--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF/Tools/bnf_lfp.ML Fri Sep 21 16:45:06 2012 +0200
@@ -0,0 +1,1838 @@
+(* Title: HOL/BNF/Tools/bnf_lfp.ML
+ Author: Dmitriy Traytel, TU Muenchen
+ Author: Andrei Popescu, TU Muenchen
+ Copyright 2012
+
+Datatype construction.
+*)
+
+signature BNF_LFP =
+sig
+ val bnf_lfp: mixfix list -> (string * sort) list option -> binding list ->
+ typ list * typ list list -> BNF_Def.BNF list -> local_theory ->
+ (term list * term list * term list * term list * thm * thm list * thm list * thm list *
+ thm list * thm list) * local_theory
+end;
+
+structure BNF_LFP : BNF_LFP =
+struct
+
+open BNF_Def
+open BNF_Util
+open BNF_Tactics
+open BNF_FP
+open BNF_FP_Sugar
+open BNF_LFP_Util
+open BNF_LFP_Tactics
+
+(*all BNFs have the same lives*)
+fun bnf_lfp mixfixes resBs bs (resDs, Dss) bnfs lthy =
+ let
+ 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 b = Binding.name (mk_common_name (map Binding.name_of bs));
+
+ (* TODO: check if m, n, etc., are sane *)
+
+ val deads = fold (union (op =)) Dss resDs;
+ val names_lthy = fold Variable.declare_typ deads lthy;
+
+ (* tvars *)
+ val (((((((passiveAs, activeAs), allAs)), (passiveBs, activeBs)),
+ activeCs), passiveXs), passiveYs) = names_lthy
+ |> mk_TFrees live
+ |> apfst (`(chop m))
+ ||> mk_TFrees live
+ ||>> apfst (chop m)
+ ||>> mk_TFrees n
+ ||>> mk_TFrees m
+ ||> fst o mk_TFrees m;
+
+ 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';
+
+ (* typs *)
+ val dead_poss =
+ (case resBs of
+ NONE => map SOME deads @ replicate m NONE
+ | SOME Ts => map (fn T => if member (op =) deads (TFree T) then SOME (TFree T) else NONE) Ts);
+ 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 (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 = map4 mk_map_of_bnf Dss Ass Ass bnfs;
+ val mapsAsBs = map4 mk_map_of_bnf Dss Ass Bss bnfs;
+ val mapsBsAs = map4 mk_map_of_bnf Dss Bss Ass bnfs;
+ val mapsBsCs' = map4 mk_map_of_bnf Dss Bss Css' bnfs;
+ val mapsAsCs' = map4 mk_map_of_bnf Dss Ass Css' bnfs;
+ val map_fsts = map4 mk_map_of_bnf Dss (replicate n (passiveAs @ prodBsAs)) Bss bnfs;
+ val map_fsts_rev = map4 mk_map_of_bnf Dss Bss (replicate n (passiveAs @ prodBsAs)) bnfs;
+ fun mk_setss Ts = map3 mk_sets_of_bnf (map (replicate live) Dss)
+ (map (replicate live) (replicate n Ts)) bnfs;
+ val setssAs = mk_setss allAs;
+ val bds = map3 mk_bd_of_bnf Dss Ass bnfs;
+ val witss = map wits_of_bnf bnfs;
+
+ val (((((((((((((((((((zs, zs'), As), 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')), (yFs, yFs')),
+ names_lthy) = lthy
+ |> mk_Frees' "z" activeAs
+ ||>> mk_Frees "A" ATs
+ ||>> 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
+ ||>> mk_Frees' "y" FTsBs;
+
+ 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 bd_card_orders = map bd_card_order_of_bnf bnfs;
+ val bd_Card_orders = map bd_Card_order_of_bnf bnfs;
+ val bd_Card_order = hd bd_Card_orders;
+ val bd_Cinfinite = bd_Cinfinite_of_bnf (hd bnfs);
+ val bd_Cnotzeros = map bd_Cnotzero_of_bnf bnfs;
+ val bd_Cnotzero = hd bd_Cnotzeros;
+ val in_bds = map in_bd_of_bnf bnfs;
+ val map_comp's = map map_comp'_of_bnf bnfs;
+ val map_congs = map map_cong_of_bnf bnfs;
+ val map_ids = map map_id_of_bnf bnfs;
+ val map_id's = map map_id'_of_bnf bnfs;
+ val map_wpulls = map map_wpull_of_bnf bnfs;
+ val set_bdss = map set_bd_of_bnf bnfs;
+ val set_natural'ss = map set_natural'_of_bnf bnfs;
+
+ val timer = time (timer "Extracted terms & thms");
+
+ (* nonemptiness check *)
+ fun new_wit X wit = 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_comp =
+ 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: fs @ all_gs) (mk_Trueprop_eq (lhs, rhs)))
+ (K (mk_map_comp_id_tac map_comp))
+ |> Thm.close_derivation
+ end;
+
+ val map_comp_id_thms = map5 mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comp's;
+
+ (*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_congL x mapAsAs sets map_cong 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 = map4 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (x :: self_fs) (Logic.list_implies (prems, goal)))
+ (K (mk_map_congL_tac m map_cong map_id'))
+ |> Thm.close_derivation
+ end;
+
+ val map_congL_thms = map5 mk_map_congL xFs mapsAsAs setssAs map_congs map_id's;
+ 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 = Binding.suffix_name ("_" ^ algN) b;
+ val alg_name = Binding.name_of alg_bind;
+ val alg_def_bind = (Thm.def_binding alg_bind, []);
+
+ (*forall i = 1 ... n: (\<forall>x \<in> Fi_in A1 .. Am B1 ... Bn. si x \<in> Bi)*)
+ val alg_spec =
+ let
+ val algT = Library.foldr (op -->) (ATs @ BTs @ sTs, HOLogic.boolT);
+
+ val ins = map3 mk_in (replicate n (As @ 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 lhs = Term.list_comb (Free (alg_name, algT), As @ Bs @ ss);
+ val rhs = Library.foldr1 HOLogic.mk_conj (map5 mk_alg_conjunct Bs ss ins xFs xFs')
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((alg_free, (_, alg_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (alg_bind, NONE, 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 = Morphism.thm phi alg_def_free;
+
+ fun mk_alg As Bs ss =
+ let
+ val args = As @ 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 As Bs ss);
+ fun mk_prem x set B = HOLogic.mk_Trueprop (mk_subset (set $ x) B);
+ fun mk_concl s x B = HOLogic.mk_Trueprop (HOLogic.mk_mem (s $ x, B));
+ val premss = map2 ((fn x => fn sets => map2 (mk_prem x) sets (As @ Bs))) xFs setssAs;
+ val concls = map3 mk_concl ss xFs Bs;
+ val goals = map3 (fn x => fn prems => fn concl =>
+ fold_rev Logic.all (x :: As @ Bs @ ss)
+ (Logic.list_implies (alg_prem :: prems, concl))) xFs premss concls;
+ in
+ map (fn goal =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_alg_set_tac alg_def)) |> Thm.close_derivation)
+ goals
+ end;
+
+ fun mk_talg ATs BTs = mk_alg (map HOLogic.mk_UNIV ATs) (map HOLogic.mk_UNIV BTs);
