar (substitute_noted_thm noted)) Ts;
in
register_fp_sugars fp_sugars
end;
(* This function could produce (fairly harmless) clashes in contrived examples (e.g., "x.A",
"x.x_A", "y.A"). *)
fun quasi_unambiguous_case_names names =
let
val ps = map (`Long_Name.base_name) names;
val dups = Library.duplicates (op =) (map fst ps);
fun underscore s =
let val ss = space_explode Long_Name.separator s in
space_implode "_" (drop (length ss - 2) ss)
end;
in
map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
end;
val mp_conj = @{thm mp_conj};
val fundefcong_attrs = @{attributes [fundef_cong]};
val nitpicksimp_attrs = @{attributes [nitpick_simp]};
val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;
val simp_attrs = @{attributes [simp]};
val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
fun flat_corec_predss_getterss qss gss = maps (op @) (qss ~~ gss);
fun flat_corec_preds_predsss_gettersss [] [qss] [gss] = flat_corec_predss_getterss qss gss
| flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (gss :: gsss) =
p :: flat_corec_predss_getterss qss gss @ flat_corec_preds_predsss_gettersss ps qsss gsss;
fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
fun flip_rels lthy n thm =
let
val Rs = Term.add_vars (prop_of thm) [];
val Rs' = rev (drop (length Rs - n) Rs);
val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
in
Drule.cterm_instantiate cRs thm
end;
fun mk_ctor_or_dtor get_T Ts t =
let val Type (_, Ts0) = get_T (fastype_of t) in
Term.subst_atomic_types (Ts0 ~~ Ts) t
end;
val mk_ctor = mk_ctor_or_dtor range_type;
val mk_dtor = mk_ctor_or_dtor domain_type;
fun mk_xtor_co_recs thy fp fpTs Cs ts0 =
let
val nn = length fpTs;
val (fpTs0, Cs0) =
map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
|> split_list;
val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
in
map (Term.subst_TVars rho) ts0
end;
fun mk_set Ts t =
subst_nonatomic_types (snd (Term.dest_Type (domain_type (fastype_of t))) ~~ Ts) t;
fun unzip_recT (Type (@{type_name prod}, _)) T = [T]
| unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts
| unzip_recT _ T = [T];
fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]
| unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts
| unzip_corecT _ T = [T];
fun liveness_of_fp_bnf n bnf =
(case T_of_bnf bnf of
Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
| _ => replicate n false);
fun cannot_merge_types fp =
error ("Mutually " ^ co_prefix fp ^ "recursive types must have the same type parameters");
fun merge_type_arg fp T T' = if T = T' then T else cannot_merge_types fp;
fun merge_type_args fp (As, As') =
if length As = length As' then map2 (merge_type_arg fp) As As' else cannot_merge_types fp;
fun type_args_named_constrained_of_spec (((((ncAs, _), _), _), _), _) = ncAs;
fun type_binding_of_spec (((((_, b), _), _), _), _) = b;
fun mixfix_of_spec ((((_, mx), _), _), _) = mx;
fun mixfixed_ctr_specs_of_spec (((_, mx_ctr_specs), _), _) = mx_ctr_specs;
fun map_binding_of_spec ((_, (b, _)), _) = b;
fun rel_binding_of_spec ((_, (_, b)), _) = b;
fun sel_default_eqs_of_spec (_, ts) = ts;
fun add_nesting_bnf_names Us =
let
fun add (Type (s, Ts)) ss =
let val (needs, ss') = fold_map add Ts ss in
if exists I needs then (true, insert (op =) s ss') else (false, ss')
end
| add T ss = (member (op =) Us T, ss);
in snd oo add end;
fun nesting_bnfs ctxt ctr_Tsss Us =
map_filter (bnf_of ctxt) (fold (fold (fold (add_nesting_bnf_names Us))) ctr_Tsss []);
fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
type lfp_sugar_thms =
(thm list * thm * Token.src list) * (thm list list * Token.src list);
fun morph_lfp_sugar_thms phi ((inducts, induct, induct_attrs), (recss, rec_attrs)) =
((map (Morphism.thm phi) inducts, Morphism.thm phi induct, induct_attrs),
(map (map (Morphism.thm phi)) recss, rec_attrs))
: lfp_sugar_thms;
val transfer_lfp_sugar_thms = morph_lfp_sugar_thms o Morphism.transfer_morphism;
type gfp_sugar_thms =
((thm list * thm) list * (Token.src list * Token.src list))
* thm list list
* thm list list
* (thm list list * Token.src list)
* (thm list list list * Token.src list);
fun morph_gfp_sugar_thms phi ((coinducts_pairs, coinduct_attrs_pair),
corecss, corec_discss, (corec_disc_iffss, corec_disc_iff_attrs),
(corec_selsss, corec_sel_attrs)) =
((map (apfst (map (Morphism.thm phi)) o apsnd (Morphism.thm phi)) coinducts_pairs,
coinduct_attrs_pair),
map (map (Morphism.thm phi)) corecss,
map (map (Morphism.thm phi)) corec_discss,
(map (map (Morphism.thm phi)) corec_disc_iffss, corec_disc_iff_attrs),
(map (map (map (Morphism.thm phi))) corec_selsss, corec_sel_attrs)) : gfp_sugar_thms;
val transfer_gfp_sugar_thms = morph_gfp_sugar_thms o Morphism.transfer_morphism;
fun mk_recs_args_types ctr_Tsss Cs absTs repTs ns mss ctor_rec_fun_Ts lthy =
let
val Css = map2 replicate ns Cs;
val x_Tssss =
map6 (fn absT => fn repT => fn n => fn ms => fn ctr_Tss => fn ctor_rec_fun_T =>
map2 (map2 unzip_recT)
ctr_Tss (dest_absumprodT absT repT n ms (domain_type ctor_rec_fun_T)))
absTs repTs ns mss ctr_Tsss ctor_rec_fun_Ts;
val x_Tsss' = map (map flat_rec_arg_args) x_Tssss;
val f_Tss = map2 (map2 (curry (op --->))) x_Tsss' Css;
val ((fss, xssss), lthy) =
lthy
|> mk_Freess "f" f_Tss
||>> mk_Freessss "x" x_Tssss;
in
((f_Tss, x_Tssss, fss, xssss), lthy)
end;
(*avoid "'a itself" arguments in corecursors*)
fun repair_nullary_single_ctr [[]] = [[HOLogic.unitT]]
| repair_nullary_single_ctr Tss = Tss;
fun mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss fun_Ts =
let
val ctr_Tsss' = map repair_nullary_single_ctr ctr_Tsss;
val g_absTs = map range_type fun_Ts;
val g_Tsss = map repair_nullary_single_ctr (map5 dest_absumprodT absTs repTs ns mss g_absTs);
val g_Tssss = map3 (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))
Cs ctr_Tsss' g_Tsss;
val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) g_Tssss;
in
(q_Tssss, g_Tsss, g_Tssss, g_absTs)
end;
fun mk_corec_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
fun mk_corec_fun_arg_types ctr_Tsss Cs absTs repTs ns mss dtor_corec =
(mk_corec_p_pred_types Cs ns,
mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss
(binder_fun_types (fastype_of dtor_corec)));
fun mk_corecs_args_types ctr_Tsss Cs absTs repTs ns mss dtor_corec_fun_Ts lthy =
let
val p_Tss = mk_corec_p_pred_types Cs ns;
val (q_Tssss, g_Tsss, g_Tssss, corec_types) =
mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss dtor_corec_fun_Ts;
val (((((Free (x, _), cs), pss), qssss), gssss), lthy) =
lthy
|> yield_singleton (mk_Frees "x") dummyT
||>> mk_Frees "a" Cs
||>> mk_Freess "p" p_Tss
||>> mk_Freessss "q" q_Tssss
||>> mk_Freessss "g" g_Tssss;
val cpss = map2 (map o rapp) cs pss;
fun build_sum_inj mk_inj = build_map lthy [] (uncurry mk_inj o dest_sumT o snd);
fun build_dtor_corec_arg _ [] [cg] = cg
| build_dtor_corec_arg T [cq] [cg, cg'] =
mk_If cq (build_sum_inj Inl_const (fastype_of cg, T) $ cg)
(build_sum_inj Inr_const (fastype_of cg', T) $ cg');
val pgss = map3 flat_corec_preds_predsss_gettersss pss qssss gssss;
val cqssss = map2 (map o map o map o rapp) cs qssss;
val cgssss = map2 (map o map o map o rapp) cs gssss;
val cqgsss = map3 (map3 (map3 build_dtor_corec_arg)) g_Tsss cqssss cgssss;
in
((x, cs, cpss, ((pgss, cqgsss), corec_types)), lthy)
end;
fun mk_co_recs_prelims fp ctr_Tsss fpTs Cs absTs repTs ns mss xtor_co_recs0 lthy =
let
val thy = Proof_Context.theory_of lthy;
val (xtor_co_rec_fun_Ts, xtor_co_recs) =
mk_xtor_co_recs thy fp fpTs Cs xtor_co_recs0 |> `(binder_fun_types o fastype_of o hd);
val ((recs_args_types, corecs_args_types), lthy') =
if fp = Least_FP then
mk_recs_args_types ctr_Tsss Cs absTs repTs ns mss xtor_co_rec_fun_Ts lthy
|>> (rpair NONE o SOME)
else
mk_corecs_args_types ctr_Tsss Cs absTs repTs ns mss xtor_co_rec_fun_Ts lthy
