(* Title: HOL/Tools/BNF/bnf_lfp.ML
Author: Dmitriy Traytel, TU Muenchen
Author: Andrei Popescu, TU Muenchen
Copyright 2012
Datatype construction.
*)
signature BNF_LFP =
sig
val construct_lfp: mixfix list -> binding list -> binding list -> binding list list ->
binding list -> (string * sort) list -> typ list * typ list list -> BNF_Def.bnf list ->
BNF_Comp.absT_info list -> local_theory -> BNF_FP_Util.fp_result * local_theory
end;
structure BNF_LFP : BNF_LFP =
struct
open BNF_Def
open BNF_Util
open BNF_Tactics
open BNF_Comp
open BNF_FP_Util
open BNF_FP_Def_Sugar
open BNF_LFP_Util
open BNF_LFP_Tactics
(*all BNFs have the same lives*)
fun construct_lfp mixfixes map_bs rel_bs set_bss0 bs resBs (resDs, Dss) bnfs _ 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 note_all = Config.get lthy bnf_note_all;
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.conceal;
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 note_all ? map Binding.conceal;
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) =
names_lthy
|> variant_tfrees passives
||>> mk_TFrees n
||>> variant_tfrees passives
||>> mk_TFrees n
||>> variant_tfrees passives
||>> mk_TFrees n
|> fst;
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 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 (curry op -->) FTsAs activeAs;
val s'Ts = map2 (curry op -->) FTsBs activeBs;
val s''Ts = map2 (curry op -->) FTsCs activeCs;
val fTs = map2 (curry op -->) activeAs activeBs;
val inv_fTs = map2 (curry op -->) activeBs activeAs;
val self_fTs = map2 (curry op -->) activeAs activeAs;
val gTs = map2 (curry op -->) activeBs activeCs;
val all_gTs = map2 (curry op -->) allBs allCs';
val prodBsAs = map2 (curry HOLogic.mk_prodT) activeBs activeAs;
val prodFTs = mk_FTs (passiveAs @ prodBsAs);
val prod_sTs = map2 (curry op -->) prodFTs activeAs;
(* 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 @ prodBsAs)) Bss bnfs;
val map_fsts_rev = @{map 4} mk_map_of_bnf Dss Bss (replicate n (passiveAs @ prodBsAs)) 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 bd0s = @{map 3} mk_bd_of_bnf Dss Ass bnfs;
val bds =
@{map 3} (fn bd0 => fn Ds => fn bnf => mk_csum bd0
(mk_card_of (HOLogic.mk_UNIV
(mk_T_of_bnf Ds (replicate live (fst (dest_relT (fastype_of bd0)))) bnf))))
bd0s Dss bnfs;
val witss = map wits_of_bnf bnfs;
val (((((((((((((((((zs, zs'), Bs), Bs_copy), B's), B''s), ss), prod_ss), s's), s''s),
fs), fs_copy), inv_fs), self_fs), gs), all_gs), (xFs, xFs')),
names_lthy) = lthy
|> mk_Frees' "z" activeAs
||>> mk_Frees "B" BTs
||>> mk_Frees "B" BTs
||>> mk_Frees "B'" B'Ts
||>> mk_Frees "B''" B''Ts
||>> mk_Frees "s" sTs
||>> mk_Frees "prods" prod_sTs
||>> mk_Frees "s'" s'Ts
||>> mk_Frees "s''" s''Ts
||>> mk_Frees "f" fTs
||>> mk_Frees "f" fTs
||>> mk_Frees "f" inv_fTs
||>> mk_Frees "f" self_fTs
||>> mk_Frees "g" gTs
||>> mk_Frees "g" all_gTs
||>> mk_Frees' "x" FTsAs;
val passive_UNIVs = map HOLogic.mk_UNIV passiveAs;
val active_UNIVs = map HOLogic.mk_UNIV activeAs;
val prod_UNIVs = map HOLogic.mk_UNIV prodBsAs;
val passive_ids = map HOLogic.id_const passiveAs;
val active_ids = map HOLogic.id_const activeAs;
val fsts = map fst_const prodBsAs;
(* thms *)
val bd0_card_orders = map bd_card_order_of_bnf bnfs;
val bd0_Card_orders = map bd_Card_order_of_bnf bnfs;
val bd0_Cinfinites = map bd_Cinfinite_of_bnf bnfs;
val set_bd0ss = map set_bd_of_bnf bnfs;
val bd_Card_order = @{thm Card_order_csum};
val bd_Card_orders = replicate n bd_Card_order;
val bd_Cinfinites = map (fn thm => thm RS @{thm Cinfinite_csum1}) bd0_Cinfinites;
val bd_Cnotzeros = map (fn thm => thm RS @{thm Cinfinite_Cnotzero}) bd_Cinfinites;
val bd_Cinfinite = hd bd_Cinfinites;
val set_bdss =
map2 (fn set_bd0s => fn bd0_Card_order =>
map (fn thm => ctrans OF [thm, bd0_Card_order RS @{thm ordLeq_csum1}]) set_bd0s)
set_bd0ss bd0_Card_orders;
val in_bds = map in_bd_of_bnf bnfs;
val sym_map_comps = map (fn bnf => map_comp0_of_bnf bnf RS sym) bnfs;
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_mapss = map set_map_of_bnf bnfs;
val rel_mono_strong0s = map rel_mono_strong0_of_bnf bnfs;
val le_rel_OOs = map le_rel_OO_of_bnf bnfs;
val timer = time (timer "Extracted terms & thms");
(* nonemptiness check *)
fun new_wit X (wit: nonemptiness_witness) = subset (op =) (#I wit, (0 upto m - 1) @ map snd X);
val all = m upto m + n - 1;
fun enrich X = map_filter (fn i =>
(case find_first (fn (_, i') => i = i') X of
NONE =>
(case find_index (new_wit X) (nth witss (i - m)) of
~1 => NONE
| j => SOME (j, i))
| SOME ji => SOME ji)) all;
val reachable = fixpoint (op =) enrich [];
val _ = (case subtract (op =) (map snd reachable) all of
[] => ()
| i :: _ => error ("Cannot define empty datatype " ^ quote (Binding.name_of (nth bs (i - m)))));
val wit_thms = flat (map2 (fn bnf => fn (j, _) => nth (wit_thmss_of_bnf bnf) j) bnfs reachable);
val timer = time (timer "Checked nonemptiness");
(* 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;
in
Goal.prove_sorry lthy [] [] (mk_Trueprop_eq (lhs, rhs))
(fn {context = ctxt, prems = _} => mk_map_comp_id_tac ctxt map_comp0)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
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);
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, goal))
(K (mk_map_cong0L_tac m map_cong0 map_id))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val map_cong0L_thms = @{map 5} mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids;
val in_mono'_thms = map (fn bnf => in_mono_of_bnf bnf OF (replicate m subset_refl)) bnfs;
val in_cong'_thms = map (fn bnf => in_cong_of_bnf bnf OF (replicate m refl)) bnfs;
val timer = time (timer "Derived simple theorems");
(* algebra *)
val alg_bind = mk_internal_b algN;
val alg_def_bind = (Thm.def_binding alg_bind, []);
(*forall i = 1 ... n: (\<forall>x \<in> Fi_in UNIV .. UNIV B1 ... Bn. si x \<in> Bi)*)
val alg_spec =
let
val ins = @{map 3} mk_in (replicate n (passive_UNIVs @ Bs)) setssAs FTsAs;
fun mk_alg_conjunct B s X x x' =
mk_Ball X (Term.absfree x' (HOLogic.mk_mem (s $ x, B)));
val rhs = Library.foldr1 HOLogic.mk_conj (@{map 5} mk_alg_conjunct Bs ss ins xFs xFs')
in
fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss) rhs
end;
val ((alg_free, (_, alg_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.define ((alg_bind, NoSyn), (alg_def_bind, alg_spec))
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val alg = fst (Term.dest_Const (Morphism.term phi alg_free));
val alg_def = mk_unabs_def (2 * n) (Morphism.thm phi alg_def_free RS meta_eq_to_obj_eq);
fun mk_alg Bs ss =
let
val args = Bs @ ss;
val Ts = map fastype_of args;
val algT = Library.foldr (op -->) (Ts, HOLogic.boolT);
in
Term.list_comb (Const (alg, algT), args)
end;
val alg_set_thms =
let
val alg_prem = HOLogic.mk_Trueprop (mk_alg Bs ss);
fun mk_prem x set B = HOLogic.mk_Trueprop (mk_leq (set $ x) B);
fun mk_concl s x B = mk_Trueprop_mem (s $ x, B);
val premss = map2 ((fn x => fn sets => map2 (mk_prem x) (drop m sets) Bs)) xFs setssAs;
val concls = @{map 3} mk_concl ss xFs Bs;
val goals = map2 (fn prems => fn concl =>
Logic.list_implies (alg_prem :: prems, concl)) premss concls;
in
map (fn goal =>
Goal.prove_sorry lthy [] [] goal (K (mk_alg_set_tac alg_def))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals
end;
val timer = time (timer "Algebra definition & thms");
val alg_not_empty_thms =
let
val alg_prem =
HOLogic.mk_Trueprop (mk_alg Bs ss);
val concls = map (HOLogic.mk_Trueprop o mk_not_empty) Bs;
val goals =
map (fn concl => Logic.mk_implies (alg_prem, concl)) concls;
in
map2 (fn goal => fn alg_set =>
Goal.prove_sorry lthy [] [] goal
(K (mk_alg_not_empty_tac lthy alg_set alg_set_thms wit_thms))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals alg_set_thms
end;
val timer = time (timer "Proved nonemptiness");
(* 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. f x \<in> B'i)*)
(*mor) forall i = 1 ... n: (\<forall>x \<in> Fi_in UNIV ... UNIV B1 ... Bn.