+
+ val talg_thm =
+ let
+ val goal = fold_rev Logic.all ss
+ (HOLogic.mk_Trueprop (mk_talg passiveAs activeAs ss))
+ in
+ Skip_Proof.prove lthy [] [] goal
+ (K (stac alg_def 1 THEN CONJ_WRAP (K (EVERY' [rtac ballI, rtac UNIV_I] 1)) ss))
+ |> Thm.close_derivation
+ end;
+
+ val timer = time (timer "Algebra definition & thms");
+
+ val alg_not_empty_thms =
+ let
+ val alg_prem =
+ HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val concls = map (HOLogic.mk_Trueprop o mk_not_empty) Bs;
+ val goals =
+ map (fn concl =>
+ fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (alg_prem, concl))) concls;
+ in
+ map2 (fn goal => fn alg_set =>
+ Skip_Proof.prove lthy [] []
+ goal (K (mk_alg_not_empty_tac alg_set alg_set_thms wit_thms))
+ |> Thm.close_derivation)
+ goals alg_set_thms
+ end;
+
+ val timer = time (timer "Proved nonemptiness");
+
+ (* morphism *)
+
+ val mor_bind = Binding.suffix_name ("_" ^ morN) b;
+ val mor_name = Binding.name_of mor_bind;
+ 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
+ val morT = Library.foldr (op -->) (BTs @ sTs @ B'Ts @ s'Ts @ fTs, HOLogic.boolT);
+
+ 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 lhs = Term.list_comb (Free (mor_name, morT), Bs @ ss @ B's @ s's @ fs);
+ val rhs = HOLogic.mk_conj
+ (Library.foldr1 HOLogic.mk_conj (map5 mk_fbetw fs Bs B's zs zs'),
+ Library.foldr1 HOLogic.mk_conj
+ (map8 mk_mor setssAs mapsAsBs fs ss s's FTsAs xFs xFs'))
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) =
+ lthy
+ |> Specification.definition (SOME (mor_bind, NONE, 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 = Morphism.thm phi mor_def_free;
+
+ 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 (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 = fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs)
+ (Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_subset (mk_image f $ B1) B2)));
+ val image_goals = map3 mk_image_goal fs Bs B's;
+ 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 =
+ fold_rev Logic.all (x :: Bs @ ss @ B's @ s's @ fs)
+ (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 = map7 mk_elim_goal setssAs mapsAsBs fs ss s's xFs FTsAs;
+ fun prove goal =
+ Skip_Proof.prove lthy [] [] goal (K (mk_mor_elim_tac mor_def)) |> Thm.close_derivation;
+ in
+ (map prove image_goals, map prove elim_goals)
+ end;
+
+ val mor_incl_thm =
+ let
+ val prems = map2 (HOLogic.mk_Trueprop oo mk_subset) Bs Bs_copy;
+ val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ Bs_copy) (Logic.list_implies (prems, concl)))
+ (K (mk_mor_incl_tac mor_def map_id's))
+ |> Thm.close_derivation
+ 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ B''s @ s''s @ fs @ gs)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_mor_comp_tac mor_def set_natural'ss map_comp_id_thms))
+ |> Thm.close_derivation
+ end;
+
+ val mor_inv_thm =
+ let
+ fun mk_inv_prem f inv_f B B' = HOLogic.mk_conj (mk_subset (mk_image inv_f $ B') B,
+ HOLogic.mk_conj (mk_inver inv_f f B, mk_inver f inv_f B'));
+ val prems = map HOLogic.mk_Trueprop
+ ([mk_mor Bs ss B's s's fs,
+ mk_alg passive_UNIVs Bs ss,
+ mk_alg passive_UNIVs B's s's] @
+ map4 mk_inv_prem fs inv_fs Bs B's);
+ val concl = HOLogic.mk_Trueprop (mk_mor B's s's Bs ss inv_fs);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ inv_fs)
+ (Logic.list_implies (prems, concl)))
+ (K (mk_mor_inv_tac alg_def mor_def
+ set_natural'ss morE_thms map_comp_id_thms map_congL_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);
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs @ fs_copy)
+ (Logic.list_implies (prems, concl)))
+ (K ((hyp_subst_tac THEN' atac) 1))
+ |> Thm.close_derivation
+ 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all ss (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
+ end;
+
+ val mor_convol_thm =
+ let
+ val maps = map3 (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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (s's @ prod_ss) (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
+ 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 (map4 mk_conjunct mapsAsBs fs ss s's);
+ in
+ Skip_Proof.prove lthy [] [] (fold_rev Logic.all (ss @ s's @ fs) (mk_Trueprop_eq (lhs, rhs)))
+ (K (mk_mor_UNIV_tac m morE_thms mor_def))
+ |> Thm.close_derivation
+ end;
+
+ val timer = time (timer "Morphism definition & thms");
+
+ (* isomorphism *)
+
+ (*mor Bs1 ss1 Bs2 ss2 fs \<and> (\<exists>gs. mor Bs2 ss2 Bs1 ss1 fs \<and>
+ forall i = 1 ... n. (inver gs[i] fs[i] Bs1[i] \<and> inver fs[i] gs[i] Bs2[i]))*)
+ fun mk_iso Bs1 ss1 Bs2 ss2 fs gs =
+ let
+ val ex_inv_mor = list_exists_free gs
+ (HOLogic.mk_conj (mk_mor Bs2 ss2 Bs1 ss1 gs,
+ Library.foldr1 HOLogic.mk_conj (map2 (curry HOLogic.mk_conj)
+ (map3 mk_inver gs fs Bs1) (map3 mk_inver fs gs Bs2))));
+ in
+ HOLogic.mk_conj (mk_mor Bs1 ss1 Bs2 ss2 fs, ex_inv_mor)
+ end;
+
+ val iso_alt_thm =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ [mk_alg passive_UNIVs Bs ss,
+ mk_alg passive_UNIVs B's s's]
+ val concl = mk_Trueprop_eq (mk_iso Bs ss B's s's fs inv_fs,
+ HOLogic.mk_conj (mk_mor Bs ss B's s's fs,
+ Library.foldr1 HOLogic.mk_conj (map3 mk_bij_betw fs Bs B's)));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ s's @ fs) (Logic.list_implies (prems, concl)))
+ (K (mk_iso_alt_tac mor_image_thms mor_inv_thm))
+ |> Thm.close_derivation
+ end;
+
+ val timer = time (timer "Isomorphism definition & thms");
+
+ (* algebra copies *)
+
+ val (copy_alg_thm, ex_copy_alg_thm) =
+ let
+ val prems = map HOLogic.mk_Trueprop
+ (mk_alg passive_UNIVs Bs ss :: map3 mk_bij_betw inv_fs B's Bs);
+ val inver_prems = map HOLogic.mk_Trueprop
+ (map3 mk_inver inv_fs fs Bs @ map3 mk_inver fs inv_fs B's);
+ val all_prems = prems @ inver_prems;
+ fun mk_s f s mapT y y' = Term.absfree y' (f $ (s $
+ (Term.list_comb (mapT, passive_ids @ inv_fs) $ y)));
