(* Title: HOL/Tools/BNF/bnf_gfp.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Codatatype construction.
*)
signature BNF_GFP =
sig
val construct_gfp: mixfix list -> binding 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
end;
structure BNF_GFP : BNF_GFP =
struct
open BNF_Def
open BNF_Util
open BNF_Tactics
open BNF_Comp
open BNF_FP_Util
open BNF_FP_Def_Sugar
open BNF_GFP_Util
open BNF_GFP_Tactics
datatype wit_tree = Wit_Leaf of int | Wit_Node of (int * int * int list) * wit_tree list;
fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts);
fun finish Iss m seen i (nwit, I) =
let
val treess = map (fn j =>
if j < m orelse member (op =) seen j then [([j], Wit_Leaf j)]
else
map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m))
|> flat
|> minimize_wits)
I;
in
map (fn (I, t) => (I, Wit_Node ((i - m, nwit, filter (fn i => i < m) I), t)))
(fold_rev (map_product mk_tree_args) treess [([], [])])
|> minimize_wits
end;
fun tree_to_ctor_wit vars _ _ (Wit_Leaf j) = ([j], nth vars j)
| tree_to_ctor_wit vars ctors witss (Wit_Node ((i, nwit, I), subtrees)) =
(I, nth ctors i $ (Term.list_comb (snd (nth (nth witss i) nwit),
map (snd o tree_to_ctor_wit vars ctors witss) subtrees)));
fun tree_to_coind_wits _ (Wit_Leaf _) = []
| tree_to_coind_wits lwitss (Wit_Node ((i, nwit, I), subtrees)) =
((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees;
(*all BNFs have the same lives*)
fun construct_gfp mixfixes map_bs rel_bs pred_bs set_bss0 bs resBs (resDs, Dss) bnfs absT_infos
lthy =
let
val time = time lthy;
val timer = time (Timer.startRealTimer ());
val live = live_of_bnf (hd bnfs);
val n = length bnfs; (*active*)
val ks = 1 upto n;
val m = live - n; (*passive, if 0 don't generate a new BNF*)
val ls = 1 upto m;
val internals = Config.get lthy bnf_internals;
val b_names = map Binding.name_of bs;
val b_name = mk_common_name b_names;
val b = Binding.name b_name;
fun mk_internal_of_b name =
Binding.prefix_name (name ^ "_") #> Binding.prefix true b_name #> Binding.concealed;
fun mk_internal_b name = mk_internal_of_b name b;
fun mk_internal_bs name = map (mk_internal_of_b name) bs;
val external_bs = map2 (Binding.prefix false) b_names bs
|> not internals ? map Binding.concealed;
val deads = fold (union (op =)) Dss resDs;
val names_lthy = fold Variable.declare_typ deads lthy;
val passives = map fst (subtract (op = o apsnd TFree) deads resBs);
(* tvars *)
val ((((((passiveAs, activeAs), passiveBs), activeBs), passiveCs), activeCs), idxT) = names_lthy
|> variant_tfrees passives
||>> mk_TFrees n
||>> variant_tfrees passives
||>> mk_TFrees n
||>> mk_TFrees m
||>> mk_TFrees n
||> fst o mk_TFrees 1
||> the_single;
val allAs = passiveAs @ activeAs;
val allBs' = passiveBs @ activeBs;
val Ass = replicate n allAs;
val allBs = passiveAs @ activeBs;
val Bss = replicate n allBs;
val allCs = passiveAs @ activeCs;
val allCs' = passiveBs @ activeCs;
val Css' = replicate n allCs';
(* types *)
val dead_poss =
map (fn x => if member (op =) deads (TFree x) then SOME (TFree x) else NONE) resBs;
fun mk_param NONE passive = (hd passive, tl passive)
| mk_param (SOME a) passive = (a, passive);
val mk_params = fold_map mk_param dead_poss #> fst;
fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs;
val (params, params') = `(map Term.dest_TFree) (mk_params passiveAs);
val FTsAs = mk_FTs allAs;
val FTsBs = mk_FTs allBs;
val FTsCs = mk_FTs allCs;
val ATs = map HOLogic.mk_setT passiveAs;
val BTs = map HOLogic.mk_setT activeAs;
val B'Ts = map HOLogic.mk_setT activeBs;
val B''Ts = map HOLogic.mk_setT activeCs;
val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs;
val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs;
val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs;
val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs;
val self_fTs = map (fn T => T --> T) activeAs;
val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs;
val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs';
val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs;
val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs;
val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs;
val setsRTs = map HOLogic.mk_setT sRTs;
val setRTs = map HOLogic.mk_setT RTs;
val all_sbisT = HOLogic.mk_tupleT setsRTs;
val setR'Ts = map HOLogic.mk_setT R'Ts;
val FRTs = mk_FTs (passiveAs @ RTs);
(* terms *)
val mapsAsAs = @{map 4} mk_map_of_bnf Dss Ass Ass bnfs;
val mapsAsBs = @{map 4} mk_map_of_bnf Dss Ass Bss bnfs;
val mapsBsCs' = @{map 4} mk_map_of_bnf Dss Bss Css' bnfs;
val mapsAsCs' = @{map 4} mk_map_of_bnf Dss Ass Css' bnfs;
val map_fsts = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs;
val map_snds = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs;
fun mk_setss Ts = @{map 3} mk_sets_of_bnf (map (replicate live) Dss)
(map (replicate live) (replicate n Ts)) bnfs;
val setssAs = mk_setss allAs;
val setssAs' = transpose setssAs;
val bis_setss = mk_setss (passiveAs @ RTs);
val relsAsBs = @{map 4} mk_rel_of_bnf Dss Ass Bss bnfs;
val bds = @{map 3} mk_bd_of_bnf Dss Ass bnfs;
val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
val sum_bdT = fst (dest_relT (fastype_of sum_bd));
val (sum_bdT_params, sum_bdT_params') = `(map TFree) (Term.add_tfreesT sum_bdT []);
val ((((((((((zs, zs'), Bs), ss), fs), self_fs), all_gs), xFs), yFs), yFs_copy), _) =
lthy
|> mk_Frees' "b" activeAs
||>> mk_Frees "B" BTs
||>> mk_Frees "s" sTs
||>> mk_Frees "f" fTs
||>> mk_Frees "f" self_fTs
||>> mk_Frees "g" all_gTs
||>> mk_Frees "x" FTsAs
||>> mk_Frees "y" FTsBs
||>> mk_Frees "y" FTsBs;
val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
val passive_eqs = map HOLogic.eq_const passiveAs;
val active_UNIVs = map HOLogic.mk_UNIV activeAs;
val passive_ids = map HOLogic.id_const passiveAs;
val active_ids = map HOLogic.id_const activeAs;
val fsts = map fst_const RTs;
val snds = map snd_const RTs;
(* thms *)
val bd_card_orders = map bd_card_order_of_bnf bnfs;
val bd_card_order = hd bd_card_orders
val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
val bd_Card_order = hd bd_Card_orders;
val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
val bd_Cinfinite = hd bd_Cinfinites;
val in_monos = map in_mono_of_bnf bnfs;
val map_comp0s = map map_comp0_of_bnf bnfs;
val sym_map_comps = map mk_sym map_comp0s;
val map_comps = map map_comp_of_bnf bnfs;
val map_cong0s = map map_cong0_of_bnf bnfs;
val map_id0s = map map_id0_of_bnf bnfs;
val map_ids = map map_id_of_bnf bnfs;
val set_bdss = map set_bd_of_bnf bnfs;
val set_mapss = map set_map_of_bnf bnfs;
val rel_congs = map rel_cong0_of_bnf bnfs;
val rel_converseps = map rel_conversep_of_bnf bnfs;
val rel_Grps = map rel_Grp_of_bnf bnfs;
val le_rel_OOs = map le_rel_OO_of_bnf bnfs;
val in_rels = map in_rel_of_bnf bnfs;
val timer = time (timer "Extracted terms & thms");
(* derived thms *)
(*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x) =
map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*)
fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp0 =
let
val lhs = Term.list_comb (mapBsCs, all_gs) $
(Term.list_comb (mapAsBs, passive_ids @ fs) $ x);
val rhs =
Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x;
val goal = mk_Trueprop_eq (lhs, rhs);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_map_comp_id_tac ctxt map_comp0)
|> Thm.close_derivation
end;
val map_comp_id_thms = @{map 5} mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comps;
(*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==>
map id ... id f(m+1) ... f(m+n) x = x*)
fun mk_map_cong0L x mapAsAs sets map_cong0 map_id =
let
fun mk_prem set f z z' =
HOLogic.mk_Trueprop
(mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z))));
val prems = @{map 4} mk_prem (drop m sets) self_fs zs zs';
val goal = mk_Trueprop_eq (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, goal))
(fn {context = ctxt, prems = _} => mk_map_cong0L_tac ctxt m map_cong0 map_id)
|> Thm.close_derivation
end;
val map_cong0L_thms = @{map 5} mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;
val in_mono'_thms = map (fn thm =>
(thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos;
val map_arg_cong_thms =
let
val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy;
val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs';
val concls =
@{map 3} (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y))
yFs yFs_copy maps;
val goals = map2 (fn prem => fn concl => Logic.mk_implies (prem, concl)) prems concls;
in
map (fn goal =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
(hyp_subst_tac ctxt THEN' rtac ctxt refl) 1))
|> Thm.close_derivation)
goals
end;
val timer = time (timer "Derived simple theorems");
(* coalgebra *)
val coalg_bind = mk_internal_b (coN ^ algN) ;
val coalg_def_bind = (Thm.def_binding coalg_bind, []);
(*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in UNIV .. UNIV B1 ... Bn)*)
val coalg_spec =
let
val ins = @{map 3} mk_in (replicate n (passive_UNIVs @ Bs)) setssAs FTsAs;
fun mk_coalg_conjunct B s X z z' =
mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X)));
val rhs = Library.foldr1 HOLogic.mk_conj (@{map 5} mk_coalg_conjunct Bs ss ins zs zs')
in
fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss) rhs
end;
val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((coalg_bind, NoSyn), (coalg_def_bind, coalg_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free));
val coalg_def = mk_unabs_def (2 * n) (Morphism.thm phi coalg_def_free RS meta_eq_to_obj_eq);
fun mk_coalg Bs ss =
let
val args = Bs @ ss;
val Ts = map fastype_of args;
val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT);
in
Term.list_comb (Const (coalg, coalgT), args)
end;
val((((((zs, zs'), Bs), B's), ss), s's), _) =
lthy
|> mk_Frees' "b" activeAs
||>> mk_Frees "B" BTs
||>> mk_Frees "B'" B'Ts
||>> mk_Frees "s" sTs
||>> mk_Frees "s'" s'Ts;
val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss);
val coalg_in_thms = map (fn i =>
coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks
val coalg_set_thmss =
let
val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss);
fun mk_prem x B = mk_Trueprop_mem (x, B);
fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B);
val prems = map2 mk_prem zs Bs;
val conclss = @{map 3} (fn s => fn x => fn sets => map2 (mk_concl s x) Bs (drop m sets))
ss zs setssAs;
val goalss = map2 (fn prem => fn concls => map (fn concl =>
Logic.list_implies (coalg_prem :: [prem], concl)) concls) prems conclss;
in
map (fn goals => map (fn goal =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_coalg_set_tac ctxt coalg_def))
|> Thm.close_derivation)
goals) goalss
end;
fun mk_tcoalg BTs = mk_coalg (map HOLogic.mk_UNIV BTs);
val tcoalg_thm =
let
val goal = HOLogic.mk_Trueprop (mk_tcoalg activeAs ss);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => (rtac ctxt (coalg_def RS iffD2) 1 THEN CONJ_WRAP
(K (EVERY' [rtac ctxt ballI, rtac ctxt CollectI,
CONJ_WRAP' (K (EVERY' [rtac ctxt @{thm subset_UNIV}])) allAs] 1)) ss))
|> Thm.close_derivation
end;
val timer = time (timer "Coalgebra definition & thms");
(* morphism *)
val mor_bind = mk_internal_b morN;
val mor_def_bind = (Thm.def_binding mor_bind, []);
(*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*)
(*mor) forall i = 1 ... n: (\<forall>x \<in> Bi.