|>> (pair NONE o SOME);
in
((xtor_co_recs, recs_args_types, corecs_args_types), lthy')
end;
fun mk_preds_getterss_join c cps absT abs cqgss =
let
val n = length cqgss;
val ts = map2 (mk_absumprod absT abs n) (1 upto n) cqgss;
in
Term.lambda c (mk_IfN absT cps ts)
end;
fun define_co_rec fp fpT Cs b rhs lthy0 =
let
val thy = Proof_Context.theory_of lthy0;
val maybe_conceal_def_binding = Thm.def_binding
#> Config.get lthy0 bnf_note_all = false ? Binding.conceal;
val ((cst, (_, def)), (lthy', lthy)) = lthy0
|> Local_Theory.define ((b, NoSyn), ((maybe_conceal_def_binding b, []), rhs))
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy lthy';
val cst' = mk_co_rec thy fp fpT Cs (Morphism.term phi cst);
val def' = Morphism.thm phi def;
in
((cst', def'), lthy')
end;
fun define_rec (_, _, fss, xssss) mk_binding fpTs Cs reps ctor_rec =
let
val nn = length fpTs;
val (ctor_rec_absTs, fpT) = strip_typeN nn (fastype_of ctor_rec)
|>> map domain_type ||> domain_type;
in
define_co_rec Least_FP fpT Cs (mk_binding recN)
(fold_rev (fold_rev Term.lambda) fss (Term.list_comb (ctor_rec,
map4 (fn ctor_rec_absT => fn rep => fn fs => fn xsss =>
mk_case_absumprod ctor_rec_absT rep fs (map (map HOLogic.mk_tuple) xsss)
(map flat_rec_arg_args xsss))
ctor_rec_absTs reps fss xssss)))
end;
fun define_corec (_, cs, cpss, ((pgss, cqgsss), f_absTs)) mk_binding fpTs Cs abss dtor_corec =
let
val nn = length fpTs;
val fpT = range_type (snd (strip_typeN nn (fastype_of dtor_corec)));
in
define_co_rec Greatest_FP fpT Cs (mk_binding corecN)
(fold_rev (fold_rev Term.lambda) pgss (Term.list_comb (dtor_corec,
map5 mk_preds_getterss_join cs cpss f_absTs abss cqgsss)))
end;
fun postproc_co_induct lthy nn prop prop_conj =
Drule.zero_var_indexes
#> `(conj_dests nn)
#>> map (fn thm => Thm.permute_prems 0 (~1) (thm RS prop))
##> (fn thm => Thm.permute_prems 0 (~nn)
(if nn = 1 then thm RS prop
else funpow nn (fn thm => unfold_thms lthy @{thms conj_assoc} (thm RS prop_conj)) thm));
fun mk_induct_attrs ctrss =
let
val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
in
[induct_case_names_attr]
end;
fun derive_rel_induct_thms_for_types lthy fpA_Ts As Bs ctrAss ctrAs_Tsss exhausts ctor_rel_induct
ctor_defss ctor_injects pre_rel_defs abs_inverses live_nesting_rel_eqs =
let
val B_ify = typ_subst_nonatomic (As ~~ Bs)
val fpB_Ts = map B_ify fpA_Ts;
val ctrBs_Tsss = map (map (map B_ify)) ctrAs_Tsss;
val ctrBss = map (map (subst_nonatomic_types (As ~~ Bs))) ctrAss;
val ((((Rs, IRs), ctrAsss), ctrBsss), names_lthy) = lthy
|> mk_Frees "R" (map2 mk_pred2T As Bs)
||>> mk_Frees "IR" (map2 mk_pred2T fpA_Ts fpB_Ts)
||>> mk_Freesss "a" ctrAs_Tsss
||>> mk_Freesss "b" ctrBs_Tsss;
val premises =
let
fun mk_prem ctrA ctrB argAs argBs =
fold_rev Logic.all (argAs @ argBs) (fold_rev (curry Logic.mk_implies)
(map2 (HOLogic.mk_Trueprop oo build_rel_app names_lthy (Rs @ IRs) fpA_Ts) argAs argBs)
(HOLogic.mk_Trueprop (build_rel_app names_lthy (Rs @ IRs) fpA_Ts
(Term.list_comb (ctrA, argAs)) (Term.list_comb (ctrB, argBs)))));
in
flat (map4 (map4 mk_prem) ctrAss ctrBss ctrAsss ctrBsss)
end;
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_leq
(map2 (build_the_rel [] lthy (Rs @ IRs) []) fpA_Ts fpB_Ts) IRs));
val rel_induct0_thm =
Goal.prove_sorry lthy [] premises goal (fn {context = ctxt, prems} =>
mk_rel_induct0_tac ctxt ctor_rel_induct prems (map (certify ctxt) IRs) exhausts ctor_defss
ctor_injects pre_rel_defs abs_inverses live_nesting_rel_eqs)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation;
in
(postproc_co_induct lthy (length fpA_Ts) @{thm predicate2D} @{thm predicate2D_conj}
rel_induct0_thm,
mk_induct_attrs ctrAss)
end;
fun derive_induct_recs_thms_for_types pre_bnfs rec_args_typess ctor_induct ctor_rec_thms
live_nesting_bnfs fp_nesting_bnfs fpTs Cs Xs ctrXs_Tsss fp_abs_inverses fp_type_definitions
abs_inverses ctrss ctr_defss recs rec_defs lthy =
let
val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
val nn = length pre_bnfs;
val ns = map length ctr_Tsss;
val mss = map (map length) ctr_Tsss;
val pre_map_defs = map map_def_of_bnf pre_bnfs;
val pre_set_defss = map set_defs_of_bnf pre_bnfs;
val live_nesting_map_ident0s = map map_ident0_of_bnf live_nesting_bnfs;
val fp_nesting_map_ident0s = map map_ident0_of_bnf fp_nesting_bnfs;
val fp_nesting_set_maps = maps set_map_of_bnf fp_nesting_bnfs;
val fp_b_names = map base_name_of_typ fpTs;
val ((((ps, ps'), xsss), us'), names_lthy) =
lthy
|> mk_Frees' "P" (map mk_pred1T fpTs)
||>> mk_Freesss "x" ctr_Tsss
||>> Variable.variant_fixes fp_b_names;
val us = map2 (curry Free) us' fpTs;
fun mk_sets bnf =
let
val Type (T_name, Us) = T_of_bnf bnf;
val lives = lives_of_bnf bnf;
val sets = sets_of_bnf bnf;
fun mk_set U =
(case find_index (curry (op =) U) lives of
~1 => Term.dummy
| i => nth sets i);
in
(T_name, map mk_set Us)
end;
val setss_fp_nesting = map mk_sets fp_nesting_bnfs;
val (induct_thms, induct_thm) =
let
fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
[([], (find_index (curry (op =) X) Xs + 1, x))]
| mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
(case AList.lookup (op =) setss_fp_nesting T_name of
NONE => []
| SOME raw_sets0 =>
let
val (Xs_Ts, (Ts, raw_sets)) =
filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
|> split_list ||> split_list;
val sets = map (mk_set Ts0) raw_sets;
val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
val xysets = map (pair x) (ys ~~ sets);
val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
in
flat (map2 (map o apfst o cons) xysets ppremss)
end)
| mk_raw_prem_prems _ _ _ = [];
fun close_prem_prem xs t =
fold_rev Logic.all (map Free (drop (nn + length xs)
(rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
fun mk_prem_prem xs (xysets, (j, x)) =
close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
mk_Trueprop_mem (y, set $ x')) xysets,
HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
let
val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
fun mk_prem (xs, raw_pprems, concl) =
fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
val goal =
Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
val kksss = map (map (map (fst o snd) o #2)) raw_premss;
val ctor_induct' = ctor_induct OF (map2 mk_absumprodE fp_type_definitions mss);
val thm =
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' fp_abs_inverses
abs_inverses fp_nesting_set_maps pre_set_defss)
|> singleton (Proof_Context.export names_lthy lthy);
in
`(conj_dests nn) thm
end;
val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
fun mk_rec_thmss (_, x_Tssss, fss, _) recs rec_defs ctor_rec_thms =
let
val frecs = map (lists_bmoc fss) recs;
fun mk_goal frec xctr f xs fxs =
fold_rev (fold_rev Logic.all) (xs :: fss)
(mk_Trueprop_eq (frec $ xctr, Term.list_comb (f, fxs)));
fun maybe_tick (T, U) u f =
if try (fst o HOLogic.dest_prodT) U = SOME T then
Term.lambda u (HOLogic.mk_prod (u, f $ u))
else
f;
fun build_rec (x as Free (_, T)) U =
if T = U then
x
else
build_map lthy [] (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
(fn kk => fn TU => maybe_tick TU (nth us kk) (nth frecs kk))) (T, U) $ x;
val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_rec))) xsss x_Tssss;
val goalss = map5 (map4 o mk_goal) frecs xctrss fss xsss fxsss;
val tacss = map4 (map ooo
mk_rec_tac pre_map_defs (fp_nesting_map_ident0s @ live_nesting_map_ident0s) rec_defs)
ctor_rec_thms fp_abs_inverses abs_inverses ctr_defss;
fun prove goal tac =
Goal.prove_sorry lthy [] [] goal (tac o #context)
|> Thm.close_derivation;
in
map2 (map2 prove) goalss tacss
end;
val rec_thmss = mk_rec_thmss (the rec_args_typess) recs rec_defs ctor_rec_thms;
in
((induct_thms, induct_thm, mk_induct_attrs ctrss),
(rec_thmss, code_nitpicksimp_attrs @ simp_attrs))
