1(* Title: HOL/Tools/BNF/bnf_gfp.ML 2 Author: Dmitriy Traytel, TU Muenchen 3 Author: Andrei Popescu, TU Muenchen 4 Author: Jasmin Blanchette, TU Muenchen 5 Copyright 2012 6 7Codatatype construction. 8*) 9 10signature BNF_GFP = 11sig 12 val construct_gfp: mixfix list -> binding list -> binding list -> binding list -> 13 binding list list -> binding list -> (string * sort) list -> typ list * typ list list -> 14 BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory -> 15 BNF_FP_Util.fp_result * local_theory 16end; 17 18structure BNF_GFP : BNF_GFP = 19struct 20 21open BNF_Def 22open BNF_Util 23open BNF_Tactics 24open BNF_Comp 25open BNF_FP_Util 26open BNF_FP_Def_Sugar 27open BNF_GFP_Util 28open BNF_GFP_Tactics 29 30datatype wit_tree = Wit_Leaf of int | Wit_Node of (int * int * int list) * wit_tree list; 31 32fun mk_tree_args (I, T) (I', Ts) = (sort_distinct int_ord (I @ I'), T :: Ts); 33 34fun finish Iss m seen i (nwit, I) = 35 let 36 val treess = map (fn j => 37 if j < m orelse member (op =) seen j then [([j], Wit_Leaf j)] 38 else 39 map_index (finish Iss m (insert (op =) j seen) j) (nth Iss (j - m)) 40 |> flat 41 |> minimize_wits) 42 I; 43 in 44 map (fn (I, t) => (I, Wit_Node ((i - m, nwit, filter (fn i => i < m) I), t))) 45 (fold_rev (map_product mk_tree_args) treess [([], [])]) 46 |> minimize_wits 47 end; 48 49fun tree_to_ctor_wit vars _ _ (Wit_Leaf j) = ([j], nth vars j) 50 | tree_to_ctor_wit vars ctors witss (Wit_Node ((i, nwit, I), subtrees)) = 51 (I, nth ctors i $ (Term.list_comb (snd (nth (nth witss i) nwit), 52 map (snd o tree_to_ctor_wit vars ctors witss) subtrees))); 53 54fun tree_to_coind_wits _ (Wit_Leaf _) = [] 55 | tree_to_coind_wits lwitss (Wit_Node ((i, nwit, I), subtrees)) = 56 ((i, I), nth (nth lwitss i) nwit) :: maps (tree_to_coind_wits lwitss) subtrees; 57 58(*all BNFs have the same lives*) 59fun construct_gfp mixfixes map_bs rel_bs pred_bs set_bss0 bs resBs (resDs, Dss) bnfs absT_infos 60 lthy = 61 let 62 val time = time lthy; 63 val timer = time (Timer.startRealTimer ()); 64 65 val live = live_of_bnf (hd bnfs); 66 val n = length bnfs; (*active*) 67 val ks = 1 upto n; 68 val m = live - n; (*passive, if 0 don't generate a new BNF*) 69 val ls = 1 upto m; 70 71 val internals = Config.get lthy bnf_internals; 72 val b_names = map Binding.name_of bs; 73 val b_name = mk_common_name b_names; 74 val b = Binding.name b_name; 75 76 fun mk_internal_of_b name = 77 Binding.prefix_name (name ^ "_") #> Binding.prefix true b_name #> Binding.concealed; 78 fun mk_internal_b name = mk_internal_of_b name b; 79 fun mk_internal_bs name = map (mk_internal_of_b name) bs; 80 val external_bs = map2 (Binding.prefix false) b_names bs 81 |> not internals ? map Binding.concealed; 82 83 val deads = fold (union (op =)) Dss resDs; 84 val names_lthy = fold Variable.declare_typ deads lthy; 85 val passives = map fst (subtract (op = o apsnd TFree) deads resBs); 86 87 (* tvars *) 88 val ((((((passiveAs, activeAs), passiveBs), activeBs), passiveCs), activeCs), idxT) = names_lthy 89 |> variant_tfrees passives 90 ||>> mk_TFrees n 91 ||>> variant_tfrees passives 92 ||>> mk_TFrees n 93 ||>> mk_TFrees m 94 ||>> mk_TFrees n 95 ||> fst o mk_TFrees 1 96 ||> the_single; 97 98 val allAs = passiveAs @ activeAs; 99 val allBs' = passiveBs @ activeBs; 100 val Ass = replicate n allAs; 101 val allBs = passiveAs @ activeBs; 102 val Bss = replicate n allBs; 103 val allCs = passiveAs @ activeCs; 104 val allCs' = passiveBs @ activeCs; 105 val Css' = replicate n allCs'; 106 107 (* types *) 108 val dead_poss = 109 map (fn x => if member (op =) deads (TFree x) then SOME (TFree x) else NONE) resBs; 110 fun mk_param NONE passive = (hd passive, tl passive) 111 | mk_param (SOME a) passive = (a, passive); 112 val mk_params = fold_map mk_param dead_poss #> fst; 113 114 fun mk_FTs Ts = map2 (fn Ds => mk_T_of_bnf Ds Ts) Dss bnfs; 115 val (params, params') = `(map Term.dest_TFree) (mk_params passiveAs); 116 val FTsAs = mk_FTs allAs; 117 val FTsBs = mk_FTs allBs; 118 val FTsCs = mk_FTs allCs; 119 val ATs = map HOLogic.mk_setT passiveAs; 120 val BTs = map HOLogic.mk_setT activeAs; 121 val B'Ts = map HOLogic.mk_setT activeBs; 122 val B''Ts = map HOLogic.mk_setT activeCs; 123 val sTs = map2 (fn T => fn U => T --> U) activeAs FTsAs; 124 val s'Ts = map2 (fn T => fn U => T --> U) activeBs FTsBs; 125 val s''Ts = map2 (fn T => fn U => T --> U) activeCs FTsCs; 126 val fTs = map2 (fn T => fn U => T --> U) activeAs activeBs; 127 val self_fTs = map (fn T => T --> T) activeAs; 128 val gTs = map2 (fn T => fn U => T --> U) activeBs activeCs; 129 val all_gTs = map2 (fn T => fn U => T --> U) allBs allCs'; 130 val RTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeBs; 131 val sRTs = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeAs activeAs; 132 val R'Ts = map2 (fn T => fn U => HOLogic.mk_prodT (T, U)) activeBs activeCs; 133 val setsRTs = map HOLogic.mk_setT sRTs; 134 val setRTs = map HOLogic.mk_setT RTs; 135 val all_sbisT = HOLogic.mk_tupleT setsRTs; 136 val setR'Ts = map HOLogic.mk_setT R'Ts; 137 val FRTs = mk_FTs (passiveAs @ RTs); 138 139 (* terms *) 140 val mapsAsAs = @{map 4} mk_map_of_bnf Dss Ass Ass bnfs; 141 val mapsAsBs = @{map 4} mk_map_of_bnf Dss Ass Bss bnfs; 142 val mapsBsCs' = @{map 4} mk_map_of_bnf Dss Bss Css' bnfs; 143 val mapsAsCs' = @{map 4} mk_map_of_bnf Dss Ass Css' bnfs; 144 val map_fsts = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Ass bnfs; 145 val map_snds = @{map 4} mk_map_of_bnf Dss (replicate n (passiveAs @ RTs)) Bss bnfs; 146 fun mk_setss Ts = @{map 3} mk_sets_of_bnf (map (replicate live) Dss) 147 (map (replicate live) (replicate n Ts)) bnfs; 148 val setssAs = mk_setss allAs; 149 val setssAs' = transpose setssAs; 150 val bis_setss = mk_setss (passiveAs @ RTs); 151 val relsAsBs = @{map 4} mk_rel_of_bnf Dss Ass Bss bnfs; 152 val bds = @{map 3} mk_bd_of_bnf Dss Ass bnfs; 153 val sum_bd = Library.foldr1 (uncurry mk_csum) bds; 154 val sum_bdT = fst (dest_relT (fastype_of sum_bd)); 155 val (sum_bdT_params, sum_bdT_params') = `(map TFree) (Term.add_tfreesT sum_bdT []); 156 157 val ((((((((((zs, zs'), Bs), ss), fs), self_fs), all_gs), xFs), yFs), yFs_copy), _) = 158 lthy 159 |> mk_Frees' "b" activeAs 160 ||>> mk_Frees "B" BTs 161 ||>> mk_Frees "s" sTs 162 ||>> mk_Frees "f" fTs 163 ||>> mk_Frees "f" self_fTs 164 ||>> mk_Frees "g" all_gTs 165 ||>> mk_Frees "x" FTsAs 166 ||>> mk_Frees "y" FTsBs 167 ||>> mk_Frees "y" FTsBs; 168 169 val passive_UNIVs = map HOLogic.mk_UNIV passiveAs; 170 val passive_eqs = map HOLogic.eq_const passiveAs; 171 val active_UNIVs = map HOLogic.mk_UNIV activeAs; 172 val passive_ids = map HOLogic.id_const passiveAs; 173 val active_ids = map HOLogic.id_const activeAs; 174 val fsts = map fst_const RTs; 175 val snds = map snd_const RTs; 176 177 (* thms *) 178 val bd_card_orders = map bd_card_order_of_bnf bnfs; 179 val bd_card_order = hd bd_card_orders 180 val bd_Card_orders = map bd_Card_order_of_bnf bnfs; 181 val bd_Card_order = hd bd_Card_orders; 182 val bd_Cinfinites = map bd_Cinfinite_of_bnf bnfs; 183 val bd_Cinfinite = hd bd_Cinfinites; 184 val in_monos = map in_mono_of_bnf bnfs; 185 val map_comp0s = map map_comp0_of_bnf bnfs; 186 val sym_map_comps = map mk_sym map_comp0s; 187 val map_comps = map map_comp_of_bnf bnfs; 188 val map_cong0s = map map_cong0_of_bnf bnfs; 189 val map_id0s = map map_id0_of_bnf bnfs; 190 val map_ids = map map_id_of_bnf bnfs; 191 val set_bdss = map set_bd_of_bnf bnfs; 192 val set_mapss = map set_map_of_bnf bnfs; 193 val rel_congs = map rel_cong0_of_bnf bnfs; 194 val rel_converseps = map rel_conversep_of_bnf bnfs; 195 val rel_Grps = map rel_Grp_of_bnf bnfs; 196 val le_rel_OOs = map le_rel_OO_of_bnf bnfs; 197 val in_rels = map in_rel_of_bnf bnfs; 198 199 val timer = time (timer "Extracted terms & thms"); 200 201 (* derived thms *) 202 203 (*map g1 ... gm g(m+1) ... g(m+n) (map id ... id f(m+1) ... f(m+n) x) = 204 map g1 ... gm (g(m+1) o f(m+1)) ... (g(m+n) o f(m+n)) x*) 205 fun mk_map_comp_id x mapAsBs mapBsCs mapAsCs map_comp0 = 206 let 207 val lhs = Term.list_comb (mapBsCs, all_gs) $ 208 (Term.list_comb (mapAsBs, passive_ids @ fs) $ x); 209 val rhs = 210 Term.list_comb (mapAsCs, take m all_gs @ map HOLogic.mk_comp (drop m all_gs ~~ fs)) $ x; 211 val goal = mk_Trueprop_eq (lhs, rhs); 212 val vars = Variable.add_free_names lthy goal []; 213 in 214 Goal.prove_sorry lthy vars [] goal 215 (fn {context = ctxt, prems = _} => mk_map_comp_id_tac ctxt map_comp0) 216 |> Thm.close_derivation \<^here> 217 end; 218 219 val map_comp_id_thms = @{map 5} mk_map_comp_id xFs mapsAsBs mapsBsCs' mapsAsCs' map_comps; 220 221 (*forall a : set(m+1) x. f(m+1) a = a; ...; forall a : set(m+n) x. f(m+n) a = a ==> 222 map id ... id f(m+1) ... f(m+n) x = x*) 223 fun mk_map_cong0L x mapAsAs sets map_cong0 map_id = 224 let 225 fun mk_prem set f z z' = 226 HOLogic.mk_Trueprop 227 (mk_Ball (set $ x) (Term.absfree z' (HOLogic.mk_eq (f $ z, z)))); 228 val prems = @{map 4} mk_prem (drop m sets) self_fs zs zs'; 229 val goal = mk_Trueprop_eq (Term.list_comb (mapAsAs, passive_ids @ self_fs) $ x, x); 230 val vars = Variable.add_free_names lthy goal []; 231 in 232 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, goal)) 233 (fn {context = ctxt, prems = _} => mk_map_cong0L_tac ctxt m map_cong0 map_id) 234 |> Thm.close_derivation \<^here> 235 end; 236 237 val map_cong0L_thms = @{map 5} mk_map_cong0L xFs mapsAsAs setssAs map_cong0s map_ids; 238 val in_mono'_thms = map (fn thm => 239 (thm OF (replicate m subset_refl)) RS @{thm set_mp}) in_monos; 240 241 val map_arg_cong_thms = 242 let 243 val prems = map2 (curry mk_Trueprop_eq) yFs yFs_copy; 244 val maps = map (fn mapx => Term.list_comb (mapx, all_gs)) mapsBsCs'; 245 val concls = 246 @{map 3} (fn x => fn y => fn mapx => mk_Trueprop_eq (mapx $ x, mapx $ y)) 247 yFs yFs_copy maps; 248 val goals = map2 (fn prem => fn concl => Logic.mk_implies (prem, concl)) prems concls; 249 in 250 map (fn goal => 251 Variable.add_free_names lthy goal [] 252 |> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 253 (hyp_subst_tac ctxt THEN' rtac ctxt refl) 1)) 254 |> Thm.close_derivation \<^here>) 255 goals 256 end; 257 258 val timer = time (timer "Derived simple theorems"); 259 260 (* coalgebra *) 261 262 val coalg_bind = mk_internal_b (coN ^ algN) ; 263 val coalg_def_bind = (Thm.def_binding coalg_bind, []); 264 265 (*forall i = 1 ... n: (\<forall>x \<in> Bi. si \<in> Fi_in UNIV .. UNIV B1 ... Bn)*) 266 val coalg_spec = 267 let 268 val ins = @{map 3} mk_in (replicate n (passive_UNIVs @ Bs)) setssAs FTsAs; 269 fun mk_coalg_conjunct B s X z z' = 270 mk_Ball B (Term.absfree z' (HOLogic.mk_mem (s $ z, X))); 271 272 val rhs = Library.foldr1 HOLogic.mk_conj (@{map 5} mk_coalg_conjunct Bs ss ins zs zs') 273 in 274 fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss) rhs 275 end; 276 277 val ((coalg_free, (_, coalg_def_free)), (lthy, lthy_old)) = 278 lthy 279 |> Local_Theory.open_target |> snd 280 |> Local_Theory.define ((coalg_bind, NoSyn), (coalg_def_bind, coalg_spec)) 281 ||> `Local_Theory.close_target; 282 283 val phi = Proof_Context.export_morphism lthy_old lthy; 284 val coalg = fst (Term.dest_Const (Morphism.term phi coalg_free)); 285 val coalg_def = mk_unabs_def (2 * n) (HOLogic.mk_obj_eq (Morphism.thm phi coalg_def_free)); 286 287 fun mk_coalg Bs ss = 288 let 289 val args = Bs @ ss; 290 val Ts = map fastype_of args; 291 val coalgT = Library.foldr (op -->) (Ts, HOLogic.boolT); 292 in 293 Term.list_comb (Const (coalg, coalgT), args) 294 end; 295 296 val((((((zs, zs'), Bs), B's), ss), s's), _) = 297 lthy 298 |> mk_Frees' "b" activeAs 299 ||>> mk_Frees "B" BTs 300 ||>> mk_Frees "B'" B'Ts 301 ||>> mk_Frees "s" sTs 302 ||>> mk_Frees "s'" s'Ts; 303 304 val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss); 305 306 val coalg_in_thms = map (fn i => 307 coalg_def RS iffD1 RS mk_conjunctN n i RS bspec) ks 308 309 val coalg_set_thmss = 310 let 311 val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss); 312 fun mk_prem x B = mk_Trueprop_mem (x, B); 313 fun mk_concl s x B set = HOLogic.mk_Trueprop (mk_leq (set $ (s $ x)) B); 314 val prems = map2 mk_prem zs Bs; 315 val conclss = @{map 3} (fn s => fn x => fn sets => map2 (mk_concl s x) Bs (drop m sets)) 316 ss zs setssAs; 317 val goalss = map2 (fn prem => fn concls => map (fn concl => 318 Logic.list_implies (coalg_prem :: [prem], concl)) concls) prems conclss; 319 in 320 map (fn goals => map (fn goal => 321 Variable.add_free_names lthy goal [] 322 |> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 323 mk_coalg_set_tac ctxt coalg_def)) 324 |> Thm.close_derivation \<^here>) 325 goals) goalss 326 end; 327 328 fun mk_tcoalg BTs = mk_coalg (map HOLogic.mk_UNIV BTs); 329 330 val tcoalg_thm = 331 let 332 val goal = HOLogic.mk_Trueprop (mk_tcoalg activeAs ss); 333 val vars = Variable.add_free_names lthy goal []; 334 in 335 Goal.prove_sorry lthy vars [] goal 336 (fn {context = ctxt, prems = _} => (rtac ctxt (coalg_def RS iffD2) 1 THEN CONJ_WRAP 337 (K (EVERY' [rtac ctxt ballI, rtac ctxt CollectI, 338 CONJ_WRAP' (K (EVERY' [rtac ctxt @{thm subset_UNIV}])) allAs] 1)) ss)) 339 |> Thm.close_derivation \<^here> 340 end; 341 342 val timer = time (timer "Coalgebra definition & thms"); 343 344 (* morphism *) 345 346 val mor_bind = mk_internal_b morN; 347 val mor_def_bind = (Thm.def_binding mor_bind, []); 348 349 (*fbetw) forall i = 1 ... n: (\<forall>x \<in> Bi. fi x \<in> B'i)*) 350 (*mor) forall i = 1 ... n: (\<forall>x \<in> Bi. 351 Fi_map id ... id f1 ... fn (si x) = si' (fi x)*) 352 val mor_spec = 353 let 354 fun mk_fbetw f B1 B2 z z' = 355 mk_Ball B1 (Term.absfree z' (HOLogic.mk_mem (f $ z, B2))); 356 fun mk_mor B mapAsBs f s s' z z' = 357 mk_Ball B (Term.absfree z' (HOLogic.mk_eq 358 (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ z]), s' $ (f $ z)))); 359 val rhs = HOLogic.mk_conj 360 (Library.foldr1 HOLogic.mk_conj (@{map 5} mk_fbetw fs Bs B's zs zs'), 361 Library.foldr1 HOLogic.mk_conj (@{map 7} mk_mor Bs mapsAsBs fs ss s's zs zs')) 362 in 363 fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ fs) rhs 364 end; 365 366 val ((mor_free, (_, mor_def_free)), (lthy, lthy_old)) = 367 lthy 368 |> Local_Theory.open_target |> snd 369 |> Local_Theory.define ((mor_bind, NoSyn), (mor_def_bind, mor_spec)) 370 ||> `Local_Theory.close_target; 371 372 val phi = Proof_Context.export_morphism lthy_old lthy; 373 val mor = fst (Term.dest_Const (Morphism.term phi mor_free)); 374 val mor_def = mk_unabs_def (5 * n) (HOLogic.mk_obj_eq (Morphism.thm phi mor_def_free)); 375 376 fun mk_mor Bs1 ss1 Bs2 ss2 fs = 377 let 378 val args = Bs1 @ ss1 @ Bs2 @ ss2 @ fs; 379 val Ts = map fastype_of (Bs1 @ ss1 @ Bs2 @ ss2 @ fs); 380 val morT = Library.foldr (op -->) (Ts, HOLogic.boolT); 381 in 382 Term.list_comb (Const (mor, morT), args) 383 end; 384 385 val ((((((((((((((zs, z's), Bs), Bs_copy), B's), B''s), ss), s's), s''s), fs), fs_copy), gs), 386 RFs), Rs), _) = 387 lthy 388 |> mk_Frees "b" activeAs 389 ||>> mk_Frees "b" activeBs 390 ||>> mk_Frees "B" BTs 391 ||>> mk_Frees "B" BTs 392 ||>> mk_Frees "B'" B'Ts 393 ||>> mk_Frees "B''" B''Ts 394 ||>> mk_Frees "s" sTs 395 ||>> mk_Frees "s'" s'Ts 396 ||>> mk_Frees "s''" s''Ts 397 ||>> mk_Frees "f" fTs 398 ||>> mk_Frees "f" fTs 399 ||>> mk_Frees "g" gTs 400 ||>> mk_Frees "x" FRTs 401 ||>> mk_Frees "R" setRTs; 402 403 val mor_prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs); 404 405 val (mor_image_thms, morE_thms) = 406 let 407 val prem = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs); 408 fun mk_image_goal f B1 B2 = 409 Logic.mk_implies (prem, HOLogic.mk_Trueprop (mk_leq (mk_image f $ B1) B2)); 410 val image_goals = @{map 3} mk_image_goal fs Bs B's; 411 fun mk_elim_goal B mapAsBs f s s' x = 412 Logic.list_implies ([prem, mk_Trueprop_mem (x, B)], 413 mk_Trueprop_eq (Term.list_comb (mapAsBs, passive_ids @ fs @ [s $ x]), s' $ (f $ x))); 414 val elim_goals = @{map 6} mk_elim_goal Bs mapsAsBs fs ss s's zs; 415 fun prove goal = 416 Variable.add_free_names lthy goal [] 417 |> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 418 mk_mor_elim_tac ctxt mor_def)) 419 |> Thm.close_derivation \<^here>; 420 in 421 (map prove image_goals, map prove elim_goals) 422 end; 423 424 val mor_image'_thms = map (fn thm => @{thm set_mp} OF [thm, imageI]) mor_image_thms; 425 426 val mor_incl_thm = 427 let 428 val prems = map2 (HOLogic.mk_Trueprop oo mk_leq) Bs Bs_copy; 429 val concl = HOLogic.mk_Trueprop (mk_mor Bs ss Bs_copy ss active_ids); 430 val vars = fold (Variable.add_free_names lthy) (concl :: prems) []; 431 in 432 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl)) 433 (fn {context = ctxt, prems = _} => mk_mor_incl_tac ctxt mor_def map_ids) 434 |> Thm.close_derivation \<^here> 435 end; 436 437 val mor_id_thm = mor_incl_thm OF (replicate n subset_refl); 438 439 val mor_comp_thm = 440 let 441 val prems = 442 [HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs), 443 HOLogic.mk_Trueprop (mk_mor B's s's B''s s''s gs)]; 444 val concl = 445 HOLogic.mk_Trueprop (mk_mor Bs ss B''s s''s (map2 (curry HOLogic.mk_comp) gs fs)); 446 val vars = fold (Variable.add_free_names lthy) (concl :: prems) []; 447 in 448 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl)) 449 (fn {context = ctxt, prems = _} => 450 mk_mor_comp_tac ctxt mor_def mor_image'_thms morE_thms map_comp_id_thms) 451 |> Thm.close_derivation \<^here> 452 end; 453 454 val mor_cong_thm = 455 let 456 val prems = map HOLogic.mk_Trueprop 457 (map2 (curry HOLogic.mk_eq) fs_copy fs @ [mk_mor Bs ss B's s's fs]) 458 val concl = HOLogic.mk_Trueprop (mk_mor Bs ss B's s's fs_copy); 459 val vars = fold (Variable.add_free_names lthy) (concl :: prems) []; 460 in 461 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl)) 462 (fn {context = ctxt, prems = _} => (hyp_subst_tac ctxt THEN' assume_tac ctxt) 1) 463 |> Thm.close_derivation \<^here> 464 end; 465 466 val mor_UNIV_thm = 467 let 468 fun mk_conjunct mapAsBs f s s' = HOLogic.mk_eq 469 (HOLogic.mk_comp (Term.list_comb (mapAsBs, passive_ids @ fs), s), 470 HOLogic.mk_comp (s', f)); 471 val lhs = mk_mor active_UNIVs ss (map HOLogic.mk_UNIV activeBs) s's fs; 472 val rhs = Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct mapsAsBs fs ss s's); 473 val vars = fold (Variable.add_free_names lthy) [lhs, rhs] []; 474 in 475 Goal.prove_sorry lthy vars [] (mk_Trueprop_eq (lhs, rhs)) 476 (fn {context = ctxt, prems = _} => mk_mor_UNIV_tac ctxt morE_thms mor_def) 477 |> Thm.close_derivation \<^here> 478 end; 479 480 val mor_str_thm = 481 let 482 val maps = map2 (fn Ds => fn bnf => Term.list_comb 483 (mk_map_of_bnf Ds allAs (passiveAs @ FTsAs) bnf, passive_ids @ ss)) Dss bnfs; 484 val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss (map HOLogic.mk_UNIV FTsAs) maps ss); 485 val vars = Variable.add_free_names lthy goal []; 486 in 487 Goal.prove_sorry lthy vars [] goal 488 (fn {context = ctxt, prems = _} => mk_mor_str_tac ctxt ks mor_UNIV_thm) 489 |> Thm.close_derivation \<^here> 490 end; 491 492 val timer = time (timer "Morphism definition & thms"); 493 494 (* bisimulation *) 495 496 val bis_bind = mk_internal_b bisN; 497 val bis_def_bind = (Thm.def_binding bis_bind, []); 498 499 fun mk_bis_le_conjunct R B1 B2 = mk_leq R (mk_Times (B1, B2)); 500 val bis_le = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_bis_le_conjunct Rs Bs B's) 501 502 val bis_spec = 503 let 504 val fst_args = passive_ids @ fsts; 505 val snd_args = passive_ids @ snds; 506 fun mk_bis R s s' b1 b2 RF map1 map2 sets = 507 list_all_free [b1, b2] (HOLogic.mk_imp 508 (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R), 509 mk_Bex (mk_in (passive_UNIVs @ Rs) sets (snd (dest_Free RF))) 510 (Term.absfree (dest_Free RF) (HOLogic.mk_conj 511 (HOLogic.mk_eq (Term.list_comb (map1, fst_args) $ RF, s $ b1), 512 HOLogic.mk_eq (Term.list_comb (map2, snd_args) $ RF, s' $ b2)))))); 513 514 val rhs = HOLogic.mk_conj 515 (bis_le, Library.foldr1 HOLogic.mk_conj 516 (@{map 9} mk_bis Rs ss s's zs z's RFs map_fsts map_snds bis_setss)) 517 in 518 fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss @ B's @ s's @ Rs) rhs 519 end; 520 521 val ((bis_free, (_, bis_def_free)), (lthy, lthy_old)) = 522 lthy 523 |> Local_Theory.open_target |> snd 524 |> Local_Theory.define ((bis_bind, NoSyn), (bis_def_bind, bis_spec)) 525 ||> `Local_Theory.close_target; 526 527 val phi = Proof_Context.export_morphism lthy_old lthy; 528 val bis = fst (Term.dest_Const (Morphism.term phi bis_free)); 529 val bis_def = mk_unabs_def (5 * n) (HOLogic.mk_obj_eq (Morphism.thm phi bis_def_free)); 530 531 fun mk_bis Bs1 ss1 Bs2 ss2 Rs = 532 let 533 val args = Bs1 @ ss1 @ Bs2 @ ss2 @ Rs; 534 val Ts = map fastype_of args; 535 val bisT = Library.foldr (op -->) (Ts, HOLogic.boolT); 536 in 537 Term.list_comb (Const (bis, bisT), args) 538 end; 539 540 val (((((((((((((((((zs, z's), Bs), B's), B''s), ss), s's), s''s), fs), (Rtuple, Rtuple')), Rs), 541 Rs_copy), R's), sRs), (idx, idx')), Idx), Ris), _) = 542 lthy 543 |> mk_Frees "b" activeAs 544 ||>> mk_Frees "b" activeBs 545 ||>> mk_Frees "B" BTs 546 ||>> mk_Frees "B'" B'Ts 547 ||>> mk_Frees "B''" B''Ts 548 ||>> mk_Frees "s" sTs 549 ||>> mk_Frees "s'" s'Ts 550 ||>> mk_Frees "s''" s''Ts 551 ||>> mk_Frees "f" fTs 552 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Rtuple") all_sbisT 553 ||>> mk_Frees "R" setRTs 554 ||>> mk_Frees "R" setRTs 555 ||>> mk_Frees "R'" setR'Ts 556 ||>> mk_Frees "R" setsRTs 557 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "i") idxT 558 ||>> yield_singleton (mk_Frees "I") (HOLogic.mk_setT idxT) 559 ||>> mk_Frees "Ri" (map (fn T => idxT --> T) setRTs); 560 561 val bis_cong_thm = 562 let 563 val prems = map HOLogic.mk_Trueprop 564 (mk_bis Bs ss B's s's Rs :: map2 (curry HOLogic.mk_eq) Rs_copy Rs) 565 val concl = HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs_copy); 566 val vars = fold (Variable.add_free_names lthy) (concl :: prems) []; 567 in 568 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl)) 569 (fn {context = ctxt, prems = _} => (hyp_subst_tac ctxt THEN' assume_tac ctxt) 1) 570 |> Thm.close_derivation \<^here> 571 end; 572 573 val bis_rel_thm = 574 let 575 fun mk_conjunct R s s' b1 b2 rel = 576 list_all_free [b1, b2] (HOLogic.mk_imp 577 (HOLogic.mk_mem (HOLogic.mk_prod (b1, b2), R), 578 Term.list_comb (rel, passive_eqs @ map mk_in_rel Rs) $ (s $ b1) $ (s' $ b2))); 579 580 val rhs = HOLogic.mk_conj 581 (bis_le, Library.foldr1 HOLogic.mk_conj 582 (@{map 6} mk_conjunct Rs ss s's zs z's relsAsBs)) 583 val goal = mk_Trueprop_eq (mk_bis Bs ss B's s's Rs, rhs); 584 val vars = Variable.add_free_names lthy goal []; 585 in 586 Goal.prove_sorry lthy vars [] goal 587 (fn {context = ctxt, prems = _} => mk_bis_rel_tac ctxt m bis_def in_rels map_comps 588 map_cong0s set_mapss) 589 |> Thm.close_derivation \<^here> 590 end; 591 592 val bis_converse_thm = 593 let 594 val goal = Logic.mk_implies (HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs), 595 HOLogic.mk_Trueprop (mk_bis B's s's Bs ss (map mk_converse Rs))); 596 val vars = Variable.add_free_names lthy goal []; 597 in 598 Goal.prove_sorry lthy vars [] goal 599 (fn {context = ctxt, prems = _} => mk_bis_converse_tac ctxt m bis_rel_thm rel_congs 600 rel_converseps) 601 |> Thm.close_derivation \<^here> 602 end; 603 604 val bis_O_thm = 605 let 606 val prems = 607 [HOLogic.mk_Trueprop (mk_bis Bs ss B's s's Rs), 608 HOLogic.mk_Trueprop (mk_bis B's s's B''s s''s R's)]; 609 val concl = 610 HOLogic.mk_Trueprop (mk_bis Bs ss B''s s''s (map2 (curry mk_rel_comp) Rs R's)); 611 val vars = fold (Variable.add_free_names lthy) (concl :: prems) []; 612 in 613 Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, concl)) 614 (fn {context = ctxt, prems = _} => mk_bis_O_tac ctxt m bis_rel_thm rel_congs le_rel_OOs) 615 |> Thm.close_derivation \<^here> 616 end; 617 618 val bis_Gr_thm = 619 let 620 val concl = HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map2 mk_Gr Bs fs)); 621 val vars = fold (Variable.add_free_names lthy) ([coalg_prem, mor_prem, concl]) []; 622 in 623 Goal.prove_sorry lthy vars [] (Logic.list_implies ([coalg_prem, mor_prem], concl)) 624 (fn {context = ctxt, prems = _} => mk_bis_Gr_tac ctxt bis_rel_thm rel_Grps mor_image_thms 625 morE_thms coalg_in_thms) 626 |> Thm.close_derivation \<^here> 627 end; 628 629 val bis_image2_thm = bis_cong_thm OF 630 ((bis_O_thm OF [bis_Gr_thm RS bis_converse_thm, bis_Gr_thm]) :: 631 replicate n @{thm image2_Gr}); 632 633 val bis_Id_on_thm = bis_cong_thm OF ((mor_id_thm RSN (2, bis_Gr_thm)) :: 634 replicate n @{thm Id_on_Gr}); 635 636 val bis_Union_thm = 637 let 638 val prem = 639 HOLogic.mk_Trueprop (mk_Ball Idx 640 (Term.absfree idx' (mk_bis Bs ss B's s's (map (fn R => R $ idx) Ris)))); 641 val concl = 642 HOLogic.mk_Trueprop (mk_bis Bs ss B's s's (map (mk_UNION Idx) Ris)); 643 val vars = fold (Variable.add_free_names lthy) [prem, concl] []; 644 in 645 Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, concl)) 646 (fn {context = ctxt, prems = _} => mk_bis_Union_tac ctxt bis_def in_mono'_thms) 647 |> Thm.close_derivation \<^here> 648 end; 649 650 (* self-bisimulation *) 651 652 fun mk_sbis Bs ss Rs = mk_bis Bs ss Bs ss Rs; 653 654 (* largest self-bisimulation *) 655 656 val lsbis_binds = mk_internal_bs lsbisN; 657 fun lsbis_bind i = nth lsbis_binds (i - 1); 658 val lsbis_def_bind = rpair [] o Thm.def_binding o lsbis_bind; 659 660 val all_sbis = HOLogic.mk_Collect (fst Rtuple', snd Rtuple', list_exists_free sRs 661 (HOLogic.mk_conj (HOLogic.mk_eq (Rtuple, HOLogic.mk_tuple sRs), mk_sbis Bs ss sRs))); 662 663 fun lsbis_spec i = 664 fold_rev (Term.absfree o Term.dest_Free) (Bs @ ss) 665 (mk_UNION all_sbis (Term.absfree Rtuple' (mk_nthN n Rtuple i))); 666 667 val ((lsbis_frees, (_, lsbis_def_frees)), (lthy, lthy_old)) = 668 lthy 669 |> Local_Theory.open_target |> snd 670 |> fold_map (fn i => Local_Theory.define 671 ((lsbis_bind i, NoSyn), (lsbis_def_bind i, lsbis_spec i))) ks 672 |>> apsnd split_list o split_list 673 ||> `Local_Theory.close_target; 674 675 val phi = Proof_Context.export_morphism lthy_old lthy; 676 677 val lsbis_defs = map (fn def => 678 mk_unabs_def (2 * n) (HOLogic.mk_obj_eq (Morphism.thm phi def))) lsbis_def_frees; 679 val lsbiss = map (fst o Term.dest_Const o Morphism.term phi) lsbis_frees; 680 681 fun mk_lsbis Bs ss i = 682 let 683 val args = Bs @ ss; 684 val Ts = map fastype_of args; 685 val RT = mk_relT (`I (HOLogic.dest_setT (fastype_of (nth Bs (i - 1))))); 686 val lsbisT = Library.foldr (op -->) (Ts, RT); 687 in 688 Term.list_comb (Const (nth lsbiss (i - 1), lsbisT), args) 689 end; 690 691 val (((((zs, zs'), Bs), ss), sRs), _) = 692 lthy 693 |> mk_Frees' "b" activeAs 694 ||>> mk_Frees "B" BTs 695 ||>> mk_Frees "s" sTs 696 ||>> mk_Frees "R" setsRTs; 697 698 val sbis_prem = HOLogic.mk_Trueprop (mk_sbis Bs ss sRs); 699 val coalg_prem = HOLogic.mk_Trueprop (mk_coalg Bs ss); 700 701 val sbis_lsbis_thm = 702 let 703 val goal = HOLogic.mk_Trueprop (mk_sbis Bs ss (map (mk_lsbis Bs ss) ks)); 704 val vars = Variable.add_free_names lthy goal []; 705 in 706 Goal.prove_sorry lthy vars [] goal 707 (fn {context = ctxt, prems = _} => 708 mk_sbis_lsbis_tac ctxt lsbis_defs bis_Union_thm bis_cong_thm) 709 |> Thm.close_derivation \<^here> 710 end; 711 712 val lsbis_incl_thms = map (fn i => sbis_lsbis_thm RS 713 (bis_def RS iffD1 RS conjunct1 RS mk_conjunctN n i)) ks; 714 val lsbisE_thms = map (fn i => (mk_specN 2 (sbis_lsbis_thm RS 715 (bis_def RS iffD1 RS conjunct2 RS mk_conjunctN n i))) RS mp) ks; 716 717 val incl_lsbis_thms = 718 let 719 fun mk_concl i R = HOLogic.mk_Trueprop (mk_leq R (mk_lsbis Bs ss i)); 720 val goals = map2 (fn i => fn R => Logic.mk_implies (sbis_prem, mk_concl i R)) ks sRs; 721 in 722 @{map 3} (fn goal => fn i => fn def => 723 Variable.add_free_names lthy goal [] 724 |> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 725 mk_incl_lsbis_tac ctxt n i def)) 726 |> Thm.close_derivation \<^here>) 727 goals ks lsbis_defs 728 end; 729 730 val equiv_lsbis_thms = 731 let 732 fun mk_concl i B = HOLogic.mk_Trueprop (mk_equiv B (mk_lsbis Bs ss i)); 733 val goals = map2 (fn i => fn B => Logic.mk_implies (coalg_prem, mk_concl i B)) ks Bs; 734 in 735 @{map 3} (fn goal => fn l_incl => fn incl_l => 736 Variable.add_free_names lthy goal [] 737 |> (fn vars => Goal.prove_sorry lthy vars [] goal 738 (fn {context = ctxt, prems = _} => mk_equiv_lsbis_tac ctxt sbis_lsbis_thm l_incl incl_l 739 bis_Id_on_thm bis_converse_thm bis_O_thm) 740 |> Thm.close_derivation \<^here>)) 741 goals lsbis_incl_thms incl_lsbis_thms 742 end; 743 744 val timer = time (timer "Bisimulations"); 745 746 (* bounds *) 747 748 val (lthy, sbd, sbdT, 749 sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) = 750 if n = 1 751 then (lthy, sum_bd, sum_bdT, bd_card_order, bd_Cinfinite, bd_Card_order, set_bdss) 752 else 753 let 754 val sbdT_bind = mk_internal_b sum_bdTN; 755 756 val ((sbdT_name, (sbdT_glob_info, sbdT_loc_info)), lthy) = 757 typedef (sbdT_bind, sum_bdT_params', NoSyn) 758 (HOLogic.mk_UNIV sum_bdT) NONE (fn ctxt => 759 EVERY' [rtac ctxt exI, rtac ctxt UNIV_I] 1) lthy; 760 761 val sbdT = Type (sbdT_name, sum_bdT_params); 762 val Abs_sbdT = Const (#Abs_name sbdT_glob_info, sum_bdT --> sbdT); 763 764 val sbd_bind = mk_internal_b sum_bdN; 765 val sbd_def_bind = (Thm.def_binding sbd_bind, []); 766 767 val sbd_spec = mk_dir_image sum_bd Abs_sbdT; 768 769 val ((sbd_free, (_, sbd_def_free)), (lthy, lthy_old)) = 770 lthy 771 |> Local_Theory.open_target |> snd 772 |> Local_Theory.define ((sbd_bind, NoSyn), (sbd_def_bind, sbd_spec)) 773 ||> `Local_Theory.close_target; 774 775 val phi = Proof_Context.export_morphism lthy_old lthy; 776 777 val sbd_def = HOLogic.mk_obj_eq (Morphism.thm phi sbd_def_free); 778 val sbd = Const (fst (Term.dest_Const (Morphism.term phi sbd_free)), mk_relT (`I sbdT)); 779 780 val Abs_sbdT_inj = mk_Abs_inj_thm (#Abs_inject sbdT_loc_info); 781 val Abs_sbdT_bij = mk_Abs_bij_thm lthy Abs_sbdT_inj (#Abs_cases sbdT_loc_info); 782 783 val sum_Cinfinite = mk_sum_Cinfinite bd_Cinfinites; 784 val sum_Card_order = sum_Cinfinite RS conjunct2; 785 val sum_card_order = mk_sum_card_order bd_card_orders; 786 787 val sbd_ordIso = @{thm ssubst_Pair_rhs} OF 788 [@{thm dir_image} OF [Abs_sbdT_inj, sum_Card_order], sbd_def]; 789 val sbd_card_order = @{thm iffD2[OF arg_cong[of _ _ card_order]]} OF 790 [sbd_def, @{thm card_order_dir_image} OF [Abs_sbdT_bij, sum_card_order]]; 791 val sbd_Cinfinite = @{thm Cinfinite_cong} OF [sbd_ordIso, sum_Cinfinite]; 792 val sbd_Card_order = sbd_Cinfinite RS conjunct2; 793 794 fun mk_set_sbd i bd_Card_order bds = 795 map (fn thm => @{thm ordLeq_ordIso_trans} OF 796 [bd_Card_order RS mk_ordLeq_csum n i thm, sbd_ordIso]) bds; 797 val set_sbdss = @{map 3} mk_set_sbd ks bd_Card_orders set_bdss; 798 in 799 (lthy, sbd, sbdT, sbd_card_order, sbd_Cinfinite, sbd_Card_order, set_sbdss) 800 end; 801 802 val sbdTs = replicate n sbdT; 803 val sum_sbdT = mk_sumTN sbdTs; 804 val sum_sbd_listT = HOLogic.listT sum_sbdT; 805 val sum_sbd_list_setT = HOLogic.mk_setT sum_sbd_listT; 806 val bdTs = passiveAs @ replicate n sbdT; 807 val to_sbd_maps = @{map 4} mk_map_of_bnf Dss Ass (replicate n bdTs) bnfs; 808 val bdFTs = mk_FTs bdTs; 809 val sbdFT = mk_sumTN bdFTs; 810 val treeT = HOLogic.mk_prodT (sum_sbd_list_setT, sum_sbd_listT --> sbdFT); 811 val treeQT = HOLogic.mk_setT treeT; 812 val treeTs = passiveAs @ replicate n treeT; 813 val treeQTs = passiveAs @ replicate n treeQT; 814 val treeFTs = mk_FTs treeTs; 815 val tree_maps = @{map 4} mk_map_of_bnf Dss (replicate n bdTs) (replicate n treeTs) bnfs; 816 val final_maps = @{map 4} mk_map_of_bnf Dss (replicate n treeTs) (replicate n treeQTs) bnfs; 817 val isNode_setss = mk_setss (passiveAs @ replicate n sbdT); 818 819 val root = HOLogic.mk_set sum_sbd_listT [HOLogic.mk_list sum_sbdT []]; 820 val Zero = HOLogic.mk_tuple (map (fn U => absdummy U root) activeAs); 821 val Lev_recT = fastype_of Zero; 822 823 val Nil = HOLogic.mk_tuple (@{map 3} (fn i => fn z => fn z'=> 824 Term.absfree z' (mk_InN activeAs z i)) ks zs zs'); 825 val rv_recT = fastype_of Nil; 826 827 val (((((((((((((((zs, zs'), zs_copy), ss), (nat, nat')), 828 (sumx, sumx')), (kks, kks')), (kl, kl')), (kl_copy, kl'_copy)), (Kl, Kl')), (lab, lab')), 829 (Kl_lab, Kl_lab')), xs), (Lev_rec, Lev_rec')), (rv_rec, rv_rec')), _) = 830 lthy 831 |> mk_Frees' "b" activeAs 832 ||>> mk_Frees "b" activeAs 833 ||>> mk_Frees "s" sTs 834 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT 835 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "sumx") sum_sbdT 836 ||>> mk_Frees' "k" sbdTs 837 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT 838 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT 839 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl") sum_sbd_list_setT 840 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "lab") (sum_sbd_listT --> sbdFT) 841 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "Kl_lab") treeT 842 ||>> mk_Frees "x" bdFTs 843 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") Lev_recT 844 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "rec") rv_recT; 845 846 val (k, k') = (hd kks, hd kks') 847 848 val timer = time (timer "Bounds"); 849 850 (* tree coalgebra *) 851 852 val isNode_binds = mk_internal_bs isNodeN; 853 fun isNode_bind i = nth isNode_binds (i - 1); 854 val isNode_def_bind = rpair [] o Thm.def_binding o isNode_bind; 855 856 val isNodeT = 857 Library.foldr (op -->) (map fastype_of [Kl, lab, kl], HOLogic.boolT); 858 859 val Succs = @{map 3} (fn i => fn k => fn k' => 860 HOLogic.mk_Collect (fst k', snd k', HOLogic.mk_mem (mk_InN sbdTs k i, mk_Succ Kl kl))) 861 ks kks kks'; 862 863 fun isNode_spec sets x i = 864 let 865 val active_sets = drop m (map (fn set => set $ x) sets); 866 val rhs = list_exists_free [x] 867 (Library.foldr1 HOLogic.mk_conj (HOLogic.mk_eq (lab $ kl, mk_InN bdFTs x i) :: 868 map2 (curry HOLogic.mk_eq) active_sets Succs)); 869 in 870 fold_rev (Term.absfree o Term.dest_Free) [Kl, lab, kl] rhs 871 end; 872 873 val ((isNode_frees, (_, isNode_def_frees)), (lthy, lthy_old)) = 874 lthy 875 |> Local_Theory.open_target |> snd 876 |> @{fold_map 3} (fn i => fn x => fn sets => Local_Theory.define 877 ((isNode_bind i, NoSyn), (isNode_def_bind i, isNode_spec sets x i))) 878 ks xs isNode_setss 879 |>> apsnd split_list o split_list 880 ||> `Local_Theory.close_target; 881 882 val phi = Proof_Context.export_morphism lthy_old lthy; 883 884 val isNode_defs = map (fn def => 885 mk_unabs_def 3 (HOLogic.mk_obj_eq (Morphism.thm phi def))) isNode_def_frees; 886 val isNodes = map (fst o Term.dest_Const o Morphism.term phi) isNode_frees; 887 888 fun mk_isNode kl i = 889 Term.list_comb (Const (nth isNodes (i - 1), isNodeT), [Kl, lab, kl]); 890 891 val isTree = 892 let 893 val empty = HOLogic.mk_mem (HOLogic.mk_list sum_sbdT [], Kl); 894 895 val tree = mk_Ball Kl (Term.absfree kl' 896 (Library.foldr1 HOLogic.mk_conj (@{map 4} (fn Succ => fn i => fn k => fn k' => 897 mk_Ball Succ (Term.absfree k' (mk_isNode 898 (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i])) i))) 899 Succs ks kks kks'))); 900 in 901 HOLogic.mk_conj (empty, tree) 902 end; 903 904 val carT_binds = mk_internal_bs carTN; 905 fun carT_bind i = nth carT_binds (i - 1); 906 val carT_def_bind = rpair [] o Thm.def_binding o carT_bind; 907 908 fun carT_spec i = 909 HOLogic.mk_Collect (fst Kl_lab', snd Kl_lab', list_exists_free [Kl, lab] 910 (HOLogic.mk_conj (HOLogic.mk_eq (Kl_lab, HOLogic.mk_prod (Kl, lab)), 911 HOLogic.mk_conj (isTree, mk_isNode (HOLogic.mk_list sum_sbdT []) i)))); 912 913 val ((carT_frees, (_, carT_def_frees)), (lthy, lthy_old)) = 914 lthy 915 |> Local_Theory.open_target |> snd 916 |> fold_map (fn i => 917 Local_Theory.define ((carT_bind i, NoSyn), (carT_def_bind i, carT_spec i))) ks 918 |>> apsnd split_list o split_list 919 ||> `Local_Theory.close_target; 920 921 val phi = Proof_Context.export_morphism lthy_old lthy; 922 923 val carT_defs = map (fn def => HOLogic.mk_obj_eq (Morphism.thm phi def)) carT_def_frees; 924 val carTs = map (fst o Term.dest_Const o Morphism.term phi) carT_frees; 925 926 fun mk_carT i = Const (nth carTs (i - 1), HOLogic.mk_setT treeT); 927 928 val strT_binds = mk_internal_bs strTN; 929 fun strT_bind i = nth strT_binds (i - 1); 930 val strT_def_bind = rpair [] o Thm.def_binding o strT_bind; 931 932 fun strT_spec mapFT FT i = 933 let 934 fun mk_f i k k' = 935 let val in_k = mk_InN sbdTs k i; 936 in Term.absfree k' (HOLogic.mk_prod (mk_Shift Kl in_k, mk_shift lab in_k)) end; 937 938 val f = Term.list_comb (mapFT, passive_ids @ @{map 3} mk_f ks kks kks'); 939 val (fTs1, fTs2) = apsnd tl (chop (i - 1) (map (fn T => T --> FT) bdFTs)); 940 val fs = map mk_undefined fTs1 @ (f :: map mk_undefined fTs2); 941 in 942 HOLogic.mk_case_prod (Term.absfree Kl' (Term.absfree lab' 943 (mk_case_sumN fs $ (lab $ HOLogic.mk_list sum_sbdT [])))) 944 end; 945 946 val ((strT_frees, (_, strT_def_frees)), (lthy, lthy_old)) = 947 lthy 948 |> Local_Theory.open_target |> snd 949 |> @{fold_map 3} (fn i => fn mapFT => fn FT => Local_Theory.define 950 ((strT_bind i, NoSyn), (strT_def_bind i, strT_spec mapFT FT i))) 951 ks tree_maps treeFTs 952 |>> apsnd split_list o split_list 953 ||> `Local_Theory.close_target; 954 955 val phi = Proof_Context.export_morphism lthy_old lthy; 956 957 val strT_defs = map (fn def => 958 trans OF [HOLogic.mk_obj_eq (Morphism.thm phi def) RS fun_cong, @{thm prod.case}]) 959 strT_def_frees; 960 val strTs = map (fst o Term.dest_Const o Morphism.term phi) strT_frees; 961 962 fun mk_strT FT i = Const (nth strTs (i - 1), treeT --> FT); 963 964 val carTAs = map mk_carT ks; 965 val strTAs = map2 mk_strT treeFTs ks; 966 967 val coalgT_thm = 968 Goal.prove_sorry lthy [] [] (HOLogic.mk_Trueprop (mk_coalg carTAs strTAs)) 969 (fn {context = ctxt, prems = _} => mk_coalgT_tac ctxt m 970 (coalg_def :: isNode_defs @ carT_defs) strT_defs set_mapss) 971 |> Thm.close_derivation \<^here>; 972 973 val timer = time (timer "Tree coalgebra"); 974 975 fun mk_to_sbd s x i i' = 976 mk_toCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd; 977 fun mk_from_sbd s x i i' = 978 mk_fromCard (nth (nth setssAs (i - 1)) (m + i' - 1) $ (s $ x)) sbd; 979 980 fun mk_to_sbd_thmss thm = map (map (fn set_sbd => 981 thm OF [set_sbd, sbd_Card_order]) o drop m) set_sbdss; 982 983 val to_sbd_inj_thmss = mk_to_sbd_thmss @{thm toCard_inj}; 984 val from_to_sbd_thmss = mk_to_sbd_thmss @{thm fromCard_toCard}; 985 986 val Lev_bind = mk_internal_b LevN; 987 val Lev_def_bind = rpair [] (Thm.def_binding Lev_bind); 988 989 val Lev_spec = 990 let 991 fun mk_Suc i s setsAs a a' = 992 let 993 val sets = drop m setsAs; 994 fun mk_set i' set b = 995 let 996 val Cons = HOLogic.mk_eq (kl_copy, 997 mk_Cons (mk_InN sbdTs (mk_to_sbd s a i i' $ b) i') kl) 998 val b_set = HOLogic.mk_mem (b, set $ (s $ a)); 999 val kl_rec = HOLogic.mk_mem (kl, mk_nthN n Lev_rec i' $ b); 1000 in 1001 HOLogic.mk_Collect (fst kl'_copy, snd kl'_copy, list_exists_free [b, kl] 1002 (HOLogic.mk_conj (Cons, HOLogic.mk_conj (b_set, kl_rec)))) 1003 end; 1004 in 1005 Term.absfree a' (Library.foldl1 mk_union (@{map 3} mk_set ks sets zs_copy)) 1006 end; 1007 1008 val Suc = Term.absdummy HOLogic.natT (Term.absfree Lev_rec' 1009 (HOLogic.mk_tuple (@{map 5} mk_Suc ks ss setssAs zs zs'))); 1010 1011 val rhs = mk_rec_nat Zero Suc; 1012 in 1013 fold_rev (Term.absfree o Term.dest_Free) ss rhs 1014 end; 1015 1016 val ((Lev_free, (_, Lev_def_free)), (lthy, lthy_old)) = 1017 lthy 1018 |> Local_Theory.open_target |> snd 1019 |> Local_Theory.define ((Lev_bind, NoSyn), (Lev_def_bind, Lev_spec)) 1020 ||> `Local_Theory.close_target; 1021 1022 val phi = Proof_Context.export_morphism lthy_old lthy; 1023 1024 val Lev_def = mk_unabs_def n (HOLogic.mk_obj_eq (Morphism.thm phi Lev_def_free)); 1025 val Lev = fst (Term.dest_Const (Morphism.term phi Lev_free)); 1026 1027 fun mk_Lev ss nat i = 1028 let 1029 val Ts = map fastype_of ss; 1030 val LevT = Library.foldr (op -->) (Ts, HOLogic.natT --> 1031 HOLogic.mk_tupleT (map (fn U => domain_type U --> sum_sbd_list_setT) Ts)); 1032 in 1033 mk_nthN n (Term.list_comb (Const (Lev, LevT), ss) $ nat) i 1034 end; 1035 1036 val Lev_0s = flat (mk_rec_simps n @{thm rec_nat_0_imp} [Lev_def]); 1037 val Lev_Sucs = flat (mk_rec_simps n @{thm rec_nat_Suc_imp} [Lev_def]); 1038 1039 val rv_bind = mk_internal_b rvN; 1040 val rv_def_bind = rpair [] (Thm.def_binding rv_bind); 1041 1042 val rv_spec = 1043 let 1044 fun mk_Cons i s b b' = 1045 let 1046 fun mk_case i' = 1047 Term.absfree k' (mk_nthN n rv_rec i' $ (mk_from_sbd s b i i' $ k)); 1048 in 1049 Term.absfree b' (mk_case_sumN (map mk_case ks) $ sumx) 1050 end; 1051 1052 val Cons = Term.absfree sumx' (Term.absdummy sum_sbd_listT (Term.absfree rv_rec' 1053 (HOLogic.mk_tuple (@{map 4} mk_Cons ks ss zs zs')))); 1054 1055 val rhs = mk_rec_list Nil Cons; 1056 in 1057 fold_rev (Term.absfree o Term.dest_Free) ss rhs 1058 end; 1059 1060 val ((rv_free, (_, rv_def_free)), (lthy, lthy_old)) = 1061 lthy 1062 |> Local_Theory.open_target |> snd 1063 |> Local_Theory.define ((rv_bind, NoSyn), (rv_def_bind, rv_spec)) 1064 ||> `Local_Theory.close_target; 1065 1066 val phi = Proof_Context.export_morphism lthy_old lthy; 1067 1068 val rv_def = mk_unabs_def n (HOLogic.mk_obj_eq (Morphism.thm phi rv_def_free)); 1069 val rv = fst (Term.dest_Const (Morphism.term phi rv_free)); 1070 1071 fun mk_rv ss kl i = 1072 let 1073 val Ts = map fastype_of ss; 1074 val As = map domain_type Ts; 1075 val rvT = Library.foldr (op -->) (Ts, fastype_of kl --> 1076 HOLogic.mk_tupleT (map (fn U => U --> mk_sumTN As) As)); 1077 in 1078 mk_nthN n (Term.list_comb (Const (rv, rvT), ss) $ kl) i 1079 end; 1080 1081 val rv_Nils = flat (mk_rec_simps n @{thm rec_list_Nil_imp} [rv_def]); 1082 val rv_Conss = flat (mk_rec_simps n @{thm rec_list_Cons_imp} [rv_def]); 1083 1084 val beh_binds = mk_internal_bs behN; 1085 fun beh_bind i = nth beh_binds (i - 1); 1086 val beh_def_bind = rpair [] o Thm.def_binding o beh_bind; 1087 1088 fun beh_spec i z = 1089 let 1090 fun mk_case i to_sbd_map s k k' = 1091 Term.absfree k' (mk_InN bdFTs 1092 (Term.list_comb (to_sbd_map, passive_ids @ map (mk_to_sbd s k i) ks) $ (s $ k)) i); 1093 1094 val Lab = Term.absfree kl' 1095 (mk_case_sumN (@{map 5} mk_case ks to_sbd_maps ss zs zs') $ (mk_rv ss kl i $ z)); 1096 1097 val rhs = HOLogic.mk_prod (mk_UNION (HOLogic.mk_UNIV HOLogic.natT) 1098 (Term.absfree nat' (mk_Lev ss nat i $ z)), Lab); 1099 in 1100 fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) rhs 1101 end; 1102 1103 val ((beh_frees, (_, beh_def_frees)), (lthy, lthy_old)) = 1104 lthy 1105 |> Local_Theory.open_target |> snd 1106 |> @{fold_map 2} (fn i => fn z => 1107 Local_Theory.define ((beh_bind i, NoSyn), (beh_def_bind i, beh_spec i z))) ks zs 1108 |>> apsnd split_list o split_list 1109 ||> `Local_Theory.close_target; 1110 1111 val phi = Proof_Context.export_morphism lthy_old lthy; 1112 1113 val beh_defs = map (fn def => 1114 mk_unabs_def (n + 1) (HOLogic.mk_obj_eq (Morphism.thm phi def))) beh_def_frees; 1115 val behs = map (fst o Term.dest_Const o Morphism.term phi) beh_frees; 1116 1117 fun mk_beh ss i = 1118 let 1119 val Ts = map fastype_of ss; 1120 val behT = Library.foldr (op -->) (Ts, nth activeAs (i - 1) --> treeT); 1121 in 1122 Term.list_comb (Const (nth behs (i - 1), behT), ss) 1123 end; 1124 1125 val ((((((zs, zs_copy), zs_copy2), ss), (nat, nat')), (kl, kl')), _) = 1126 lthy 1127 |> mk_Frees "b" activeAs 1128 ||>> mk_Frees "b" activeAs 1129 ||>> mk_Frees "b" activeAs 1130 ||>> mk_Frees "s" sTs 1131 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT 1132 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "kl") sum_sbd_listT; 1133 1134 val (length_Lev_thms, length_Lev'_thms) = 1135 let 1136 fun mk_conjunct i z = HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), 1137 HOLogic.mk_eq (mk_size kl, nat)); 1138 val goal = list_all_free (kl :: zs) 1139 (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs)); 1140 val vars = Variable.add_free_names lthy goal []; 1141 1142 val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat]; 1143 1144 val length_Lev = 1145 Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 1146 (fn {context = ctxt, prems = _} => mk_length_Lev_tac ctxt cts Lev_0s Lev_Sucs) 1147 |> Thm.close_derivation \<^here>; 1148 1149 val length_Lev' = mk_specN (n + 1) length_Lev; 1150 val length_Levs = map (fn i => length_Lev' RS mk_conjunctN n i RS mp) ks; 1151 1152 fun mk_goal i z = Logic.mk_implies 1153 (mk_Trueprop_mem (kl, mk_Lev ss nat i $ z), 1154 mk_Trueprop_mem (kl, mk_Lev ss (mk_size kl) i $ z)); 1155 val goals = map2 mk_goal ks zs; 1156 1157 val length_Levs' = 1158 map2 (fn goal => fn length_Lev => 1159 Variable.add_free_names lthy goal [] 1160 |> (fn vars => Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 1161 mk_length_Lev'_tac ctxt length_Lev)) 1162 |> Thm.close_derivation \<^here>) 1163 goals length_Levs; 1164 in 1165 (length_Levs, length_Levs') 1166 end; 1167 1168 val rv_last_thmss = 1169 let 1170 fun mk_conjunct i z i' z_copy = list_exists_free [z_copy] 1171 (HOLogic.mk_eq 1172 (mk_rv ss (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i'])) i $ z, 1173 mk_InN activeAs z_copy i')); 1174 val goal = list_all_free (k :: zs) 1175 (Library.foldr1 HOLogic.mk_conj (map2 (fn i => fn z => 1176 Library.foldr1 HOLogic.mk_conj 1177 (map2 (mk_conjunct i z) ks zs_copy)) ks zs)); 1178 val vars = Variable.add_free_names lthy goal []; 1179 1180 val cTs = [SOME (Thm.ctyp_of lthy sum_sbdT)]; 1181 val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree kl' goal, kl]; 1182 1183 val rv_last = 1184 Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 1185 (fn {context = ctxt, prems = _} => mk_rv_last_tac ctxt cTs cts rv_Nils rv_Conss) 1186 |> Thm.close_derivation \<^here>; 1187 1188 val rv_last' = mk_specN (n + 1) rv_last; 1189 in 1190 map (fn i => map (fn i' => rv_last' RS mk_conjunctN n i RS mk_conjunctN n i') ks) ks 1191 end; 1192 1193 val set_Lev_thmsss = 1194 let 1195 fun mk_conjunct i z = 1196 let 1197 fun mk_conjunct' i' sets s z' = 1198 let 1199 fun mk_conjunct'' i'' set z'' = HOLogic.mk_imp 1200 (HOLogic.mk_mem (z'', set $ (s $ z')), 1201 HOLogic.mk_mem (mk_append (kl, 1202 HOLogic.mk_list sum_sbdT [mk_InN sbdTs (mk_to_sbd s z' i' i'' $ z'') i'']), 1203 mk_Lev ss (HOLogic.mk_Suc nat) i $ z)); 1204 in 1205 HOLogic.mk_imp (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z' i'), 1206 (Library.foldr1 HOLogic.mk_conj 1207 (@{map 3} mk_conjunct'' ks (drop m sets) zs_copy2))) 1208 end; 1209 in 1210 HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), 1211 Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct' ks setssAs ss zs_copy)) 1212 end; 1213 1214 val goal = list_all_free (kl :: zs @ zs_copy @ zs_copy2) 1215 (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs)); 1216 val vars = Variable.add_free_names lthy goal []; 1217 1218 val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat]; 1219 1220 val set_Lev = 1221 Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 1222 (fn {context = ctxt, prems = _} => 1223 mk_set_Lev_tac ctxt cts Lev_0s Lev_Sucs rv_Nils rv_Conss from_to_sbd_thmss) 1224 |> Thm.close_derivation \<^here>; 1225 1226 val set_Lev' = mk_specN (3 * n + 1) set_Lev; 1227 in 1228 map (fn i => map (fn i' => map (fn i'' => set_Lev' RS 1229 mk_conjunctN n i RS mp RS 1230 mk_conjunctN n i' RS mp RS 1231 mk_conjunctN n i'' RS mp) ks) ks) ks 1232 end; 1233 1234 val set_image_Lev_thmsss = 1235 let 1236 fun mk_conjunct i z = 1237 let 1238 fun mk_conjunct' i' sets = 1239 let 1240 fun mk_conjunct'' i'' set s z'' = HOLogic.mk_imp 1241 (HOLogic.mk_eq (mk_rv ss kl i $ z, mk_InN activeAs z'' i''), 1242 HOLogic.mk_mem (k, mk_image (mk_to_sbd s z'' i'' i') $ (set $ (s $ z'')))); 1243 in 1244 HOLogic.mk_imp (HOLogic.mk_mem 1245 (mk_append (kl, HOLogic.mk_list sum_sbdT [mk_InN sbdTs k i']), 1246 mk_Lev ss (HOLogic.mk_Suc nat) i $ z), 1247 (Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct'' ks sets ss zs_copy))) 1248 end; 1249 in 1250 HOLogic.mk_imp (HOLogic.mk_mem (kl, mk_Lev ss nat i $ z), 1251 Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct' ks (drop m setssAs'))) 1252 end; 1253 1254 val goal = list_all_free (kl :: k :: zs @ zs_copy) 1255 (Library.foldr1 HOLogic.mk_conj (map2 mk_conjunct ks zs)); 1256 val vars = Variable.add_free_names lthy goal []; 1257 1258 val cts = map (SOME o Thm.cterm_of lthy) [Term.absfree nat' goal, nat]; 1259 1260 val set_image_Lev = 1261 Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 1262 (fn {context = ctxt, prems = _} => 1263 