+
+ val alg = HOLogic.mk_Trueprop
+ (mk_alg passive_UNIVs B's (map5 mk_s fs ss mapsBsAs yFs yFs'));
+ val copy_str_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (all_prems, alg)))
+ (K (mk_copy_str_tac set_natural'ss alg_def alg_set_thms))
+ |> Thm.close_derivation;
+
+ val iso = HOLogic.mk_Trueprop
+ (mk_iso B's (map5 mk_s fs ss mapsBsAs yFs yFs') Bs ss inv_fs fs_copy);
+ val copy_alg_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (all_prems, iso)))
+ (K (mk_copy_alg_tac set_natural'ss alg_set_thms mor_def iso_alt_thm copy_str_thm))
+ |> Thm.close_derivation;
+
+ val ex = HOLogic.mk_Trueprop
+ (list_exists_free s's
+ (HOLogic.mk_conj (mk_alg passive_UNIVs B's s's,
+ mk_iso B's s's Bs ss inv_fs fs_copy)));
+ val ex_copy_alg_thm = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss @ B's @ inv_fs @ fs)
+ (Logic.list_implies (prems, ex)))
+ (K (mk_ex_copy_alg_tac n copy_str_thm copy_alg_thm))
+ |> Thm.close_derivation;
+ in
+ (copy_alg_thm, ex_copy_alg_thm)
+ end;
+
+ val timer = time (timer "Copy thms");
+
+
+ (* bounds *)
+
+ val sum_Card_order = if n = 1 then bd_Card_order else @{thm Card_order_csum};
+ val sum_Cnotzero = if n = 1 then bd_Cnotzero else bd_Cnotzero RS @{thm csum_Cnotzero1};
+ val sum_Cinfinite = if n = 1 then bd_Cinfinite else bd_Cinfinite RS @{thm Cinfinite_csum1};
+ fun mk_set_bd_sums i bd_Card_order bds =
+ if n = 1 then bds
+ else map (fn thm => bd_Card_order RS mk_ordLeq_csum n i thm) bds;
+ val set_bd_sumss = map3 mk_set_bd_sums ks bd_Card_orders set_bdss;
+
+ fun mk_in_bd_sum i Co Cnz bd =
+ if n = 1 then bd
+ else Cnz RS ((Co RS mk_ordLeq_csum n i (Co RS @{thm ordLeq_refl})) RS
+ (bd RS @{thm ordLeq_transitive[OF _
+ cexp_mono2_Cnotzero[OF _ csum_Cnotzero2[OF ctwo_Cnotzero]]]}));
+ val in_bd_sums = map4 mk_in_bd_sum ks bd_Card_orders bd_Cnotzeros in_bds;
+
+ val sum_bd = Library.foldr1 (uncurry mk_csum) bds;
+ val suc_bd = mk_cardSuc sum_bd;
+ 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 = if n = 1 then bd_Card_order RS @{thm cardSuc_Card_order}
+ else @{thm cardSuc_Card_order[OF Card_order_csum]};
+ val suc_bd_Cinfinite = if n = 1 then bd_Cinfinite RS @{thm Cinfinite_cardSuc}
+ else bd_Cinfinite RS @{thm Cinfinite_cardSuc[OF Cinfinite_csum1]};
+ 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 Asuc_bd_Cnotzero = Asuc_bd_Cinfinite RS @{thm Cinfinite_Cnotzero};
+
+ val suc_bd_Asuc_bd = @{thm ordLess_ordLeq_trans[OF
+ ordLess_ctwo_cexp
+ cexp_mono1_Cnotzero[OF _ ctwo_Cnotzero]]} OF
+ [suc_bd_Card_order, basis_Asuc, suc_bd_Card_order];
+
+ val Asuc_bdT = fst (dest_relT (fastype_of (mk_Asuc_bd As)));
+ 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all idxs (Logic.list_implies ([prem], concl)))
+ (K (mk_bd_limit_tac n suc_bd_Cinfinite))
+ |> Thm.close_derivation
+ 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 As Asi i sets Ts s k =
+ HOLogic.mk_binop @{const_name "sup"}
+ (mk_minG Asi i k, mk_image s $ mk_in (As @ map (mk_minG Asi i) ks) sets Ts);
+
+ fun mk_min_algs As ss =
+ let
+ val BTs = map (range_type o fastype_of) ss;
+ val Ts = map (HOLogic.dest_setT o fastype_of) As @ 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
+ (map4 (mk_minH_component As 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 As 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
+ (map4 (mk_minH_component As min_algs idx) setssAs FTsAs ss ks)));
+ val goal = fold_rev Logic.all (idx :: As @ ss)
+ (Logic.mk_implies (HOLogic.mk_Trueprop i_field, concl));
+
+ val min_algs_thm = Skip_Proof.prove lthy [] [] goal
+ (K (mk_min_algs_tac suc_bd_worel in_cong'_thms))
+ |> Thm.close_derivation;
+
+ val min_algs_thms = map (fn k => min_algs_thm RS mk_nthI n k) ks;
+
+ fun mk_mono_goal min_alg =
+ fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_relChain suc_bd
+ (Term.absfree idx' min_alg)));
+
+ val monos =
+ map2 (fn goal => fn min_algs =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_min_algs_mono_tac min_algs))
+ |> Thm.close_derivation)
+ (map mk_mono_goal min_algss) min_algs_thms;
+
+ val Asuc_bd = mk_Asuc_bd As;
+
+ 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 = certifyT lthy suc_bdT;
+ val card_ct = certify lthy (Term.absfree idx' card_conjunction);
+
+ val card_of = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove 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_bd_sums
+ sum_Card_order sum_Cnotzero suc_bd_Card_order suc_bd_Cinfinite suc_bd_Cnotzero
+ suc_bd_Asuc_bd Asuc_bd_Cinfinite Asuc_bd_Cnotzero)))
+ |> Thm.close_derivation;
+
+ val least_prem = HOLogic.mk_Trueprop (mk_alg As Bs ss);
+ val least_conjunction = Library.foldr1 HOLogic.mk_conj (map2 mk_subset min_algss Bs);
+ val least_cT = certifyT lthy suc_bdT;
+ val least_ct = certify lthy (Term.absfree idx' least_conjunction);
+
+ val least = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove 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;
+ in
+ (min_algs_thms, monos, card_of, least)
+ end;
+
+ val timer = time (timer "min_algs definition & thms");
+
+ fun min_alg_bind i = Binding.suffix_name
+ ("_" ^ min_algN ^ (if n = 1 then "" else string_of_int i)) b;
+ val min_alg_name = Binding.name_of o min_alg_bind;
+ val min_alg_def_bind = rpair [] o Thm.def_binding o min_alg_bind;
+
+ fun min_alg_spec i =
+ let
+ val min_algT =
+ Library.foldr (op -->) (ATs @ sTs, HOLogic.mk_setT (nth activeAs (i - 1)));
+
+ val lhs = Term.list_comb (Free (min_alg_name i, min_algT), As @ ss);
+ val rhs = mk_UNION (field_suc_bd)
+ (Term.absfree idx' (mk_nthN n (mk_min_algs As ss $ idx) i));
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((min_alg_frees, (_, min_alg_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => Specification.definition
+ (SOME (min_alg_bind i, NONE, 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 (Morphism.thm phi) min_alg_def_frees;