Fi_map id ... id f1 ... fn (si x) = si' (fi x)*)
val mor_spec =
let
fun mk_fbetw f B1 B2 z z' =
mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2)));
fun mk_mor B mapAsBs f s s' z z' =
mk_Ball B (Term.absfree z' (HOLogic.mk_eq
(Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z))));
val rhs = HOLogic.mk_conj
(Library.foldr1 HOLogic.mk_conj (@{map 5} mk_fbetw fs Bs B's zs zs'),
Library.foldr1 HOLogic.mk_conj (@{map 7} mk_mor Bs mapsAsBs fs ss s's zs zs'))
in
fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ fs) rhs
end;
val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((mor_bind, NoSyn), (mor_def_bind, mor_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val mor = fst (Term.dest_Const (Morphism.term phi mor_free));
val mor_def = mk_unabs_def (5 * n) (Morphism.thm phi mor_def_free RS meta_eq_to_obj_eq);
fun mk_mor Bs1 ss1 Bs2 ss2 fs =
let
val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs;
val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs);
val morT = Library.foldr (op -->) (Ts, HOLogic.boolT);
in
Term.list_comb (Const (mor, morT), args)
end;
val ((((((((((((((zs, z's), Bs), Bs_copy), B's), B''s), ss), s's), s''s), fs), fs_copy), gs),
RFs), Rs), _) =
lthy
|> mk_Frees "b" activeAs
||>> mk_Frees "b" activeBs
||>> mk_Frees "B" BTs
||>> mk_Frees "B" BTs
||>> mk_Frees "B'" B'Ts
||>> mk_Frees "B''" B''Ts
||>> mk_Frees "s" sTs
||>> mk_Frees "s'" s'Ts
||>> mk_Frees "s''" s''Ts
||>> mk_Frees "f" fTs
||>> mk_Frees "f" fTs
||>> mk_Frees "g" gTs
||>> mk_Frees "x" FRTs
||>> mk_Frees "R" setRTs;
val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
val (mor_image_thms, morE_thms) =
let
val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
fun mk_image_goal f B1 B2 =
Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2));
val image_goals = @{map 3} mk_image_goal fs Bs B's;
fun mk_elim_goal B mapAsBs f s s' x =
Logic.list_implies ([prem, mk_Trueprop_mem (x, B)],
mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x)));
val elim_goals = @{map 6} mk_elim_goal Bs mapsAsBs fs ss s's zs;
fun prove goal =
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_mor_elim_tac ctxt mor_def))
|> Thm.close_derivation;
in
(map prove image_goals, map prove elim_goals)
end;
val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms;
val mor_incl_thm =
let
val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy;
val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
val vars = fold (Variable.add_free_names lthy) (concl :: prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, prems = _} => mk_mor_incl_tac ctxt mor_def map_ids)
|> Thm.close_derivation
end;
val mor_id_thm = mor_incl_thm OF (replicate n subset_refl);
val mor_comp_thm =
let
val prems =
[HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs),
HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)];
val concl =
HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs));
val vars = fold (Variable.add_free_names lthy) (concl :: prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, prems = _} =>
mk_mor_comp_tac ctxt mor_def mor_image'_thms morE_thms map_comp_id_thms)
|> Thm.close_derivation
end;
val mor_cong_thm =
let
val prems = map HOLogic.mk_Trueprop
(map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs])
val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy);
val vars = fold (Variable.add_free_names lthy) (concl :: prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, prems = _} => (hyp_subst_tac ctxt THEN' assume_tac ctxt) 1)
|> Thm.close_derivation
end;
val mor_UNIV_thm =
let
fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
(HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s),
HOLogic.mk_comp (s', f));
val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs;
val rhs = Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct mapsAsBs fs ss s's);
val vars = fold (Variable.add_free_names lthy) [lhs, rhs] [];
in
Goal.prove_sorry lthy vars [] (mk_Trueprop_eq (lhs, rhs))
(fn {context = ctxt, prems = _} => mk_mor_UNIV_tac ctxt morE_thms mor_def)
|> Thm.close_derivation
end;
val mor_str_thm =
let
val maps = map2 (fn Ds => fn bnf => Term.list_comb
(mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs;
val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_mor_str_tac ctxt ks mor_UNIV_thm)
|> Thm.close_derivation
end;
val timer = time (timer "Morphism definition & thms");
(* bisimulation *)
val bis_bind = mk_internal_b bisN;
val bis_def_bind = (Thm.def_binding bis_bind, []);
fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2));
val bis_le = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_bis_le_conjunct Rs Bs B's)
val bis_spec =
let
val fst_args = passive_ids @ fsts;
val snd_args = passive_ids @ snds;
fun mk_bis R s s' b1 b2 RF map1 map2 sets =
list_all_free [b1, b2] (HOLogic.mk_imp
(HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
mk_Bex (mk_in (passive_UNIVs @ Rs) sets (snd (dest_Free RF)))
(Term.absfree (dest_Free RF) (HOLogic.mk_conj
(HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1),
HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2))))));
val rhs = HOLogic.mk_conj
(bis_le, Library.foldr1 HOLogic.mk_conj
(@{map 9} mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss))
in
fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ Rs) rhs
end;
val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((bis_bind, NoSyn), (bis_def_bind, bis_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val bis = fst (Term.dest_Const (Morphism.term phi bis_free));
val bis_def = mk_unabs_def (5 * n) (Morphism.thm phi bis_def_free RS meta_eq_to_obj_eq);
fun mk_bis Bs1 ss1 Bs2 ss2 Rs =
let
val args = Bs1 @ ss1 @ Bs2 @ ss2 @ Rs;
val Ts = map fastype_of args;
val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT);
in
Term.list_comb (Const (bis, bisT), args)
end;
val (((((((((((((((((zs, z's), Bs), B's), B''s), ss), s's), s''s), fs), (Rtuple, Rtuple')), Rs),
Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), _) =
lthy
|> mk_Frees "b" activeAs
||>> mk_Frees "b" activeBs
||>> mk_Frees "B" BTs
||>> mk_Frees "B'" B'Ts
||>> mk_Frees "B''" B''Ts
||>> mk_Frees "s" sTs
||>> mk_Frees "s'" s'Ts
||>> mk_Frees "s''" s''Ts
||>> mk_Frees "f" fTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "Rtuple") all_sbisT
||>> mk_Frees "R" setRTs
||>> mk_Frees "R" setRTs
||>> mk_Frees "R'" setR'Ts
||>> mk_Frees "R" setsRTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "i") idxT
||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT)
||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs);
val bis_cong_thm =
let
val prems = map HOLogic.mk_Trueprop
(mk_bis Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs)
val concl = HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs_copy);
val vars = fold (Variable.add_free_names lthy) (concl :: prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, prems = _} => (hyp_subst_tac ctxt THEN' assume_tac ctxt) 1)
|> Thm.close_derivation
end;
val bis_rel_thm =
let
fun mk_conjunct R s s' b1 b2 rel =
list_all_free [b1, b2] (HOLogic.mk_imp
(HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R),
Term.list_comb (rel, passive_eqs @ map mk_in_rel Rs) $ (s $ b1) $ (s' $ b2)));
val rhs = HOLogic.mk_conj
(bis_le, Library.foldr1 HOLogic.mk_conj
(@{map 6} mk_conjunct Rs ss s's zs z's relsAsBs))
val goal = mk_Trueprop_eq (mk_bis Bs ss B's s's Rs, rhs);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_bis_rel_tac ctxt m bis_def in_rels map_comps
map_cong0s set_mapss)
|> Thm.close_derivation
end;
val bis_converse_thm =
let
val goal = Logic.mk_implies (HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs),
HOLogic.mk_Trueprop (mk_bis B's s's Bs ss (map mk_converse Rs)));
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_bis_converse_tac ctxt m bis_rel_thm rel_congs
rel_converseps)
|> Thm.close_derivation
end;
val bis_O_thm =
let
val prems =
[HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs),
HOLogic.mk_Trueprop (mk_bis B's s's B''s s''s R's)];
val concl =
HOLogic.mk_Trueprop (mk_bis Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's));
val vars = fold (Variable.add_free_names lthy) (concl :: prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, prems = _} => mk_bis_O_tac ctxt m bis_rel_thm rel_congs le_rel_OOs)
|> Thm.close_derivation
end;
val bis_Gr_thm =
let
val concl = HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map2 mk_Gr Bs fs));
val vars = fold (Variable.add_free_names lthy) ([coalg_prem, mor_prem, concl]) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies ([coalg_prem, mor_prem], concl))
(fn {context = ctxt, prems = _} => mk_bis_Gr_tac ctxt bis_rel_thm rel_Grps mor_image_thms
morE_thms coalg_in_thms)
|> Thm.close_derivation
end;
val bis_image2_thm = bis_cong_thm OF
((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) ::
replicate n @{thm image2_Gr});
val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) ::
replicate n @{thm Id_on_Gr});
val bis_Union_thm =
let
val prem =
HOLogic.mk_Trueprop (mk_Ball Idx
(Term.absfree idx' (mk_bis Bs ss B's s's (map (fn R => R $ idx) Ris))));
val concl =
HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map (mk_UNION Idx) Ris));
val vars = fold (Variable.add_free_names lthy) [prem, concl] [];
in
Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, concl))
(fn {context = ctxt, prems = _} => mk_bis_Union_tac ctxt bis_def in_mono'_thms)
|> Thm.close_derivation
end;
(* self-bisimulation *)
fun mk_sbis Bs ss Rs = mk_bis Bs ss Bs ss Rs;
(* largest self-bisimulation *)
val lsbis_binds = mk_internal_bs lsbisN;
fun lsbis_bind i = nth lsbis_binds (i - 1);
val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind;
val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs
(HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis Bs ss sRs)));
fun lsbis_spec i =
fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss)
(mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i)));
val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> fold_map (fn i => Local_Theory.define
((lsbis_bind i, NoSyn), (lsbis_def_bind i, lsbis_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val lsbis_defs = map (fn def =>
mk_unabs_def (2 * n) (Morphism.thm phi def RS meta_eq_to_obj_eq)) lsbis_def_frees;
val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees;
fun mk_lsbis Bs ss i =
let
val args = Bs @ ss;
val Ts = map fastype_of args;
val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1)))));
val lsbisT = Library.foldr (op -->) (Ts, RT);
in
Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args)
end;
val (((((zs, zs'), Bs), ss), sRs), _) =
lthy
|> mk_Frees' "b" activeAs
||>> mk_Frees "B" BTs
||>> mk_Frees "s" sTs
||>> mk_Frees "R" setsRTs;
val sbis_prem = HOLogic.mk_Trueprop (mk_sbis Bs ss sRs);
val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss);
val sbis_lsbis_thm =
let
val goal = HOLogic.mk_Trueprop (mk_sbis Bs ss (map (mk_lsbis Bs ss) ks));