mk_set_image_Lev_tac ctxt cts Lev_0s Lev_Sucs rv_Nils rv_Conss 1264 from_to_sbd_thmss to_sbd_inj_thmss) 1265 |> Thm.close_derivation \<^here>; 1266 1267 val set_image_Lev' = mk_specN (2 * n + 2) set_image_Lev; 1268 in 1269 map (fn i => map (fn i' => map (fn i'' => set_image_Lev' RS 1270 mk_conjunctN n i RS mp RS 1271 mk_conjunctN n i'' RS mp RS 1272 mk_conjunctN n i' RS mp) ks) ks) ks 1273 end; 1274 1275 val mor_beh_thm = 1276 let 1277 val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss carTAs strTAs (map (mk_beh ss) ks)); 1278 val vars = Variable.add_free_names lthy goal []; 1279 in 1280 Goal.prove_sorry lthy vars [] goal 1281 (fn {context = ctxt, prems = _} => mk_mor_beh_tac ctxt m mor_def mor_cong_thm 1282 beh_defs carT_defs strT_defs isNode_defs 1283 to_sbd_inj_thmss from_to_sbd_thmss Lev_0s Lev_Sucs rv_Nils rv_Conss 1284 length_Lev_thms length_Lev'_thms rv_last_thmss set_Lev_thmsss 1285 set_image_Lev_thmsss set_mapss map_comp_id_thms map_cong0s) 1286 |> Thm.close_derivation \<^here> 1287 end; 1288 1289 val timer = time (timer "Behavioral morphism"); 1290 1291 val lsbisAs = map (mk_lsbis carTAs strTAs) ks; 1292 1293 fun mk_str_final i = 1294 mk_univ (HOLogic.mk_comp (Term.list_comb (nth final_maps (i - 1), 1295 passive_ids @ map mk_proj lsbisAs), nth strTAs (i - 1))); 1296 1297 val car_finals = map2 mk_quotient carTAs lsbisAs; 1298 val str_finals = map mk_str_final ks; 1299 1300 val coalgT_set_thmss = map (map (fn thm => coalgT_thm RS thm)) coalg_set_thmss; 1301 val equiv_LSBIS_thms = map (fn thm => coalgT_thm RS thm) equiv_lsbis_thms; 1302 1303 val congruent_str_final_thms = 1304 let 1305 fun mk_goal R final_map strT = 1306 HOLogic.mk_Trueprop (mk_congruent R (HOLogic.mk_comp 1307 (Term.list_comb (final_map, passive_ids @ map mk_proj lsbisAs), strT))); 1308 1309 val goals = @{map 3} mk_goal lsbisAs final_maps strTAs; 1310 in 1311 @{map 4} (fn goal => fn lsbisE => fn map_comp_id => fn map_cong0 => 1312 Goal.prove_sorry lthy [] [] goal 1313 (fn {context = ctxt, prems = _} => mk_congruent_str_final_tac ctxt m lsbisE map_comp_id 1314 map_cong0 equiv_LSBIS_thms) 1315 |> Thm.close_derivation \<^here>) 1316 goals lsbisE_thms map_comp_id_thms map_cong0s 1317 end; 1318 1319 val coalg_final_thm = Goal.prove_sorry lthy [] [] 1320 (HOLogic.mk_Trueprop (mk_coalg car_finals str_finals)) 1321 (fn {context = ctxt, prems = _} => mk_coalg_final_tac ctxt m coalg_def 1322 congruent_str_final_thms equiv_LSBIS_thms set_mapss coalgT_set_thmss) 1323 |> Thm.close_derivation \<^here>; 1324 1325 val mor_T_final_thm = Goal.prove_sorry lthy [] [] 1326 (HOLogic.mk_Trueprop (mk_mor carTAs strTAs car_finals str_finals (map mk_proj lsbisAs))) 1327 (fn {context = ctxt, prems = _} => mk_mor_T_final_tac ctxt mor_def congruent_str_final_thms 1328 equiv_LSBIS_thms) 1329 |> Thm.close_derivation \<^here>; 1330 1331 val mor_final_thm = mor_comp_thm OF [mor_beh_thm, mor_T_final_thm]; 1332 val in_car_final_thms = map (fn thm => thm OF [mor_final_thm, UNIV_I]) mor_image'_thms; 1333 1334 val timer = time (timer "Final coalgebra"); 1335 1336 val ((T_names, (T_glob_infos, T_loc_infos)), lthy) = 1337 lthy 1338 |> @{fold_map 4} (fn b => fn mx => fn car_final => fn in_car_final => 1339 typedef (b, params, mx) car_final NONE 1340 (fn ctxt => EVERY' [rtac ctxt exI, rtac ctxt in_car_final] 1)) 1341 bs mixfixes car_finals in_car_final_thms 1342 |>> apsnd split_list o split_list; 1343 1344 val Ts = map (fn name => Type (name, params')) T_names; 1345 fun mk_Ts passive = map (Term.typ_subst_atomic (passiveAs ~~ passive)) Ts; 1346 val Ts' = mk_Ts passiveBs; 1347 val Rep_Ts = map2 (fn info => fn T => Const (#Rep_name info, T --> treeQT)) T_glob_infos Ts; 1348 val Abs_Ts = map2 (fn info => fn T => Const (#Abs_name info, treeQT --> T)) T_glob_infos Ts; 1349 1350 val Reps = map #Rep T_loc_infos; 1351 val Rep_injects = map #Rep_inject T_loc_infos; 1352 val Abs_inverses = map #Abs_inverse T_loc_infos; 1353 1354 val timer = time (timer "THE TYPEDEFs & Rep/Abs thms"); 1355 1356 val UNIVs = map HOLogic.mk_UNIV Ts; 1357 val FTs = mk_FTs (passiveAs @ Ts); 1358 val FTs_setss = mk_setss (passiveAs @ Ts); 1359 val map_FTs = map2 (fn Ds => mk_map_of_bnf Ds treeQTs (passiveAs @ Ts)) Dss bnfs; 1360 val unfold_fTs = map2 (curry op -->) activeAs Ts; 1361 1362 val emptys = map (fn T => HOLogic.mk_set T []) passiveAs; 1363 val Zeros = map (fn empty => 1364 HOLogic.mk_tuple (map (fn U => absdummy U empty) Ts)) emptys; 1365 val hrecTs = map fastype_of Zeros; 1366 1367 val (((zs, ss), (Jzs, Jzs')), _) = 1368 lthy 1369 |> mk_Frees "b" activeAs 1370 ||>> mk_Frees "s" sTs 1371 ||>> mk_Frees' "z" Ts; 1372 1373 fun dtor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtorN ^ "_"); 1374 val dtor_def_bind = rpair [] o Binding.concealed o Thm.def_binding o dtor_bind; 1375 1376 fun dtor_spec rep str map_FT Jz Jz' = 1377 Term.absfree Jz' 1378 (Term.list_comb (map_FT, map HOLogic.id_const passiveAs @ Abs_Ts) $ (str $ (rep $ Jz))); 1379 1380 val ((dtor_frees, (_, dtor_def_frees)), (lthy, lthy_old)) = 1381 lthy 1382 |> Local_Theory.open_target |> snd 1383 |> @{fold_map 6} (fn i => fn rep => fn str => fn mapx => fn Jz => fn Jz' => 1384 Local_Theory.define ((dtor_bind i, NoSyn), 1385 (dtor_def_bind i, dtor_spec rep str mapx Jz Jz'))) 1386 ks Rep_Ts str_finals map_FTs Jzs Jzs' 1387 |>> apsnd split_list o split_list 1388 ||> `Local_Theory.close_target; 1389 1390 val phi = Proof_Context.export_morphism lthy_old lthy; 1391 fun mk_dtors passive = 1392 map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ (mk_params passive)) o 1393 Morphism.term phi) dtor_frees; 1394 val dtors = mk_dtors passiveAs; 1395 val dtor's = mk_dtors passiveBs; 1396 val dtor_defs = 1397 map (fn def => HOLogic.mk_obj_eq (Morphism.thm phi def) RS fun_cong) dtor_def_frees; 1398 1399 val coalg_final_set_thmss = map (map (fn thm => coalg_final_thm RS thm)) coalg_set_thmss; 1400 val (mor_Rep_thm, mor_Abs_thm) = 1401 let 1402 val mor_Rep = 1403 Goal.prove_sorry lthy [] [] 1404 (HOLogic.mk_Trueprop (mk_mor UNIVs dtors car_finals str_finals Rep_Ts)) 1405 (fn {context = ctxt, prems = _} => mk_mor_Rep_tac ctxt (mor_def :: dtor_defs) Reps 1406 Abs_inverses coalg_final_set_thmss map_comp_id_thms map_cong0L_thms) 1407 |> Thm.close_derivation \<^here>; 1408 1409 val mor_Abs = 1410 Goal.prove_sorry lthy [] [] 1411 (HOLogic.mk_Trueprop (mk_mor car_finals str_finals UNIVs dtors Abs_Ts)) 1412 (fn {context = ctxt, prems = _} => mk_mor_Abs_tac ctxt (mor_def :: dtor_defs) 1413 Abs_inverses) 1414 |> Thm.close_derivation \<^here>; 1415 in 1416 (mor_Rep, mor_Abs) 1417 end; 1418 1419 val timer = time (timer "dtor definitions & thms"); 1420 1421 fun unfold_bind i = nth external_bs (i - 1) |> Binding.prefix_name (dtor_unfoldN ^ "_"); 1422 val unfold_def_bind = rpair [] o Binding.concealed o Thm.def_binding o unfold_bind; 1423 1424 fun unfold_spec abs f z = fold_rev (Term.absfree o Term.dest_Free) (ss @ [z]) (abs $ (f $ z)); 1425 1426 val ((unfold_frees, (_, unfold_def_frees)), (lthy, lthy_old)) = 1427 lthy 1428 |> Local_Theory.open_target |> snd 1429 |> @{fold_map 4} (fn i => fn abs => fn f => fn z => 1430 Local_Theory.define ((unfold_bind i, NoSyn), (unfold_def_bind i, unfold_spec abs f z))) 1431 ks Abs_Ts (map (fn i => HOLogic.mk_comp 1432 (mk_proj (nth lsbisAs (i - 1)), mk_beh ss i)) ks) zs 1433 |>> apsnd split_list o split_list 1434 ||> `Local_Theory.close_target; 1435 1436 val phi = Proof_Context.export_morphism lthy_old lthy; 1437 val unfolds = map (Morphism.term phi) unfold_frees; 1438 val unfold_names = map (fst o dest_Const) unfolds; 1439 fun mk_unfolds passives actives = 1440 @{map 3} (fn name => fn T => fn active => 1441 Const (name, Library.foldr (op -->) 1442 (map2 (curry op -->) actives (mk_FTs (passives @ actives)), active --> T))) 1443 unfold_names (mk_Ts passives) actives; 1444 fun mk_unfold Ts ss i = Term.list_comb (Const (nth unfold_names (i - 1), Library.foldr (op -->) 1445 (map fastype_of ss, domain_type (fastype_of (nth ss (i - 1))) --> nth Ts (i - 1))), ss); 1446 val unfold_defs = map (fn def => 1447 mk_unabs_def (n + 1) (HOLogic.mk_obj_eq (Morphism.thm phi def))) unfold_def_frees; 1448 1449 val (((ss, TRs), unfold_fs), _) = 1450 lthy 1451 |> mk_Frees "s" sTs 1452 ||>> mk_Frees "r" (map (mk_relT o `I) Ts) 1453 ||>> mk_Frees "f" unfold_fTs; 1454 1455 val mor_unfold_thm = 1456 let 1457 val Abs_inverses' = map2 (curry op RS) in_car_final_thms Abs_inverses; 1458 val morEs' = map (fn thm => (thm OF [mor_final_thm, UNIV_I]) RS sym) morE_thms; 1459 val goal = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors (map (mk_unfold Ts ss) ks)); 1460 val vars = Variable.add_free_names lthy goal []; 1461 in 1462 Goal.prove_sorry lthy vars [] goal 1463 (fn {context = ctxt, prems = _} => mk_mor_unfold_tac ctxt m mor_UNIV_thm dtor_defs 1464 unfold_defs Abs_inverses' morEs' map_comp_id_thms map_cong0s) 1465 |> Thm.close_derivation \<^here> 1466 end; 1467 val dtor_unfold_thms = map (fn thm => (thm OF [mor_unfold_thm, UNIV_I]) RS sym) morE_thms; 1468 1469 val (raw_coind_thms, raw_coind_thm) = 1470 let 1471 val prem = HOLogic.mk_Trueprop (mk_sbis UNIVs dtors TRs); 1472 val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 1473 (map2 (fn R => fn T => mk_leq R (Id_const T)) TRs Ts)); 1474 val vars = fold (Variable.add_free_names lthy) [prem, concl] []; 1475 in 1476 `split_conj_thm (Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, concl)) 1477 (fn {context = ctxt, prems = _} => mk_raw_coind_tac ctxt bis_def bis_cong_thm bis_O_thm 1478 bis_converse_thm bis_Gr_thm tcoalg_thm coalgT_thm mor_T_final_thm sbis_lsbis_thm 1479 lsbis_incl_thms incl_lsbis_thms equiv_LSBIS_thms mor_Rep_thm Rep_injects) 1480 |> Thm.close_derivation \<^here>) 