+
+ fun mk_min_alg As ss i =
+ let
+ val T = HOLogic.mk_setT (range_type (fastype_of (nth ss (i - 1))))
+ val args = As @ ss;
+ val Ts = map fastype_of args;
+ val min_algT = Library.foldr (op -->) (Ts, T);
+ in
+ Term.list_comb (Const (nth min_algs (i - 1), min_algT), args)
+ end;
+
+ val (alg_min_alg_thm, card_of_min_alg_thms, least_min_alg_thms, mor_incl_min_alg_thm) =
+ let
+ val min_algs = map (mk_min_alg As ss) ks;
+
+ val goal = fold_rev Logic.all (As @ ss) (HOLogic.mk_Trueprop (mk_alg As min_algs ss));
+ val alg_min_alg = Skip_Proof.prove lthy [] [] goal
+ (K (mk_alg_min_alg_tac m alg_def min_alg_defs suc_bd_limit_thm sum_Cinfinite
+ set_bd_sumss min_algs_thms min_algs_mono_thms))
+ |> Thm.close_derivation;
+
+ val Asuc_bd = mk_Asuc_bd As;
+ fun mk_card_of_thm min_alg def = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ ss)
+ (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 As Bs ss);
+ fun mk_least_thm min_alg B def = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (As @ Bs @ ss)
+ (Logic.mk_implies (least_prem, HOLogic.mk_Trueprop (mk_subset min_alg B))))
+ (K (mk_least_min_alg_tac def least_min_algs_thm))
+ |> Thm.close_derivation;
+
+ val leasts = map3 mk_least_thm min_algs Bs min_alg_defs;
+
+ val incl_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val incl_min_algs = map (mk_min_alg passive_UNIVs ss) ks;
+ val incl = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss)
+ (Logic.mk_implies (incl_prem,
+ HOLogic.mk_Trueprop (mk_mor incl_min_algs ss Bs ss active_ids))))
+ (K (EVERY' (rtac mor_incl_thm :: map etac leasts) 1))
+ |> Thm.close_derivation;
+ 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 = Binding.suffix_name ("_" ^ IITN) b;
+
+ val ((IIT_name, (IIT_glob_info, IIT_loc_info)), lthy) =
+ typedef false NONE (IIT_bind, params, NoSyn)
+ (HOLogic.mk_UNIV II_repT) NONE (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 passive_UNIVs 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;
+
+ fun str_init_bind i = Binding.suffix_name ("_" ^ str_initN ^ (if n = 1 then "" else
+ string_of_int i)) b;
+ val str_init_name = Binding.name_of o str_init_bind;
+ val str_init_def_bind = rpair [] o Thm.def_binding o str_init_bind;
+
+ fun str_init_spec i =
+ let
+ val T = nth init_FTs (i - 1);
+ 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 str_initT = T --> IIT --> Asuc_bdT;
+
+ val lhs = Term.list_comb (Free (str_init_name i, str_initT), [init_xF, iidx]);
+ val rhs = select_s $ (Term.list_comb (map, map_args) $ init_xF);
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((str_init_frees, (_, str_init_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map (fn i => Specification.definition
+ (SOME (str_init_bind i, NONE, 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 (Morphism.thm phi) str_init_def_frees;
+
+ val car_inits = map (mk_min_alg passive_UNIVs str_inits) ks;
+
+ (*TODO: replace with instantiate? (problem: figure out right type instantiation)*)
+ val alg_init_thm = Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_alg passive_UNIVs car_inits str_inits))
+ (K (rtac alg_min_alg_thm 1))
+ |> Thm.close_derivation;
+
+ val alg_select_thm = Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_Ball II
+ (Term.absfree iidx' (mk_alg passive_UNIVs select_Bs select_ss))))
+ (mk_alg_select_tac Abs_IIT_inverse_thm)
+ |> Thm.close_derivation;
+
+ val mor_select_thm =
+ let
+ val alg_prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val i_prem = HOLogic.mk_Trueprop (HOLogic.mk_mem (iidx, II));
+ val mor_prem = HOLogic.mk_Trueprop (mk_mor select_Bs select_ss Bs ss Asuc_fs);
+ val prems = [alg_prem, i_prem, mor_prem];
+ val concl = HOLogic.mk_Trueprop
+ (mk_mor car_inits str_inits Bs ss
+ (map (fn f => HOLogic.mk_comp (f, mk_rapp iidx Asuc_bdT)) Asuc_fs));
+ in
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (iidx :: Bs @ ss @ Asuc_fs) (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_natural'ss str_init_defs))
+ |> Thm.close_derivation
+ end;
+
+ val (init_ex_mor_thm, init_unique_mor_thms) =
+ let
+ val prem = HOLogic.mk_Trueprop (mk_alg passive_UNIVs Bs ss);
+ val concl = HOLogic.mk_Trueprop
+ (list_exists_free init_fs (mk_mor car_inits str_inits Bs ss init_fs));
+ val ex_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (Bs @ ss) (Logic.mk_implies (prem, concl)))
+ (mk_init_ex_mor_tac Abs_IIT_inverse_thm ex_copy_alg_thm alg_min_alg_thm
+ card_of_min_alg_thms mor_comp_thm mor_select_thm mor_incl_min_alg_thm)
+ |> Thm.close_derivation;
+
+ 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 (map3 mk_fun_eq init_fs init_fs_copy init_xs));
+ val unique_mor = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (init_xs @ Bs @ ss @ init_fs @ init_fs_copy)
+ (Logic.list_implies (prems @ mor_prems, unique)))
+ (K (mk_init_unique_mor_tac m alg_def alg_init_thm least_min_alg_thms
+ in_mono'_thms alg_set_thms morE_thms map_congs))
+ |> Thm.close_derivation;
+ in
+ (ex_mor, 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
+ (map6 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
+ Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all init_phis (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
+ end;
+
+ val timer = time (timer "Initiality definition & thms");
+
+ val ((T_names, (T_glob_infos, T_loc_infos)), lthy) =
+ lthy
+ |> fold_map3 (fn b => fn mx => fn car_init => typedef false NONE (b, params, mx) car_init NONE
+ (EVERY' [rtac ssubst, rtac @{thm ex_in_conv}, resolve_tac 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_casess = map #Rep_cases T_loc_infos;
+ val Rep_injects = map #Rep_inject T_loc_infos;
+ val Rep_inverses = map #Rep_inverse T_loc_infos;
+ val Abs_inverses = map #Abs_inverse T_loc_infos;
+
+ fun mk_inver_thm mk_tac rep abs X thm =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_inver rep abs X))