f (s1 x) = s1' (Fi_map id ... id f1 ... fn 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 sets mapAsBs f s s' T x x' =
mk_Ball (mk_in (passive_UNIVs @ Bs) sets T)
(Term.absfree x' (HOLogic.mk_eq (f $ (s $ x), s' $
(Term.list_comb (mapAsBs, passive_ids @ fs) $ x))));
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 8} mk_mor setssAs mapsAsBs fs ss s's FTsAs xFs xFs'))
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.define ((mor_bind, NoSyn), (mor_def_bind, mor_spec))
||> `Local_Theory.restore;
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 morE_thms =
let
val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs);
fun mk_elim_prem sets x T = HOLogic.mk_Trueprop
(HOLogic.mk_mem (x, mk_in (passive_UNIVs @ Bs) sets T));
fun mk_elim_goal sets mapAsBs f s s' x T =
Logic.list_implies ([prem, mk_elim_prem sets x T],
mk_Trueprop_eq (f $ (s $ x), s' $ Term.list_comb (mapAsBs, passive_ids @ fs @ [x])));
val elim_goals = @{map 7} mk_elim_goal setssAs mapsAsBs fs ss s's xFs FTsAs;
fun prove goal = Goal.prove_sorry lthy [] [] goal (K (mk_mor_elim_tac mor_def))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
map prove elim_goals
end;
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);
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
(fn {context = ctxt, ...} => mk_mor_incl_tac ctxt mor_def map_ids)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
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));
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
(K (mk_mor_comp_tac mor_def set_mapss map_comp_id_thms))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
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);
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
(K ((hyp_subst_tac lthy THEN' assume_tac lthy) 1))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val mor_str_thm =
let
val maps = map2 (fn Ds => fn bnf => Term.list_comb
(mk_map_of_bnf Ds (passiveAs @ FTsAs) allAs bnf, passive_ids @ ss)) Dss bnfs;
in
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop
(mk_mor (map HOLogic.mk_UNIV FTsAs) maps active_UNIVs ss ss))
(K (mk_mor_str_tac ks mor_def))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val mor_convol_thm =
let
val maps = @{map 3} (fn s => fn prod_s => fn mapx =>
mk_convol (HOLogic.mk_comp (s, Term.list_comb (mapx, passive_ids @ fsts)), prod_s))
s's prod_ss map_fsts;
in
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop
(mk_mor prod_UNIVs maps (map HOLogic.mk_UNIV activeBs) s's fsts))
(K (mk_mor_convol_tac ks mor_def))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val mor_UNIV_thm =
let
fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq
(HOLogic.mk_comp (f, s),
HOLogic.mk_comp (s', Term.list_comb (mapAsBs, passive_ids @ fs)));
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);
in
Goal.prove_sorry lthy [] [] (mk_Trueprop_eq (lhs, rhs))
(K (mk_mor_UNIV_tac m morE_thms mor_def))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val timer = time (timer "Morphism definition & thms");
(* bounds *)
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 (lthy, sbd, sbd_Cinfinite, sbd_Card_order, set_sbdss, in_sbds) =
if n = 1
then (lthy, sum_bd, bd_Cinfinite, bd_Card_order, set_bdss, in_bds)
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 _ => EVERY' [rtac exI, rtac 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.define ((sbd_bind, NoSyn), (sbd_def_bind, sbd_spec))
||> `Local_Theory.restore;
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 sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites;
val sum_Card_order = sum_Cinfinite RS conjunct2;
val sbd_ordIso = @{thm ssubst_Pair_rhs} OF
[@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order], sbd_def];
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;
fun mk_in_bd_sum i Co Cnz bd =
Cnz RS ((@{thm ordLeq_ordIso_trans} OF
[Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl}), sbd_ordIso]) RS
(bd RS @{thm ordLeq_transitive[OF _ cexp_mono2_Cnotzero[OF _ Card_order_csum]]}));
val in_sbds = @{map 4} mk_in_bd_sum ks bd_Card_orders bd_Cnotzeros in_bds;
in
(lthy, sbd, sbd_Cinfinite, sbd_Card_order, set_sbdss, in_sbds)
end;
val sbd_Cnotzero = sbd_Cinfinite RS @{thm Cinfinite_Cnotzero};
val suc_bd = mk_cardSuc sbd;
val field_suc_bd = mk_Field suc_bd;
val suc_bdT = fst (dest_relT (fastype_of suc_bd));
fun mk_Asuc_bd [] = mk_cexp ctwo suc_bd
| mk_Asuc_bd As =
mk_cexp (mk_csum (Library.foldr1 (uncurry mk_csum) (map mk_card_of As)) ctwo) suc_bd;
val suc_bd_Card_order = sbd_Card_order RS @{thm cardSuc_Card_order};
val suc_bd_Cinfinite = sbd_Cinfinite RS @{thm Cinfinite_cardSuc};
val suc_bd_Cnotzero = suc_bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
val suc_bd_worel = suc_bd_Card_order RS @{thm Card_order_wo_rel}
val basis_Asuc = if m = 0 then @{thm ordLeq_refl[OF Card_order_ctwo]}
else @{thm ordLeq_csum2[OF Card_order_ctwo]};
val Asuc_bd_Cinfinite = suc_bd_Cinfinite RS (basis_Asuc RS @{thm Cinfinite_cexp});
val suc_bd_Asuc_bd = @{thm ordLess_ordLeq_trans[OF ordLess_ctwo_cexp cexp_mono1]} OF
[suc_bd_Card_order, basis_Asuc, suc_bd_Card_order];
val Asuc_bd = mk_Asuc_bd passive_UNIVs;
val Asuc_bdT = fst (dest_relT (fastype_of Asuc_bd));
val II_BTs = replicate n (HOLogic.mk_setT Asuc_bdT);
val II_sTs = map2 (fn Ds => fn bnf =>
mk_T_of_bnf Ds (passiveAs @ replicate n Asuc_bdT) bnf --> Asuc_bdT) Dss bnfs;
val (((((((idxs, Asi_name), (idx, idx')), (jdx, jdx')), II_Bs), II_ss), Asuc_fs),
names_lthy) = names_lthy
|> mk_Frees "i" (replicate n suc_bdT)
||>> (fn ctxt => apfst the_single (mk_fresh_names ctxt 1 "Asi"))
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") suc_bdT
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "j") suc_bdT
||>> mk_Frees "IIB" II_BTs
||>> mk_Frees "IIs" II_sTs
||>> mk_Frees "f" (map (fn T => Asuc_bdT --> T) activeAs);
val suc_bd_limit_thm =
let
val prem = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map (fn idx => HOLogic.mk_mem (idx, field_suc_bd)) idxs));
fun mk_conjunct idx = HOLogic.mk_conj (mk_not_eq idx jdx,
HOLogic.mk_mem (HOLogic.mk_prod (idx, jdx), suc_bd));
val concl = HOLogic.mk_Trueprop (mk_Bex field_suc_bd
(Term.absfree jdx' (Library.foldr1 HOLogic.mk_conj (map mk_conjunct idxs))));
in
Goal.prove_sorry lthy [] [] (Logic.list_implies ([prem], concl))
(K (mk_bd_limit_tac n suc_bd_Cinfinite))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val timer = time (timer "Bounds");
(* minimal algebra *)
fun mk_minG Asi i k = mk_UNION (mk_underS suc_bd $ i)
(Term.absfree jdx' (mk_nthN n (Asi $ jdx) k));
fun mk_minH_component Asi i sets Ts s k =
HOLogic.mk_binop @{const_name "sup"}
(mk_minG Asi i k, mk_image s $ mk_in (passive_UNIVs @ map (mk_minG Asi i) ks) sets Ts);
fun mk_min_algs ss =
let
val BTs = map (range_type o fastype_of) ss;
val Ts = passiveAs @ BTs;
val (Asi, Asi') = `Free (Asi_name, suc_bdT -->
Library.foldr1 HOLogic.mk_prodT (map HOLogic.mk_setT BTs));
in
mk_worec suc_bd (Term.absfree Asi' (Term.absfree idx' (HOLogic.mk_tuple
(@{map 4} (mk_minH_component Asi idx) (mk_setss Ts) (mk_FTs Ts) ss ks))))
end;
val (min_algs_thms, min_algs_mono_thms, card_of_min_algs_thm, least_min_algs_thm) =
let
val i_field = HOLogic.mk_mem (idx, field_suc_bd);
val min_algs = mk_min_algs ss;
val min_algss = map (fn k => mk_nthN n (min_algs $ idx) k) ks;
val concl = HOLogic.mk_Trueprop
(HOLogic.mk_eq (min_algs $ idx, HOLogic.mk_tuple