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_sbis_lsbis_tac ctxt lsbis_defs bis_Union_thm bis_cong_thm)
|> Thm.close_derivation
end;
val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS
(bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks;
val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS
(bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks;
val incl_lsbis_thms =
let
fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis Bs ss i));
val goals = map2 (fn i => fn R => Logic.mk_implies (sbis_prem, mk_concl i R)) ks sRs;
in
@{map 3} (fn goal => fn i => fn def =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_incl_lsbis_tac ctxt n i def))
|> Thm.close_derivation)
goals ks lsbis_defs
end;
val equiv_lsbis_thms =
let
fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis Bs ss i));
val goals = map2 (fn i => fn B => Logic.mk_implies (coalg_prem, mk_concl i B)) ks Bs;
in
@{map 3} (fn goal => fn l_incl => fn incl_l =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_equiv_lsbis_tac ctxt sbis_lsbis_thm l_incl incl_l
bis_Id_on_thm bis_converse_thm bis_O_thm)
|> Thm.close_derivation))
goals lsbis_incl_thms incl_lsbis_thms
end;
val timer = time (timer "Bisimulations");
(* bounds *)
val (lthy, sbd, sbdT,
sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) =
if n = 1
then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss)
else
let
val sbdT_bind = mk_internal_b sum_bdTN;
val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) =
typedef (sbdT_bind, sum_bdT_params', NoSyn)
(HOLogic.mk_UNIV sum_bdT) NONE (fn ctxt =>
EVERY' [rtac ctxt exI, rtac ctxt UNIV_I] 1) lthy;
val sbdT = Type (sbdT_name, sum_bdT_params);
val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT);
val sbd_bind = mk_internal_b sum_bdN;
val sbd_def_bind = (Thm.def_binding sbd_bind, []);
val sbd_spec = mk_dir_image sum_bd Abs_sbdT;
val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((sbd_bind, NoSyn), (sbd_def_bind, sbd_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val sbd_def = Morphism.thm phi sbd_def_free RS meta_eq_to_obj_eq;
val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT));
val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info);
val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info);
val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
val sum_Card_order = sum_Cinfinite RS conjunct2;
val sum_card_order = mk_sum_card_order bd_card_orders;
val sbd_ordIso = @{thm ssubst_Pair_rhs} OF
[@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order], sbd_def];
val sbd_card_order = @{thm iffD2[OF arg_cong[of _ _ card_order]]} OF
[sbd_def, @{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]];
val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite];
val sbd_Card_order = sbd_Cinfinite RS conjunct2;
fun mk_set_sbd i bd_Card_order bds =
map (fn thm => @{thm ordLeq_ordIso_trans} OF
[bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds;
val set_sbdss = @{map 3} mk_set_sbd ks bd_Card_orders set_bdss;
in
(lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss)
end;
val sbdTs = replicate n sbdT;
val sum_sbdT = mk_sumTN sbdTs;
val sum_sbd_listT = HOLogic.listT sum_sbdT;
val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT;
val bdTs = passiveAs @ replicate n sbdT;
val to_sbd_maps = @{map 4} mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs;
val bdFTs = mk_FTs bdTs;
val sbdFT = mk_sumTN bdFTs;
val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT);
val treeQT = HOLogic.mk_setT treeT;
val treeTs = passiveAs @ replicate n treeT;
val treeQTs = passiveAs @ replicate n treeQT;
val treeFTs = mk_FTs treeTs;
val tree_maps = @{map 4} mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs;
val final_maps = @{map 4} mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs;
val isNode_setss = mk_setss (passiveAs @ replicate n sbdT);
val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []];
val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs);
val Lev_recT = fastype_of Zero;
val Nil = HOLogic.mk_tuple (@{map 3} (fn i => fn z => fn z'=>
Term.absfree z' (mk_InN activeAs z i)) ks zs zs');
val rv_recT = fastype_of Nil;
val (((((((((((((((zs, zs'), zs_copy), ss), (nat, nat')),
(sumx, sumx')), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')), (lab, lab')),
(Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')), _) =
lthy
|> mk_Frees' "b" activeAs
||>> mk_Frees "b" activeAs
||>> mk_Frees "s" sTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "n") HOLogic.natT
||>> yield_singleton (apfst (~~) oo mk_Frees' "sumx") sum_sbdT
||>> mk_Frees' "k" sbdTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "kl") sum_sbd_listT
||>> yield_singleton (apfst (~~) oo mk_Frees' "kl") sum_sbd_listT
||>> yield_singleton (apfst (~~) oo mk_Frees' "Kl") sum_sbd_list_setT
||>> yield_singleton (apfst (~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT)
||>> yield_singleton (apfst (~~) oo mk_Frees' "Kl_lab") treeT
||>> mk_Frees "x" bdFTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "rec") Lev_recT
||>> yield_singleton (apfst (~~) oo mk_Frees' "rec") rv_recT;
val (k, k') = (hd kks, hd kks')
val timer = time (timer "Bounds");
(* tree coalgebra *)
val isNode_binds = mk_internal_bs isNodeN;
fun isNode_bind i = nth isNode_binds (i - 1);
val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind;
val isNodeT =
Library.foldr (op -->) (map fastype_of [Kl, lab, kl], HOLogic.boolT);
val Succs = @{map 3} (fn i => fn k => fn k' =>
HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl)))
ks kks kks';
fun isNode_spec sets x i =
let
val active_sets = drop m (map (fn set => set $ x) sets);
val rhs = list_exists_free [x]
(Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) ::
map2 (curry HOLogic.mk_eq) active_sets Succs));
in
fold_rev (Term.absfree o Term.dest_Free) [Kl, lab, kl] rhs
end;
val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 3} (fn i => fn x => fn sets => Local_Theory.define
((isNode_bind i, NoSyn), (isNode_def_bind i, isNode_spec sets x i)))
ks xs isNode_setss
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val isNode_defs = map (fn def =>
mk_unabs_def 3 (Morphism.thm phi def RS meta_eq_to_obj_eq)) isNode_def_frees;
val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees;
fun mk_isNode kl i =
Term.list_comb (Const (nth isNodes (i - 1), isNodeT), [Kl, lab, kl]);
val isTree =
let
val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl);
val tree = mk_Ball Kl (Term.absfree kl'
(Library.foldr1 HOLogic.mk_conj (@{map 4} (fn Succ => fn i => fn k => fn k' =>
mk_Ball Succ (Term.absfree k' (mk_isNode
(mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i)))
Succs ks kks kks')));
in
HOLogic.mk_conj (empty, tree)
end;
val carT_binds = mk_internal_bs carTN;
fun carT_bind i = nth carT_binds (i - 1);
val carT_def_bind = rpair [] o Thm.def_binding o carT_bind;
fun carT_spec i =
HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab]
(HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)),
HOLogic.mk_conj (isTree, mk_isNode (HOLogic.mk_list sum_sbdT []) i))));
val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> fold_map (fn i =>
Local_Theory.define ((carT_bind i, NoSyn), (carT_def_bind i, carT_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val carT_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) carT_def_frees;
val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees;
fun mk_carT i = Const (nth carTs (i - 1), HOLogic.mk_setT treeT);
val strT_binds = mk_internal_bs strTN;
fun strT_bind i = nth strT_binds (i - 1);
val strT_def_bind = rpair [] o Thm.def_binding o strT_bind;
fun strT_spec mapFT FT i =
let
fun mk_f i k k' =
let val in_k = mk_InN sbdTs k i;
in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end;
val f = Term.list_comb (mapFT, passive_ids @ @{map 3} mk_f ks kks kks');
val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs));
val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2);
in
HOLogic.mk_case_prod (Term.absfree Kl' (Term.absfree lab'
(mk_case_sumN fs $ (lab $ HOLogic.mk_list sum_sbdT []))))
end;
val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 3} (fn i => fn mapFT => fn FT => Local_Theory.define
((strT_bind i, NoSyn), (strT_def_bind i, strT_spec mapFT FT i)))
ks tree_maps treeFTs
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val strT_defs = map (fn def =>
trans OF [Morphism.thm phi def RS meta_eq_to_obj_eq RS fun_cong, @{thm prod.case}])
strT_def_frees;
val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees;
fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT);
val carTAs = map mk_carT ks;
val strTAs = map2 mk_strT treeFTs ks;
val coalgT_thm =
Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop (mk_coalg carTAs strTAs))
(fn {context = ctxt, prems = _} => mk_coalgT_tac ctxt m
(coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss)
|> Thm.close_derivation;
val timer = time (timer "Tree coalgebra");
fun mk_to_sbd s x i i' =
mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
fun mk_from_sbd s x i i' =
mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd;
fun mk_to_sbd_thmss thm = map (map (fn set_sbd =>
thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss;
val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj};
val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard};
val Lev_bind = mk_internal_b LevN;
val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind);
val Lev_spec =
let
fun mk_Suc i s setsAs a a' =
let
val sets = drop m setsAs;
fun mk_set i' set b =
let
val Cons = HOLogic.mk_eq (kl_copy,
mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl)
val b_set = HOLogic.mk_mem (b, set $ (s $ a));
val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b);
in
HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl]
(HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec))))
end;
in
Term.absfree a' (Library.foldl1 mk_union (@{map 3} mk_set ks sets zs_copy))
end;
val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec'
(HOLogic.mk_tuple (@{map 5} mk_Suc ks ss setssAs zs zs')));
val rhs = mk_rec_nat Zero Suc;
in
fold_rev (Term.absfree o Term.dest_Free) ss rhs
end;
val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((Lev_bind, NoSyn), (Lev_def_bind, Lev_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val Lev_def = mk_unabs_def n (Morphism.thm phi Lev_def_free RS meta_eq_to_obj_eq);
val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free));
fun mk_Lev ss nat i =
let
val Ts = map fastype_of ss;
val LevT = Library.foldr (op -->) (Ts, HOLogic.natT -->
HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts));
in
mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i
end;
val Lev_0s = flat (mk_rec_simps n @{thm rec_nat_0_imp} [Lev_def]);
val Lev_Sucs = flat (mk_rec_simps n @{thm rec_nat_Suc_imp} [Lev_def]);
val rv_bind = mk_internal_b rvN;