end;
fun mk_coinduct_attributes fpTs ctrss discss mss =
let
val nn = length fpTs;
val fp_b_names = map base_name_of_typ fpTs;
fun mk_coinduct_concls ms discs ctrs =
let
fun mk_disc_concl disc = [name_of_disc disc];
fun mk_ctr_concl 0 _ = []
| mk_ctr_concl _ ctor = [name_of_ctr ctor];
val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
val ctr_concls = map2 mk_ctr_concl ms ctrs;
in
flat (map2 append disc_concls ctr_concls)
end;
val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
val coinduct_conclss =
map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
val common_coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes 1));
val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
val coinduct_case_concl_attrs =
map2 (fn casex => fn concls =>
Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
coinduct_cases coinduct_conclss;
val common_coinduct_attrs =
common_coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
val coinduct_attrs =
coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
in
(coinduct_attrs, common_coinduct_attrs)
end;
fun derive_rel_coinduct_thm_for_types lthy fpA_Ts ns As Bs mss (ctr_sugars : ctr_sugar list)
abs_inverses abs_injects ctor_injects dtor_ctors rel_pre_defs ctor_defss dtor_rel_coinduct
live_nesting_rel_eqs =
let
val fpB_Ts = map (typ_subst_nonatomic (As ~~ Bs)) fpA_Ts;
val (Rs, IRs, fpAs, fpBs, names_lthy) =
let
val fp_names = map base_name_of_typ fpA_Ts;
val ((((Rs, IRs), fpAs_names), fpBs_names), names_lthy) = lthy
|> mk_Frees "R" (map2 mk_pred2T As Bs)
||>> mk_Frees "IR" (map2 mk_pred2T fpA_Ts fpB_Ts)
||>> Variable.variant_fixes fp_names
||>> Variable.variant_fixes (map (suffix "'") fp_names);
in
(Rs, IRs,
map2 (curry Free) fpAs_names fpA_Ts,
map2 (curry Free) fpBs_names fpB_Ts,
names_lthy)
end;
val ((discA_tss, selA_tsss), (discB_tss, selB_tsss)) =
let
val discss = map #discs ctr_sugars;
val selsss = map #selss ctr_sugars;
fun mk_discss ts Ts = map2 (map o rapp) ts (map (map (mk_disc_or_sel Ts)) discss);
fun mk_selsss ts Ts = map2 (map o map o rapp) ts (map (map (map (mk_disc_or_sel Ts)))
selsss);
in
((mk_discss fpAs As, mk_selsss fpAs As),
(mk_discss fpBs Bs, mk_selsss fpBs Bs))
end;
val premises =
let
fun mk_prem_ctr_concls n k discA_t selA_ts discB_t selB_ts =
(if k = n then [] else [HOLogic.mk_eq (discA_t, discB_t)]) @
(case (selA_ts, selB_ts) of
([], []) => []
| (_ :: _, _ :: _) =>
[Library.foldr HOLogic.mk_imp
(if n = 1 then [] else [discA_t, discB_t],
Library.foldr1 HOLogic.mk_conj
(map2 (build_rel_app lthy (Rs @ IRs) fpA_Ts) selA_ts selB_ts))]);
fun mk_prem_concl n discA_ts selA_tss discB_ts selB_tss =
Library.foldr1 HOLogic.mk_conj (flat (map5 (mk_prem_ctr_concls n)
(1 upto n) discA_ts selA_tss discB_ts selB_tss))
handle List.Empty => @{term True};
fun mk_prem IR tA tB n discA_ts selA_tss discB_ts selB_tss =
fold_rev Logic.all [tA, tB] (Logic.mk_implies (HOLogic.mk_Trueprop (IR $ tA $ tB),
HOLogic.mk_Trueprop (mk_prem_concl n discA_ts selA_tss discB_ts selB_tss)));
in
map8 mk_prem IRs fpAs fpBs ns discA_tss selA_tsss discB_tss selB_tsss
end;
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_leq
IRs (map2 (build_the_rel [] lthy (Rs @ IRs) []) fpA_Ts fpB_Ts)));
val rel_coinduct0_thm =
Goal.prove_sorry lthy [] premises goal (fn {context = ctxt, prems} =>
mk_rel_coinduct0_tac ctxt dtor_rel_coinduct (map (certify ctxt) IRs) prems
(map #exhaust ctr_sugars) (map (flat o #disc_thmss) ctr_sugars)
(map (flat o #sel_thmss) ctr_sugars) ctor_defss dtor_ctors ctor_injects abs_injects
rel_pre_defs abs_inverses live_nesting_rel_eqs)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation;
in
(postproc_co_induct lthy (length fpA_Ts) @{thm predicate2D} @{thm predicate2D_conj}
rel_coinduct0_thm,
mk_coinduct_attributes fpA_Ts (map #ctrs ctr_sugars) (map #discs ctr_sugars) mss)
end;
fun derive_set_induct_thms_for_types lthy nn fpTs ctrss setss dtor_set_inducts exhausts
set_pre_defs ctor_defs dtor_ctors Abs_pre_inverses =
let
fun mk_prems A Ps ctr_args t ctxt =
(case fastype_of t of
Type (type_name, innerTs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
let
fun seq_assm a set ctxt =
let
val X = HOLogic.dest_setT (range_type (fastype_of set));
val (x, ctxt') = yield_singleton (mk_Frees "x") X ctxt;
val assm = mk_Trueprop_mem (x, set $ a);
in
(case build_binary_fun_app Ps x a of
NONE =>
mk_prems A Ps ctr_args x ctxt'
|>> map (Logic.all x o Logic.mk_implies o pair assm)
| SOME f =>
([Logic.all x
(Logic.mk_implies (assm,
Logic.mk_implies (HOLogic.mk_Trueprop f,
HOLogic.mk_Trueprop (the (build_binary_fun_app Ps x ctr_args)))))],
ctxt'))
end;
in
fold_map (seq_assm t o mk_set innerTs) (sets_of_bnf bnf) ctxt
|>> flat
end)
| T =>
if T = A then ([HOLogic.mk_Trueprop (the (build_binary_fun_app Ps t ctr_args))], ctxt)
else ([], ctxt));
fun mk_prems_for_ctr A Ps ctr ctxt =
let
val (args, ctxt') = mk_Frees "z" (binder_types (fastype_of ctr)) ctxt;
in
fold_map (mk_prems A Ps (list_comb (ctr, args))) args ctxt'
|>> map (fold_rev Logic.all args) o flat
|>> (fn prems => (prems, mk_names (length prems) (name_of_ctr ctr)))
end;
fun mk_prems_and_concl_for_type A Ps ((fpT, ctrs), set) ctxt =
let
val ((x, fp), ctxt') = ctxt
|> yield_singleton (mk_Frees "x") A
||>> yield_singleton (mk_Frees "a") fpT;
val concl = mk_Ball (set $ fp) (Term.absfree (dest_Free x)
(the (build_binary_fun_app Ps x fp)));
in
fold_map (mk_prems_for_ctr A Ps) ctrs ctxt'
|>> split_list
|>> map_prod flat flat
|>> apfst (rpair concl)
end;
fun mk_thm ctxt fpTs ctrss sets =
let
val A = HOLogic.dest_setT (range_type (fastype_of (hd sets)));
val (Ps, ctxt') = mk_Frees "P" (map (fn fpT => A --> fpT --> HOLogic.boolT) fpTs) ctxt;
val (((premises, conclusion), case_names), ctxt'') =
(fold_map (mk_prems_and_concl_for_type A Ps) (fpTs ~~ ctrss ~~ sets) ctxt')
|>> apfst split_list o split_list
|>> apfst (apfst flat)
|>> apfst (apsnd (Library.foldr1 HOLogic.mk_conj))
|>> apsnd flat;
val thm =
Goal.prove_sorry lthy [] premises (HOLogic.mk_Trueprop conclusion)
(fn {context = ctxt, prems} =>
mk_set_induct0_tac ctxt (map (certify ctxt'') Ps) prems dtor_set_inducts exhausts
set_pre_defs ctor_defs dtor_ctors Abs_pre_inverses)
|> singleton (Proof_Context.export ctxt'' ctxt)
|> Thm.close_derivation;
val case_names_attr =
Attrib.internal (K (Rule_Cases.case_names (quasi_unambiguous_case_names case_names)));
val induct_set_attrs = map (Attrib.internal o K o Induct.induct_pred o name_of_set) sets;
in
(thm, case_names_attr :: induct_set_attrs)
end
val consumes_attr = Attrib.internal (K (Rule_Cases.consumes 1));
in
map (fn Asets =>
let
fun massage_thm thm = rotate_prems (~1) (thm RS bspec);
in
mk_thm lthy fpTs ctrss Asets |> nn = 1 ? map_prod massage_thm (cons consumes_attr)
end) (transpose setss)
end;
fun derive_coinduct_corecs_thms_for_types pre_bnfs (x, cs, cpss, ((pgss, cqgsss), _))
dtor_coinduct dtor_injects dtor_ctors dtor_corec_thms live_nesting_bnfs fpTs Cs Xs ctrXs_Tsss
kss mss ns fp_abs_inverses abs_inverses mk_vimage2p ctr_defss (ctr_sugars : ctr_sugar list)
corecs corec_defs export_args lthy =
let
fun mk_ctor_dtor_corec_thm dtor_inject dtor_ctor corec =
iffD1 OF [dtor_inject, trans OF [corec, dtor_ctor RS sym]];
val ctor_dtor_corec_thms = map3 mk_ctor_dtor_corec_thm dtor_injects dtor_ctors dtor_corec_thms;
val nn = length pre_bnfs;
val pre_map_defs = map map_def_of_bnf pre_bnfs;
val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
val live_nesting_map_ident0s = map map_ident0_of_bnf live_nesting_bnfs;
val live_nesting_rel_eqs = map rel_eq_of_bnf live_nesting_bnfs;
val fp_b_names = map base_name_of_typ fpTs;
val ctrss = map #ctrs ctr_sugars;
val discss = map #discs ctr_sugars;
val selsss = map #selss ctr_sugars;
val exhausts = map #exhaust ctr_sugars;