(@{map 4} (mk_minH_component min_algs idx) setssAs FTsAs ss ks)));
val goal = Logic.mk_implies (HOLogic.mk_Trueprop i_field, concl);
val min_algs_thm = Goal.prove_sorry lthy [] [] goal
(K (mk_min_algs_tac suc_bd_worel in_cong'_thms))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
val min_algs_thms = map (fn k => min_algs_thm RS mk_nthI n k) ks;
fun mk_mono_goal min_alg =
HOLogic.mk_Trueprop (mk_relChain suc_bd (Term.absfree idx' min_alg));
val monos =
map2 (fn goal => fn min_algs =>
Goal.prove_sorry lthy [] [] goal (K (mk_min_algs_mono_tac lthy min_algs))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
(map mk_mono_goal min_algss) min_algs_thms;
fun mk_card_conjunct min_alg = mk_ordLeq (mk_card_of min_alg) Asuc_bd;
val card_conjunction = Library.foldr1 HOLogic.mk_conj (map mk_card_conjunct min_algss);
val card_cT = Thm.ctyp_of lthy suc_bdT;
val card_ct = Thm.cterm_of lthy (Term.absfree idx' card_conjunction);
val card_of =
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, card_conjunction)))
(K (mk_min_algs_card_of_tac card_cT card_ct
m suc_bd_worel min_algs_thms in_sbds
sbd_Card_order sbd_Cnotzero suc_bd_Card_order suc_bd_Cinfinite suc_bd_Cnotzero
suc_bd_Asuc_bd Asuc_bd_Cinfinite))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
val least_prem = HOLogic.mk_Trueprop (mk_alg Bs ss);
val least_conjunction = Library.foldr1 HOLogic.mk_conj (map2 mk_leq min_algss Bs);
val least_cT = Thm.ctyp_of lthy suc_bdT;
val least_ct = Thm.cterm_of lthy (Term.absfree idx' least_conjunction);
val least =
(Goal.prove_sorry lthy [] []
(Logic.mk_implies (least_prem,
HOLogic.mk_Trueprop (HOLogic.mk_imp (i_field, least_conjunction))))
(K (mk_min_algs_least_tac least_cT least_ct
suc_bd_worel min_algs_thms alg_set_thms)))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
(min_algs_thms, monos, card_of, least)
end;
val timer = time (timer "min_algs definition & thms");
val min_alg_binds = mk_internal_bs min_algN;
fun min_alg_bind i = nth min_alg_binds (i - 1);
val min_alg_def_bind = rpair [] o Thm.def_binding o min_alg_bind;
fun min_alg_spec i =
let
val rhs = mk_UNION (field_suc_bd)
(Term.absfree idx' (mk_nthN n (mk_min_algs ss $ idx) i));
in
fold_rev (Term.absfree o Term.dest_Free) ss rhs
end;
val ((min_alg_frees, (_, min_alg_def_frees)), (lthy, lthy_old)) =
lthy
|> fold_map (fn i => Local_Theory.define
((min_alg_bind i, NoSyn), (min_alg_def_bind i, min_alg_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val min_algs = map (fst o Term.dest_Const o Morphism.term phi) min_alg_frees;
val min_alg_defs = map (fn def =>
mk_unabs_def n (Morphism.thm phi def RS meta_eq_to_obj_eq)) min_alg_def_frees;
fun mk_min_alg ss i =
let
val T = HOLogic.mk_setT (range_type (fastype_of (nth ss (i - 1))))
val Ts = map fastype_of ss;
val min_algT = Library.foldr (op -->) (Ts, T);
in
Term.list_comb (Const (nth min_algs (i - 1), min_algT), ss)
end;
val min_algs = map (mk_min_alg ss) ks;
val (alg_min_alg_thm, card_of_min_alg_thms, least_min_alg_thms, mor_incl_min_alg_thm) =
let
val goal = HOLogic.mk_Trueprop (mk_alg min_algs ss);
val alg_min_alg = Goal.prove_sorry lthy [] [] goal
(K (mk_alg_min_alg_tac m alg_def min_alg_defs suc_bd_limit_thm sbd_Cinfinite
set_sbdss min_algs_thms min_algs_mono_thms))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
fun mk_card_of_thm min_alg def = Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_ordLeq (mk_card_of min_alg) Asuc_bd))
(K (mk_card_of_min_alg_tac def card_of_min_algs_thm
suc_bd_Card_order suc_bd_Asuc_bd Asuc_bd_Cinfinite))
|> Thm.close_derivation;
val least_prem = HOLogic.mk_Trueprop (mk_alg Bs ss);
fun mk_least_thm min_alg B def = Goal.prove_sorry lthy [] []
(Logic.mk_implies (least_prem, HOLogic.mk_Trueprop (mk_leq min_alg B)))
(K (mk_least_min_alg_tac def least_min_algs_thm))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
val leasts = @{map 3} mk_least_thm min_algs Bs min_alg_defs;
val incl_prem = HOLogic.mk_Trueprop (mk_alg Bs ss);
val incl = Goal.prove_sorry lthy [] []
(Logic.mk_implies (incl_prem,
HOLogic.mk_Trueprop (mk_mor min_algs ss Bs ss active_ids)))
(K (EVERY' (rtac mor_incl_thm :: map etac leasts) 1))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
(alg_min_alg, map2 mk_card_of_thm min_algs min_alg_defs, leasts, incl)
end;
val timer = time (timer "Minimal algebra definition & thms");
val II_repT = HOLogic.mk_prodT (HOLogic.mk_tupleT II_BTs, HOLogic.mk_tupleT II_sTs);
val IIT_bind = mk_internal_b IITN;
val ((IIT_name, (IIT_glob_info, IIT_loc_info)), lthy) =
typedef (IIT_bind, params, NoSyn)
(HOLogic.mk_UNIV II_repT) NONE (fn _ => EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
val IIT = Type (IIT_name, params');
val Abs_IIT = Const (#Abs_name IIT_glob_info, II_repT --> IIT);
val Rep_IIT = Const (#Rep_name IIT_glob_info, IIT --> II_repT);
val Abs_IIT_inverse_thm = UNIV_I RS #Abs_inverse IIT_loc_info;
val initT = IIT --> Asuc_bdT;
val active_initTs = replicate n initT;
val init_FTs = map2 (fn Ds => mk_T_of_bnf Ds (passiveAs @ active_initTs)) Dss bnfs;
val init_fTs = map (fn T => initT --> T) activeAs;
val (((((((iidx, iidx'), init_xs), (init_xFs, init_xFs')),
init_fs), init_fs_copy), init_phis), names_lthy) = names_lthy
|> yield_singleton (apfst (op ~~) oo mk_Frees' "i") IIT
||>> mk_Frees "ix" active_initTs
||>> mk_Frees' "x" init_FTs
||>> mk_Frees "f" init_fTs
||>> mk_Frees "f" init_fTs
||>> mk_Frees "P" (replicate n (mk_pred1T initT));
val II = HOLogic.mk_Collect (fst iidx', IIT, list_exists_free (II_Bs @ II_ss)
(HOLogic.mk_conj (HOLogic.mk_eq (iidx,
Abs_IIT $ (HOLogic.mk_prod (HOLogic.mk_tuple II_Bs, HOLogic.mk_tuple II_ss))),
mk_alg II_Bs II_ss)));
val select_Bs = map (mk_nthN n (HOLogic.mk_fst (Rep_IIT $ iidx))) ks;
val select_ss = map (mk_nthN n (HOLogic.mk_snd (Rep_IIT $ iidx))) ks;
val str_init_binds = mk_internal_bs str_initN;
fun str_init_bind i = nth str_init_binds (i - 1);
val str_init_def_bind = rpair [] o Thm.def_binding o str_init_bind;
fun str_init_spec i =
let
val init_xF = nth init_xFs (i - 1)
val select_s = nth select_ss (i - 1);
val map = mk_map_of_bnf (nth Dss (i - 1))
(passiveAs @ active_initTs) (passiveAs @ replicate n Asuc_bdT)
(nth bnfs (i - 1));
val map_args = passive_ids @ replicate n (mk_rapp iidx Asuc_bdT);
val rhs = select_s $ (Term.list_comb (map, map_args) $ init_xF);
in
fold_rev (Term.absfree o Term.dest_Free) [init_xF, iidx] rhs
end;
val ((str_init_frees, (_, str_init_def_frees)), (lthy, lthy_old)) =
lthy
|> fold_map (fn i => Local_Theory.define
((str_init_bind i, NoSyn), (str_init_def_bind i, str_init_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val str_inits =
map (Term.subst_atomic_types (map (`(Morphism.typ phi)) params') o Morphism.term phi)
str_init_frees;
val str_init_defs = map (fn def =>
mk_unabs_def 2 (Morphism.thm phi def RS meta_eq_to_obj_eq)) str_init_def_frees;
val car_inits = map (mk_min_alg str_inits) ks;
val alg_init_thm = cterm_instantiate_pos (map (SOME o Thm.cterm_of lthy) str_inits) alg_min_alg_thm;
val alg_select_thm = Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_Ball II
(Term.absfree iidx' (mk_alg select_Bs select_ss))))
(fn {context = ctxt, prems = _} => mk_alg_select_tac ctxt Abs_IIT_inverse_thm)
|> Thm.close_derivation;
val mor_select_thm =
let
val i_prem = mk_Trueprop_mem (iidx, II);
val mor_prem = HOLogic.mk_Trueprop (mk_mor select_Bs select_ss active_UNIVs ss Asuc_fs);