val rv_def_bind = rpair [] (Thm.def_binding rv_bind);
val rv_spec =
let
fun mk_Cons i s b b' =
let
fun mk_case i' =
Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k));
in
Term.absfree b' (mk_case_sumN (map mk_case ks) $ sumx)
end;
val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec'
(HOLogic.mk_tuple (@{map 4} mk_Cons ks ss zs zs'))));
val rhs = mk_rec_list Nil Cons;
in
fold_rev (Term.absfree o Term.dest_Free) ss rhs
end;
val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> Local_Theory.define ((rv_bind, NoSyn), (rv_def_bind, rv_spec))
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val rv_def = mk_unabs_def n (Morphism.thm phi rv_def_free RS meta_eq_to_obj_eq);
val rv = fst (Term.dest_Const (Morphism.term phi rv_free));
fun mk_rv ss kl i =
let
val Ts = map fastype_of ss;
val As = map domain_type Ts;
val rvT = Library.foldr (op -->) (Ts, fastype_of kl -->
HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As));
in
mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i
end;
val rv_Nils = flat (mk_rec_simps n @{thm rec_list_Nil_imp} [rv_def]);
val rv_Conss = flat (mk_rec_simps n @{thm rec_list_Cons_imp} [rv_def]);
val beh_binds = mk_internal_bs behN;
fun beh_bind i = nth beh_binds (i - 1);
val beh_def_bind = rpair [] o Thm.def_binding o beh_bind;
fun beh_spec i z =
let
fun mk_case i to_sbd_map s k k' =
Term.absfree k' (mk_InN bdFTs
(Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i);
val Lab = Term.absfree kl'
(mk_case_sumN (@{map 5} mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z));
val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
(Term.absfree nat' (mk_Lev ss nat i $ z)), Lab);
in
fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) rhs
end;
val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 2} (fn i => fn z =>
Local_Theory.define ((beh_bind i, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val beh_defs = map (fn def =>
mk_unabs_def (n + 1) (Morphism.thm phi def RS meta_eq_to_obj_eq)) beh_def_frees;
val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees;
fun mk_beh ss i =
let
val Ts = map fastype_of ss;
val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT);
in
Term.list_comb (Const (nth behs (i - 1), behT), ss)
end;
val ((((((zs, zs_copy), zs_copy2), ss), (nat, nat')), (kl, kl')), _) =
lthy
|> mk_Frees "b" activeAs
||>> mk_Frees "b" activeAs
||>> mk_Frees "b" activeAs
||>> mk_Frees "s" sTs
||>> yield_singleton (apfst (~~) oo mk_Frees' "n") HOLogic.natT
||>> yield_singleton (apfst (~~) oo mk_Frees' "kl") sum_sbd_listT;
val (length_Lev_thms, length_Lev'_thms) =
let
fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
HOLogic.mk_eq (mk_size kl, nat));
val goal = list_all_free (kl :: zs)
(Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
val vars = Variable.add_free_names lthy goal [];
val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat];
val length_Lev =
Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} => mk_length_Lev_tac ctxt cts Lev_0s Lev_Sucs)
|> Thm.close_derivation;
val length_Lev' = mk_specN (n + 1) length_Lev;
val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks;
fun mk_goal i z = Logic.mk_implies
(mk_Trueprop_mem (kl, mk_Lev ss nat i $ z),
mk_Trueprop_mem (kl, mk_Lev ss (mk_size kl) i $ z));
val goals = map2 mk_goal ks zs;
val length_Levs' =
map2 (fn goal => fn length_Lev =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_length_Lev'_tac ctxt length_Lev))
|> Thm.close_derivation)
goals length_Levs;
in
(length_Levs, length_Levs')
end;
val rv_last_thmss =
let
fun mk_conjunct i z i' z_copy = list_exists_free [z_copy]
(HOLogic.mk_eq
(mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z,
mk_InN activeAs z_copy i'));
val goal = list_all_free (k :: zs)
(Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z =>
Library.foldr1 HOLogic.mk_conj
(map2 (mk_conjunct i z) ks zs_copy)) ks zs));
val vars = Variable.add_free_names lthy goal [];
val cTs = [SOME (Thm.ctyp_of lthy sum_sbdT)];
val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree kl' goal, kl];
val rv_last =
Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} => mk_rv_last_tac ctxt cTs cts rv_Nils rv_Conss)
|> Thm.close_derivation;
val rv_last' = mk_specN (n + 1) rv_last;
in
map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks
end;
val set_Lev_thmsss =
let
fun mk_conjunct i z =
let
fun mk_conjunct' i' sets s z' =
let
fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp
(HOLogic.mk_mem (z'', set $ (s $ z')),
HOLogic.mk_mem (mk_append (kl,
HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']),
mk_Lev ss (HOLogic.mk_Suc nat) i $ z));
in
HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'),
(Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_conjunct'' ks (drop m sets) zs_copy2)))
end;
in
HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct' ks setssAs ss zs_copy))
end;
val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2)
(Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
val vars = Variable.add_free_names lthy goal [];
val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat];
val set_Lev =
Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} =>
mk_set_Lev_tac ctxt cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss)
|> Thm.close_derivation;
val set_Lev' = mk_specN (3 * n + 1) set_Lev;
in
map (fn i => map (fn i' => map (fn i'' => set_Lev' RS
mk_conjunctN n i RS mp RS
mk_conjunctN n i' RS mp RS
mk_conjunctN n i'' RS mp) ks) ks) ks
end;
val set_image_Lev_thmsss =
let
fun mk_conjunct i z =
let
fun mk_conjunct' i' sets =
let
fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp
(HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''),
HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z''))));
in
HOLogic.mk_imp (HOLogic.mk_mem
(mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']),
mk_Lev ss (HOLogic.mk_Suc nat) i $ z),
(Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct'' ks sets ss zs_copy)))
end;
in
HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z),
Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs')))
end;
val goal = list_all_free (kl :: k :: zs @ zs_copy)
(Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs));
val vars = Variable.add_free_names lthy goal [];
val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat];
val set_image_Lev =
Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} =>
mk_set_image_Lev_tac ctxt cts Lev_0s Lev_Sucs rv_Nils rv_Conss
from_to_sbd_thmss to_sbd_inj_thmss)
|> Thm.close_derivation;
val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev;
in
map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS
mk_conjunctN n i RS mp RS
mk_conjunctN n i'' RS mp RS
mk_conjunctN n i' RS mp) ks) ks) ks
end;
val mor_beh_thm =
let
val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss carTAs strTAs (map (mk_beh ss) ks));
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_mor_beh_tac ctxt m mor_def mor_cong_thm
beh_defs carT_defs strT_defs isNode_defs
to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss
length_Lev_thms length_Lev'_thms rv_last_thmss set_Lev_thmsss
set_image_Lev_thmsss set_mapss map_comp_id_thms map_cong0s)
|> Thm.close_derivation
end;
val timer = time (timer "Behavioral morphism");
val lsbisAs = map (mk_lsbis carTAs strTAs) ks;
fun mk_str_final i =
mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1),
passive_ids @ map mk_proj lsbisAs), nth strTAs (i - 1)));
val car_finals = map2 mk_quotient carTAs lsbisAs;
val str_finals = map mk_str_final ks;
val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss;
val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms;
val congruent_str_final_thms =
let
fun mk_goal R final_map strT =
HOLogic.mk_Trueprop (mk_congruent R (HOLogic.mk_comp
(Term.list_comb (final_map, passive_ids @ map mk_proj lsbisAs), strT)));
val goals = @{map 3} mk_goal lsbisAs final_maps strTAs;
in
@{map 4} (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_congruent_str_final_tac ctxt m lsbisE map_comp_id
map_cong0 equiv_LSBIS_thms)
|> Thm.close_derivation)
goals lsbisE_thms map_comp_id_thms map_cong0s
end;
val coalg_final_thm = Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_coalg car_finals str_finals))
(fn {context = ctxt, prems = _} => mk_coalg_final_tac ctxt m coalg_def
congruent_str_final_thms equiv_LSBIS_thms set_mapss coalgT_set_thmss)
|> Thm.close_derivation;
val mor_T_final_thm = Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finals str_finals (map mk_proj lsbisAs)))
(fn {context = ctxt, prems = _} => mk_mor_T_final_tac ctxt mor_def congruent_str_final_thms
equiv_LSBIS_thms)
|> Thm.close_derivation;
val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm];
val in_car_final_thms = map (fn thm => thm OF [mor_final_thm, UNIV_I]) mor_image'_thms;
val timer = time (timer "Final coalgebra");
val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
lthy
|> @{fold_map 4} (fn b => fn mx => fn car_final => fn in_car_final =>
typedef (b, params, mx) car_final NONE
(fn ctxt => EVERY' [rtac ctxt exI, rtac ctxt in_car_final] 1))
bs mixfixes car_finals in_car_final_thms
|>> apsnd split_list o split_list;
val Ts = map (fn name => Type (name, params')) T_names;
fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts;
val Ts' = mk_Ts passiveBs;
val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts;
val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts;
val Reps = map #Rep T_loc_infos;
val Rep_injects = map #Rep_inject T_loc_infos;
val Abs_inverses = map #Abs_inverse T_loc_infos;
val timer = time (timer "THE TYPEDEFs & Rep/Abs thms");
val UNIVs = map HOLogic.mk_UNIV Ts;
val FTs = mk_FTs (passiveAs @ Ts);
val FTs_setss = mk_setss (passiveAs @ Ts);
val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs;
val unfold_fTs = map2 (curry op -->) activeAs Ts;
val emptys = map (fn T => HOLogic.mk_set T []) passiveAs;
val Zeros = map (fn empty =>
HOLogic.mk_tuple (map (fn U => absdummy U empty) Ts)) emptys;
val hrecTs = map fastype_of Zeros;
val (((zs, ss), (Jzs, Jzs')), _) =
lthy
|> mk_Frees "b" activeAs
||>> mk_Frees "s" sTs
||>> mk_Frees' "z" Ts;
fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
val dtor_def_bind = rpair [] o Binding.concealed o Thm.def_binding o dtor_bind;
fun dtor_spec rep str map_FT Jz Jz' =
Term.absfree Jz'
(Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $ (str $ (rep $ Jz)));
val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 6} (fn i => fn rep => fn str => fn mapx => fn Jz => fn Jz' =>
Local_Theory.define ((dtor_bind i, NoSyn),
(dtor_def_bind i, dtor_spec rep str mapx Jz Jz')))
ks Rep_Ts str_finals map_FTs Jzs Jzs'
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
fun mk_dtors passive =
map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o
Morphism.term phi) dtor_frees;
val dtors = mk_dtors passiveAs;
val dtor's = mk_dtors passiveBs;
val dtor_defs = map (fn def =>