val disc_thmsss = map #disc_thmss ctr_sugars;
val discIss = map #discIs ctr_sugars;
val sel_thmsss = map #sel_thmss ctr_sugars;
val (((rs, us'), vs'), names_lthy) =
lthy
|> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
||>> Variable.variant_fixes fp_b_names
||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
val us = map2 (curry Free) us' fpTs;
val udiscss = map2 (map o rapp) us discss;
val uselsss = map2 (map o map o rapp) us selsss;
val vs = map2 (curry Free) vs' fpTs;
val vdiscss = map2 (map o rapp) vs discss;
val vselsss = map2 (map o map o rapp) vs selsss;
val coinduct_thms_pairs =
let
val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
val strong_rs =
map4 (fn u => fn v => fn uvr => fn uv_eq =>
fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
fun build_the_rel rs' T Xs_T =
build_rel [] lthy [] (fn (_, X) => nth rs' (find_index (curry (op =) X) Xs)) (T, Xs_T)
|> Term.subst_atomic_types (Xs ~~ fpTs);
fun build_rel_app rs' usel vsel Xs_T =
fold rapp [usel, vsel] (build_the_rel rs' (fastype_of usel) Xs_T);
fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels ctrXs_Ts =
(if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
(if null usels then
[]
else
[Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
Library.foldr1 HOLogic.mk_conj (map3 (build_rel_app rs') usels vsels ctrXs_Ts))]);
fun mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss =
Library.foldr1 HOLogic.mk_conj (flat (map6 (mk_prem_ctr_concls rs' n)
(1 upto n) udiscs uselss vdiscs vselss ctrXs_Tss))
handle List.Empty => @{term True};
fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss ctrXs_Tss =
fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss)));
val concl =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
uvrs us vs));
fun mk_goal rs' =
Logic.list_implies (map9 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss
ctrXs_Tsss, concl);
val goals = map mk_goal [rs, strong_rs];
fun prove dtor_coinduct' goal =
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
mk_coinduct_tac ctxt live_nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs
fp_abs_inverses abs_inverses dtor_ctors exhausts ctr_defss disc_thmsss sel_thmsss)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation;
val rel_eqs = map rel_eq_of_bnf pre_bnfs;
val rel_monos = map rel_mono_of_bnf pre_bnfs;
val dtor_coinducts =
[dtor_coinduct, mk_coinduct_strong_thm dtor_coinduct rel_eqs rel_monos mk_vimage2p lthy]
in
map2 (postproc_co_induct lthy nn mp mp_conj oo prove) dtor_coinducts goals
end;
fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
val gcorecs = map (lists_bmoc pgss) corecs;
val corec_thmss =
let
fun mk_goal c cps gcorec n k ctr m cfs' =
fold_rev (fold_rev Logic.all) ([c] :: pgss)
(Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
mk_Trueprop_eq (gcorec $ c, Term.list_comb (ctr, take m cfs'))));
val mk_U = typ_subst_nonatomic (map2 (fn C => fn fpT => (mk_sumT (fpT, C), fpT)) Cs fpTs);
fun tack (c, u) f =
let val x' = Free (x, mk_sumT (fastype_of u, fastype_of c)) in
Term.lambda x' (mk_case_sum (Term.lambda u u, Term.lambda c (f $ c)) $ x')
end;
fun build_corec cqg =
let val T = fastype_of cqg in
if exists_subtype_in Cs T then
let val U = mk_U T in
build_map lthy [] (indexify fst (map2 (curry mk_sumT) fpTs Cs) (fn kk => fn _ =>
tack (nth cs kk, nth us kk) (nth gcorecs kk))) (T, U) $ cqg
end
else
cqg
end;
val cqgsss' = map (map (map build_corec)) cqgsss;
val goalss = map8 (map4 oooo mk_goal) cs cpss gcorecs ns kss ctrss mss cqgsss';
val tacss =
map4 (map ooo mk_corec_tac corec_defs live_nesting_map_ident0s)
ctor_dtor_corec_thms pre_map_defs abs_inverses ctr_defss;
fun prove goal tac =
Goal.prove_sorry lthy [] [] goal (tac o #context)
|> Thm.close_derivation;
in
map2 (map2 prove) goalss tacss
|> map (map (unfold_thms lthy @{thms case_sum_if}))
end;
val corec_disc_iff_thmss =
let
fun mk_goal c cps gcorec n k disc =
mk_Trueprop_eq (disc $ (gcorec $ c),
if n = 1 then @{const True}
else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
val goalss = map6 (map2 oooo mk_goal) cs cpss gcorecs ns kss discss;
fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
val case_splitss' = map (map mk_case_split') cpss;
val tacss = map3 (map oo mk_corec_disc_iff_tac) case_splitss' corec_thmss disc_thmsss;
fun prove goal tac =
Goal.prove_sorry lthy [] [] goal (tac o #context)
|> singleton export_args
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation;
fun proves [_] [_] = []
| proves goals tacs = map2 prove goals tacs;
in
map2 proves goalss tacss
end;
fun mk_corec_disc_thms corecs discIs = map (op RS) (corecs ~~ discIs);
val corec_disc_thmss = map2 mk_corec_disc_thms corec_thmss discIss;
fun mk_corec_sel_thm corec_thm sel sel_thm =
let
val (domT, ranT) = dest_funT (fastype_of sel);
val arg_cong' =
Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
[NONE, NONE, SOME (certify lthy sel)] arg_cong
|> Thm.varifyT_global;
val sel_thm' = sel_thm RSN (2, trans);
in
corec_thm RS arg_cong' RS sel_thm'
end;
fun mk_corec_sel_thms corec_thmss =
map3 (map3 (map2 o mk_corec_sel_thm)) corec_thmss selsss sel_thmsss;
val corec_sel_thmsss = mk_corec_sel_thms corec_thmss;
in
((coinduct_thms_pairs,
mk_coinduct_attributes fpTs (map #ctrs ctr_sugars) (map #discs ctr_sugars) mss),
corec_thmss,
corec_disc_thmss,
(corec_disc_iff_thmss, simp_attrs),
(corec_sel_thmsss, simp_attrs))
end;
fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
((discs_sels0, no_code), specs) no_defs_lthy0 =
let
(* TODO: sanity checks on arguments *)
val discs_sels = discs_sels0 orelse fp = Greatest_FP;
val nn = length specs;
val fp_bs = map type_binding_of_spec specs;
val fp_b_names = map Binding.name_of fp_bs;
val fp_common_name = mk_common_name fp_b_names;
val map_bs = map map_binding_of_spec specs;
val rel_bs = map rel_binding_of_spec specs;
fun prepare_type_arg (_, (ty, c)) =
let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
TFree (s, prepare_constraint no_defs_lthy0 c)
end;
val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of_spec) specs;
val unsorted_Ass0 = map (map (resort_tfree @{sort type})) Ass0;
val unsorted_As = Library.foldr1 (merge_type_args fp) unsorted_Ass0;
val num_As = length unsorted_As;
val set_boss = map (map fst o type_args_named_constrained_of_spec) specs;
val set_bss = map (map (the_default Binding.empty)) set_boss;
val ((((Bs0, Cs as C1 :: _), Es as E1 :: _), Xs), no_defs_lthy) =
no_defs_lthy0
|> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
|> mk_TFrees num_As
||>> mk_TFrees nn
||>> mk_TFrees nn
||>> variant_tfrees fp_b_names;
fun add_fake_type spec =
Typedecl.basic_typedecl (type_binding_of_spec spec, num_As, mixfix_of_spec spec);
val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
val qsoty = quote o Syntax.string_of_typ fake_lthy;
val _ = (case Library.duplicates (op =) unsorted_As of [] => ()
| A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
"datatype specification"));
val bad_args =
map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
|> filter_out Term.is_TVar;
val _ = null bad_args orelse
error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
"datatype specification");
val mixfixes = map mixfix_of_spec specs;
val _ = (case Library.duplicates Binding.eq_name fp_bs of [] => ()
| b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
val mx_ctr_specss = map mixfixed_ctr_specs_of_spec specs;
val ctr_specss = map (map fst) mx_ctr_specss;
val ctr_mixfixess = map (map snd) mx_ctr_specss;
val disc_bindingss = map (map disc_of_ctr_spec) ctr_specss;
val ctr_bindingss =
map2 (fn fp_b_name => map (Binding.qualify false fp_b_name o ctr_of_ctr_spec)) fp_b_names
ctr_specss;
val ctr_argsss = map (map args_of_ctr_spec) ctr_specss;