val prems = [i_prem, mor_prem];
val concl = HOLogic.mk_Trueprop
(mk_mor car_inits str_inits active_UNIVs ss
(map (fn f => HOLogic.mk_comp (f, mk_rapp iidx Asuc_bdT)) Asuc_fs));
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
(K (mk_mor_select_tac mor_def mor_cong_thm mor_comp_thm mor_incl_min_alg_thm alg_def
alg_select_thm alg_set_thms set_mapss str_init_defs))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val init_unique_mor_thms =
let
val prems = map2 (HOLogic.mk_Trueprop oo curry HOLogic.mk_mem) init_xs car_inits
val mor_prems = map HOLogic.mk_Trueprop
[mk_mor car_inits str_inits Bs ss init_fs,
mk_mor car_inits str_inits Bs ss init_fs_copy];
fun mk_fun_eq f g x = HOLogic.mk_eq (f $ x, g $ x);
val unique = HOLogic.mk_Trueprop
(Library.foldr1 HOLogic.mk_conj (@{map 3} mk_fun_eq init_fs init_fs_copy init_xs));
val cts = map (Thm.cterm_of lthy) ss;
val unique_mor =
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems @ mor_prems, unique))
(K (mk_init_unique_mor_tac cts m alg_def alg_init_thm least_min_alg_thms
in_mono'_thms alg_set_thms morE_thms map_cong0s))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
split_conj_thm unique_mor
end;
val init_setss = mk_setss (passiveAs @ active_initTs);
val active_init_setss = map (drop m) init_setss;
val init_ins = map2 (fn sets => mk_in (passive_UNIVs @ car_inits) sets) init_setss init_FTs;
fun mk_closed phis =
let
fun mk_conjunct phi str_init init_sets init_in x x' =
let
val prem = Library.foldr1 HOLogic.mk_conj
(map2 (fn set => mk_Ball (set $ x)) init_sets phis);
val concl = phi $ (str_init $ x);
in
mk_Ball init_in (Term.absfree x' (HOLogic.mk_imp (prem, concl)))
end;
in
Library.foldr1 HOLogic.mk_conj
(@{map 6} mk_conjunct phis str_inits active_init_setss init_ins init_xFs init_xFs')
end;
val init_induct_thm =
let
val prem = HOLogic.mk_Trueprop (mk_closed init_phis);
val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map2 mk_Ball car_inits init_phis));
in
Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, concl))
(K (mk_init_induct_tac m alg_def alg_init_thm least_min_alg_thms alg_set_thms))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val timer = time (timer "Initiality definition & thms");
val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
lthy
|> @{fold_map 3} (fn b => fn mx => fn car_init =>
typedef (b, params, mx) car_init NONE
(fn ctxt =>
EVERY' [rtac iffD2, rtac @{thm ex_in_conv}, resolve_tac ctxt alg_not_empty_thms,
rtac alg_init_thm] 1)) bs mixfixes car_inits
|>> 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 --> initT)) T_glob_infos Ts;
val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, initT --> T)) T_glob_infos Ts;
val type_defs = map #type_definition T_loc_infos;
val Reps = map #Rep T_loc_infos;
val Rep_inverses = map #Rep_inverse 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' = mk_FTs (passiveBs @ Ts');
fun mk_set_Ts T = passiveAs @ replicate n (HOLogic.mk_setT T);
val setFTss = map (mk_FTs o mk_set_Ts) passiveAs;
val FTs_setss = mk_setss (passiveAs @ Ts);
val FTs'_setss = mk_setss (passiveBs @ Ts');
val map_FT_inits = map2 (fn Ds =>
mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ active_initTs)) Dss bnfs;
val fTs = map2 (curry op -->) Ts activeAs;
val foldT = Library.foldr1 HOLogic.mk_prodT (map2 (curry op -->) Ts activeAs);
val rec_sTs = map (Term.typ_subst_atomic (activeBs ~~ Ts)) prod_sTs;
val rec_maps = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts;
val rec_maps_rev = map (Term.subst_atomic_types (activeBs ~~ Ts)) map_fsts_rev;
val rec_fsts = map (Term.subst_atomic_types (activeBs ~~ Ts)) fsts;
val rec_UNIVs = map2 (HOLogic.mk_UNIV oo curry HOLogic.mk_prodT) Ts activeAs;
val (((((((((Izs1, Izs1'), (Izs2, Izs2')), xFs), yFs), (AFss, AFss')),
(fold_f, fold_f')), fs), rec_ss), names_lthy) = names_lthy
|> mk_Frees' "z1" Ts
||>> mk_Frees' "z2" Ts'
||>> mk_Frees "x" FTs
||>> mk_Frees "y" FTs'
||>> mk_Freess' "z" setFTss
||>> yield_singleton (apfst (op ~~) oo mk_Frees' "f") foldT
||>> mk_Frees "f" fTs
||>> mk_Frees "s" rec_sTs;
val Izs = map2 retype_const_or_free Ts zs;
val phis = map2 retype_const_or_free (map mk_pred1T Ts) init_phis;
val phi2s = map2 retype_const_or_free (map2 mk_pred2T Ts Ts') init_phis;
fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_");
val ctor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o ctor_bind;
fun ctor_spec abs str map_FT_init =
Library.foldl1 HOLogic.mk_comp [abs, str,
Term.list_comb (map_FT_init, map HOLogic.id_const passiveAs @ Rep_Ts)];
val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
lthy
|> @{fold_map 4} (fn i => fn abs => fn str => fn mapx =>
Local_Theory.define
((ctor_bind i, NoSyn), (ctor_def_bind i, ctor_spec abs str mapx)))
ks Abs_Ts str_inits map_FT_inits
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
fun mk_ctors passive =
map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o
Morphism.term phi) ctor_frees;
val ctors = mk_ctors passiveAs;
val ctor's = mk_ctors passiveBs;
val ctor_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) ctor_def_frees;
val (mor_Rep_thm, mor_Abs_thm) =
let
val defs = mor_def :: ctor_defs;
val mor_Rep =
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor UNIVs ctors car_inits str_inits Rep_Ts))
(fn {context = ctxt, prems = _} => mk_mor_Rep_tac ctxt m defs Reps Abs_inverses
alg_min_alg_thm alg_set_thms set_mapss)
|> Thm.close_derivation;
fun mk_ct initFT str abs = Term.absdummy initFT (abs $ (str $ Bound 0))
val cts = @{map 3} (Thm.cterm_of lthy ooo mk_ct) init_FTs str_inits Abs_Ts;
val mor_Abs =
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor car_inits str_inits UNIVs ctors Abs_Ts))
(fn {context = ctxt, prems = _} => mk_mor_Abs_tac ctxt cts defs Abs_inverses
map_comp_id_thms map_cong0L_thms)
|> Thm.close_derivation;
in
(mor_Rep, mor_Abs)
end;
val timer = time (timer "ctor definitions & thms");
val fold_fun = Term.absfree fold_f'
(mk_mor UNIVs ctors active_UNIVs ss (map (mk_nthN n fold_f) ks));
val foldx = HOLogic.choice_const foldT $ fold_fun;
fun fold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctor_foldN ^ "_");
val fold_def_bind = rpair [] o Binding.conceal o Thm.def_binding o fold_bind;
fun fold_spec i = fold_rev (Term.absfree o Term.dest_Free) ss (mk_nthN n foldx i);
val ((fold_frees, (_, fold_def_frees)), (lthy, lthy_old)) =
lthy
|> fold_map (fn i =>
Local_Theory.define ((fold_bind i, NoSyn), (fold_def_bind i, fold_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val folds = map (Morphism.term phi) fold_frees;
val fold_names = map (fst o dest_Const) folds;
fun mk_folds passives actives =
@{map 3} (fn name => fn T => fn active =>
Const (name, Library.foldr (op -->)
(map2 (curry op -->) (mk_FTs (passives @ actives)) actives, T --> active)))
fold_names (mk_Ts passives) actives;
fun mk_fold Ts ss i = Term.list_comb (Const (nth fold_names (i - 1), Library.foldr (op -->)
(map fastype_of ss, nth Ts (i - 1) --> range_type (fastype_of (nth ss (i - 1))))), ss);
val fold_defs = map (fn def =>
mk_unabs_def n (Morphism.thm phi def RS meta_eq_to_obj_eq)) fold_def_frees;
(* algebra copies *)
val copy_thm =
let
val prems = HOLogic.mk_Trueprop (mk_alg Bs ss) ::
@{map 3} (HOLogic.mk_Trueprop ooo mk_bij_betw) inv_fs B's Bs;
val concl = HOLogic.mk_Trueprop (list_exists_free s's
(HOLogic.mk_conj (mk_alg B's s's, mk_mor B's s's Bs ss inv_fs)));
in
Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, concl))
(K (mk_copy_tac m alg_def mor_def alg_set_thms set_mapss))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val init_ex_mor_thm =
let
val goal = HOLogic.mk_Trueprop
(list_exists_free fs (mk_mor UNIVs ctors active_UNIVs ss fs));
in
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} =>
mk_init_ex_mor_tac ctxt Abs_IIT_inverse_thm (alg_min_alg_thm RS copy_thm)
card_of_min_alg_thms mor_Rep_thm mor_comp_thm mor_select_thm mor_incl_thm)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val mor_fold_thm =
let
val mor_cong = mor_cong_thm OF (map (mk_nth_conv n) ks);
val cT = Thm.ctyp_of lthy foldT;
val ct = Thm.cterm_of lthy fold_fun
in
Goal.prove_sorry lthy [] []
(HOLogic.mk_Trueprop (mk_mor UNIVs ctors active_UNIVs ss (map (mk_fold Ts ss) ks)))
(fn {context = ctxt, ...} =>
mk_mor_fold_tac ctxt cT ct fold_defs init_ex_mor_thm mor_cong)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val ctor_fold_thms = map (fn morE => rule_by_tactic lthy
((rtac CollectI THEN' CONJ_WRAP' (K (rtac @{thm subset_UNIV})) (1 upto m + n)) 1)
(mor_fold_thm RS morE)) morE_thms;
val (fold_unique_mor_thms, fold_unique_mor_thm) =
let
val prem = HOLogic.mk_Trueprop (mk_mor UNIVs ctors active_UNIVs ss fs);
fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_fold Ts ss i);
val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_fun_eq fs ks));
val unique_mor = Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, unique))
(K (mk_fold_unique_mor_tac type_defs init_unique_mor_thms Reps
mor_comp_thm mor_Abs_thm mor_fold_thm))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
`split_conj_thm unique_mor
end;
val (ctor_fold_unique_thms, ctor_fold_unique_thm) =
`split_conj_thm (mk_conjIN n RS
(mor_UNIV_thm RS iffD2 RS fold_unique_mor_thm))
val fold_ctor_thms =
map (fn thm => (mor_incl_thm OF replicate n @{thm subset_UNIV}) RS thm RS sym)
fold_unique_mor_thms;
val ctor_o_fold_thms =
let
val mor = mor_comp_thm OF [mor_fold_thm, mor_str_thm];
in
map2 (fn unique => fn fold_ctor =>
trans OF [mor RS unique, fold_ctor]) fold_unique_mor_thms fold_ctor_thms
end;
val timer = time (timer "fold definitions & thms");
val map_ctors = map2 (fn Ds => fn bnf =>
Term.list_comb (mk_map_of_bnf Ds (passiveAs @ FTs) (passiveAs @ Ts) bnf,
map HOLogic.id_const passiveAs @ ctors)) Dss bnfs;
fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_");
val dtor_def_bind = rpair [] o Binding.conceal o Thm.def_binding o dtor_bind;
fun dtor_spec i = mk_fold Ts map_ctors i;
val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
lthy
|> fold_map (fn i =>
Local_Theory.define ((dtor_bind i, NoSyn), (dtor_def_bind i, dtor_spec i))) ks
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
fun mk_dtors params =
map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi)
dtor_frees;
val dtors = mk_dtors params';
val dtor_defs = map (fn def => Morphism.thm phi def RS meta_eq_to_obj_eq) dtor_def_frees;
val ctor_o_dtor_thms = map2 (fold_thms lthy o single) dtor_defs ctor_o_fold_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 dtor_def => fn foldx => fn map_comp_id => fn map_cong0L =>
Goal.prove_sorry lthy [] [] goal
(K (mk_dtor_o_ctor_tac dtor_def foldx map_comp_id map_cong0L ctor_o_fold_thms))
|> Thm.close_derivation)
goals dtor_defs ctor_fold_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 "dtor definitions & thms");
val fst_rec_pair_thms =
let
val mor = mor_comp_thm OF [mor_fold_thm, mor_convol_thm];
in
map2 (fn unique => fn fold_ctor =>
trans OF [mor RS unique, fold_ctor]) fold_unique_mor_thms fold_ctor_thms
end;
fun rec_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctor_recN ^ "_");
val rec_def_bind = rpair [] o Binding.conceal o Thm.def_binding o rec_bind;
val rec_strs =
@{map 3} (fn ctor => fn prod_s => fn mapx =>
mk_convol (HOLogic.mk_comp (ctor, Term.list_comb (mapx, passive_ids @ rec_fsts)), prod_s))
ctors rec_ss rec_maps;
fun rec_spec i T AT =
fold_rev (Term.absfree o Term.dest_Free) rec_ss
(HOLogic.mk_comp (snd_const (HOLogic.mk_prodT (T, AT)), mk_fold Ts rec_strs i));
val ((rec_frees, (_, rec_def_frees)), (lthy, lthy_old)) =
lthy
|> @{fold_map 3} (fn i => fn T => fn AT =>
Local_Theory.define ((rec_bind i, NoSyn), (rec_def_bind i, rec_spec i T AT))) ks Ts activeAs
|>> apsnd split_list o split_list
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val recs = map (Morphism.term phi) rec_frees;
val rec_names = map (fst o dest_Const) recs;
fun mk_recs Ts passives actives =
let val Tactives = map2 (curry HOLogic.mk_prodT) Ts actives;
in
@{map 3} (fn name => fn T => fn active =>
Const (name, Library.foldr (op -->)
(map2 (curry op -->) (mk_FTs (passives @ Tactives)) actives, T --> active)))
rec_names Ts actives
end;
fun mk_rec ss i = Term.list_comb (Const (nth rec_names (i - 1), Library.foldr (op -->)
(map fastype_of ss, nth Ts (i - 1) --> range_type (fastype_of (nth ss (i - 1))))), ss);
val rec_defs = map (fn def =>
mk_unabs_def n (Morphism.thm phi def RS meta_eq_to_obj_eq)) rec_def_frees;
val convols = map2 (fn T => fn i => mk_convol (HOLogic.id_const T, mk_rec rec_ss i)) Ts ks;
val ctor_rec_thms =
let
fun mk_goal i rec_s rec_map ctor x =
let
val lhs = mk_rec rec_ss i $ (ctor $ x);
val rhs = rec_s $ (Term.list_comb (rec_map, passive_ids @ convols) $ x);
in
mk_Trueprop_eq (lhs, rhs)
end;
val goals = @{map 5} mk_goal ks rec_ss rec_maps_rev ctors xFs;
in
map2 (fn goal => fn foldx =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_rec_tac ctxt rec_defs foldx fst_rec_pair_thms)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals ctor_fold_thms
end;
val rec_unique_mor_thm =
let
val id_fs = map2 (fn T => fn f => mk_convol (HOLogic.id_const T, f)) Ts fs;
val prem = HOLogic.mk_Trueprop (mk_mor UNIVs ctors rec_UNIVs rec_strs id_fs);
fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_rec rec_ss i);
val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_fun_eq fs ks));
in
Goal.prove_sorry lthy [] [] (Logic.mk_implies (prem, unique))
(fn {context = ctxt, prems = _} => mk_rec_unique_mor_tac ctxt rec_defs fst_rec_pair_thms
fold_unique_mor_thm)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val (ctor_rec_unique_thms, ctor_rec_unique_thm) =
`split_conj_thm (split_conj_prems n
(mor_UNIV_thm RS iffD2 RS rec_unique_mor_thm)
|> unfold_thms lthy (@{thms convol_o comp_id id_comp comp_assoc fst_convol} @
map_id0s @ sym_map_comps) OF replicate n @{thm arg_cong2[of _ _ _ _ convol, OF refl]});
val timer = time (timer "rec definitions & thms");
val (ctor_induct_thm, induct_params) =
let
fun mk_prem phi ctor sets x =
let
fun mk_IH phi set z =
let
val prem = mk_Trueprop_mem (z, set $ x);
val concl = HOLogic.mk_Trueprop (phi $ z);
in
Logic.all z (Logic.mk_implies (prem, concl))
end;
val IHs = @{map 3} mk_IH phis (drop m sets) Izs;
val concl = HOLogic.mk_Trueprop (phi $ (ctor $ x));
in
Logic.all x (Logic.list_implies (IHs, concl))
end;
val prems = @{map 4} mk_prem phis ctors FTs_setss xFs;
fun mk_concl phi z = phi $ z;
val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 mk_concl phis Izs));
val goal = Logic.list_implies (prems, concl);
in
(Goal.prove_sorry lthy [] [] goal
(K (mk_ctor_induct_tac lthy m set_mapss init_induct_thm morE_thms mor_Abs_thm
Rep_inverses Abs_inverses Reps))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy),
rev (Term.add_tfrees goal []))
end;
val cTs = map (SOME o Thm.ctyp_of lthy o TFree) induct_params;