Morphism.thm phi def RS meta_eq_to_obj_eq RS fun_cong) dtor_def_frees;
val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss;
val (mor_Rep_thm, mor_Abs_thm) =
let
val mor_Rep =
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts))
(fn {context = ctxt, prems = _} => mk_mor_Rep_tac ctxt (mor_def :: dtor_defs) Reps
Abs_inverses coalg_final_set_thmss map_comp_id_thms map_cong0L_thms)
|> Thm.close_derivation;
val mor_Abs =
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts))
(fn {context = ctxt, prems = _} => mk_mor_Abs_tac ctxt (mor_def :: dtor_defs)
Abs_inverses)
|> Thm.close_derivation;
in
(mor_Rep, mor_Abs)
end;
val timer = time (timer "dtor definitions & thms");
fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_");
val unfold_def_bind = rpair [] o Binding.concealed o Thm.def_binding o unfold_bind;
fun unfold_spec abs f z = fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) (abs $ (f $ z));
val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 4} (fn i => fn abs => fn f => fn z =>
Local_Theory.define ((unfold_bind i, NoSyn), (unfold_def_bind i, unfold_spec abs f z)))
ks Abs_Ts (map (fn i => HOLogic.mk_comp
(mk_proj (nth lsbisAs (i - 1)), mk_beh ss i)) ks) zs
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val unfolds = map (Morphism.term phi) unfold_frees;
val unfold_names = map (fst o dest_Const) unfolds;
fun mk_unfolds passives actives =
@{map 3} (fn name => fn T => fn active =>
Const (name, Library.foldr (op -->)
(map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T)))
unfold_names (mk_Ts passives) actives;
fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->)
(map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss);
val unfold_defs = map (fn def =>
mk_unabs_def (n + 1) (Morphism.thm phi def RS meta_eq_to_obj_eq)) unfold_def_frees;
val (((ss, TRs), unfold_fs), _) =
lthy
|> mk_Frees "s" sTs
||>> mk_Frees "r" (map (mk_relT o `I) Ts)
||>> mk_Frees "f" unfold_fTs;
val mor_unfold_thm =
let
val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses;
val morEs' = map (fn thm => (thm OF [mor_final_thm, UNIV_I]) RS sym) morE_thms;
val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks));
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_mor_unfold_tac ctxt m mor_UNIV_thm dtor_defs
unfold_defs Abs_inverses' morEs' map_comp_id_thms map_cong0s)
|> Thm.close_derivation
end;
val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms;
val (raw_coind_thms, raw_coind_thm) =
let
val prem = HOLogic.mk_Trueprop (mk_sbis UNIVs dtors TRs);
val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts));
val vars = fold (Variable.add_free_names lthy) [prem, concl] [];
in
`split_conj_thm (Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, concl))
(fn {context = ctxt, prems = _} => mk_raw_coind_tac ctxt bis_def bis_cong_thm bis_O_thm
bis_converse_thm bis_Gr_thm tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm
lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects)
|> Thm.close_derivation)
end;
val (unfold_unique_mor_thms, unfold_unique_mor_thm) =
let
val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs);
fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i);
val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map2 mk_fun_eq unfold_fs ks));
val vars = fold (Variable.add_free_names lthy) [prem, unique] [];
val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm);
val mor_thm = mor_comp_thm OF [mor_final_thm, mor_Abs_thm];
val unique_mor = Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, unique))
(fn {context = ctxt, prems = _} => mk_unfold_unique_mor_tac ctxt raw_coind_thms
bis_thm mor_thm unfold_defs)
|> Thm.close_derivation;
in
`split_conj_thm unique_mor
end;
val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n
(mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm));
val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms;
val unfold_o_dtor_thms =
let
val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm];
in
map2 (fn unique => fn unfold_ctor =>
trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms
end;
val timer = time (timer "unfold definitions & thms");
val map_dtors = map2 (fn Ds => fn bnf =>
Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf,
map HOLogic.id_const passiveAs @ dtors)) Dss bnfs;
fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
val ctor_def_bind = rpair [] o Binding.concealed o Thm.def_binding o ctor_bind;
fun ctor_spec i = mk_unfold Ts map_dtors i;
val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> fold_map (fn i =>
Local_Theory.define ((ctor_bind i, NoSyn), (ctor_def_bind i, ctor_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
fun mk_ctors params =
map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
ctor_frees;
val ctors = mk_ctors params';
val ctor_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) ctor_def_frees;
val ctor_o_dtor_thms = map2 (Local_Defs.fold lthy o single) ctor_defs unfold_o_dtor_thms;
val dtor_o_ctor_thms =
let
fun mk_goal dtor ctor FT =
mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT);
val goals = @{map 3} mk_goal dtors ctors FTs;
in
@{map 5} (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_dtor_o_ctor_tac ctxt ctor_def unfold map_comp_id
map_cong0L unfold_o_dtor_thms)
|> Thm.close_derivation)
goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms
end;
val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms;
val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms;
val bij_dtor_thms =
map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms;
val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms;
val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms;
val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms;
val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms;
val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms;
val bij_ctor_thms =
map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms;
val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms;
val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms;
val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms;
val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms;
val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms;
val timer = time (timer "ctor definitions & thms");
val (((((Jzs, Jzs_copy), Jzs1), Jzs2), phis), _) =
lthy
|> mk_Frees "z" Ts
||>> mk_Frees "z'" Ts
||>> mk_Frees "z1" Ts
||>> mk_Frees "z2" Ts
||>> mk_Frees "P" (map2 mk_pred2T Ts Ts);
val (coinduct_params, dtor_coinduct_thm) =
let
val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs;
fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2));
val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_concl phis Jzs1 Jzs2));
fun mk_rel_prem phi dtor rel Jz Jz_copy =
let
val concl = Term.list_comb (rel, passive_eqs @ phis) $
(dtor $ Jz) $ (dtor $ Jz_copy);
in
HOLogic.mk_Trueprop
(list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl)))
end;
val rel_prems = @{map 5} mk_rel_prem phis dtors rels Jzs Jzs_copy;
val dtor_coinduct_goal = Logic.list_implies (rel_prems, concl);
val dtor_coinduct =
Variable.add_free_names lthy dtor_coinduct_goal []
|> (fn vars => Goal.prove_sorry lthy vars [] dtor_coinduct_goal
(fn {context = ctxt, prems = _} => mk_dtor_coinduct_tac ctxt m raw_coind_thm bis_rel_thm
rel_congs))
|> Thm.close_derivation;
in
(rev (Term.add_tfrees dtor_coinduct_goal []), dtor_coinduct)
end;
val timer = time (timer "coinduction");
fun mk_dtor_map_DEADID_thm dtor_inject map_id0 =
trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym];
fun mk_dtor_map_unique_DEADID_thm () =
let
val (funs, algs) =
HOLogic.conjuncts (HOLogic.dest_Trueprop (Thm.concl_of dtor_unfold_unique_thm))
|> map_split HOLogic.dest_eq
||> snd o strip_comb o hd
|> @{apply 2} (map (fst o dest_Var));
fun mk_fun_insts T ix = Thm.cterm_of lthy (Var (ix, T --> T));
val theta =
(funs ~~ @{map 2} mk_fun_insts Ts funs) @ (algs ~~ map (Thm.cterm_of lthy) dtors);
val dtor_unfold_dtors = (dtor_unfold_unique_thm OF
map (fn thm => mk_trans (thm RS @{thm arg_cong2[of _ _ _ _ "(\<circ>)", OF _ refl]})
@{thm trans[OF id_o o_id[symmetric]]}) map_id0s)
|> split_conj_thm |> map mk_sym;
in
infer_instantiate lthy theta dtor_unfold_unique_thm
|> Morphism.thm (Local_Theory.target_morphism lthy)
|> unfold_thms lthy dtor_unfold_dtors
|> (fn thm => thm OF replicate n sym)
end;
(*
thm trans[OF x.dtor_unfold_unique x.dtor_unfold_unique[symmetric,
OF trans[OF arg_cong2[of _ _ _ _ "(o)", OF pre_x.map_id0 refl] trans[OF id_o o_id[symmetric]]]],
OF sym]
*)
fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf =
trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym;
val JphiTs = map2 mk_pred2T passiveAs passiveBs;
val Jpsi1Ts = map2 mk_pred2T passiveAs passiveCs;
val Jpsi2Ts = map2 mk_pred2T passiveCs passiveBs;
val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts';
val fstsTsTs' = map fst_const prodTsTs';
val sndsTsTs' = map snd_const prodTsTs';
val activephiTs = map2 mk_pred2T activeAs activeBs;
val activeJphiTs = map2 mk_pred2T Ts Ts';
val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
val ((((Jzs, Jz's), Jphis), activeJphis), _) =
lthy
|> mk_Frees "z" Ts
||>> mk_Frees "y" Ts'
||>> mk_Frees "R" JphiTs
||>> mk_Frees "JR" activeJphiTs;
fun mk_Jrel_DEADID_coinduct_thm () =
mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis (map HOLogic.eq_const Ts) Jphis
Jzs Jz's dtors dtor's (fn {context = ctxt, prems} =>
(unfold_thms_tac ctxt @{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} THEN
REPEAT_DETERM (rtac ctxt allI 1) THEN rtac ctxt (dtor_coinduct_thm OF prems) 1)) lthy;
(*register new codatatypes as BNFs*)
val (timer, Jbnfs, (dtor_Jmap_o_thms, dtor_Jmap_thms), dtor_Jmap_unique_thm, dtor_Jset_thmss',
dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_notes, dtor_Jset_induct_thms, lthy) =
if m = 0 then
(timer, replicate n DEADID_bnf,
map_split (`(mk_pointfree2 lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids),
mk_dtor_map_unique_DEADID_thm (),
replicate n [],
map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs,
mk_Jrel_DEADID_coinduct_thm (), [], [], lthy)
else let
val fTs = map2 (curry op -->) passiveAs passiveBs;
val gTs = map2 (curry op -->) passiveBs passiveCs;
val uTs = map2 (curry op -->) Ts Ts';
val (((((nat, nat'), (Jzs, Jzs')), (hrecs, hrecs')), (fs, fs')), _) =
lthy
|> yield_singleton (apfst (~~) oo mk_Frees' "n") HOLogic.natT
||>> mk_Frees' "z" Ts
||>> mk_Frees' "rec" hrecTs
||>> mk_Frees' "f" fTs;
val map_FTFT's = map2 (fn Ds =>
mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
fun mk_maps ATs BTs Ts mk_T =
map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs;
fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts);
fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps =
mk_unfold Ts' (map2 (fn dtor => fn Fmap =>
HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T));
val mk_map_id = mk_map HOLogic.id_const I;