val sel_bindingsss = map (map (map fst)) ctr_argsss;
val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
val raw_sel_default_eqss = map sel_default_eqs_of_spec specs;
val (As :: _) :: fake_ctr_Tsss =
burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
val As' = map dest_TFree As;
val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
val _ = (case subtract (op =) As' rhs_As' of [] => ()
| extras => error ("Extra type variables on right-hand side: " ^
commas (map (qsoty o TFree) extras)));
val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
s = s' andalso (Ts = Ts' orelse
error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
" (expected " ^ qsoty T' ^ ")"))
| eq_fpT_check _ _ = false;
fun freeze_fp (T as Type (s, Ts)) =
(case find_index (eq_fpT_check T) fake_Ts of
~1 => Type (s, map freeze_fp Ts)
| kk => nth Xs kk)
| freeze_fp T = T;
val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
val ctrXs_repTs = map mk_sumprodT_balanced ctrXs_Tsss;
val fp_eqs =
map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_repTs;
val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
| extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
val killed_As =
map_filter (fn (A, set_bos) => if exists is_none set_bos then SOME A else NONE)
(unsorted_As ~~ transpose set_boss);
val ((pre_bnfs, absT_infos), (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0,
dtors = dtors0, xtor_co_recs = xtor_co_recs0, xtor_co_induct, dtor_ctors,
ctor_dtors, ctor_injects, dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms,
xtor_co_rec_thms, rel_xtor_co_induct_thm, dtor_set_induct_thms, ...},
lthy)) =
fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As)
(map dest_TFree killed_As) fp_eqs no_defs_lthy0
handle BAD_DEAD (X, X_backdrop) =>
(case X_backdrop of
Type (bad_tc, _) =>
let
val fake_T = qsoty (unfreeze_fp X);
val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
fun register_hint () =
"\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
\it";
in
if is_some (bnf_of no_defs_lthy bad_tc) orelse
is_some (fp_sugar_of no_defs_lthy bad_tc) then
error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
" in type expression " ^ fake_T_backdrop)
else if is_some (Old_Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
bad_tc) then
error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
" via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
fake_T_backdrop ^ register_hint ())
else
error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
" via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
register_hint ())
end);
val abss = map #abs absT_infos;
val reps = map #rep absT_infos;
val absTs = map #absT absT_infos;
val repTs = map #repT absT_infos;
val abs_injects = map #abs_inject absT_infos;
val abs_inverses = map #abs_inverse absT_infos;
val type_definitions = map #type_definition absT_infos;
val time = time lthy;
val timer = time (Timer.startRealTimer ());
val fp_nesting_bnfs = nesting_bnfs lthy ctrXs_Tsss Xs;
val live_nesting_bnfs = nesting_bnfs lthy ctrXs_Tsss As;
val pre_map_defs = map map_def_of_bnf pre_bnfs;
val pre_set_defss = map set_defs_of_bnf pre_bnfs;
val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
val fp_nesting_set_maps = maps set_map_of_bnf fp_nesting_bnfs;
val live_nesting_set_maps = maps set_map_of_bnf live_nesting_bnfs;
val live = live_of_bnf any_fp_bnf;
val _ =
if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
warning "Map function and relator names ignored"
else
();
val Bs =
map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
(liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
val B_ify = Term.typ_subst_atomic (As ~~ Bs);
val ctors = map (mk_ctor As) ctors0;
val dtors = map (mk_dtor As) dtors0;
val fpTs = map (domain_type o fastype_of) dtors;
val fpBTs = map B_ify fpTs;
fun massage_simple_notes base =
filter_out (null o #2)
#> map (fn (thmN, thms, f_attrs) =>
((Binding.qualify true base (Binding.name thmN), []), map_index (fn (i, thm) => ([thm], f_attrs i)) thms));
val massage_multi_notes =
maps (fn (thmN, thmss, attrs) =>
map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
((Binding.qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
fp_b_names fpTs thmss)
#> filter_out (null o fst o hd o snd);
val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
val ns = map length ctr_Tsss;
val kss = map (fn n => 1 upto n) ns;
val mss = map (map length) ctr_Tsss;
val ((xtor_co_recs, recs_args_types, corecs_args_types), lthy') =
mk_co_recs_prelims fp ctr_Tsss fpTs Cs absTs repTs ns mss xtor_co_recs0 lthy;
fun define_ctrs_dtrs_for_type (((((((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
xtor_co_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), abs),
abs_inject), abs_inverse), type_definition), ctr_bindings), ctr_mixfixes), ctr_Tss),
disc_bindings), sel_bindingss), raw_sel_default_eqs) no_defs_lthy =
let
val fp_b_name = Binding.name_of fp_b;
val fpBT = B_ify fpT;
val ctr_absT = domain_type (fastype_of ctor);
val ((((w, xss), yss), u'), names_lthy) =
no_defs_lthy
|> yield_singleton (mk_Frees "w") ctr_absT
||>> mk_Freess "x" ctr_Tss
||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
||>> yield_singleton Variable.variant_fixes fp_b_name;
val u = Free (u', fpT);
val ctr_rhss =
map2 (fn k => fn xs => fold_rev Term.lambda xs (ctor $ mk_absumprod ctr_absT abs n k xs))
ks xss;
val maybe_conceal_def_binding = Thm.def_binding
#> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;
val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
|> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)
ctr_bindings ctr_mixfixes ctr_rhss
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy lthy';
val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
val ctr_defs' =
map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
val ctrs0 = map (Morphism.term phi) raw_ctrs;
val ctrs = map (mk_ctr As) ctrs0;
fun wrap_ctrs lthy =
let
fun exhaust_tac {context = ctxt, prems = _} =
let
val ctor_iff_dtor_thm =
let
val goal =
fold_rev Logic.all [w, u]
(mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
in
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [ctr_absT, fpT])
(certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
|> Morphism.thm phi
|> Thm.close_derivation
end;
val sumEN_thm' =
unfold_thms lthy @{thms unit_all_eq1} (mk_absumprodE type_definition ms)
|> Morphism.thm phi;
in
mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
end;
val inject_tacss =
map2 (fn ctr_def => fn 0 => [] | _ => [fn {context = ctxt, ...} =>
mk_inject_tac ctxt ctr_def ctor_inject abs_inject]) ctr_defs ms;
val half_distinct_tacss =
map (map (fn (def, def') => fn {context = ctxt, ...} =>
mk_half_distinct_tac ctxt ctor_inject abs_inject [def, def']))
(mk_half_pairss (`I ctr_defs));
val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
val sel_Tss = map (map (curry (op -->) fpT)) ctr_Tss;
val sel_bTs =
flat sel_bindingss ~~ flat sel_Tss
|> filter_out (Binding.is_empty o fst)
|> distinct (Binding.eq_name o pairself fst);
val sel_default_lthy = fake_local_theory_for_sel_defaults sel_bTs lthy;
val sel_default_eqs = map (prepare_term sel_default_lthy) raw_sel_default_eqs;
fun ctr_spec_of disc_b ctr0 sel_bs = ((disc_b, ctr0), sel_bs);
val ctr_specs = map3 ctr_spec_of disc_bindings ctrs0 sel_bindingss;
in
free_constructors tacss ((((discs_sels, no_code), standard_binding), ctr_specs),
sel_default_eqs) lthy
end;
fun derive_maps_sets_rels (ctr_sugar as {casex, case_cong, case_thms, discs, selss, ctrs,
exhaust, exhaust_discs, disc_thmss, sel_thmss, injects, distincts, distinct_discsss,
...} : ctr_sugar, lthy) =
if live = 0 then
((([], [], [], []), ctr_sugar), lthy)
else
let
val rel_flip = rel_flip_of_bnf fp_bnf;
val nones = replicate live NONE;