val weak_ctor_induct_thms =
let fun insts i = (replicate (i - 1) TrueI) @ (asm_rl :: replicate (n - i) TrueI);
in map (fn i => (ctor_induct_thm OF insts i) RS mk_conjunctN n i) ks end;
val (ctor_induct2_thm, induct2_params) =
let
fun mk_prem phi ctor ctor' sets sets' x y =
let
fun mk_IH phi set set' z1 z2 =
let
val prem1 = mk_Trueprop_mem (z1, (set $ x));
val prem2 = mk_Trueprop_mem (z2, (set' $ y));
val concl = HOLogic.mk_Trueprop (phi $ z1 $ z2);
in
fold_rev Logic.all [z1, z2] (Logic.list_implies ([prem1, prem2], concl))
end;
val IHs = @{map 5} mk_IH phi2s (drop m sets) (drop m sets') Izs1 Izs2;
val concl = HOLogic.mk_Trueprop (phi $ (ctor $ x) $ (ctor' $ y));
in
fold_rev Logic.all [x, y] (Logic.list_implies (IHs, concl))
end;
val prems = @{map 7} mk_prem phi2s ctors ctor's FTs_setss FTs'_setss xFs yFs;
fun mk_concl phi z1 z2 = phi $ z1 $ z2;
val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_concl phi2s Izs1 Izs2));
fun mk_t phi (z1, z1') (z2, z2') =
Term.absfree z1' (HOLogic.mk_all (fst z2', snd z2', phi $ z1 $ z2));
val cts = @{map 3} (SOME o Thm.cterm_of lthy ooo mk_t) phi2s (Izs1 ~~ Izs1') (Izs2 ~~ Izs2');
val goal = Logic.list_implies (prems, concl);
in
(Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_ctor_induct2_tac ctxt cTs cts ctor_induct_thm
weak_ctor_induct_thms)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy),
rev (Term.add_tfrees goal []))
end;
val timer = time (timer "induction");
fun mk_ctor_map_DEADID_thm ctor_inject map_id0 =
trans OF [id_apply, iffD2 OF [ctor_inject, map_id0 RS sym]];
fun mk_ctor_Irel_DEADID_thm ctor_inject bnf =
trans OF [ctor_inject, rel_eq_of_bnf bnf RS @{thm predicate2_eqD} RS sym];
val IphiTs = map2 mk_pred2T passiveAs passiveBs;
val Ipsi1Ts = map2 mk_pred2T passiveAs passiveCs;
val Ipsi2Ts = map2 mk_pred2T passiveCs passiveBs;
val activephiTs = map2 mk_pred2T activeAs activeBs;
val activeIphiTs = map2 mk_pred2T Ts Ts';
val (((((Iphis, Ipsi1s), Ipsi2s), activephis), activeIphis), names_lthy) = names_lthy
|> mk_Frees "R" IphiTs
||>> mk_Frees "R" Ipsi1Ts
||>> mk_Frees "Q" Ipsi2Ts
||>> mk_Frees "S" activephiTs
||>> mk_Frees "IR" activeIphiTs;
val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
(*register new datatypes as BNFs*)
val (timer, Ibnfs, (ctor_Imap_o_thms, ctor_Imap_thms), ctor_Iset_thmss',
ctor_Irel_thms, Ibnf_notes, lthy) =
if m = 0 then
(timer, replicate n DEADID_bnf,
map_split (`(mk_pointfree lthy)) (map2 mk_ctor_map_DEADID_thm ctor_inject_thms map_ids),
replicate n [], map2 mk_ctor_Irel_DEADID_thm ctor_inject_thms bnfs, [], lthy)
else let
val fTs = map2 (curry op -->) passiveAs passiveBs;
val uTs = map2 (curry op -->) Ts Ts';
val (((((fs, fs'), fs_copy), us), (ys, ys')),
names_lthy) = names_lthy
|> mk_Frees' "f" fTs
||>> mk_Frees "f" fTs
||>> mk_Frees "u" uTs
||>> mk_Frees' "y" passiveAs;
val map_FTFT's = map2 (fn Ds =>
mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
fun mk_passive_maps ATs BTs Ts =
map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ Ts)) Dss bnfs;
fun mk_map_fold_arg fs Ts ctor fmap =
HOLogic.mk_comp (ctor, Term.list_comb (fmap, fs @ map HOLogic.id_const Ts));
fun mk_map Ts fs Ts' ctors mk_maps =
mk_fold Ts (map2 (mk_map_fold_arg fs Ts') ctors (mk_maps Ts'));
val pmapsABT' = mk_passive_maps passiveAs passiveBs;
val fs_maps = map (mk_map Ts fs Ts' ctor's pmapsABT') ks;
val ls = 1 upto m;
val setsss = map (mk_setss o mk_set_Ts) passiveAs;
fun mk_col l T z z' sets =
let
fun mk_UN set = mk_Union T $ (set $ z);
in
Term.absfree z'
(mk_union (nth sets (l - 1) $ z,
Library.foldl1 mk_union (map mk_UN (drop m sets))))
end;
val colss = @{map 5} (fn l => fn T => @{map 3} (mk_col l T)) ls passiveAs AFss AFss' setsss;
val setss_by_range = map (fn cols => map (mk_fold Ts cols) ks) colss;
val setss_by_bnf = transpose setss_by_range;
val set_bss =
map (flat o map2 (fn B => fn b =>
if member (op =) deads (TFree B) then [] else [b]) resBs) set_bss0;
val ctor_witss =
let
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;
fun close_wit (I, wit) = fold_rev Term.absfree (map (nth ys') I) wit;
fun wit_apply (arg_I, arg_wit) (fun_I, fun_wit) =
(union (op =) arg_I fun_I, fun_wit $ arg_wit);
fun gen_arg support i =
if i < m then [([i], nth ys i)]
else maps (mk_wit support (nth ctors (i - m)) (i - m)) (nth support (i - m))
and mk_wit support ctor i (I, wit) =
let val args = map (gen_arg (nth_map i (remove (op =) (I, wit)) support)) I;
in
(args, [([], wit)])
|-> fold (map_product wit_apply)
|> map (apsnd (fn t => ctor $ t))
|> minimize_wits
end;
in
@{map 3} (fn ctor => fn i => map close_wit o minimize_wits o maps (mk_wit witss ctor i))
ctors (0 upto n - 1) witss
end;
val (lthy, sbd0, sbd0_card_order, sbd0_Cinfinite, set_sbd0ss) =
if n = 1
then (lthy, hd bd0s, hd bd0_card_orders, hd bd0_Cinfinites, set_bd0ss)
else
let
val sum_bd0 = Library.foldr1 (uncurry mk_csum) bd0s;
val sum_bd0T = fst (dest_relT (fastype_of sum_bd0));
val (sum_bd0T_params, sum_bd0T_params') = `(map TFree) (Term.add_tfreesT sum_bd0T []);
val sbd0T_bind = mk_internal_b (sum_bdTN ^ "0");
val ((sbd0T_name, (sbd0T_glob_info, sbd0T_loc_info)), lthy) =
typedef (sbd0T_bind, sum_bd0T_params', NoSyn)
(HOLogic.mk_UNIV sum_bd0T) NONE (fn _ => EVERY' [rtac exI, rtac UNIV_I] 1) lthy;
val sbd0T = Type (sbd0T_name, sum_bd0T_params);
val Abs_sbd0T = Const (#Abs_name sbd0T_glob_info, sum_bd0T --> sbd0T);
val sbd0_bind = mk_internal_b (sum_bdN ^ "0");
val sbd0_def_bind = (Thm.def_binding sbd0_bind, []);
val sbd0_spec = mk_dir_image sum_bd0 Abs_sbd0T;
val ((sbd0_free, (_, sbd0_def_free)), (lthy, lthy_old)) =
lthy
|> Local_Theory.define ((sbd0_bind, NoSyn), (sbd0_def_bind, sbd0_spec))
||> `Local_Theory.restore;
val phi = Proof_Context.export_morphism lthy_old lthy;
val sbd0_def = Morphism.thm phi sbd0_def_free RS meta_eq_to_obj_eq;
val sbd0 = Const (fst (Term.dest_Const (Morphism.term phi sbd0_free)),
mk_relT (`I sbd0T));
val Abs_sbd0T_inj = mk_Abs_inj_thm (#Abs_inject sbd0T_loc_info);
val Abs_sbd0T_bij = mk_Abs_bij_thm lthy Abs_sbd0T_inj (#Abs_cases sbd0T_loc_info);
val sum_Cinfinite = mk_sum_Cinfinite bd0_Cinfinites;
val sum_Card_order = sum_Cinfinite RS conjunct2;
val sum_card_order = mk_sum_card_order bd0_card_orders;
val sbd0_ordIso = @{thm ssubst_Pair_rhs} OF
[@{thm dir_image} OF [Abs_sbd0T_inj, sum_Card_order], sbd0_def];
val sbd0_Cinfinite = @{thm Cinfinite_cong} OF [sbd0_ordIso, sum_Cinfinite];
val sbd0_card_order = @{thm iffD2[OF arg_cong[of _ _ card_order]]} OF
[sbd0_def, @{thm card_order_dir_image} OF [Abs_sbd0T_bij, sum_card_order]];
fun mk_set_sbd0 i bd0_Card_order bd0s =
map (fn thm => @{thm ordLeq_ordIso_trans} OF
[bd0_Card_order RS mk_ordLeq_csum n i thm, sbd0_ordIso]) bd0s;
val set_sbd0ss = @{map 3} mk_set_sbd0 ks bd0_Card_orders set_bd0ss;
in
(lthy, sbd0, sbd0_card_order, sbd0_Cinfinite, set_sbd0ss)
end;
val (Ibnf_consts, lthy) =
@{fold_map 8} (fn b => fn map_b => fn rel_b => fn set_bs => fn mapx => fn sets =>
fn wits => fn T => fn lthy =>
define_bnf_consts Hardly_Inline (user_policy Note_Some lthy) false (SOME deads)
map_b rel_b set_bs
((((((b, T), fold_rev Term.absfree fs' mapx), sets), sbd0), wits), NONE) lthy)
bs map_bs rel_bs set_bss fs_maps setss_by_bnf ctor_witss Ts lthy;
val (_, Iconsts, Iconst_defs, mk_Iconsts) = @{split_list 4} Ibnf_consts;
val (_, Isetss, Ibds_Ds, Iwitss_Ds, _) = @{split_list 5} Iconsts;