val mk_mapsAB = mk_maps passiveAs passiveBs;
val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks;
val set_bss =
map (flat o map2 (fn B => fn b =>
if member (op =) resDs (TFree B) then [] else [b]) resBs) set_bss0;
fun col_bind j = mk_internal_b (colN ^ (if m = 1 then "" else string_of_int j));
val col_def_bind = rpair [] o Thm.def_binding o col_bind;
fun col_spec j Zero hrec hrec' =
let
fun mk_Suc dtor sets z z' =
let
val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m sets);
fun mk_UN set k = mk_UNION (set $ (dtor $ z)) (mk_nthN n hrec k);
in
Term.absfree z'
(mk_union (set $ (dtor $ z), Library.foldl1 mk_union (map2 mk_UN sets ks)))
end;
val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec'
(HOLogic.mk_tuple (@{map 4} mk_Suc dtors FTs_setss Jzs Jzs')));
in
mk_rec_nat Zero Suc
end;
val ((col_frees, (_, col_def_frees)), (lthy, lthy_old)) =
lthy
|> Local_Theory.open_target |> snd
|> @{fold_map 4} (fn j => fn Zero => fn hrec => fn hrec' => Local_Theory.define
((col_bind j, NoSyn), (col_def_bind j, col_spec j Zero hrec hrec')))
ls Zeros hrecs hrecs'
|>> apsnd split_list o split_list
||> `Local_Theory.close_target;
val phi = Proof_Context.export_morphism lthy_old lthy;
val col_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) col_def_frees;
val cols = map (fst o Term.dest_Const o Morphism.term phi) col_frees;
fun mk_col Ts nat i j T =
let
val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) Ts)
val colT = HOLogic.natT --> hrecT;
in
mk_nthN n (Term.list_comb (Const (nth cols (j - 1), colT), [nat])) i
end;
val col_0ss = mk_rec_simps n @{thm rec_nat_0_imp} col_defs;
val col_Sucss = mk_rec_simps n @{thm rec_nat_Suc_imp} col_defs;
val col_0ss' = transpose col_0ss;
val col_Sucss' = transpose col_Sucss;
fun mk_set Ts i j T =
Abs (Name.uu, nth Ts (i - 1), mk_UNION (HOLogic.mk_UNIV HOLogic.natT)
(Term.absfree nat' (mk_col Ts nat i j T $ Bound 1)));
val setss = map (fn i => map2 (mk_set Ts i) ls passiveAs) ks;
val (Jbnf_consts, lthy) =
@{fold_map 8} (fn b => fn map_b => fn rel_b => fn pred_b => fn set_bs => fn mapx =>
fn sets => fn T => fn lthy =>
define_bnf_consts Hardly_Inline (user_policy Note_Some lthy) false (SOME deads)
map_b rel_b pred_b set_bs
(((((((b, T), fold_rev Term.absfree fs' mapx), sets), sbd),
[Const (@{const_name undefined}, T)]), NONE), NONE) lthy)
bs map_bs rel_bs pred_bs set_bss fs_maps setss Ts lthy;
val (_, Jconsts, Jconst_defs, mk_Jconsts) = @{split_list 4} Jbnf_consts;
val (_, Jsetss, Jbds_Ds, _, _, _) = @{split_list 6} Jconsts;
val (Jmap_defs, Jset_defss, Jbd_defs, _, Jrel_defs, Jpred_defs) =
@{split_list 6} Jconst_defs;
val (mk_Jmaps_Ds, mk_Jt_Ds, _, mk_Jrels_Ds, mk_Jpreds_Ds, _, _) =
@{split_list 7} mk_Jconsts;
val Jrel_unabs_defs = map (fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) Jrel_defs;
val Jpred_unabs_defs = map (fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) Jpred_defs;
val Jset_defs = flat Jset_defss;
fun mk_Jmaps As Bs = map (fn mk => mk deads As Bs) mk_Jmaps_Ds;
fun mk_Jsetss As = map2 (fn mk => fn Jsets => map (mk deads As) Jsets) mk_Jt_Ds Jsetss;
val Jbds = map2 (fn mk => mk deads passiveAs) mk_Jt_Ds Jbds_Ds;
fun mk_Jrels As Bs = map (fn mk => mk deads As Bs) mk_Jrels_Ds;
fun mk_Jpreds As = map (fn mk => mk deads As) mk_Jpreds_Ds;
val Jmaps = mk_Jmaps passiveAs passiveBs;
val (Jsetss_by_range, Jsetss_by_bnf) = `transpose (mk_Jsetss passiveAs);
val timer = time (timer "bnf constants for the new datatypes");
val ((((((((((((((((((((ys, ys'), (nat, nat')), (Jzs, Jzs')), Jz's), Jzs_copy), Jz's_copy),
dtor_set_induct_phiss), Jphis), Jpsi1s), Jpsi2s), activeJphis), fs), fs_copy), gs), us),
(Jys, Jys')), (Jys_copy, Jys'_copy)), (ys_copy, ys'_copy)), Kss), names_lthy) =
lthy
|> mk_Frees' "y" passiveAs
||>> yield_singleton (apfst (~~) oo mk_Frees' "n") HOLogic.natT
||>> mk_Frees' "z" Ts
||>> mk_Frees "y" Ts'
||>> mk_Frees "z'" Ts
||>> mk_Frees "y'" Ts'
||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs)
||>> mk_Frees "R" JphiTs
||>> mk_Frees "R" Jpsi1Ts
||>> mk_Frees "Q" Jpsi2Ts
||>> mk_Frees "JR" activeJphiTs
||>> mk_Frees "f" fTs
||>> mk_Frees "f" fTs
||>> mk_Frees "g" gTs
||>> mk_Frees "u" uTs
||>> mk_Frees' "b" Ts'
||>> mk_Frees' "b" Ts'
||>> mk_Frees' "y" passiveAs
||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) Ts) ATs);
val fs_Jmaps = map (fn m => Term.list_comb (m, fs)) Jmaps;
val fs_copy_Jmaps = map (fn m => Term.list_comb (m, fs_copy)) Jmaps;
val gs_Jmaps = map (fn m => Term.list_comb (m, gs)) (mk_Jmaps passiveBs passiveCs);
val fgs_Jmaps = map (fn m => Term.list_comb (m, map2 (curry HOLogic.mk_comp) gs fs))
(mk_Jmaps passiveAs passiveCs);
val (dtor_Jmap_thms, Jmap_thms) =
let
fun mk_goal fs_Jmap map dtor dtor' = mk_Trueprop_eq (HOLogic.mk_comp (dtor', fs_Jmap),
HOLogic.mk_comp (Term.list_comb (map, fs @ fs_Jmaps), dtor));
val goals = @{map 4} mk_goal fs_Jmaps map_FTFT's dtors dtor's;
val maps =
@{map 5} (fn goal => fn unfold => fn map_comp => fn map_cong0 => fn map_arg_cong =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN
mk_map_tac ctxt m n map_arg_cong unfold map_comp map_cong0))
|> Thm.close_derivation)
goals dtor_unfold_thms map_comps map_cong0s map_arg_cong_thms;
in
map_split (fn thm => (thm RS @{thm comp_eq_dest}, thm)) maps
end;
val (dtor_Jmap_unique_thms, dtor_Jmap_unique_thm) =
let
fun mk_prem u map dtor dtor' =
mk_Trueprop_eq (HOLogic.mk_comp (dtor', u),
HOLogic.mk_comp (Term.list_comb (map, fs @ us), dtor));
val prems = @{map 4} mk_prem us map_FTFT's dtors dtor's;
val goal =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map2 (curry HOLogic.mk_eq) us fs_Jmaps));
val vars = fold (Variable.add_free_names lthy) (goal :: prems) [];
in
`split_conj_thm (Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, goal))
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN
mk_dtor_map_unique_tac ctxt dtor_unfold_unique_thm sym_map_comps)
|> Thm.close_derivation)
end;
val Jmap_comp0_thms =
let
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} (fn fmap => fn gmap => fn fgmap =>
HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap))
fs_Jmaps gs_Jmaps fgs_Jmaps))
val vars = Variable.add_free_names lthy goal [];
in
split_conj_thm (Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_map_comp0_tac ctxt Jmap_thms map_comp0s dtor_Jmap_unique_thm)
|> Thm.close_derivation)
end;
val timer = time (timer "map functions for the new codatatypes");
val Jset_minimal_thms =
let
fun mk_passive_prem set dtor x K =
Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (K $ x)));
fun mk_active_prem dtor x1 K1 set x2 K2 =
fold_rev Logic.all [x1, x2]
(Logic.mk_implies (mk_Trueprop_mem (x2, set $ (dtor $ x1)),
HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1))));
val premss = map2 (fn j => fn Ks =>
@{map 4} mk_passive_prem (map (fn xs => nth xs (j - 1)) FTs_setss) dtors Jzs Ks @
flat (@{map 4} (fn sets => fn s => fn x1 => fn K1 =>
@{map 3} (mk_active_prem s x1 K1) (drop m sets) Jzs_copy Ks) FTs_setss dtors Jzs Ks))
ls Kss;
val col_minimal_thms =
let
fun mk_conjunct j T i K x = mk_leq (mk_col Ts nat i j T $ x) (K $ x);
fun mk_concl j T Ks = list_all_free Jzs
(Library.foldr1 HOLogic.mk_conj (@{map 3} (mk_conjunct j T) ks Ks Jzs));
val concls = @{map 3} mk_concl ls passiveAs Kss;
val goals = map2 (fn prems => fn concl =>
Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls
val ctss =
map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat]) concls;
in
@{map 4} (fn goal => fn cts => fn col_0s => fn col_Sucs =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_col_minimal_tac ctxt m cts col_0s
col_Sucs))
|> Thm.close_derivation)
goals ctss col_0ss' col_Sucss'
end;
fun mk_conjunct set K x = mk_leq (set $ x) (K $ x);
fun mk_concl sets Ks = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_conjunct sets Ks Jzs);
val concls = map2 mk_concl Jsetss_by_range Kss;
val goals = map2 (fn prems => fn concl =>
Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls;
in
map2 (fn goal => fn col_minimal =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
mk_Jset_minimal_tac ctxt n col_minimal))
|> Thm.close_derivation)
goals col_minimal_thms
end;
val (dtor_Jset_incl_thmss, dtor_set_Jset_incl_thmsss) =
let
fun mk_set_incl_Jset dtor x set Jset =
HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (Jset $ x));
fun mk_set_Jset_incl_Jset dtor x y set Jset1 Jset2 =
Logic.mk_implies (mk_Trueprop_mem (x, set $ (dtor $ y)),
HOLogic.mk_Trueprop (mk_leq (Jset1 $ x) (Jset2 $ y)));
val set_incl_Jset_goalss =
@{map 4} (fn dtor => fn x => fn sets => fn Jsets =>
map2 (mk_set_incl_Jset dtor x) (take m sets) Jsets)
dtors Jzs FTs_setss Jsetss_by_bnf;
(*x(k) : F(i)set(m+k) (dtor(i) y(i)) ==> J(k)set(j) x(k) <= J(i)set(j) y(i)*)
val set_Jset_incl_Jset_goalsss =
@{map 4} (fn dtori => fn yi => fn sets => fn Jsetsi =>
@{map 3} (fn xk => fn set => fn Jsetsk =>
map2 (mk_set_Jset_incl_Jset dtori xk yi set) Jsetsk Jsetsi)
Jzs_copy (drop m sets) Jsetss_by_bnf)
dtors Jzs FTs_setss Jsetss_by_bnf;
in
(map2 (fn goals => fn rec_Sucs =>
map2 (fn goal => fn rec_Suc =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
mk_set_incl_Jset_tac ctxt rec_Suc))
|> Thm.close_derivation)
goals rec_Sucs)
set_incl_Jset_goalss col_Sucss,
map2 (fn goalss => fn rec_Sucs =>
map2 (fn k => fn goals =>
map2 (fn goal => fn rec_Suc =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN
mk_set_Jset_incl_Jset_tac ctxt n rec_Suc k))
|> Thm.close_derivation)
goals rec_Sucs)
ks goalss)
set_Jset_incl_Jset_goalsss col_Sucss)
end;
val set_incl_Jset_thmss' = transpose dtor_Jset_incl_thmss;
val set_Jset_incl_Jset_thmsss' = transpose (map transpose dtor_set_Jset_incl_thmsss);
val set_Jset_thmss = map (map (fn thm => thm RS @{thm set_mp})) dtor_Jset_incl_thmss;
val set_Jset_Jset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp})))
dtor_set_Jset_incl_thmsss;
val set_Jset_thmss' = transpose set_Jset_thmss;
val set_Jset_Jset_thmsss' = transpose (map transpose set_Jset_Jset_thmsss);
val dtor_Jset_induct_thms =
let
val incls =
maps (map (fn thm => thm RS @{thm subset_Collect_iff})) dtor_Jset_incl_thmss @
@{thms subset_Collect_iff[OF subset_refl]};
val cTs = map (SOME o Thm.ctyp_of lthy) params';
fun mk_induct_tinst phis jsets y y' =
@{map 4} (fn phi => fn jset => fn Jz => fn Jz' =>
SOME (Thm.cterm_of lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y',
HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz))))))
phis jsets Jzs Jzs';
in
@{map 6} (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis =>
((set_minimal
|> Thm.instantiate' cTs (mk_induct_tinst phis jsets y y')
|> unfold_thms lthy incls) OF
(replicate n ballI @
maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss))
|> singleton (Proof_Context.export names_lthy lthy)
|> rule_by_tactic lthy (ALLGOALS (TRY o etac lthy asm_rl)))
Jset_minimal_thms set_Jset_incl_Jset_thmsss' Jsetss_by_range ys ys' dtor_set_induct_phiss
end;
val (dtor_Jset_thmss', dtor_Jset_thmss) =