val ctor_cong =
if fp = Least_FP then
Drule.dummy_thm
else
let val ctor' = mk_ctor Bs ctor in
cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
end;
fun mk_cIn ctor k xs =
let val absT = domain_type (fastype_of ctor) in
mk_absumprod absT abs n k xs
|> fp = Greatest_FP ? curry (op $) ctor
|> certify lthy
end;
val cxIns = map2 (mk_cIn ctor) ks xss;
val cyIns = map2 (mk_cIn (Term.map_types B_ify ctor)) ks yss;
fun mk_map_thm ctr_def' cxIn =
fold_thms lthy [ctr_def']
(unfold_thms lthy (o_apply :: pre_map_def ::
(if fp = Least_FP then [] else [dtor_ctor]) @ sumprod_thms_map @
abs_inverses)
(cterm_instantiate_pos (nones @ [SOME cxIn])
(if fp = Least_FP then fp_map_thm
else fp_map_thm RS ctor_cong RS (ctor_dtor RS sym RS trans))))
|> singleton (Proof_Context.export names_lthy no_defs_lthy);
fun mk_set0_thm fp_set_thm ctr_def' cxIn =
fold_thms lthy [ctr_def']
(unfold_thms lthy (pre_set_defs @ fp_nesting_set_maps @ live_nesting_set_maps @
(if fp = Least_FP then [] else [dtor_ctor]) @ basic_sumprod_thms_set @
@{thms UN_Un sup_assoc[THEN sym]} @ abs_inverses)
(cterm_instantiate_pos [SOME cxIn] fp_set_thm))
|> singleton (Proof_Context.export names_lthy no_defs_lthy);
fun mk_set0_thms fp_set_thm = map2 (mk_set0_thm fp_set_thm) ctr_defs' cxIns;
val map_thms = map2 mk_map_thm ctr_defs' cxIns;
val set0_thmss = map mk_set0_thms fp_set_thms;
val set0_thms = flat set0_thmss;
val set_thms = set0_thms
|> map (unfold_thms lthy @{thms insert_is_Un[THEN sym] Un_empty_left
Un_insert_left});
val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
fold_thms lthy ctr_defs'
(unfold_thms lthy (pre_rel_def :: abs_inverse ::
(if fp = Least_FP then [] else [dtor_ctor]) @ sumprod_thms_rel @
@{thms vimage2p_def sum.inject sum.distinct(1)[THEN eq_False[THEN iffD2]]})
(cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
|> postproc
|> singleton (Proof_Context.export names_lthy no_defs_lthy);
fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
fun mk_rel_intro_thm thm =
let
fun impl thm = rotate_prems (~1) (impl (rotate_prems 1 (conjI RS thm)))
handle THM _ => thm
in
impl (thm RS iffD2)
handle THM _ => thm
end;
fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
cxIn cyIn;
fun mk_other_half_rel_distinct_thm thm =
flip_rels lthy live thm
RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
val rel_intro_thms = map mk_rel_intro_thm rel_inject_thms;
val half_rel_distinct_thmss =
map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
val other_half_rel_distinct_thmss =
map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
val (rel_distinct_thms, _) =
join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
val rel_eq_thms =
map (fn th => th RS @{thm eq_False[THEN iffD2]}) rel_distinct_thms @
map2 (fn th => fn 0 => th RS @{thm eq_True[THEN iffD2]} | _ => th)
rel_inject_thms ms;
val (map_disc_iff_thms, map_sel_thms, set_sel_thms, rel_sel_thms, set_intros_thms,
case_transfer_thms, ctr_transfer_thms, disc_transfer_thms,
(set_cases_thms, set_cases_attrss), (rel_cases_thm, rel_cases_attrs)) =
let
val live_AsBs = filter (op <>) (As ~~ Bs);
val fTs = map (op -->) live_AsBs;
val (((((fs, Rs), ta), tb), thesis), names_lthy) = names_lthy
|> mk_Frees "f" fTs
||>> mk_Frees "R" (map (uncurry mk_pred2T) live_AsBs)
||>> yield_singleton (mk_Frees "a") fpT
||>> yield_singleton (mk_Frees "b") fpBT
||>> apfst HOLogic.mk_Trueprop o
yield_singleton (mk_Frees "thesis") HOLogic.boolT;
val map_term = mk_map live As Bs (map_of_bnf fp_bnf);
val ctrAs = map (mk_ctr As) ctrs;
val ctrBs = map (mk_ctr Bs) ctrs;
val relAsBs = mk_rel live As Bs (rel_of_bnf fp_bnf);
val setAs = map (mk_set As) (sets_of_bnf fp_bnf);
val discAs = map (mk_disc_or_sel As) discs;
val discBs = map (mk_disc_or_sel Bs) discs;
val selAss = map (map (mk_disc_or_sel As)) selss;
val discAs_selAss = flat (map2 (map o pair) discAs selAss);
val ctr_transfer_thms =
let
val goals = map2 (mk_parametricity_goal names_lthy Rs) ctrAs ctrBs;
in
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(K (mk_ctr_transfer_tac rel_intro_thms))
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val (set_cases_thms, set_cases_attrss) =
let
fun mk_prems assms elem t ctxt =
(case fastype_of t of
Type (type_name, xs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
apfst flat (fold_map (fn set => fn ctxt =>
let
val X = HOLogic.dest_setT (range_type (fastype_of set));
val new_var = not (X = fastype_of elem);
val (x, ctxt') =
if new_var then yield_singleton (mk_Frees "x") X ctxt
else (elem, ctxt);
in
mk_prems (mk_Trueprop_mem (x, set $ t) :: assms) elem x ctxt'
|>> map (if new_var then Logic.all x else I)
end) (map (mk_set xs) (sets_of_bnf bnf)) ctxt))
| T => rpair ctxt
(if T = fastype_of elem then [fold (curry Logic.mk_implies) assms thesis]
else []));
in
split_list (map (fn set =>
let
val A = HOLogic.dest_setT (range_type (fastype_of set));
val (elem, names_lthy) = yield_singleton (mk_Frees "e") A names_lthy;
val premss =
map (fn ctr =>
let
val (args, names_lthy) =
mk_Frees "z" (binder_types (fastype_of ctr)) names_lthy;
in
flat (zipping_map (fn (prev_args, arg, next_args) =>
let
val (args_with_elem, args_without_elem) =
if fastype_of arg = A then
(prev_args @ [elem] @ next_args, prev_args @ next_args)
else
`I (prev_args @ [arg] @ next_args);
in
mk_prems
[mk_Trueprop_eq (ta, Term.list_comb (ctr, args_with_elem))]
elem arg names_lthy
|> fst
|> map (fold_rev Logic.all args_without_elem)
end) args)
end) ctrAs;
val goal = Logic.mk_implies (mk_Trueprop_mem (elem, set $ ta), thesis);
val thm =
Goal.prove_sorry lthy [] (flat premss) goal
(fn {context = ctxt, prems} =>
mk_set_cases_tac ctxt (certify ctxt ta) prems exhaust set_thms)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation
|> rotate_prems ~1;
val consumes_attr = Attrib.internal (K (Rule_Cases.consumes 1));
val cases_set_attr =
Attrib.internal (K (Induct.cases_pred (name_of_set set)));
in
(* TODO: @{attributes [elim?]} *)
(thm, [consumes_attr, cases_set_attr])
end) setAs)
end;
val set_intros_thms =
let
fun mk_goals A setA ctr_args t ctxt =
(case fastype_of t of
Type (type_name, innerTs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
apfst flat (fold_map (fn set => fn ctxt =>
let
val X = HOLogic.dest_setT (range_type (fastype_of set));
val (x, ctxt') = yield_singleton (mk_Frees "x") X ctxt;
val assm = mk_Trueprop_mem (x, set $ t);
in
apfst (map (Logic.mk_implies o pair assm))
(mk_goals A setA ctr_args x ctxt')
end) (map (mk_set innerTs) (sets_of_bnf bnf)) ctxt))
| T =>
(if T = A then [mk_Trueprop_mem (t, setA $ ctr_args)] else [], ctxt));
val (goals, names_lthy) =
apfst flat (fold_map (fn set => fn ctxt =>
let
val A = HOLogic.dest_setT (range_type (fastype_of set));
in
apfst flat (fold_map (fn ctr => fn ctxt =>
let
val (args, ctxt') =
mk_Frees "arg" (binder_types (fastype_of ctr)) ctxt;
val ctr_args = Term.list_comb (ctr, args);
in
apfst flat (fold_map (mk_goals A set ctr_args) args ctxt')
end) ctrAs ctxt)
end) setAs lthy);
in
if null goals then []
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} => mk_set_intros_tac ctxt set0_thms)
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val rel_sel_thms =
let
val selBss = map (map (mk_disc_or_sel Bs)) selss;
val n = length discAs;
fun mk_rhs n k discA selAs discB selBs =
(if k = n then [] else [HOLogic.mk_eq (discA $ ta, discB $ tb)]) @
(case (selAs, selBs) of
([], []) => []
| (_ :: _, _ :: _) => [Library.foldr HOLogic.mk_imp
(if n = 1 then [] else [discA $ ta, discB $ tb],
Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app names_lthy Rs [])
(map (rapp ta) selAs) (map (rapp tb) selBs)))]);
val goals = if n = 0 then []
else [mk_Trueprop_eq
(build_rel_app names_lthy Rs [] ta tb,
Library.foldr1 HOLogic.mk_conj