val (Imap_defs, Iset_defss, Ibd_defs, Iwit_defss, Irel_defs) = @{split_list 5} Iconst_defs;
val (mk_Imaps_Ds, mk_It_Ds, _, mk_Irels_Ds, _) = @{split_list 5} mk_Iconsts;
val Irel_unabs_defs = map (fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) Irel_defs;
val Iset_defs = flat Iset_defss;
fun mk_Imaps As Bs = map (fn mk => mk deads As Bs) mk_Imaps_Ds;
fun mk_Isetss As = map2 (fn mk => fn Isets => map (mk deads As) Isets) mk_It_Ds Isetss;
val Ibds = map2 (fn mk => mk deads passiveAs) mk_It_Ds Ibds_Ds;
val Iwitss =
map2 (fn mk => fn Iwits => map (mk deads passiveAs o snd) Iwits) mk_It_Ds Iwitss_Ds;
fun mk_Irels As Bs = map (fn mk => mk deads As Bs) mk_Irels_Ds;
val Imaps = mk_Imaps passiveAs passiveBs;
val fs_Imaps = map (fn m => Term.list_comb (m, fs)) Imaps;
val fs_copy_Imaps = map (fn m => Term.list_comb (m, fs_copy)) Imaps;
val (Isetss_by_range, Isetss_by_bnf) = `transpose (mk_Isetss passiveAs);
val map_setss = map (fn T => map2 (fn Ds =>
mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) passiveAs;
val timer = time (timer "bnf constants for the new datatypes");
val (ctor_Imap_thms, ctor_Imap_o_thms) =
let
fun mk_goal fs_map map ctor ctor' =
mk_Trueprop_eq (HOLogic.mk_comp (fs_map, ctor),
HOLogic.mk_comp (ctor', Term.list_comb (map, fs @ fs_Imaps)));
val goals = @{map 4} mk_goal fs_Imaps map_FTFT's ctors ctor's;
val maps =
@{map 4} (fn goal => fn foldx => fn map_comp_id => fn map_cong0 =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Imap_defs THEN
mk_map_tac m n foldx map_comp_id map_cong0)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals ctor_fold_thms map_comp_id_thms map_cong0s;
in
`(map (fn thm => thm RS @{thm comp_eq_dest})) maps
end;
val (ctor_Imap_unique_thms, ctor_Imap_unique_thm) =
let
fun mk_prem u map ctor ctor' =
mk_Trueprop_eq (HOLogic.mk_comp (u, ctor),
HOLogic.mk_comp (ctor', Term.list_comb (map, fs @ us)));
val prems = @{map 4} mk_prem us map_FTFT's ctors ctor's;
val goal =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(map2 (curry HOLogic.mk_eq) us fs_Imaps));
val unique = Goal.prove_sorry lthy [] [] (Logic.list_implies (prems, goal))
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Imap_defs THEN
mk_ctor_map_unique_tac ctxt ctor_fold_unique_thm sym_map_comps)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
`split_conj_thm unique
end;
val timer = time (timer "map functions for the new datatypes");
val ctor_Iset_thmss =
let
fun mk_goal sets ctor set col map =
mk_Trueprop_eq (HOLogic.mk_comp (set, ctor),
HOLogic.mk_comp (col, Term.list_comb (map, passive_ids @ sets)));
val goalss =
@{map 3} (fn sets => @{map 4} (mk_goal sets) ctors sets)
Isetss_by_range colss map_setss;
val setss = map (map2 (fn foldx => fn goal =>
Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, prems = _} =>
unfold_thms_tac ctxt Iset_defs THEN mk_set_tac foldx)
|> Thm.close_derivation)
ctor_fold_thms) goalss;
fun mk_simp_goal pas_set act_sets sets ctor z set =
mk_Trueprop_eq (set $ (ctor $ z),
mk_union (pas_set $ z,
Library.foldl1 mk_union (map2 (fn X => mk_UNION (X $ z)) act_sets sets)));
val simp_goalss =
map2 (fn i => fn sets =>
@{map 4} (fn Fsets => mk_simp_goal (nth Fsets (i - 1)) (drop m Fsets) sets)
FTs_setss ctors xFs sets)
ls Isetss_by_range;
val ctor_setss = @{map 3} (fn i => @{map 3} (fn set_nats => fn goal => fn set =>
Goal.prove_sorry lthy [] [] goal
(K (mk_ctor_set_tac set (nth set_nats (i - 1)) (drop m set_nats)))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
set_mapss) ls simp_goalss setss;
in
ctor_setss
end;
fun mk_set_thms ctor_set = (@{thm xt1(3)} OF [ctor_set, @{thm Un_upper1}]) ::
map (fn i => (@{thm xt1(3)} OF [ctor_set, @{thm Un_upper2}]) RS
(mk_Un_upper n i RS subset_trans) RSN
(2, @{thm UN_upper} RS subset_trans))
(1 upto n);
val set_Iset_thmsss = transpose (map (map mk_set_thms) ctor_Iset_thmss);
val timer = time (timer "set functions for the new datatypes");
val cxs = map (SOME o Thm.cterm_of lthy) Izs;
val Isetss_by_range' =
map (map (Term.subst_atomic_types (passiveAs ~~ passiveBs))) Isetss_by_range;
val Iset_Imap0_thmss =
let
fun mk_set_map0 f map z set set' =
HOLogic.mk_eq (mk_image f $ (set $ z), set' $ (map $ z));
fun mk_cphi f map z set set' = Thm.cterm_of lthy
(Term.absfree (dest_Free z) (mk_set_map0 f map z set set'));
val csetss = map (map (Thm.cterm_of lthy)) Isetss_by_range';
val cphiss = @{map 3} (fn f => fn sets => fn sets' =>
(@{map 4} (mk_cphi f) fs_Imaps Izs sets sets')) fs Isetss_by_range Isetss_by_range';
val inducts = map (fn cphis =>
Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm) cphiss;
val goals =
@{map 3} (fn f => fn sets => fn sets' =>
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 4} (mk_set_map0 f) fs_Imaps Izs sets sets')))
fs Isetss_by_range Isetss_by_range';
fun mk_tac ctxt induct = mk_set_nat_tac ctxt m (rtac induct) set_mapss ctor_Imap_thms;
val thms =
@{map 5} (fn goal => fn csets => fn ctor_sets => fn induct => fn i =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_tac ctxt induct csets ctor_sets i)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals csetss ctor_Iset_thmss inducts ls;
in
map split_conj_thm thms
end;
val Iset_bd_thmss =
let
fun mk_set_bd z bd set = mk_ordLeq (mk_card_of (set $ z)) bd;
fun mk_cphi z set = Thm.cterm_of lthy (Term.absfree (dest_Free z) (mk_set_bd z sbd0 set));
val cphiss = map (map2 mk_cphi Izs) Isetss_by_range;
val inducts = map (fn cphis =>
Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm) cphiss;
val goals =
map (fn sets =>
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 3} mk_set_bd Izs Ibds sets))) Isetss_by_range;
fun mk_tac ctxt induct = mk_set_bd_tac ctxt m (rtac induct) sbd0_Cinfinite set_sbd0ss;
val thms =
@{map 4} (fn goal => fn ctor_sets => fn induct => fn i =>
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Ibd_defs THEN
mk_tac ctxt induct ctor_sets i)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
goals ctor_Iset_thmss inducts ls;
in
map split_conj_thm thms
end;
val Imap_cong0_thms =
let
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_map_cong0 sets z fmap gmap =
HOLogic.mk_imp
(Library.foldr1 HOLogic.mk_conj (@{map 5} (mk_prem z) sets fs fs_copy ys ys'),
HOLogic.mk_eq (fmap $ z, gmap $ z));
fun mk_cphi sets z fmap gmap =
Thm.cterm_of lthy (Term.absfree (dest_Free z) (mk_map_cong0 sets z fmap gmap));
val cphis = @{map 4} mk_cphi Isetss_by_bnf Izs fs_Imaps fs_copy_Imaps;
val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm;
val goal =
HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
(@{map 4} mk_map_cong0 Isetss_by_bnf Izs fs_Imaps fs_copy_Imaps));
val thm = Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_mcong_tac ctxt (rtac induct) set_Iset_thmsss
map_cong0s ctor_Imap_thms)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy);
in
split_conj_thm thm
end;
val in_rels = map in_rel_of_bnf bnfs;
val in_Irels = map (fn def => trans OF [def, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD})
Irel_unabs_defs;
val ctor_Iset_incl_thmss = map (map hd) set_Iset_thmsss;
val ctor_set_Iset_incl_thmsss = map (transpose o map tl) set_Iset_thmsss;
val ctor_Iset_thmss' = transpose ctor_Iset_thmss;
val Irels = mk_Irels passiveAs passiveBs;
val Irelphis = map (fn rel => Term.list_comb (rel, Iphis)) Irels;
val relphis = map (fn rel => Term.list_comb (rel, Iphis @ Irelphis)) rels;
val Irelpsi1s = map (fn rel => Term.list_comb (rel, Ipsi1s)) (mk_Irels passiveAs passiveCs);