+ (K (EVERY' [rtac ssubst, rtac @{thm inver_def}, rtac ballI, mk_tac thm] 1))
+ |> Thm.close_derivation;
+
+ val inver_Reps = map4 (mk_inver_thm rtac) Abs_Ts Rep_Ts (map HOLogic.mk_UNIV Ts) Rep_inverses;
+ val inver_Abss = map4 (mk_inver_thm etac) Rep_Ts Abs_Ts car_inits Abs_inverses;
+
+ 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 (((((((((Izs1, Izs1'), (Izs2, Izs2')), (xFs, 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_free Ts zs;
+ val phis = map2 retype_free (map mk_pred1T Ts) init_phis;
+ val phi2s = map2 retype_free (map2 mk_pred2T Ts Ts') init_phis;
+
+ fun ctor_bind i = Binding.suffix_name ("_" ^ ctorN) (nth bs (i - 1));
+ val ctor_name = Binding.name_of o ctor_bind;
+ val ctor_def_bind = rpair [] o Thm.def_binding o ctor_bind;
+
+ fun ctor_spec i abs str map_FT_init x x' =
+ let
+ val ctorT = nth FTs (i - 1) --> nth Ts (i - 1);
+
+ val lhs = Free (ctor_name i, ctorT);
+ val rhs = Term.absfree x' (abs $ (str $
+ (Term.list_comb (map_FT_init, map HOLogic.id_const passiveAs @ Rep_Ts) $ x)));
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map6 (fn i => fn abs => fn str => fn mapx => fn x => fn x' =>
+ Specification.definition
+ (SOME (ctor_bind i, NONE, NoSyn), (ctor_def_bind i, ctor_spec i abs str mapx x x')))
+ ks Abs_Ts str_inits map_FT_inits xFs xFs'
+ |>> 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 (Morphism.thm phi) ctor_def_frees;
+
+ val (mor_Rep_thm, mor_Abs_thm) =
+ let
+ val copy = alg_init_thm RS copy_alg_thm;
+ fun mk_bij inj Rep cases = @{thm bij_betwI'} OF [inj, Rep, cases];
+ val bijs = map3 mk_bij Rep_injects Reps Rep_casess;
+ val mor_Rep =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor UNIVs ctors car_inits str_inits Rep_Ts))
+ (mk_mor_Rep_tac ctor_defs copy bijs inver_Abss inver_Reps)
+ |> Thm.close_derivation;
+
+ val inv = mor_inv_thm OF [mor_Rep, talg_thm, alg_init_thm];
+ val mor_Abs =
+ Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor car_inits str_inits UNIVs ctors Abs_Ts))
+ (K (mk_mor_Abs_tac inv inver_Abss inver_Reps))
+ |> 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 = Binding.suffix_name ("_" ^ ctor_foldN) (nth bs (i - 1));
+ val fold_name = Binding.name_of o fold_bind;
+ val fold_def_bind = rpair [] o Thm.def_binding o fold_bind;
+
+ fun fold_spec i T AT =
+ let
+ val foldT = Library.foldr (op -->) (sTs, T --> AT);
+
+ val lhs = Term.list_comb (Free (fold_name i, foldT), ss);
+ val rhs = mk_nthN n foldx i;
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((fold_frees, (_, fold_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn T => fn AT =>
+ Specification.definition
+ (SOME (fold_bind i, NONE, NoSyn), (fold_def_bind i, fold_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 folds = map (Morphism.term phi) fold_frees;
+ val fold_names = map (fst o dest_Const) folds;
+ 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 (Morphism.thm phi) fold_def_frees;
+
+ val mor_fold_thm =
+ let
+ val ex_mor = talg_thm RS init_ex_mor_thm;
+ val mor_cong = mor_cong_thm OF (map (mk_nth_conv n) ks);
+ val mor_comp = mor_Rep_thm RS mor_comp_thm;
+ val cT = certifyT lthy foldT;
+ val ct = certify lthy fold_fun
+ in
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] []
+ (HOLogic.mk_Trueprop (mk_mor UNIVs ctors active_UNIVs ss (map (mk_fold Ts ss) ks)))
+ (K (mk_mor_fold_tac cT ct fold_defs ex_mor (mor_comp RS mor_cong))))
+ |> Thm.close_derivation
+ 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 = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (ss @ fs) (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;
+ in
+ `split_conj_thm unique_mor
+ end;
+
+ val ctor_fold_unique_thms =
+ split_conj_thm (mk_conjIN n RS
+ (mor_UNIV_thm RS @{thm ssubst[of _ _ "%x. x"]} 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 = Binding.suffix_name ("_" ^ dtorN) (nth bs (i - 1));
+ val dtor_name = Binding.name_of o dtor_bind;
+ val dtor_def_bind = rpair [] o Thm.def_binding o dtor_bind;
+
+ fun dtor_spec i FT T =
+ let
+ val dtorT = T --> FT;
+
+ val lhs = Free (dtor_name i, dtorT);
+ val rhs = mk_fold Ts map_ctors i;
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn FT => fn T =>
+ Specification.definition
+ (SOME (dtor_bind i, NONE, NoSyn), (dtor_def_bind i, dtor_spec i FT T))) ks FTs Ts
+ |>> 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 (Morphism.thm phi) 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 = map3 mk_goal dtors ctors FTs;
+ in
+ map5 (fn goal => fn dtor_def => fn foldx => fn map_comp_id => fn map_congL =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_dtor_o_ctor_tac dtor_def foldx map_comp_id map_congL ctor_o_fold_thms))
+ |> Thm.close_derivation)
+ goals dtor_defs ctor_fold_thms map_comp_id_thms map_congL_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 = Binding.suffix_name ("_" ^ ctor_recN) (nth bs (i - 1));
+ val rec_name = Binding.name_of o rec_bind;
+ val rec_def_bind = rpair [] o Thm.def_binding o rec_bind;
+
+ fun rec_spec i T AT =
+ let
+ val recT = Library.foldr (op -->) (rec_sTs, T --> AT);
+ val maps = map3 (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;
+
+ val lhs = Term.list_comb (Free (rec_name i, recT), rec_ss);
+ val rhs = HOLogic.mk_comp (snd_const (HOLogic.mk_prodT (T, AT)), mk_fold Ts maps i);
+ in
+ mk_Trueprop_eq (lhs, rhs)
+ end;
+
+ val ((rec_frees, (_, rec_def_frees)), (lthy, lthy_old)) =
+ lthy
+ |> fold_map3 (fn i => fn T => fn AT =>
+ Specification.definition
+ (SOME (rec_bind i, NONE, 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_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 (Morphism.thm phi) 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
+ fold_rev Logic.all (x :: rec_ss) (mk_Trueprop_eq (lhs, rhs))
+ end;
+ val goals = map5 mk_goal ks rec_ss rec_maps_rev ctors xFs;
+ in
+ map2 (fn goal => fn foldx =>