1481 end; 1482 1483 val (unfold_unique_mor_thms, unfold_unique_mor_thm) = 1484 let 1485 val prem = HOLogic.mk_Trueprop (mk_mor active_UNIVs ss UNIVs dtors unfold_fs); 1486 fun mk_fun_eq f i = HOLogic.mk_eq (f, mk_unfold Ts ss i); 1487 val unique = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 1488 (map2 mk_fun_eq unfold_fs ks)); 1489 val vars = fold (Variable.add_free_names lthy) [prem, unique] []; 1490 1491 val bis_thm = tcoalg_thm RSN (2, tcoalg_thm RS bis_image2_thm); 1492 val mor_thm = mor_comp_thm OF [mor_final_thm, mor_Abs_thm]; 1493 1494 val unique_mor = Goal.prove_sorry lthy vars [] (Logic.mk_implies (prem, unique)) 1495 (fn {context = ctxt, prems = _} => mk_unfold_unique_mor_tac ctxt raw_coind_thms 1496 bis_thm mor_thm unfold_defs) 1497 |> Thm.close_derivation \<^here>; 1498 in 1499 `split_conj_thm unique_mor 1500 end; 1501 1502 val (dtor_unfold_unique_thms, dtor_unfold_unique_thm) = `split_conj_thm (split_conj_prems n 1503 (mor_UNIV_thm RS iffD2 RS unfold_unique_mor_thm)); 1504 1505 val unfold_dtor_thms = map (fn thm => mor_id_thm RS thm RS sym) unfold_unique_mor_thms; 1506 1507 val unfold_o_dtor_thms = 1508 let 1509 val mor = mor_comp_thm OF [mor_str_thm, mor_unfold_thm]; 1510 in 1511 map2 (fn unique => fn unfold_ctor => 1512 trans OF [mor RS unique, unfold_ctor]) unfold_unique_mor_thms unfold_dtor_thms 1513 end; 1514 1515 val timer = time (timer "unfold definitions & thms"); 1516 1517 val map_dtors = map2 (fn Ds => fn bnf => 1518 Term.list_comb (mk_map_of_bnf Ds (passiveAs @ Ts) (passiveAs @ FTs) bnf, 1519 map HOLogic.id_const passiveAs @ dtors)) Dss bnfs; 1520 1521 fun ctor_bind i = nth external_bs (i - 1) |> Binding.prefix_name (ctorN ^ "_"); 1522 val ctor_def_bind = rpair [] o Binding.concealed o Thm.def_binding o ctor_bind; 1523 1524 fun ctor_spec i = mk_unfold Ts map_dtors i; 1525 1526 val ((ctor_frees, (_, ctor_def_frees)), (lthy, lthy_old)) = 1527 lthy 1528 |> Local_Theory.open_target |> snd 1529 |> fold_map (fn i => 1530 Local_Theory.define ((ctor_bind i, NoSyn), (ctor_def_bind i, ctor_spec i))) ks 1531 |>> apsnd split_list o split_list 1532 ||> `Local_Theory.close_target; 1533 1534 val phi = Proof_Context.export_morphism lthy_old lthy; 1535 fun mk_ctors params = 1536 map (Term.subst_atomic_types (map (Morphism.typ phi) params' ~~ params) o Morphism.term phi) 1537 ctor_frees; 1538 val ctors = mk_ctors params'; 1539 val ctor_defs = map (fn def => HOLogic.mk_obj_eq (Morphism.thm phi def)) ctor_def_frees; 1540 1541 val ctor_o_dtor_thms = map2 (Local_Defs.fold lthy o single) ctor_defs unfold_o_dtor_thms; 1542 1543 val dtor_o_ctor_thms = 1544 let 1545 fun mk_goal dtor ctor FT = 1546 mk_Trueprop_eq (HOLogic.mk_comp (dtor, ctor), HOLogic.id_const FT); 1547 val goals = @{map 3} mk_goal dtors ctors FTs; 1548 in 1549 @{map 5} (fn goal => fn ctor_def => fn unfold => fn map_comp_id => fn map_cong0L => 1550 Goal.prove_sorry lthy [] [] goal 1551 (fn {context = ctxt, prems = _} => mk_dtor_o_ctor_tac ctxt ctor_def unfold map_comp_id 1552 map_cong0L unfold_o_dtor_thms) 1553 |> Thm.close_derivation \<^here>) 1554 goals ctor_defs dtor_unfold_thms map_comp_id_thms map_cong0L_thms 1555 end; 1556 1557 val dtor_ctor_thms = map (fn thm => thm RS @{thm pointfree_idE}) dtor_o_ctor_thms; 1558 val ctor_dtor_thms = map (fn thm => thm RS @{thm pointfree_idE}) ctor_o_dtor_thms; 1559 1560 val bij_dtor_thms = 1561 map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) ctor_o_dtor_thms dtor_o_ctor_thms; 1562 val inj_dtor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_dtor_thms; 1563 val surj_dtor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_dtor_thms; 1564 val dtor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_dtor_thms; 1565 val dtor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_dtor_thms; 1566 val dtor_exhaust_thms = map (fn thm => thm RS exE) dtor_nchotomy_thms; 1567 1568 val bij_ctor_thms = 1569 map2 (fn thm1 => fn thm2 => @{thm o_bij} OF [thm1, thm2]) dtor_o_ctor_thms ctor_o_dtor_thms; 1570 val inj_ctor_thms = map (fn thm => thm RS @{thm bij_is_inj}) bij_ctor_thms; 1571 val surj_ctor_thms = map (fn thm => thm RS @{thm bij_is_surj}) bij_ctor_thms; 1572 val ctor_nchotomy_thms = map (fn thm => thm RS @{thm surjD}) surj_ctor_thms; 1573 val ctor_inject_thms = map (fn thm => thm RS @{thm inj_eq}) inj_ctor_thms; 1574 val ctor_exhaust_thms = map (fn thm => thm RS exE) ctor_nchotomy_thms; 1575 1576 val timer = time (timer "ctor definitions & thms"); 1577 1578 val (((((Jzs, Jzs_copy), Jzs1), Jzs2), phis), _) = 1579 lthy 1580 |> mk_Frees "z" Ts 1581 ||>> mk_Frees "z'" Ts 1582 ||>> mk_Frees "z1" Ts 1583 ||>> mk_Frees "z2" Ts 1584 ||>> mk_Frees "P" (map2 mk_pred2T Ts Ts); 1585 1586 val (coinduct_params, dtor_coinduct_thm) = 1587 let 1588 val rels = map (Term.subst_atomic_types ((activeAs ~~ Ts) @ (activeBs ~~ Ts))) relsAsBs; 1589 1590 fun mk_concl phi z1 z2 = HOLogic.mk_imp (phi $ z1 $ z2, HOLogic.mk_eq (z1, z2)); 1591 val concl = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 1592 (@{map 3} mk_concl phis Jzs1 Jzs2)); 1593 1594 fun mk_rel_prem phi dtor rel Jz Jz_copy = 1595 let 1596 val concl = Term.list_comb (rel, passive_eqs @ phis) $ 1597 (dtor $ Jz) $ (dtor $ Jz_copy); 1598 in 1599 HOLogic.mk_Trueprop 1600 (list_all_free [Jz, Jz_copy] (HOLogic.mk_imp (phi $ Jz $ Jz_copy, concl))) 1601 end; 1602 1603 val rel_prems = @{map 5} mk_rel_prem phis dtors rels Jzs Jzs_copy; 1604 val dtor_coinduct_goal = Logic.list_implies (rel_prems, concl); 1605 1606 val dtor_coinduct = 1607 Variable.add_free_names lthy dtor_coinduct_goal [] 1608 |> (fn vars => Goal.prove_sorry lthy vars [] dtor_coinduct_goal 1609 (fn {context = ctxt, prems = _} => mk_dtor_coinduct_tac ctxt m raw_coind_thm bis_rel_thm 1610 rel_congs)) 1611 |> Thm.close_derivation \<^here>; 1612 in 1613 (rev (Term.add_tfrees dtor_coinduct_goal []), dtor_coinduct) 1614 end; 1615 1616 val timer = time (timer "coinduction"); 1617 1618 fun mk_dtor_map_DEADID_thm dtor_inject map_id0 = 1619 trans OF [iffD2 OF [dtor_inject, id_apply], map_id0 RS sym]; 1620 1621 fun mk_dtor_map_unique_DEADID_thm () = 1622 let 1623 val (funs, algs) = 1624 HOLogic.conjuncts (HOLogic.dest_Trueprop (Thm.concl_of dtor_unfold_unique_thm)) 1625 |> map_split HOLogic.dest_eq 1626 ||> snd o strip_comb o hd 1627 |> @{apply 2} (map (fst o dest_Var)); 1628 fun mk_fun_insts T ix = Thm.cterm_of lthy (Var (ix, T --> T)); 1629 val theta = 1630 (funs ~~ @{map 2} mk_fun_insts Ts funs) @ (algs ~~ map (Thm.cterm_of lthy) dtors); 1631 val dtor_unfold_dtors = (dtor_unfold_unique_thm OF 1632 map (fn thm => mk_trans (thm RS @{thm arg_cong2[of _ _ _ _ "(\<circ>)", OF _ refl]}) 1633 @{thm trans[OF id_o o_id[symmetric]]}) map_id0s) 1634 |> split_conj_thm |> map mk_sym; 1635 in 1636 infer_instantiate lthy theta dtor_unfold_unique_thm 1637 |> Morphism.thm (Local_Theory.target_morphism lthy) 1638 |> unfold_thms lthy dtor_unfold_dtors 1639 |> (fn thm => thm OF replicate n sym) 1640 end; 1641(* 1642thm trans[OF x.dtor_unfold_unique x.dtor_unfold_unique[symmetric, 1643 OF trans[OF arg_cong2[of _ _ _ _ "(o)", OF pre_x.map_id0 refl] trans[OF id_o o_id[symmetric]]]], 1644 OF sym] 1645*) 1646 1647 fun mk_dtor_Jrel_DEADID_thm dtor_inject bnf = 1648 trans OF [rel_eq_of_bnf bnf RS @{thm predicate2_eqD}, dtor_inject] RS sym; 1649 1650 val JphiTs = map2 mk_pred2T passiveAs passiveBs; 1651 val Jpsi1Ts = map2 mk_pred2T passiveAs passiveCs; 1652 val Jpsi2Ts = map2 mk_pred2T passiveCs passiveBs; 1653 val prodTsTs' = map2 (curry HOLogic.mk_prodT) Ts Ts'; 1654 val fstsTsTs' = map fst_const prodTsTs'; 1655 val sndsTsTs' = map snd_const prodTsTs'; 1656 val activephiTs = map2 mk_pred2T activeAs activeBs; 1657 val activeJphiTs = map2 mk_pred2T Ts Ts'; 1658 1659 val rels = map2 (fn Ds => mk_rel_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs; 1660 1661 val ((((Jzs, Jz's), Jphis), activeJphis), _) = 1662 lthy 1663 |> mk_Frees "z" Ts 1664 ||>> mk_Frees "y" Ts' 1665 ||>> mk_Frees "R" JphiTs 1666 ||>> mk_Frees "JR" activeJphiTs; 1667 1668 fun mk_Jrel_DEADID_coinduct_thm () = 1669 mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis (map HOLogic.eq_const Ts) Jphis 1670 Jzs Jz's dtors dtor's (fn {context = ctxt, prems} => 1671 (unfold_thms_tac ctxt @{thms le_fun_def le_bool_def all_simps(1,2)[symmetric]} THEN 1672 REPEAT_DETERM (rtac ctxt allI 1) THEN rtac ctxt (dtor_coinduct_thm OF prems) 1)) lthy; 1673 1674 (*register new codatatypes as BNFs*) 1675 val (timer, Jbnfs, (dtor_Jmap_o_thms, dtor_Jmap_thms), dtor_Jmap_unique_thm, dtor_Jset_thmss', 1676 dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_notes, dtor_Jset_induct_thms, lthy) = 1677 if m = 0 then 1678 (timer, replicate n DEADID_bnf, 1679 map_split (`(mk_pointfree2 lthy)) (map2 mk_dtor_map_DEADID_thm dtor_inject_thms map_ids), 1680 mk_dtor_map_unique_DEADID_thm (), 1681 replicate n [], 1682 map2 mk_dtor_Jrel_DEADID_thm dtor_inject_thms bnfs, 1683 mk_Jrel_DEADID_coinduct_thm (), [], [], lthy) 1684 else let 1685 val fTs = map2 (curry op -->) passiveAs passiveBs; 1686 val gTs = map2 (curry op -->) passiveBs passiveCs; 1687 val uTs = map2 (curry op -->) Ts Ts'; 1688 val (((((nat, nat'), (Jzs, Jzs')), (hrecs, hrecs')), (fs, fs')), _) = 1689 lthy 1690 |> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT 1691 ||>> mk_Frees' "z" Ts 1692 ||>> mk_Frees' "rec" hrecTs 1693 ||>> mk_Frees' "f" fTs; 1694 1695 val map_FTFT's = map2 (fn Ds => 1696 mk_map_of_bnf Ds (passiveAs @ Ts) (passiveBs @ Ts')) Dss bnfs; 1697 1698 fun mk_maps ATs BTs Ts mk_T = 1699 map2 (fn Ds => mk_map_of_bnf Ds (ATs @ Ts) (BTs @ map mk_T Ts)) Dss bnfs; 1700 fun mk_Fmap mk_const fs Ts Fmap = Term.list_comb (Fmap, fs @ map mk_const Ts); 1701 fun mk_map mk_const mk_T Ts fs Ts' dtors