val Irelpsi2s = map (fn rel => Term.list_comb (rel, Ipsi2s)) (mk_Irels passiveCs passiveBs);
val Irelpsi12s = map (fn rel =>
Term.list_comb (rel, map2 (curry mk_rel_compp) Ipsi1s Ipsi2s)) Irels;
val ctor_Irel_thms =
let
fun mk_goal xF yF ctor ctor' Irelphi relphi =
mk_Trueprop_eq (Irelphi $ (ctor $ xF) $ (ctor' $ yF), relphi $ xF $ yF);
val goals = @{map 6} mk_goal xFs yFs ctors ctor's Irelphis relphis;
in
@{map 12} (fn i => fn goal => fn in_rel => fn map_comp0 => fn map_cong0 =>
fn ctor_map => fn ctor_sets => fn ctor_inject => fn ctor_dtor =>
fn set_map0s => fn ctor_set_incls => fn ctor_set_set_inclss =>
Goal.prove_sorry lthy [] [] goal
(K (mk_ctor_rel_tac lthy in_Irels i in_rel map_comp0 map_cong0 ctor_map ctor_sets
ctor_inject ctor_dtor set_map0s ctor_set_incls ctor_set_set_inclss))
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy))
ks goals in_rels map_comps map_cong0s ctor_Imap_thms ctor_Iset_thmss'
ctor_inject_thms ctor_dtor_thms set_mapss ctor_Iset_incl_thmss
ctor_set_Iset_incl_thmsss
end;
val le_Irel_OO_thm =
let
fun mk_le_Irel_OO Irelpsi1 Irelpsi2 Irelpsi12 Iz1 Iz2 =
HOLogic.mk_imp (mk_rel_compp (Irelpsi1, Irelpsi2) $ Iz1 $ Iz2,
Irelpsi12 $ Iz1 $ Iz2);
val goals = @{map 5} mk_le_Irel_OO Irelpsi1s Irelpsi2s Irelpsi12s Izs1 Izs2;
val cTs = map (SOME o Thm.ctyp_of lthy o TFree) induct2_params;
val cxs = map (SOME o Thm.cterm_of lthy) (splice Izs1 Izs2);
fun mk_cphi z1 z2 goal = SOME (Thm.cterm_of lthy (Term.absfree z1 (Term.absfree z2 goal)));
val cphis = @{map 3} mk_cphi Izs1' Izs2' goals;
val induct = Drule.instantiate' cTs (cphis @ cxs) ctor_induct2_thm;
val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals);
in
Goal.prove_sorry lthy [] [] goal
(fn {context = ctxt, prems = _} => mk_le_rel_OO_tac ctxt m induct ctor_nchotomy_thms
ctor_Irel_thms rel_mono_strong0s le_rel_OOs)
|> Thm.close_derivation
|> singleton (Proof_Context.export names_lthy lthy)
end;
val timer = time (timer "helpers for BNF properties");
val map_id0_tacs = map (K o mk_map_id0_tac map_id0s) ctor_Imap_unique_thms;
val map_comp0_tacs =
map2 (K oo mk_map_comp0_tac map_comps ctor_Imap_thms) ctor_Imap_unique_thms ks;
val map_cong0_tacs = map (fn thm => fn ctxt => mk_map_cong0_tac ctxt m thm) Imap_cong0_thms;
val set_map0_tacss = map (map (K o mk_set_map0_tac)) (transpose Iset_Imap0_thmss);
val bd_co_tacs = replicate n (fn ctxt =>
unfold_thms_tac ctxt Ibd_defs THEN rtac sbd0_card_order 1);
val bd_cinf_tacs = replicate n (fn ctxt =>
unfold_thms_tac ctxt Ibd_defs THEN rtac (sbd0_Cinfinite RS conjunct1) 1);
val set_bd_tacss = map (map (fn thm => K (rtac thm 1))) (transpose Iset_bd_thmss);
val le_rel_OO_tacs = map (fn i =>
K ((rtac @{thm predicate2I} THEN' etac (le_Irel_OO_thm RS mk_conjunctN n i RS mp)) 1)) ks;
val rel_OO_Grp_tacs = map (fn def => K (rtac def 1)) Irel_unabs_defs;
val tacss = @{map 9} 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;
fun wit_tac ctxt = unfold_thms_tac ctxt (flat Iwit_defss) THEN
mk_wit_tac ctxt n (flat ctor_Iset_thmss) (maps wit_thms_of_bnf bnfs);
val (Ibnfs, lthy) =
@{fold_map 5} (fn tacs => fn map_b => fn rel_b => fn set_bs => fn consts =>
bnf_def Do_Inline (user_policy Note_Some) false I tacs wit_tac (SOME deads)
map_b rel_b set_bs consts)
tacss map_bs rel_bs set_bss
((((((replicate n Binding.empty ~~ Ts) ~~ Imaps) ~~ Isetss_by_bnf) ~~ Ibds) ~~
Iwitss) ~~ map SOME Irels) lthy;
val timer = time (timer "registered new datatypes as BNFs");
val ls' = if m = 1 then [0] else ls
val Ibnf_common_notes =
[(ctor_map_uniqueN, [ctor_Imap_unique_thm])]
|> map (fn (thmN, thms) =>
((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
val Ibnf_notes =
[(ctor_mapN, map single ctor_Imap_thms),
(ctor_relN, map single ctor_Irel_thms),
(ctor_set_inclN, ctor_Iset_incl_thmss),
(ctor_set_set_inclN, map flat ctor_set_Iset_incl_thmsss)] @
map2 (fn i => fn thms => (mk_ctor_setN i, map single thms)) ls' ctor_Iset_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, Ibnfs, (ctor_Imap_o_thms, ctor_Imap_thms), ctor_Iset_thmss',
ctor_Irel_thms, Ibnf_common_notes @ Ibnf_notes, lthy)
end;
val ctor_fold_o_Imap_thms = mk_xtor_un_fold_o_map_thms Least_FP false m ctor_fold_unique_thm
ctor_Imap_o_thms (map (mk_pointfree lthy) ctor_fold_thms) sym_map_comps map_cong0s;
val ctor_rec_o_Imap_thms = mk_xtor_un_fold_o_map_thms Least_FP true m ctor_rec_unique_thm
ctor_Imap_o_thms (map (mk_pointfree lthy) ctor_rec_thms) sym_map_comps map_cong0s;
val Irels = if m = 0 then map HOLogic.eq_const Ts
else map (mk_rel_of_bnf deads passiveAs passiveBs) Ibnfs;
val Irel_induct_thm =
mk_xtor_rel_co_induct_thm Least_FP rels activeIphis Irels Iphis xFs yFs ctors ctor's
(fn {context = ctxt, prems = IHs} => mk_rel_induct_tac ctxt IHs m ctor_induct2_thm ks
ctor_Irel_thms rel_mono_strong0s) lthy;
val rels = map2 (fn Ds => mk_rel_of_bnf Ds allAs allBs') Dss bnfs;
val ctor_fold_transfer_thms =
mk_co_iter_transfer_thms Least_FP rels activephis activephis Irels Iphis
(mk_folds passiveAs activeAs) (mk_folds passiveBs activeBs)
(fn {context = ctxt, prems = _} => mk_fold_transfer_tac ctxt m Irel_induct_thm
(map map_transfer_of_bnf bnfs) ctor_fold_thms)
lthy;
val rec_allAs = passiveAs @ map2 (curry HOLogic.mk_prodT) Ts activeAs;
val rec_allBs = passiveBs @ map2 (curry HOLogic.mk_prodT) Ts' activeBs;
val rec_rels = map2 (fn Ds => mk_rel_of_bnf Ds rec_allAs rec_allBs) Dss bnfs;
val rec_activephis =
map2 (fn Irel => mk_rel_prod (Term.list_comb (Irel, Iphis))) Irels activephis;
val ctor_rec_transfer_thms =
mk_co_iter_transfer_thms Least_FP rec_rels rec_activephis activephis Irels Iphis
(mk_recs Ts passiveAs activeAs) (mk_recs Ts' passiveBs activeBs)
(fn {context = ctxt, prems = _} => mk_ctor_rec_transfer_tac ctxt n m rec_defs
ctor_fold_transfer_thms (map map_transfer_of_bnf bnfs) ctor_Irel_thms)
lthy;
val timer = time (timer "relator induction");
val common_notes =
[(ctor_inductN, [ctor_induct_thm]),
(ctor_induct2N, [ctor_induct2_thm]),
(ctor_fold_transferN, ctor_fold_transfer_thms),
(ctor_rec_transferN, ctor_rec_transfer_thms),
(ctor_rel_inductN, [Irel_induct_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_foldN, ctor_fold_thms),
(ctor_fold_uniqueN, ctor_fold_unique_thms),
(ctor_rec_uniqueN, ctor_rec_unique_thms),
(ctor_fold_o_mapN, ctor_fold_o_Imap_thms),
(ctor_rec_o_mapN, ctor_rec_o_Imap_thms),
(ctor_injectN, ctor_inject_thms),
(ctor_recN, ctor_rec_thms),
(dtor_ctorN, dtor_ctor_thms),
(dtor_exhaustN, dtor_exhaust_thms),
(dtor_injectN, dtor_inject_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 |> note_all ? snd o Local_Theory.notes (common_notes @ notes @ Ibnf_notes);
val fp_res =
{Ts = Ts, bnfs = Ibnfs, ctors = ctors, dtors = dtors, xtor_co_recs = recs,
xtor_co_induct = ctor_induct_thm, dtor_ctors = dtor_ctor_thms, ctor_dtors = ctor_dtor_thms,
ctor_injects = ctor_inject_thms, dtor_injects = dtor_inject_thms,
xtor_maps = ctor_Imap_thms, xtor_setss = ctor_Iset_thmss',
xtor_rels = ctor_Irel_thms, xtor_co_rec_thms = ctor_rec_thms,
xtor_co_rec_o_maps = ctor_rec_o_Imap_thms, xtor_rel_co_induct = Irel_induct_thm,
dtor_set_inducts = [], xtor_co_rec_transfers = ctor_rec_transfer_thms};
in
timer; (fp_res, lthy')
end;
val _ =
Outer_Syntax.local_theory @{command_spec "datatype"} "define inductive datatypes"
(parse_co_datatype_cmd Least_FP construct_lfp);
val _ = Theory.setup (fp_antiquote_setup @{binding datatype});
end;