+ Skip_Proof.prove lthy [] [] goal (mk_rec_tac rec_defs foldx fst_rec_pair_thms)
+ |> Thm.close_derivation)
+ goals ctor_fold_thms
+ end;
+
+ 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 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z, set $ x));
+ val concl = HOLogic.mk_Trueprop (phi $ z);
+ in
+ Logic.all z (Logic.mk_implies (prem, concl))
+ end;
+
+ val IHs = map3 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 = map4 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
+ (Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (phis @ Izs) goal)
+ (K (mk_ctor_induct_tac m set_natural'ss init_induct_thm morE_thms mor_Abs_thm
+ Rep_inverses Abs_inverses Reps))
+ |> Thm.close_derivation,
+ rev (Term.add_tfrees goal []))
+ end;
+
+ val cTs = map (SOME o certifyT lthy o TFree) induct_params;
+
+ val weak_ctor_induct_thms =
+ let fun insts i = (replicate (i - 1) TrueI) @ (@{thm 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 = HOLogic.mk_Trueprop (HOLogic.mk_mem (z1, (set $ x)));
+ val prem2 = HOLogic.mk_Trueprop (HOLogic.mk_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 = map5 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 = map7 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
+ (map3 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 = map3 (SOME o certify lthy ooo mk_t) phi2s (Izs1 ~~ Izs1') (Izs2 ~~ Izs2');
+ val goal = Logic.list_implies (prems, concl);
+ in
+ (singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_ctor_induct2_tac cTs cts ctor_induct_thm weak_ctor_induct_thms))
+ |> Thm.close_derivation,
+ rev (Term.add_tfrees goal []))
+ end;
+
+ val timer = time (timer "induction");
+
+ (*register new datatypes as BNFs*)
+ val lthy = if m = 0 then lthy else
+ let
+ val fTs = map2 (curry op -->) passiveAs passiveBs;
+ val f1Ts = map2 (curry op -->) passiveAs passiveYs;
+ val f2Ts = map2 (curry op -->) passiveBs passiveYs;
+ val p1Ts = map2 (curry op -->) passiveXs passiveAs;
+ val p2Ts = map2 (curry op -->) passiveXs passiveBs;
+ val uTs = map2 (curry op -->) Ts Ts';
+ val B1Ts = map HOLogic.mk_setT passiveAs;
+ val B2Ts = map HOLogic.mk_setT passiveBs;
+ val AXTs = map HOLogic.mk_setT passiveXs;
+ val XTs = mk_Ts passiveXs;
+ val YTs = mk_Ts passiveYs;
+ val IRTs = map2 (curry mk_relT) passiveAs passiveBs;
+ val IphiTs = map2 mk_pred2T passiveAs passiveBs;
+
+ val (((((((((((((((fs, fs'), fs_copy), us),
+ B1s), B2s), AXs), (xs, xs')), f1s), f2s), p1s), p2s), (ys, ys')), IRs), Iphis),
+ names_lthy) = names_lthy
+ |> mk_Frees' "f" fTs
+ ||>> mk_Frees "f" fTs
+ ||>> mk_Frees "u" uTs
+ ||>> mk_Frees "B1" B1Ts
+ ||>> mk_Frees "B2" B2Ts
+ ||>> mk_Frees "A" AXTs
+ ||>> mk_Frees' "x" XTs
+ ||>> mk_Frees "f1" f1Ts
+ ||>> mk_Frees "f2" f2Ts
+ ||>> mk_Frees "p1" p1Ts
+ ||>> mk_Frees "p2" p2Ts
+ ||>> mk_Frees' "y" passiveAs
+ ||>> mk_Frees "R" IRTs
+ ||>> mk_Frees "P" IphiTs;
+
+ 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 fs_copy_maps = map (mk_map Ts fs_copy Ts' ctor's pmapsABT') ks;
+ val Yctors = mk_ctors passiveYs;
+ val f1s_maps = map (mk_map Ts f1s YTs Yctors (mk_passive_maps passiveAs passiveYs)) ks;
+ val f2s_maps = map (mk_map Ts' f2s YTs Yctors (mk_passive_maps passiveBs passiveYs)) ks;
+ val p1s_maps = map (mk_map XTs p1s Ts ctors (mk_passive_maps passiveXs passiveAs)) ks;
+ val p2s_maps = map (mk_map XTs p2s Ts' ctor's (mk_passive_maps passiveXs passiveBs)) ks;
+
+ val map_simp_thms =
+ let
+ fun mk_goal fs_map map ctor ctor' = fold_rev Logic.all fs
+ (mk_Trueprop_eq (HOLogic.mk_comp (fs_map, ctor),
+ HOLogic.mk_comp (ctor', Term.list_comb (map, fs @ fs_maps))));
+ val goals = map4 mk_goal fs_maps map_FTFT's ctors ctor's;
+ val maps =
+ map4 (fn goal => fn foldx => fn map_comp_id => fn map_cong =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_map_tac m n foldx map_comp_id map_cong))
+ |> Thm.close_derivation)
+ goals ctor_fold_thms map_comp_id_thms map_congs;
+ in
+ map (fn thm => thm RS @{thm pointfreeE}) maps
+ end;
+
+ val (map_unique_thms, map_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 = map4 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_maps));
+ val unique = Skip_Proof.prove lthy [] []
+ (fold_rev Logic.all (us @ fs) (Logic.list_implies (prems, goal)))
+ (K (mk_map_unique_tac m mor_def fold_unique_mor_thm map_comp_id_thms map_congs))
+ |> Thm.close_derivation;
+ in
+ `split_conj_thm unique
+ end;
+
+ val timer = time (timer "map functions for the new datatypes");
+
+ val bd = mk_cpow sum_bd;
+ val bd_Cinfinite = sum_Cinfinite RS @{thm Cinfinite_cpow};
+ fun mk_cpow_bd thm = @{thm ordLeq_transitive} OF
+ [thm, sum_Card_order RS @{thm cpow_greater_eq}];
+ val set_bd_cpowss = map (map mk_cpow_bd) set_bd_sumss;
+
+ val timer = time (timer "bounds for the new datatypes");
+
+ val ls = 1 upto m;
+ val setsss = map (mk_setss o mk_set_Ts) passiveAs;
+ val map_setss = map (fn T => map2 (fn Ds =>
+ mk_map_of_bnf Ds (passiveAs @ Ts) (mk_set_Ts T)) Dss bnfs) 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 = map5 (fn l => fn T => map3 (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_simp_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 =
+ map3 (fn sets => map4 (mk_goal sets) ctors sets) setss_by_range colss map_setss;
+ val setss = map (map2 (fn foldx => fn goal =>
+ Skip_Proof.prove lthy [] [] goal (K (mk_set_tac foldx)) |> Thm.close_derivation)
+ ctor_fold_thms) goalss;
+
+ fun mk_simp_goal pas_set act_sets sets ctor z set =
+ Logic.all z (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 =>
+ map4 (fn Fsets => mk_simp_goal (nth Fsets (i - 1)) (drop m Fsets) sets)
+ FTs_setss ctors xFs sets)
+ ls setss_by_range;
+
+ val set_simpss = map3 (fn i => map3 (fn set_nats => fn goal => fn set =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_set_simp_tac set (nth set_nats (i - 1)) (drop m set_nats)))
+ |> Thm.close_derivation)