let
fun mk_simp_goal relate pas_set act_sets sets dtor z set =
relate (set $ z, mk_union (pas_set $ (dtor $ z),
Library.foldl1 mk_union
(map2 (fn X => mk_UNION (X $ (dtor $ z))) act_sets sets)));
fun mk_goals eq =
map2 (fn i => fn sets =>
@{map 4} (fn Fsets =>
mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets)
FTs_setss dtors Jzs sets)
ls Jsetss_by_range;
val le_goals = map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj)
(mk_goals (uncurry mk_leq));
val set_le_thmss = map split_conj_thm
(@{map 4} (fn goal => fn Jset_minimal => fn set_Jsets => fn set_Jset_Jsetss =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_set_le_tac ctxt n Jset_minimal set_Jsets set_Jset_Jsetss))
|> Thm.close_derivation)
le_goals Jset_minimal_thms set_Jset_thmss' set_Jset_Jset_thmsss');
val ge_goalss = map (map HOLogic.mk_Trueprop) (mk_goals (uncurry mk_leq o swap));
val set_ge_thmss =
@{map 3} (@{map 3} (fn goal => fn set_incl_Jset => fn set_Jset_incl_Jsets =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_set_ge_tac ctxt n set_incl_Jset set_Jset_incl_Jsets))
|> Thm.close_derivation))
ge_goalss set_incl_Jset_thmss' set_Jset_incl_Jset_thmsss'
in
map2 (map2 (fn le => fn ge => equalityI OF [le, ge])) set_le_thmss set_ge_thmss
|> `transpose
end;
val timer = time (timer "set functions for the new codatatypes");
val colss = map2 (fn j => fn T =>
map (fn i => mk_col Ts nat i j T) ks) ls passiveAs;
val colss' = map2 (fn j => fn T =>
map (fn i => mk_col Ts' nat i j T) ks) ls passiveBs;
val col_natural_thmss =
let
fun mk_col_natural f map z col col' =
HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z));
fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
(@{map 4} (mk_col_natural f) fs_Jmaps Jzs cols cols'));
val goals = @{map 3} mk_goal fs colss colss';
val ctss =
map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat]) goals;
val thms =
@{map 4} (fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} => mk_col_natural_tac ctxt cts rec_0s rec_Sucs
dtor_Jmap_thms set_mapss))
|> Thm.close_derivation)
goals ctss col_0ss' col_Sucss';
in
map (split_conj_thm o mk_specN n) thms
end;
val col_bd_thmss =
let
fun mk_col_bd z col bd = mk_ordLeq (mk_card_of (col $ z)) bd;
fun mk_goal bds cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_col_bd Jzs cols bds));
val goals = map (mk_goal Jbds) colss;
val ctss = map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat])
(map (mk_goal (replicate n sbd)) colss);
val thms =
@{map 5} (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal)
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jbd_defs THEN
mk_col_bd_tac ctxt m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss))
|> Thm.close_derivation)
ls goals ctss col_0ss' col_Sucss';
in
map (split_conj_thm o mk_specN n) thms
end;
val map_cong0_thms =
let
val cTs = map (SOME o Thm.ctyp_of lthy o
Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params;
fun mk_prem z set f g y y' =
mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y)));
fun mk_prems sets z =
Library.foldr1 HOLogic.mk_conj (@{map 5} (mk_prem z) sets fs fs_copy ys ys')
fun mk_map_cong0 sets z fmap gmap =
HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z));
fun mk_coind_body sets (x, T) z fmap gmap y y_copy =
HOLogic.mk_conj
(HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)),
HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z),
HOLogic.mk_eq (y_copy, gmap $ z)))
fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy =
HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy)
|> Term.absfree y'_copy
|> Term.absfree y'
|> Thm.cterm_of lthy;
val cphis = @{map 9} mk_cphi
Jsetss_by_bnf Jzs' Jzs fs_Jmaps fs_copy_Jmaps Jys' Jys Jys'_copy Jys_copy;
val coinduct = Thm.instantiate' cTs (map SOME cphis) dtor_coinduct_thm;
val goal =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 4} mk_map_cong0 Jsetss_by_bnf Jzs fs_Jmaps fs_copy_Jmaps));
val vars = Variable.add_free_names lthy goal [];
val thm =
Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_mcong_tac ctxt m (rtac ctxt coinduct) map_comps
dtor_Jmap_thms map_cong0s
set_mapss set_Jset_thmss set_Jset_Jset_thmsss in_rels)
|> Thm.close_derivation;
in
split_conj_thm thm
end;
val in_Jrels = map (fn def => trans OF [def, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD})
Jrel_unabs_defs;
val Jrels = mk_Jrels passiveAs passiveBs;
val Jpreds = mk_Jpreds passiveAs;
val Jrelphis = map (fn rel => Term.list_comb (rel, Jphis)) Jrels;
val relphis = map (fn rel => Term.list_comb (rel, Jphis @ Jrelphis)) rels;
val Jrelpsi1s = map (fn rel => Term.list_comb (rel, Jpsi1s)) (mk_Jrels passiveAs passiveCs);
val Jrelpsi2s = map (fn rel => Term.list_comb (rel, Jpsi2s)) (mk_Jrels passiveCs passiveBs);
val Jrelpsi12s = map (fn rel =>
Term.list_comb (rel, map2 (curry mk_rel_compp) Jpsi1s Jpsi2s)) Jrels;
val dtor_Jrel_thms =
let
fun mk_goal Jz Jz' dtor dtor' Jrelphi relphi =
mk_Trueprop_eq (Jrelphi $ Jz $ Jz', relphi $ (dtor $ Jz) $ (dtor' $ Jz'));
val goals = @{map 6} mk_goal Jzs Jz's dtors dtor's Jrelphis relphis;
in
@{map 12} (fn i => fn goal => fn in_rel => fn map_comp0 => fn map_cong0 =>
fn dtor_map => fn dtor_sets => fn dtor_inject => fn dtor_ctor =>
fn set_map0s => fn dtor_set_incls => fn dtor_set_set_inclss =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} =>
mk_dtor_rel_tac ctxt in_Jrels i in_rel map_comp0 map_cong0 dtor_map dtor_sets
dtor_inject dtor_ctor set_map0s dtor_set_incls dtor_set_set_inclss))
|> Thm.close_derivation)
ks goals in_rels map_comps map_cong0s dtor_Jmap_thms dtor_Jset_thmss'
dtor_inject_thms dtor_ctor_thms set_mapss dtor_Jset_incl_thmss
dtor_set_Jset_incl_thmsss
end;
val passiveABs = map2 (curry HOLogic.mk_prodT) passiveAs passiveBs;
val zip_ranTs = passiveABs @ prodTsTs';
val allJphis = Jphis @ activeJphis;
val zipFTs = mk_FTs zip_ranTs;
val zipTs = @{map 3} (fn T => fn T' => fn FT => T --> T' --> FT) Ts Ts' zipFTs;
val zip_zTs = mk_Ts passiveABs;
val (((zips, (abs, abs')), (zip_zs, zip_zs')), _) =
names_lthy
|> mk_Frees "zip" zipTs
||>> mk_Frees' "ab" passiveABs
||>> mk_Frees' "z" zip_zTs;
val Iphi_sets =
map2 (fn phi => fn T => HOLogic.Collect_const T $ HOLogic.mk_case_prod phi) allJphis zip_ranTs;
val in_phis = map2 (mk_in Iphi_sets) (mk_setss zip_ranTs) zipFTs;
val fstABs = map fst_const passiveABs;
val all_fsts = fstABs @ fstsTsTs';
val map_all_fsts = map2 (fn Ds => fn bnf =>
Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveAs @ Ts) bnf, all_fsts)) Dss bnfs;
val Jmap_fsts = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T
else Term.list_comb (map, fstABs)) (mk_Jmaps passiveABs passiveAs) Ts;
val sndABs = map snd_const passiveABs;
val all_snds = sndABs @ sndsTsTs';
val map_all_snds = map2 (fn Ds => fn bnf =>
Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveBs @ Ts') bnf, all_snds)) Dss bnfs;
val Jmap_snds = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T
else Term.list_comb (map, sndABs)) (mk_Jmaps passiveABs passiveBs) Ts;
val zip_unfolds = map (mk_unfold zip_zTs (map HOLogic.mk_case_prod zips)) ks;
val zip_setss = mk_Jsetss passiveABs |> transpose;
fun Jrel_coinduct_tac {context = ctxt, prems = CIHs} =
let
fun mk_helper_prem phi in_phi zip x y map map' dtor dtor' =
let
val zipxy = zip $ x $ y;
in
HOLogic.mk_Trueprop (list_all_free [x, y]
(HOLogic.mk_imp (phi $ x $ y, HOLogic.mk_conj (HOLogic.mk_mem (zipxy, in_phi),
HOLogic.mk_conj (HOLogic.mk_eq (map $ zipxy, dtor $ x),
HOLogic.mk_eq (map' $ zipxy, dtor' $ y))))))
end;
val helper_prems = @{map 9} mk_helper_prem
activeJphis in_phis zips Jzs Jz's map_all_fsts map_all_snds dtors dtor's;
fun mk_helper_coind_phi fst phi x alt y map zip_unfold =
list_exists_free [if fst then y else x] (HOLogic.mk_conj (phi $ x $ y,
HOLogic.mk_eq (alt, map $ (zip_unfold $ HOLogic.mk_prod (x, y)))))
val coind1_phis = @{map 6} (mk_helper_coind_phi true)
activeJphis Jzs Jzs_copy Jz's Jmap_fsts zip_unfolds;
val coind2_phis = @{map 6} (mk_helper_coind_phi false)
activeJphis Jzs Jz's_copy Jz's Jmap_snds zip_unfolds;
fun mk_cts zs z's phis =
@{map 3} (fn z => fn z' => fn phi =>
SOME (Thm.cterm_of lthy (fold_rev (Term.absfree o Term.dest_Free) [z', z] phi)))
zs z's phis @
map (SOME o Thm.cterm_of lthy) (splice z's zs);
val cts1 = mk_cts Jzs Jzs_copy coind1_phis;
val cts2 = mk_cts Jz's Jz's_copy coind2_phis;
fun mk_helper_coind_concl z alt coind_phi =
HOLogic.mk_imp (coind_phi, HOLogic.mk_eq (alt, z));
val helper_coind1_concl =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_helper_coind_concl Jzs Jzs_copy coind1_phis));
val helper_coind2_concl =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_helper_coind_concl Jz's Jz's_copy coind2_phis));
fun mk_helper_coind_thms fst concl cts =
let
val vars = fold (Variable.add_free_names lthy) (concl :: helper_prems) [];
in
Goal.prove_sorry lthy vars [] (Logic.list_implies (helper_prems, concl))
(fn {context = ctxt, prems = _} =>
mk_rel_coinduct_coind_tac ctxt fst m
(infer_instantiate' ctxt cts dtor_coinduct_thm) ks map_comps map_cong0s
map_arg_cong_thms set_mapss dtor_unfold_thms dtor_Jmap_thms in_rels)
|> Thm.close_derivation
|> split_conj_thm
end;
val helper_coind1_thms = mk_helper_coind_thms true helper_coind1_concl cts1;
val helper_coind2_thms = mk_helper_coind_thms false helper_coind2_concl cts2;
fun mk_helper_ind_phi phi ab fst snd z active_phi x y zip_unfold =
list_all_free [x, y] (HOLogic.mk_imp
(HOLogic.mk_conj (active_phi $ x $ y,
HOLogic.mk_eq (z, zip_unfold $ HOLogic.mk_prod (x, y))),
phi $ (fst $ ab) $ (snd $ ab)));
val helper_ind_phiss =
@{map 4} (fn Jphi => fn ab => fn fst => fn snd =>
@{map 5} (mk_helper_ind_phi Jphi ab fst snd)
zip_zs activeJphis Jzs Jz's zip_unfolds)
Jphis abs fstABs sndABs;
val ctss = map2 (fn ab' => fn phis =>
map2 (fn z' => fn phi =>
SOME (Thm.cterm_of lthy (Term.absfree ab' (Term.absfree z' phi))))
zip_zs' phis @
map (SOME o Thm.cterm_of lthy) zip_zs)
abs' helper_ind_phiss;
fun mk_helper_ind_concl ab' z ind_phi set =
mk_Ball (set $ z) (Term.absfree ab' ind_phi);
val mk_helper_ind_concls =
@{map 3} (fn ab' => fn ind_phis => fn zip_sets =>
@{map 3} (mk_helper_ind_concl ab') zip_zs ind_phis zip_sets)
abs' helper_ind_phiss zip_setss
|> map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj);
val helper_ind_thmss = if m = 0 then replicate n [] else
@{map 4} (fn concl => fn j => fn set_induct => fn cts =>
fold (Variable.add_free_names lthy) (concl :: helper_prems) []
|> (fn vars => Goal.prove_sorry lthy vars [] (Logic.list_implies (helper_prems, concl))
(fn {context = ctxt, prems = _} =>
mk_rel_coinduct_ind_tac ctxt m ks
dtor_unfold_thms set_mapss j (infer_instantiate' ctxt cts set_induct)))
|> Thm.close_derivation
|> split_conj_thm)
mk_helper_ind_concls ls dtor_Jset_induct_thms ctss
|> transpose;
in
mk_rel_coinduct_tac ctxt CIHs in_rels in_Jrels
helper_ind_thmss helper_coind1_thms helper_coind2_thms
end;
val Jrel_coinduct_thm =
mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis Jrels Jphis Jzs Jz's dtors dtor's
Jrel_coinduct_tac lthy;
val le_Jrel_OO_thm =
let
fun mk_le_Jrel_OO Jrelpsi1 Jrelpsi2 Jrelpsi12 =
mk_leq (mk_rel_compp (Jrelpsi1, Jrelpsi2)) Jrelpsi12;
val goals = @{map 3} mk_le_Jrel_OO Jrelpsi1s Jrelpsi2s Jrelpsi12s;
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals);
val vars = Variable.add_free_names lthy goal [];
in
Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} =>
mk_le_rel_OO_tac ctxt Jrel_coinduct_thm dtor_Jrel_thms le_rel_OOs)
|> Thm.close_derivation
end;
val timer = time (timer "helpers for BNF properties");
fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit);
val all_unitTs = replicate live HOLogic.unitT;
val unitTs = replicate n HOLogic.unitT;
val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit);
fun mk_map_args I =
map (fn i =>
if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i)
else mk_undefined (HOLogic.unitT --> nth passiveAs i))
(0 upto (m - 1));
fun mk_nat_wit Ds bnf (I, wit) () =
let
val passiveI = filter (fn i => i < m) I;
val map_args = mk_map_args passiveI;
in
Term.absdummy HOLogic.unitT (Term.list_comb
(mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit)
end;
fun mk_dummy_wit Ds bnf I =
let
val map_args = mk_map_args I;
in
Term.absdummy HOLogic.unitT (Term.list_comb
(mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $
mk_undefined (mk_T_of_bnf Ds all_unitTs bnf))
end;
val nat_witss =
map2 (fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds)
(replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf
|> map (fn (I, wit) =>
(I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I))))))
Dss bnfs;
val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs)
val Iss = map (map fst) nat_witss;
fun filter_wits (I, wit) =
let val J = filter (fn i => i < m) I;
in (J, (length J < length I, wit)) end;
val wit_treess = map_index (fn (i, Is) =>
map_index (finish Iss m [i+m] (i+m)) Is) Iss
|> map (minimize_wits o map filter_wits o minimize_wits o flat);
val coind_wit_argsss =
map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess;
val nonredundant_coind_wit_argsss =
fold (fn i => fn argsss =>
nth_map (i - 1) (filter_out (fn xs =>
exists (fn ys =>
let
val xs' = (map (fst o fst) xs, snd (fst (hd xs)));
val ys' = (map (fst o fst) ys, snd (fst (hd ys)));
in
eq_pair (subset (op =)) (eq_set (op =)) (xs', ys') andalso not (fst xs' = fst ys')
end)
(flat argsss)))
argsss)
ks coind_wit_argsss;
fun prepare_args args =
let
val I = snd (fst (hd args));
val (dummys, args') =
map_split (fn i =>
(case find_first (fn arg => fst (fst arg) = i - 1) args of
SOME (_, ((_, wit), thms)) => (NONE, (Lazy.force wit, thms))
| NONE =>
(SOME (i - 1), (mk_dummy_wit (nth Dss (i - 1)) (nth bnfs (i - 1)) I, []))))
ks;
in
((I, dummys), apsnd flat (split_list args'))
end;
fun mk_coind_wits ((I, dummys), (args, thms)) =
((I, dummys), (map (fn i => mk_unfold Ts args i $ HOLogic.unit) ks, thms));
val coind_witss =
maps (map (mk_coind_wits o prepare_args)) nonredundant_coind_wit_argsss;
val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf
(replicate (nwits_of_bnf bnf) Ds)
(replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs;
val ctor_witss =
map (map (uncurry close_wit o tree_to_ctor_wit ys ctors witss o snd o snd) o
filter_out (fst o snd)) wit_treess;
fun mk_coind_wit_thms ((I, dummys), (wits, wit_thms)) =
let
fun mk_goal sets y y_copy y'_copy j =
let
fun mk_conjunct set z dummy wit =
mk_Ball (set $ z) (Term.absfree y'_copy
(if dummy = NONE orelse member (op =) I (j - 1) then
HOLogic.mk_imp (HOLogic.mk_eq (z, wit),
if member (op =) I (j - 1) then HOLogic.mk_eq (y_copy, y)
else @{term False})
else @{term True}));
in
HOLogic.mk_Trueprop
(Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct sets Jzs dummys wits))
end;
val goals = @{map 5} mk_goal Jsetss_by_range ys ys_copy ys'_copy ls;
in
map2 (fn goal => fn induct =>
Variable.add_free_names lthy goal []
|> (fn vars => Goal.prove_sorry lthy vars [] goal
(fn {context = ctxt, prems = _} => mk_coind_wit_tac ctxt induct dtor_unfold_thms
(flat set_mapss) wit_thms))
|> Thm.close_derivation)
goals dtor_Jset_induct_thms
|> map split_conj_thm
|> transpose
|> map (map_filter (try (fn thm => thm RS bspec RS mp)))
|> curry op ~~ (map_index Library.I (map (close_wit I) wits))
|> filter (fn (_, thms) => length thms = m)
end;
val coind_wit_thms = maps mk_coind_wit_thms coind_witss;
val (wit_thmss, all_witss) =
fold (fn ((i, wit), thms) => fn witss =>
nth_map i (fn (thms', wits) => (thms @ thms', wit :: wits)) witss)
coind_wit_thms (map (pair []) ctor_witss)
|> map (apsnd (map snd o minimize_wits))
|> split_list;
val timer = time (timer "witnesses");
val map_id0_tacs =
map2 (fn thm => fn thm' => fn ctxt =>
mk_map_id0_tac ctxt Jmap_thms thm thm')
dtor_unfold_unique_thms unfold_dtor_thms;
val map_comp0_tacs = map (fn thm => fn ctxt => rtac ctxt (thm RS sym) 1) Jmap_comp0_thms;
val map_cong0_tacs = map (fn thm => fn ctxt => mk_map_cong0_tac ctxt m thm) map_cong0_thms;
val set_map0_tacss =
map (map (fn col => fn ctxt =>
unfold_thms_tac ctxt Jset_defs THEN mk_set_map0_tac ctxt col))
(transpose col_natural_thmss);
val Jbd_card_orders = map (fn def => Local_Defs.fold lthy [def] sbd_card_order) Jbd_defs;
val Jbd_Cinfinites = map (fn def => Local_Defs.fold lthy [def] sbd_Cinfinite) Jbd_defs;
val bd_co_tacs = map (fn thm => fn ctxt => rtac ctxt thm 1) Jbd_card_orders;
val bd_cinf_tacs = map (fn thm => fn ctxt => rtac ctxt (thm RS conjunct1) 1) Jbd_Cinfinites;
val set_bd_tacss =
map2 (fn Cinf => map (fn col => fn ctxt =>
unfold_thms_tac ctxt Jset_defs THEN mk_set_bd_tac ctxt Cinf col))
Jbd_Cinfinites (transpose col_bd_thmss);
val le_rel_OO_tacs = map (fn i => fn ctxt =>
rtac ctxt (le_Jrel_OO_thm RS mk_conjunctN n i) 1) ks;
val rel_OO_Grp_tacs = map (fn def => fn ctxt => rtac ctxt def 1) Jrel_unabs_defs;
val pred_set_tacs = map (fn def => fn ctxt => rtac ctxt def 1) Jpred_unabs_defs;
val tacss = @{map 10} zip_axioms map_id0_tacs map_comp0_tacs map_cong0_tacs set_map0_tacss
bd_co_tacs bd_cinf_tacs set_bd_tacss le_rel_OO_tacs rel_OO_Grp_tacs pred_set_tacs;
fun wit_tac thms ctxt =
mk_wit_tac ctxt n dtor_ctor_thms (flat dtor_Jset_thmss) (maps wit_thms_of_bnf bnfs) thms;
val (Jbnfs, lthy) =
@{fold_map 7} (fn tacs => fn map_b => fn rel_b => fn pred_b => fn set_bs => fn wit_thms =>
fn consts =>
bnf_def Hardly_Inline (user_policy Note_Some) false I tacs (wit_tac wit_thms)
(SOME deads) map_b rel_b pred_b set_bs consts)
tacss map_bs rel_bs pred_bs set_bss wit_thmss
(((((((replicate n Binding.empty ~~ Ts) ~~ Jmaps) ~~ Jsetss_by_bnf) ~~ Jbds) ~~
all_witss) ~~ map SOME Jrels) ~~ map SOME Jpreds)
lthy;
val timer = time (timer "registered new codatatypes as BNFs");
val ls' = if m = 1 then [0] else ls;
val Jbnf_common_notes =
map2 (fn i => fn thm => (mk_dtor_set_inductN i, [thm])) ls' dtor_Jset_induct_thms
|> map (fn (thmN, thms) =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
val Jbnf_notes =
[(dtor_mapN, map single dtor_Jmap_thms),
(dtor_map_uniqueN, map single dtor_Jmap_unique_thms),
(dtor_relN, map single dtor_Jrel_thms),
(dtor_set_inclN, dtor_Jset_incl_thmss),
(dtor_set_set_inclN, map flat dtor_set_Jset_incl_thmsss)] @
map2 (fn i => fn thms => (mk_dtor_setN i, map single thms)) ls' dtor_Jset_thmss
|> maps (fn (thmN, thmss) =>
map2 (fn b => fn thms =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
bs thmss)
in
(timer, Jbnfs, (Jmap_thms, dtor_Jmap_thms), dtor_Jmap_unique_thm, dtor_Jset_thmss',
dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_common_notes @ Jbnf_notes, dtor_Jset_induct_thms,
lthy)
end;
val ((Jphis, activephis), _) =
lthy
|> mk_Frees "R" JphiTs
||>> mk_Frees "S" activephiTs;
val dtor_unfold_o_Jmap_thms = mk_xtor_co_iter_o_map_thms Greatest_FP false m
dtor_unfold_unique_thm dtor_Jmap_o_thms (map (mk_pointfree2 lthy) dtor_unfold_thms)
sym_map_comps map_cong0s;
val rels = map2 (fn Ds => mk_rel_of_bnf Ds allAs allBs') Dss bnfs;
val Jrels = if m = 0 then map HOLogic.eq_const Ts
else map (mk_rel_of_bnf deads passiveAs passiveBs) Jbnfs;
val dtor_unfold_transfer_thms =
mk_xtor_co_iter_transfer_thms Greatest_FP rels activephis activephis Jrels Jphis
(mk_unfolds passiveAs activeAs) (mk_unfolds passiveBs activeBs)
(fn {context = ctxt, prems = _} => mk_unfold_transfer_tac ctxt m Jrel_coinduct_thm
(map map_transfer_of_bnf bnfs) dtor_unfold_thms)
lthy;
val timer = time (timer "relator coinduction");
fun mk_Ts As = map (typ_subst_atomic (passiveAs ~~ As)) Ts;
val export = map (Morphism.term (Local_Theory.target_morphism lthy))
val ((corecs, (dtor_corec_thms, dtor_corec_unique_thm, dtor_corec_o_Jmap_thms,
dtor_corec_transfer_thms)), lthy) = lthy
|> derive_xtor_co_recs Greatest_FP external_bs mk_Ts (Dss, resDs) bnfs
(export dtors) (export unfolds)
dtor_unfold_unique_thm dtor_unfold_thms dtor_unfold_transfer_thms
dtor_Jmap_thms dtor_Jrel_thms (replicate n NONE);
val timer = time (timer "recursor");
val common_notes =
[(dtor_coinductN, [dtor_coinduct_thm]),
(dtor_rel_coinductN, [Jrel_coinduct_thm])]
|> map (fn (thmN, thms) =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
val notes =
[(ctor_dtorN, ctor_dtor_thms),
(ctor_exhaustN, ctor_exhaust_thms),
(ctor_injectN, ctor_inject_thms),
(dtor_ctorN, dtor_ctor_thms),
(dtor_exhaustN, dtor_exhaust_thms),
(dtor_injectN, dtor_inject_thms),
(dtor_unfoldN, dtor_unfold_thms),
(dtor_unfold_o_mapN, dtor_unfold_o_Jmap_thms),
(dtor_unfold_transferN, dtor_unfold_transfer_thms),
(dtor_unfold_uniqueN, dtor_unfold_unique_thms)]
|> map (apsnd (map single))
|> maps (fn (thmN, thmss) =>
map2 (fn b => fn thms =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
bs thmss);
val lthy' = lthy |> internals ? snd o Local_Theory.notes (common_notes @ notes @ Jbnf_notes);
val fp_res =
{Ts = Ts, bnfs = Jbnfs, pre_bnfs = bnfs, absT_infos = absT_infos,
ctors = ctors, dtors = dtors, xtor_un_folds = unfolds, xtor_co_recs = export corecs,
xtor_co_induct = dtor_coinduct_thm, dtor_ctors = dtor_ctor_thms, ctor_dtors = ctor_dtor_thms,
ctor_injects = ctor_inject_thms, dtor_injects = dtor_inject_thms, xtor_maps = dtor_Jmap_thms,
xtor_map_unique = dtor_Jmap_unique_thm, xtor_setss = dtor_Jset_thmss',
xtor_rels = dtor_Jrel_thms, xtor_un_fold_thms = dtor_unfold_thms,
xtor_co_rec_thms = dtor_corec_thms, xtor_un_fold_unique = dtor_unfold_unique_thm,
xtor_co_rec_unique = dtor_corec_unique_thm,
xtor_un_fold_o_maps = dtor_unfold_o_Jmap_thms,
xtor_co_rec_o_maps = dtor_corec_o_Jmap_thms,
xtor_un_fold_transfers = dtor_unfold_transfer_thms,
xtor_co_rec_transfers = dtor_corec_transfer_thms, xtor_rel_co_induct = Jrel_coinduct_thm,
dtor_set_inducts = dtor_Jset_induct_thms};
in
timer; (fp_res, lthy')
end;
val _ =
Outer_Syntax.local_theory @{command_keyword codatatype} "define coinductive datatypes"
(parse_co_datatype_cmd Greatest_FP construct_gfp);
val _ = Theory.setup (fp_antiquote_setup @{binding codatatype});
end;