(flat (map5 (mk_rhs n) (1 upto n) discAs selAss discBs selBss)))];
in
if null goals then
[]
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} =>
mk_rel_sel_tac ctxt (certify ctxt ta) (certify ctxt tb) exhaust
(flat disc_thmss) (flat sel_thmss) rel_inject_thms distincts
rel_distinct_thms)
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val (rel_cases_thm, rel_cases_attrs) =
let
val rel_Rs_a_b = list_comb (relAsBs, Rs) $ ta $ tb;
val ctrBs = map (mk_ctr Bs) ctrs;
fun mk_assms ctrA ctrB ctxt =
let
val argA_Ts = binder_types (fastype_of ctrA);
val argB_Ts = binder_types (fastype_of ctrB);
val ((argAs, argBs), names_ctxt) = ctxt
|> mk_Frees "x" argA_Ts
||>> mk_Frees "y" argB_Ts;
val ctrA_args = list_comb (ctrA, argAs);
val ctrB_args = list_comb (ctrB, argBs);
in
(fold_rev Logic.all (argAs @ argBs) (Logic.list_implies
(mk_Trueprop_eq (ta, ctrA_args) ::
mk_Trueprop_eq (tb, ctrB_args) ::
map2 (HOLogic.mk_Trueprop oo build_rel_app lthy Rs [])
argAs argBs,
thesis)),
names_ctxt)
end;
val (assms, names_lthy) = fold_map2 mk_assms ctrAs ctrBs names_lthy;
val goal =
Logic.list_implies (HOLogic.mk_Trueprop rel_Rs_a_b :: assms, thesis);
val thm =
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} =>
mk_rel_cases_tac ctxt (certify ctxt ta) (certify ctxt tb) exhaust
injects rel_inject_thms distincts rel_distinct_thms
(map rel_eq_of_bnf live_nesting_bnfs))
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation;
val ctr_names = quasi_unambiguous_case_names ((map name_of_ctr) ctrAs);
val case_names_attr = Attrib.internal (K (Rule_Cases.case_names ctr_names));
val consumes_attr = Attrib.internal (K (Rule_Cases.consumes 1));
val cases_pred_attr = Attrib.internal o K o Induct.cases_pred;
in
(thm, [consumes_attr, case_names_attr, cases_pred_attr ""])
end;
val case_transfer_thms =
let
val (R, names_lthy) =
yield_singleton (mk_Frees "R") (mk_pred2T C1 E1) names_lthy;
val caseA = mk_case As C1 casex;
val caseB = mk_case Bs E1 casex;
val goal = mk_parametricity_goal names_lthy (R :: Rs) caseA caseB;
in
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} =>
mk_case_transfer_tac ctxt rel_cases_thm case_thms)
|> singleton (Proof_Context.export names_lthy lthy)
|> Thm.close_derivation
end;
val disc_transfer_thms =
let
val goals = map2 (mk_parametricity_goal names_lthy Rs) discAs discBs;
in
if null goals then []
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} =>
mk_disc_transfer_tac ctxt (the_single rel_sel_thms)
(the_single exhaust_discs) (flat (flat distinct_discsss)))
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val map_disc_iff_thms =
let
val discsB = map (mk_disc_or_sel Bs) discs;
val discsA_t = map (fn disc1 => Term.betapply (disc1, ta)) discAs;
fun mk_goal (discA_t, discB) =
if head_of discA_t aconv HOLogic.Not orelse is_refl_bool discA_t then
NONE
else
SOME (mk_Trueprop_eq
(betapply (discB, (Term.list_comb (map_term, fs) $ ta)), discA_t));
val goals = map_filter mk_goal (discsA_t ~~ discsB);
in
if null goals then
[]
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} =>
mk_map_disc_iff_tac ctxt (certify ctxt ta) exhaust (flat disc_thmss)
map_thms)
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val map_sel_thms =
let
fun mk_goal (discA, selA) =
let
val prem = Term.betapply (discA, ta);
val selB = mk_disc_or_sel Bs selA;
val lhs = selB $ (Term.list_comb (map_term, fs) $ ta);
val lhsT = fastype_of lhs;
val map_rhsT =
map_atyps (perhaps (AList.lookup (op =) (map swap live_AsBs))) lhsT;
val map_rhs = build_map lthy []
(the o (AList.lookup (op =) (live_AsBs ~~ fs))) (map_rhsT, lhsT);
val rhs = (case map_rhs of
Const (@{const_name id}, _) => selA $ ta
| _ => map_rhs $ (selA $ ta));
val concl = mk_Trueprop_eq (lhs, rhs);
in
if is_refl_bool prem then concl
else Logic.mk_implies (HOLogic.mk_Trueprop prem, concl)
end;
val goals = map mk_goal discAs_selAss;
in
if null goals then
[]
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} =>
mk_map_sel_tac ctxt (certify ctxt ta) exhaust (flat disc_thmss)
map_thms (flat sel_thmss))
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
val set_sel_thms =
let
fun mk_goal discA selA setA ctxt =
let
val prem = Term.betapply (discA, ta);
val sel_rangeT = range_type (fastype_of selA);
val A = HOLogic.dest_setT (range_type (fastype_of setA));
fun travese_nested_types t ctxt =
(case fastype_of t of
Type (type_name, innerTs) =>
(case bnf_of ctxt type_name of
NONE => ([], ctxt)
| SOME bnf =>
let
fun seq_assm a set ctxt =
let
val X = HOLogic.dest_setT (range_type (fastype_of set));
val (x, ctxt') = yield_singleton (mk_Frees "x") X ctxt;
val assm = mk_Trueprop_mem (x, set $ a);
in
travese_nested_types x ctxt'
|>> map (Logic.mk_implies o pair assm)
end;
in
fold_map (seq_assm t o mk_set innerTs) (sets_of_bnf bnf) ctxt
|>> flat
end)
| T =>
if T = A then
([mk_Trueprop_mem (t, setA $ ta)], ctxt)
else
([], ctxt));
val (concls, ctxt') =
if sel_rangeT = A then
([mk_Trueprop_mem (selA $ ta, setA $ ta)], ctxt)
else
travese_nested_types (selA $ ta) ctxt;
in
if exists_subtype_in [A] sel_rangeT then
if is_refl_bool prem then
(concls, ctxt')
else
(map (Logic.mk_implies o pair (HOLogic.mk_Trueprop prem)) concls,
ctxt')
else
([], ctxt)
end;
val (goals, names_lthy) = apfst (flat o flat) (fold_map (fn (disc, sel) =>
fold_map (mk_goal disc sel) setAs) discAs_selAss names_lthy);
in
if null goals then
[]
else
Goal.prove_sorry lthy [] [] (Logic.mk_conjunction_balanced goals)
(fn {context = ctxt, prems = _} =>
mk_set_sel_tac ctxt (certify ctxt ta) exhaust (flat disc_thmss)
(flat sel_thmss) set0_thms)
|> Conjunction.elim_balanced (length goals)
|> Proof_Context.export names_lthy lthy
|> map Thm.close_derivation
end;
in
(map_disc_iff_thms, map_sel_thms, set_sel_thms, rel_sel_thms, set_intros_thms,
case_transfer_thms, ctr_transfer_thms, disc_transfer_thms,
(set_cases_thms, set_cases_attrss), (rel_cases_thm, rel_cases_attrs))
end;
val anonymous_notes =
[([case_cong], fundefcong_attrs),
(rel_eq_thms, code_nitpicksimp_attrs)]
|> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
val notes =
[(case_transferN, [case_transfer_thms], K []),
(ctr_transferN, ctr_transfer_thms, K []),
(disc_transferN, disc_transfer_thms, K []),
(mapN, map_thms, K (code_nitpicksimp_attrs @ simp_attrs)),
(map_disc_iffN, map_disc_iff_thms, K simp_attrs),
(map_selN, map_sel_thms, K []),
(rel_casesN, [rel_cases_thm], K rel_cases_attrs),
(rel_distinctN, rel_distinct_thms, K simp_attrs),
(rel_injectN, rel_inject_thms, K simp_attrs),
(rel_introsN, rel_intro_thms, K []),
(rel_selN, rel_sel_thms, K []),
(setN, set_thms, K (code_nitpicksimp_attrs @ simp_attrs)),
(set_casesN, set_cases_thms, nth set_cases_attrss),
(set_introsN, set_intros_thms, K []),
(set_selN, set_sel_thms, K [])]
|> massage_simple_notes fp_b_name;
val (noted, lthy') =
lthy
|> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) map_thms)
|> fp = Least_FP
? Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) rel_eq_thms)
|> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) set0_thms)
|> Local_Theory.notes (anonymous_notes @ notes);
val subst = Morphism.thm (substitute_noted_thm noted);
in
(((map subst map_thms, map subst rel_inject_thms, map subst rel_distinct_thms,
map (map subst) set0_thmss), ctr_sugar), lthy')
end;
fun mk_binding pre = Binding.qualify false fp_b_name (Binding.prefix_name (pre ^ "_") fp_b);
fun massage_res (((maps_sets_rels, ctr_sugar), co_rec_res), lthy) =
(((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_rec_res), lthy);
in
(wrap_ctrs
#> derive_maps_sets_rels
##>>
(if fp = Least_FP then define_rec (the recs_args_types) mk_binding fpTs Cs reps
else define_corec (the corecs_args_types) mk_binding fpTs Cs abss) xtor_co_rec
#> massage_res, lthy')
end;
fun wrap_types_etc (wrap_types_etcs, lthy) =