+ set_natural'ss) ls simp_goalss setss;
+ in
+ set_simpss
+ end;
+
+ fun mk_set_thms set_simp = (@{thm xt1(3)} OF [set_simp, @{thm Un_upper1}]) ::
+ map (fn i => (@{thm xt1(3)} OF [set_simp, @{thm Un_upper2}]) RS
+ (mk_Un_upper n i RS subset_trans) RSN
+ (2, @{thm UN_upper} RS subset_trans))
+ (1 upto n);
+ val Fset_set_thmsss = transpose (map (map mk_set_thms) set_simp_thmss);
+
+ val timer = time (timer "set functions for the new datatypes");
+
+ val cxs = map (SOME o certify lthy) Izs;
+ val setss_by_bnf' =
+ map (map (Term.subst_atomic_types (passiveAs ~~ passiveBs))) setss_by_bnf;
+ val setss_by_range' = transpose setss_by_bnf';
+
+ val set_natural_thmss =
+ let
+ fun mk_set_natural f map z set set' =
+ HOLogic.mk_eq (mk_image f $ (set $ z), set' $ (map $ z));
+
+ fun mk_cphi f map z set set' = certify lthy
+ (Term.absfree (dest_Free z) (mk_set_natural f map z set set'));
+
+ val csetss = map (map (certify lthy)) setss_by_range';
+
+ val cphiss = map3 (fn f => fn sets => fn sets' =>
+ (map4 (mk_cphi f) fs_maps Izs sets sets')) fs setss_by_range setss_by_range';
+
+ val inducts = map (fn cphis =>
+ Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm) cphiss;
+
+ val goals =
+ map3 (fn f => fn sets => fn sets' =>
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 (mk_set_natural f) fs_maps Izs sets sets')))
+ fs setss_by_range setss_by_range';
+
+ fun mk_tac induct = mk_set_nat_tac m (rtac induct) set_natural'ss map_simp_thms;
+ val thms =
+ map5 (fn goal => fn csets => fn set_simps => fn induct => fn i =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal (mk_tac induct csets set_simps i))
+ |> Thm.close_derivation)
+ goals csetss set_simp_thmss inducts ls;
+ in
+ map split_conj_thm thms
+ end;
+
+ val set_bd_thmss =
+ let
+ fun mk_set_bd z set = mk_ordLeq (mk_card_of (set $ z)) bd;
+
+ fun mk_cphi z set = certify lthy (Term.absfree (dest_Free z) (mk_set_bd z set));
+
+ val cphiss = map (map2 mk_cphi Izs) setss_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
+ (map2 mk_set_bd Izs sets))) setss_by_range;
+
+ fun mk_tac induct = mk_set_bd_tac m (rtac induct) bd_Cinfinite set_bd_cpowss;
+ val thms =
+ map4 (fn goal => fn set_simps => fn induct => fn i =>
+ singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal (mk_tac induct set_simps i))
+ |> Thm.close_derivation)
+ goals set_simp_thmss inducts ls;
+ in
+ map split_conj_thm thms
+ end;
+
+ val map_cong_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_cong sets z fmap gmap =
+ HOLogic.mk_imp
+ (Library.foldr1 HOLogic.mk_conj (map5 (mk_prem z) sets fs fs_copy ys ys'),
+ HOLogic.mk_eq (fmap $ z, gmap $ z));
+
+ fun mk_cphi sets z fmap gmap =
+ certify lthy (Term.absfree (dest_Free z) (mk_map_cong sets z fmap gmap));
+
+ val cphis = map4 mk_cphi setss_by_bnf Izs fs_maps fs_copy_maps;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm;
+
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map4 mk_map_cong setss_by_bnf Izs fs_maps fs_copy_maps));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_mcong_tac (rtac induct) Fset_set_thmsss map_congs map_simp_thms))
+ |> Thm.close_derivation;
+ in
+ split_conj_thm thm
+ end;
+
+ val in_incl_min_alg_thms =
+ let
+ fun mk_prem z sets =
+ HOLogic.mk_mem (z, mk_in As sets (fastype_of z));
+
+ fun mk_incl z sets i =
+ HOLogic.mk_imp (mk_prem z sets, HOLogic.mk_mem (z, mk_min_alg As ctors i));
+
+ fun mk_cphi z sets i =
+ certify lthy (Term.absfree (dest_Free z) (mk_incl z sets i));
+
+ val cphis = map3 mk_cphi Izs setss_by_bnf ks;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct_thm;
+
+ val goal =
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
+ (map3 mk_incl Izs setss_by_bnf ks));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (mk_incl_min_alg_tac (rtac induct) Fset_set_thmsss alg_set_thms alg_min_alg_thm))
+ |> Thm.close_derivation;
+ in
+ split_conj_thm thm
+ end;
+
+ val Xsetss = map (map (Term.subst_atomic_types (passiveAs ~~ passiveXs))) setss_by_bnf;
+
+ val map_wpull_thms =
+ let
+ val cTs = map (SOME o certifyT lthy o TFree) induct2_params;
+ val cxs = map (SOME o certify lthy) (interleave Izs1 Izs2);
+
+ fun mk_prem z1 z2 sets1 sets2 map1 map2 =
+ HOLogic.mk_conj
+ (HOLogic.mk_mem (z1, mk_in B1s sets1 (fastype_of z1)),
+ HOLogic.mk_conj
+ (HOLogic.mk_mem (z2, mk_in B2s sets2 (fastype_of z2)),
+ HOLogic.mk_eq (map1 $ z1, map2 $ z2)));
+
+ val prems = map6 mk_prem Izs1 Izs2 setss_by_bnf setss_by_bnf' f1s_maps f2s_maps;
+
+ fun mk_concl z1 z2 sets map1 map2 T x x' =
+ mk_Bex (mk_in AXs sets T) (Term.absfree x'
+ (HOLogic.mk_conj (HOLogic.mk_eq (map1 $ x, z1), HOLogic.mk_eq (map2 $ x, z2))));
+
+ val concls = map8 mk_concl Izs1 Izs2 Xsetss p1s_maps p2s_maps XTs xs xs';
+
+ val goals = map2 (curry HOLogic.mk_imp) prems concls;
+
+ fun mk_cphi z1 z2 goal = certify lthy (Term.absfree z1 (Term.absfree z2 goal));
+
+ val cphis = map3 mk_cphi Izs1' Izs2' goals;
+
+ val induct = Drule.instantiate' cTs (map SOME cphis @ cxs) ctor_induct2_thm;
+
+ val goal = Logic.list_implies (map HOLogic.mk_Trueprop
+ (map8 mk_wpull AXs B1s B2s f1s f2s (replicate m NONE) p1s p2s),
+ HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals));
+
+ val thm = singleton (Proof_Context.export names_lthy lthy)
+ (Skip_Proof.prove lthy [] [] goal
+ (K (mk_lfp_map_wpull_tac m (rtac induct) map_wpulls map_simp_thms
+ (transpose set_simp_thmss) Fset_set_thmsss ctor_inject_thms)))
+ |> Thm.close_derivation;
+ in
+ split_conj_thm thm
+ end;
+
+ val timer = time (timer "helpers for BNF properties");
+
+ val map_id_tacs = map (K o mk_map_id_tac map_ids) map_unique_thms;
+ val map_comp_tacs =
+ map2 (K oo mk_map_comp_tac map_comp's map_simp_thms) map_unique_thms ks;
+ val map_cong_tacs = map (mk_map_cong_tac m) map_cong_thms;
+ val set_nat_tacss = map (map (K o mk_set_natural_tac)) (transpose set_natural_thmss);