fold_map I wrap_types_etcs lthy
|>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
o split_list;
fun mk_simp_thms ({injects, distincts, case_thms, ...} : ctr_sugar) co_recs mapsx rel_injects
rel_distincts setss =
injects @ distincts @ case_thms @ co_recs @ mapsx @ rel_injects @ rel_distincts @ flat setss;
fun derive_note_induct_recs_thms_for_types
((((mapss, rel_injectss, rel_distinctss, setss), (ctrss, _, ctr_defss, ctr_sugars)),
(recs, rec_defs)), lthy) =
let
val ((induct_thms, induct_thm, induct_attrs), (rec_thmss, rec_attrs)) =
derive_induct_recs_thms_for_types pre_bnfs recs_args_types xtor_co_induct xtor_co_rec_thms
live_nesting_bnfs fp_nesting_bnfs fpTs Cs Xs ctrXs_Tsss abs_inverses type_definitions
abs_inverses ctrss ctr_defss recs rec_defs lthy;
val induct_type_attr = Attrib.internal o K o Induct.induct_type;
val induct_pred_attr = Attrib.internal o K o Induct.induct_pred;
val ((rel_induct_thmss, common_rel_induct_thms),
(rel_induct_attrs, common_rel_induct_attrs)) =
if live = 0 then
((replicate nn [], []), ([], []))
else
let
val ((rel_induct_thms, common_rel_induct_thm), rel_induct_attrs) =
derive_rel_induct_thms_for_types lthy fpTs As Bs ctrss ctr_Tsss
(map #exhaust ctr_sugars) rel_xtor_co_induct_thm ctr_defss ctor_injects
pre_rel_defs abs_inverses (map rel_eq_of_bnf live_nesting_bnfs);
in
((map single rel_induct_thms, single common_rel_induct_thm),
(rel_induct_attrs, rel_induct_attrs))
end;
val simp_thmss =
map6 mk_simp_thms ctr_sugars rec_thmss mapss rel_injectss rel_distinctss setss;
val common_notes =
(if nn > 1 then
[(inductN, [induct_thm], K induct_attrs),
(rel_inductN, common_rel_induct_thms, K common_rel_induct_attrs)]
else
[])
|> massage_simple_notes fp_common_name;
val notes =
[(inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
(recN, rec_thmss, K rec_attrs),
(rel_inductN, rel_induct_thmss, K (rel_induct_attrs @ [induct_pred_attr ""])),
(simpsN, simp_thmss, K [])]
|> massage_multi_notes;
in
lthy
|> Spec_Rules.add Spec_Rules.Equational (recs, flat rec_thmss)
|> Local_Theory.notes (common_notes @ notes)
(* for "datatype_realizer.ML": *)
|>> name_noted_thms
(fst (dest_Type (hd fpTs)) ^ (implode (map (prefix "_") (tl fp_b_names)))) inductN
|-> register_as_fp_sugars fpTs fpBTs Xs Least_FP pre_bnfs absT_infos fp_nesting_bnfs
live_nesting_bnfs fp_res ctrXs_Tsss ctr_defss ctr_sugars recs rec_defs mapss [induct_thm]
(map single induct_thms) rec_thmss (replicate nn []) (replicate nn []) rel_injectss
rel_distinctss
end;
fun derive_note_coinduct_corecs_thms_for_types
((((mapss, rel_injectss, rel_distinctss, setss), (_, _, ctr_defss, ctr_sugars)),
(corecs, corec_defs)), lthy) =
let
val (([(coinduct_thms, coinduct_thm), (coinduct_strong_thms, coinduct_strong_thm)],
(coinduct_attrs, common_coinduct_attrs)),
corec_thmss, corec_disc_thmss,
(corec_disc_iff_thmss, corec_disc_iff_attrs), (corec_sel_thmsss, corec_sel_attrs)) =
derive_coinduct_corecs_thms_for_types pre_bnfs (the corecs_args_types) xtor_co_induct
dtor_injects dtor_ctors xtor_co_rec_thms live_nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss
ns abs_inverses abs_inverses I ctr_defss ctr_sugars corecs corec_defs
(Proof_Context.export lthy' no_defs_lthy) lthy;
fun distinct_prems ctxt th =
Goal.prove (*no sorry*) ctxt [] []
(th |> Thm.prop_of |> Logic.strip_horn |>> distinct (op aconv) |> Logic.list_implies)
(fn _ => HEADGOAL (cut_tac th THEN' atac) THEN ALLGOALS eq_assume_tac);
fun eq_ifIN _ [thm] = thm
| eq_ifIN ctxt (thm :: thms) =
distinct_prems ctxt (@{thm eq_ifI} OF
(map (unfold_thms ctxt @{thms atomize_imp[of _ "t = u" for t u]})
[thm, eq_ifIN ctxt thms]));
val corec_code_thms = map (eq_ifIN lthy) corec_thmss;
val corec_sel_thmss = map flat corec_sel_thmsss;
val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
val coinduct_pred_attr = Attrib.internal o K o Induct.coinduct_pred;
val flat_corec_thms = append oo append;
val ((rel_coinduct_thmss, common_rel_coinduct_thms),
(rel_coinduct_attrs, common_rel_coinduct_attrs)) =
if live = 0 then
((replicate nn [], []), ([], []))
else
let
val ((rel_coinduct_thms, common_rel_coinduct_thm),
(rel_coinduct_attrs, common_rel_coinduct_attrs)) =
derive_rel_coinduct_thm_for_types lthy fpTs ns As Bs mss ctr_sugars abs_inverses
abs_injects ctor_injects dtor_ctors pre_rel_defs ctr_defss rel_xtor_co_induct_thm
(map rel_eq_of_bnf live_nesting_bnfs)
in
((map single rel_coinduct_thms, single common_rel_coinduct_thm),
(rel_coinduct_attrs, common_rel_coinduct_attrs))
end;
val (set_induct_thms, set_induct_attrss) =
derive_set_induct_thms_for_types lthy nn fpTs (map #ctrs ctr_sugars)
(map (map (mk_set As)) (map sets_of_bnf fp_bnfs)) dtor_set_induct_thms
(map #exhaust ctr_sugars) (flat pre_set_defss) (flat ctr_defss)
dtor_ctors abs_inverses
|> split_list;
val simp_thmss =
map6 mk_simp_thms ctr_sugars
(map3 flat_corec_thms corec_disc_thmss corec_disc_iff_thmss corec_sel_thmss)
mapss rel_injectss rel_distinctss setss;
val common_notes =
(set_inductN, set_induct_thms, nth set_induct_attrss) ::
(if nn > 1 then
[(coinductN, [coinduct_thm], K common_coinduct_attrs),
(coinduct_strongN, [coinduct_strong_thm], K common_coinduct_attrs),
(rel_coinductN, common_rel_coinduct_thms, K common_rel_coinduct_attrs)]
else [])
|> massage_simple_notes fp_common_name;
val notes =
[(coinductN, map single coinduct_thms,
fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
(coinduct_strongN, map single coinduct_strong_thms, K coinduct_attrs),
(corecN, corec_thmss, K []),
(corec_codeN, map single corec_code_thms, K code_nitpicksimp_attrs),
(corec_discN, corec_disc_thmss, K []),
(corec_disc_iffN, corec_disc_iff_thmss, K corec_disc_iff_attrs),
(corec_selN, corec_sel_thmss, K corec_sel_attrs),
(rel_coinductN, rel_coinduct_thmss, K (rel_coinduct_attrs @ [coinduct_pred_attr ""])),
(simpsN, simp_thmss, K [])]
|> massage_multi_notes;
in
lthy
|> fold (curry (Spec_Rules.add Spec_Rules.Equational) corecs)
[flat corec_sel_thmss, flat corec_thmss]
|> Local_Theory.notes (common_notes @ notes)
|-> register_as_fp_sugars fpTs fpBTs Xs Greatest_FP pre_bnfs absT_infos fp_nesting_bnfs
live_nesting_bnfs fp_res ctrXs_Tsss ctr_defss ctr_sugars corecs corec_defs mapss
[coinduct_thm, coinduct_strong_thm] (transpose [coinduct_thms, coinduct_strong_thms])
corec_thmss corec_disc_thmss corec_sel_thmsss rel_injectss rel_distinctss
end;
val lthy'' = lthy'
|> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
xtor_co_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~ pre_set_defss ~~
pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~ kss ~~ mss ~~
abss ~~ abs_injects ~~ abs_inverses ~~ type_definitions ~~ ctr_bindingss ~~ ctr_mixfixess ~~
ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~ raw_sel_default_eqss)
|> wrap_types_etc
|> case_fp fp derive_note_induct_recs_thms_for_types
derive_note_coinduct_corecs_thms_for_types;
val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
co_prefix fp ^ "datatype"));
in
timer; lthy''
end;
fun co_datatypes x = define_co_datatypes (K I) (K I) (K I) x;
fun co_datatype_cmd x =
define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term x;
val parse_ctr_arg =
@{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"}
|| Parse.typ >> pair Binding.empty;
val parse_ctr_specs =
Parse.enum1 "|" (parse_ctr_spec Parse.binding parse_ctr_arg -- Parse.opt_mixfix);
val parse_spec =
parse_type_args_named_constrained -- Parse.binding -- Parse.opt_mixfix --
(@{keyword "="} |-- parse_ctr_specs) -- parse_map_rel_bindings -- parse_sel_default_eqs;
val parse_co_datatype = parse_ctr_options -- Parse.and_list1 parse_spec;
fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
val _ = Context.>> (Context.map_theory FP_Sugar_Interpretation.init);
end;