+ val bd_co_tacs = replicate n (K (mk_bd_card_order_tac bd_card_orders));
+ val bd_cinf_tacs = replicate n (K (rtac (bd_Cinfinite RS conjunct1) 1));
+ val set_bd_tacss = map (map (fn thm => K (rtac thm 1))) (transpose set_bd_thmss);
+ val in_bd_tacs = map2 (K oo mk_in_bd_tac sum_Card_order suc_bd_Cnotzero)
+ in_incl_min_alg_thms card_of_min_alg_thms;
+ val map_wpull_tacs = map (K o mk_wpull_tac) map_wpull_thms;
+
+ val srel_O_Gr_tacs = replicate n (simple_srel_O_Gr_tac o #context);
+
+ val tacss = map10 zip_axioms map_id_tacs map_comp_tacs map_cong_tacs set_nat_tacss
+ bd_co_tacs bd_cinf_tacs set_bd_tacss in_bd_tacs map_wpull_tacs srel_O_Gr_tacs;
+
+ 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
+ map3 (fn ctor => fn i => map close_wit o minimize_wits o maps (mk_wit witss ctor i))
+ ctors (0 upto n - 1) witss
+ end;
+
+ fun wit_tac _ = mk_wit_tac n (flat set_simp_thmss) (maps wit_thms_of_bnf bnfs);
+
+ val policy = user_policy Derive_All_Facts_Note_Most;
+
+ val (Ibnfs, lthy) =
+ fold_map6 (fn tacs => fn b => fn mapx => fn sets => fn T => fn wits => fn lthy =>
+ bnf_def Dont_Inline policy I tacs wit_tac (SOME deads)
+ (((((b, fold_rev Term.absfree fs' mapx), sets), absdummy T bd), wits), NONE) lthy
+ |> register_bnf (Local_Theory.full_name lthy b))
+ tacss bs fs_maps setss_by_bnf Ts ctor_witss lthy;
+
+ val fold_maps = fold_thms lthy (map (fn bnf =>
+ mk_unabs_def m (map_def_of_bnf bnf RS @{thm meta_eq_to_obj_eq})) Ibnfs);
+
+ val fold_sets = fold_thms lthy (maps (fn bnf =>
+ map (fn thm => thm RS @{thm meta_eq_to_obj_eq}) (set_defs_of_bnf bnf)) Ibnfs);
+
+ val timer = time (timer "registered new datatypes as BNFs");
+
+ val srels = map2 (fn Ds => mk_srel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Isrels = map (mk_srel_of_bnf deads passiveAs passiveBs) Ibnfs;
+ val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs;
+ val Irels = map (mk_rel_of_bnf deads passiveAs passiveBs) Ibnfs;
+
+ val IrelRs = map (fn Isrel => Term.list_comb (Isrel, IRs)) Isrels;
+ val relRs = map (fn srel => Term.list_comb (srel, IRs @ IrelRs)) srels;
+ val Ipredphis = map (fn Isrel => Term.list_comb (Isrel, Iphis)) Irels;
+ val predphis = map (fn srel => Term.list_comb (srel, Iphis @ Ipredphis)) rels;
+
+ val in_srels = map in_srel_of_bnf bnfs;
+ val in_Isrels = map in_srel_of_bnf Ibnfs;
+ val srel_defs = map srel_def_of_bnf bnfs;
+ val Isrel_defs = map srel_def_of_bnf Ibnfs;
+ val Irel_defs = map rel_def_of_bnf Ibnfs;
+
+ val set_incl_thmss = map (map (fold_sets o hd)) Fset_set_thmsss;
+ val set_set_incl_thmsss = map (transpose o map (map fold_sets o tl)) Fset_set_thmsss;
+ val folded_map_simp_thms = map fold_maps map_simp_thms;
+ val folded_set_simp_thmss = map (map fold_sets) set_simp_thmss;
+ val folded_set_simp_thmss' = transpose folded_set_simp_thmss;
+
+ val Isrel_simp_thms =
+ let
+ fun mk_goal xF yF ctor ctor' IrelR relR = fold_rev Logic.all (xF :: yF :: IRs)
+ (mk_Trueprop_eq (HOLogic.mk_mem (HOLogic.mk_prod (ctor $ xF, ctor' $ yF), IrelR),
+ HOLogic.mk_mem (HOLogic.mk_prod (xF, yF), relR)));
+ val goals = map6 mk_goal xFs yFs ctors ctor's IrelRs relRs;
+ in
+ map12 (fn i => fn goal => fn in_srel => fn map_comp => fn map_cong =>
+ fn map_simp => fn set_simps => fn ctor_inject => fn ctor_dtor =>
+ fn set_naturals => fn set_incls => fn set_set_inclss =>
+ Skip_Proof.prove lthy [] [] goal
+ (K (mk_srel_simp_tac in_Isrels i in_srel map_comp map_cong map_simp set_simps
+ ctor_inject ctor_dtor set_naturals set_incls set_set_inclss))
+ |> Thm.close_derivation)
+ ks goals in_srels map_comp's map_congs folded_map_simp_thms folded_set_simp_thmss'
+ ctor_inject_thms ctor_dtor_thms set_natural'ss set_incl_thmss set_set_incl_thmsss
+ end;
+
+ val Irel_simp_thms =
+ let
+ fun mk_goal xF yF ctor ctor' Ipredphi predphi = fold_rev Logic.all (xF :: yF :: Iphis)
+ (mk_Trueprop_eq (Ipredphi $ (ctor $ xF) $ (ctor' $ yF), predphi $ xF $ yF));
+ val goals = map6 mk_goal xFs yFs ctors ctor's Ipredphis predphis;
+ in
+ map3 (fn goal => fn srel_def => fn Isrel_simp =>
+ Skip_Proof.prove lthy [] [] goal
+ (mk_rel_simp_tac srel_def Irel_defs Isrel_defs Isrel_simp)
+ |> Thm.close_derivation)
+ goals srel_defs Isrel_simp_thms
+ end;
+
+ val timer = time (timer "additional properties");
+
+ val ls' = if m = 1 then [0] else ls
+
+ val Ibnf_common_notes =
+ [(map_uniqueN, [fold_maps map_unique_thm])]
+ |> map (fn (thmN, thms) =>
+ ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]));
+
+ val Ibnf_notes =
+ [(map_simpsN, map single folded_map_simp_thms),
+ (set_inclN, set_incl_thmss),
+ (set_set_inclN, map flat set_set_incl_thmsss),
+ (srel_simpN, map single Isrel_simp_thms),
+ (rel_simpN, map single Irel_simp_thms)] @
+ map2 (fn i => fn thms => (mk_set_simpsN i, map single thms)) ls' folded_set_simp_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; lthy |> Local_Theory.notes (Ibnf_common_notes @ Ibnf_notes) |> snd
+ end;
+
+ val common_notes =
+ [(ctor_inductN, [ctor_induct_thm]),
+ (ctor_induct2N, [ctor_induct2_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_fold_uniqueN, ctor_fold_unique_thms),
+ (ctor_foldsN, ctor_fold_thms),
+ (ctor_injectN, ctor_inject_thms),
+ (ctor_recsN, 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)
+ in
+ ((dtors, ctors, folds, recs, ctor_induct_thm, dtor_ctor_thms, ctor_dtor_thms, ctor_inject_thms,
+ ctor_fold_thms, ctor_rec_thms),
+ lthy |> Local_Theory.notes (common_notes @ notes) |> snd)
+ end;
+
+val _ =
+ Outer_Syntax.local_theory @{command_spec "data_raw"}
+ "define BNF-based inductive datatypes (low-level)"
+ (Parse.and_list1
+ ((Parse.binding --| @{keyword ":"}) -- (Parse.typ --| @{keyword "="} -- Parse.typ)) >>
+ (snd oo fp_bnf_cmd bnf_lfp o apsnd split_list o split_list));
+
+val _ =
+ Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"
+ (parse_datatype_cmd true bnf_lfp);
+
+end;