mk_maps = 1702 mk_unfold Ts' (map2 (fn dtor => fn Fmap => 1703 HOLogic.mk_comp (mk_Fmap mk_const fs Ts Fmap, dtor)) dtors (mk_maps Ts mk_T)); 1704 val mk_map_id = mk_map HOLogic.id_const I; 1705 val mk_mapsAB = mk_maps passiveAs passiveBs; 1706 val fs_maps = map (mk_map_id Ts fs Ts' dtors mk_mapsAB) ks; 1707 1708 val set_bss = 1709 map (flat o map2 (fn B => fn b => 1710 if member (op =) resDs (TFree B) then [] else [b]) resBs) set_bss0; 1711 1712 fun col_bind j = mk_internal_b (colN ^ (if m = 1 then "" else string_of_int j)); 1713 val col_def_bind = rpair [] o Thm.def_binding o col_bind; 1714 1715 fun col_spec j Zero hrec hrec' = 1716 let 1717 fun mk_Suc dtor sets z z' = 1718 let 1719 val (set, sets) = apfst (fn xs => nth xs (j - 1)) (chop m sets); 1720 fun mk_UN set k = mk_UNION (set $ (dtor $ z)) (mk_nthN n hrec k); 1721 in 1722 Term.absfree z' 1723 (mk_union (set $ (dtor $ z), Library.foldl1 mk_union (map2 mk_UN sets ks))) 1724 end; 1725 1726 val Suc = Term.absdummy HOLogic.natT (Term.absfree hrec' 1727 (HOLogic.mk_tuple (@{map 4} mk_Suc dtors FTs_setss Jzs Jzs'))); 1728 in 1729 mk_rec_nat Zero Suc 1730 end; 1731 1732 val ((col_frees, (_, col_def_frees)), (lthy, lthy_old)) = 1733 lthy 1734 |> Local_Theory.open_target |> snd 1735 |> @{fold_map 4} (fn j => fn Zero => fn hrec => fn hrec' => Local_Theory.define 1736 ((col_bind j, NoSyn), (col_def_bind j, col_spec j Zero hrec hrec'))) 1737 ls Zeros hrecs hrecs' 1738 |>> apsnd split_list o split_list 1739 ||> `Local_Theory.close_target; 1740 1741 val phi = Proof_Context.export_morphism lthy_old lthy; 1742 1743 val col_defs = map (fn def => HOLogic.mk_obj_eq (Morphism.thm phi def)) col_def_frees; 1744 val cols = map (fst o Term.dest_Const o Morphism.term phi) col_frees; 1745 1746 fun mk_col Ts nat i j T = 1747 let 1748 val hrecT = HOLogic.mk_tupleT (map (fn U => U --> HOLogic.mk_setT T) Ts) 1749 val colT = HOLogic.natT --> hrecT; 1750 in 1751 mk_nthN n (Term.list_comb (Const (nth cols (j - 1), colT), [nat])) i 1752 end; 1753 1754 val col_0ss = mk_rec_simps n @{thm rec_nat_0_imp} col_defs; 1755 val col_Sucss = mk_rec_simps n @{thm rec_nat_Suc_imp} col_defs; 1756 val col_0ss' = transpose col_0ss; 1757 val col_Sucss' = transpose col_Sucss; 1758 1759 fun mk_set Ts i j T = 1760 Abs (Name.uu, nth Ts (i - 1), mk_UNION (HOLogic.mk_UNIV HOLogic.natT) 1761 (Term.absfree nat' (mk_col Ts nat i j T $ Bound 1))); 1762 1763 val setss = map (fn i => map2 (mk_set Ts i) ls passiveAs) ks; 1764 1765 val (Jbnf_consts, lthy) = 1766 @{fold_map 8} (fn b => fn map_b => fn rel_b => fn pred_b => fn set_bs => fn mapx => 1767 fn sets => fn T => fn lthy => 1768 define_bnf_consts Hardly_Inline (user_policy Note_Some lthy) false (SOME deads) 1769 map_b rel_b pred_b set_bs 1770 (((((((b, T), fold_rev Term.absfree fs' mapx), sets), sbd), 1771 [Const (\<^const_name>\<open>undefined\<close>, T)]), NONE), NONE) lthy) 1772 bs map_bs rel_bs pred_bs set_bss fs_maps setss Ts lthy; 1773 1774 val (_, Jconsts, Jconst_defs, mk_Jconsts) = @{split_list 4} Jbnf_consts; 1775 val (_, Jsetss, Jbds_Ds, _, _, _) = @{split_list 6} Jconsts; 1776 val (Jmap_defs, Jset_defss, Jbd_defs, _, Jrel_defs, Jpred_defs) = 1777 @{split_list 6} Jconst_defs; 1778 val (mk_Jmaps_Ds, mk_Jt_Ds, _, mk_Jrels_Ds, mk_Jpreds_Ds, _, _) = 1779 @{split_list 7} mk_Jconsts; 1780 1781 val Jrel_unabs_defs = map (fn def => mk_unabs_def m (HOLogic.mk_obj_eq def)) Jrel_defs; 1782 val Jpred_unabs_defs = map (fn def => mk_unabs_def m (HOLogic.mk_obj_eq def)) Jpred_defs; 1783 val Jset_defs = flat Jset_defss; 1784 1785 fun mk_Jmaps As Bs = map (fn mk => mk deads As Bs) mk_Jmaps_Ds; 1786 fun mk_Jsetss As = map2 (fn mk => fn Jsets => map (mk deads As) Jsets) mk_Jt_Ds Jsetss; 1787 val Jbds = map2 (fn mk => mk deads passiveAs) mk_Jt_Ds Jbds_Ds; 1788 fun mk_Jrels As Bs = map (fn mk => mk deads As Bs) mk_Jrels_Ds; 1789 fun mk_Jpreds As = map (fn mk => mk deads As) mk_Jpreds_Ds; 1790 1791 val Jmaps = mk_Jmaps passiveAs passiveBs; 1792 val (Jsetss_by_range, Jsetss_by_bnf) = `transpose (mk_Jsetss passiveAs); 1793 1794 val timer = time (timer "bnf constants for the new datatypes"); 1795 1796 val ((((((((((((((((((((ys, ys'), (nat, nat')), (Jzs, Jzs')), Jz's), Jzs_copy), Jz's_copy), 1797 dtor_set_induct_phiss), Jphis), Jpsi1s), Jpsi2s), activeJphis), fs), fs_copy), gs), us), 1798 (Jys, Jys')), (Jys_copy, Jys'_copy)), (ys_copy, ys'_copy)), Kss), names_lthy) = 1799 lthy 1800 |> mk_Frees' "y" passiveAs 1801 ||>> yield_singleton (apfst (op ~~) oo mk_Frees' "n") HOLogic.natT 1802 ||>> mk_Frees' "z" Ts 1803 ||>> mk_Frees "y" Ts' 1804 ||>> mk_Frees "z'" Ts 1805 ||>> mk_Frees "y'" Ts' 1806 ||>> mk_Freess "P" (map (fn A => map (mk_pred2T A) Ts) passiveAs) 1807 ||>> mk_Frees "R" JphiTs 1808 ||>> mk_Frees "R" Jpsi1Ts 1809 ||>> mk_Frees "Q" Jpsi2Ts 1810 ||>> mk_Frees "JR" activeJphiTs 1811 ||>> mk_Frees "f" fTs 1812 ||>> mk_Frees "f" fTs 1813 ||>> mk_Frees "g" gTs 1814 ||>> mk_Frees "u" uTs 1815 ||>> mk_Frees' "b" Ts' 1816 ||>> mk_Frees' "b" Ts' 1817 ||>> mk_Frees' "y" passiveAs 1818 ||>> mk_Freess "K" (map (fn AT => map (fn T => T --> AT) Ts) ATs); 1819 1820 val fs_Jmaps = map (fn m => Term.list_comb (m, fs)) Jmaps; 1821 val fs_copy_Jmaps = map (fn m => Term.list_comb (m, fs_copy)) Jmaps; 1822 val gs_Jmaps = map (fn m => Term.list_comb (m, gs)) (mk_Jmaps passiveBs passiveCs); 1823 val fgs_Jmaps = map (fn m => Term.list_comb (m, map2 (curry HOLogic.mk_comp) gs fs)) 1824 (mk_Jmaps passiveAs passiveCs); 1825 1826 val (dtor_Jmap_thms, Jmap_thms) = 1827 let 1828 fun mk_goal fs_Jmap map dtor dtor' = mk_Trueprop_eq (HOLogic.mk_comp (dtor', fs_Jmap), 1829 HOLogic.mk_comp (Term.list_comb (map, fs @ fs_Jmaps), dtor)); 1830 val goals = @{map 4} mk_goal fs_Jmaps map_FTFT's dtors dtor's; 1831 val maps = 1832 @{map 5} (fn goal => fn unfold => fn map_comp => fn map_cong0 => fn map_arg_cong => 1833 Variable.add_free_names lthy goal [] 1834 |> (fn vars => Goal.prove_sorry lthy vars [] goal 1835 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN 1836 mk_map_tac ctxt m n map_arg_cong unfold map_comp map_cong0)) 1837 |> Thm.close_derivation \<^here>) 1838 goals dtor_unfold_thms map_comps map_cong0s map_arg_cong_thms; 1839 in 1840 map_split (fn thm => (thm RS @{thm comp_eq_dest}, thm)) maps 1841 end; 1842 1843 val (dtor_Jmap_unique_thms, dtor_Jmap_unique_thm) = 1844 let 1845 fun mk_prem u map dtor dtor' = 1846 mk_Trueprop_eq (HOLogic.mk_comp (dtor', u), 1847 HOLogic.mk_comp (Term.list_comb (map, fs @ us), dtor)); 1848 val prems = @{map 4} mk_prem us map_FTFT's dtors dtor's; 1849 val goal = 1850 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 1851 (map2 (curry HOLogic.mk_eq) us fs_Jmaps)); 1852 val vars = fold (Variable.add_free_names lthy) (goal :: prems) []; 1853 in 1854 `split_conj_thm (Goal.prove_sorry lthy vars [] (Logic.list_implies (prems, goal)) 1855 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jmap_defs THEN 1856 mk_dtor_map_unique_tac ctxt dtor_unfold_unique_thm sym_map_comps) 1857 |> Thm.close_derivation \<^here>) 1858 end; 1859 1860 val Jmap_comp0_thms = 1861 let 1862 val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 1863 (@{map 3} (fn fmap => fn gmap => fn fgmap => 1864 HOLogic.mk_eq (HOLogic.mk_comp (gmap, fmap), fgmap)) 1865 fs_Jmaps gs_Jmaps fgs_Jmaps)) 1866 val vars = Variable.add_free_names lthy goal []; 1867 in 1868 split_conj_thm (Goal.prove_sorry lthy vars [] goal 1869 (fn {context = ctxt, prems = _} => 1870 mk_map_comp0_tac ctxt Jmap_thms map_comp0s dtor_Jmap_unique_thm) 1871 |> Thm.close_derivation \<^here>) 1872 end; 1873 1874 val timer = time (timer "map functions for the new codatatypes"); 1875 1876 val Jset_minimal_thms = 1877 let 1878 fun mk_passive_prem set dtor x K = 1879 Logic.all x (HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (K $ x))); 1880 1881 fun mk_active_prem dtor x1 K1 set x2 K2 = 1882 fold_rev Logic.all [x1, x2] 1883 (Logic.mk_implies (mk_Trueprop_mem (x2, set $ (dtor $ x1)), 1884 HOLogic.mk_Trueprop (mk_leq (K2 $ x2) (K1 $ x1)))); 1885 1886 val premss = map2 (fn j => fn Ks => 1887 @{map 4} mk_passive_prem (map (fn xs => nth xs (j - 1)) FTs_setss) dtors Jzs Ks @ 1888 flat (@{map 4} (fn sets => fn s => fn x1 => fn K1 => 1889 @{map 3} (mk_active_prem s x1 K1) (drop m sets) Jzs_copy Ks) FTs_setss dtors Jzs Ks)) 1890 ls Kss; 1891 1892 val col_minimal_thms = 1893 let 1894 fun mk_conjunct j T i K x = mk_leq (mk_col Ts nat i j T $ x) (K $ x); 1895 fun mk_concl j T Ks = list_all_free Jzs 1896 (Library.foldr1 HOLogic.mk_conj (@{map 3} (mk_conjunct j T) ks Ks Jzs)); 1897 val concls = @{map 3} mk_concl ls passiveAs Kss; 1898 1899 val goals = map2 (fn prems => fn concl => 1900 Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls 1901 1902 val ctss = 1903 map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat]) concls; 1904 in 1905 @{map 4} (fn goal => fn cts => fn col_0s => fn col_Sucs => 1906 Variable.add_free_names lthy goal [] 1907 |> (fn vars => Goal.prove_sorry lthy vars [] goal 1908 (fn {context = ctxt, prems = _} => mk_col_minimal_tac ctxt m cts col_0s 1909 col_Sucs)) 1910 |> Thm.close_derivation \<^here>) 1911 goals ctss col_0ss' col_Sucss' 1912 end; 1913 1914 fun mk_conjunct set K x = mk_leq (set $ x) (K $ x); 1915 fun mk_concl sets Ks = Library.foldr1 HOLogic.mk_conj (@{map 3} mk_conjunct sets Ks Jzs); 1916 val concls = map2 mk_concl Jsetss_by_range Kss; 1917 1918 val goals = map2 (fn prems => fn concl => 1919 Logic.list_implies (prems, HOLogic.mk_Trueprop concl)) premss concls; 1920 in 1921 map2 (fn goal => fn col_minimal => 1922 Variable.add_free_names lthy goal [] 1923 |> (fn vars => Goal.prove_sorry lthy vars [] goal 1924 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN 1925 mk_Jset_minimal_tac ctxt n col_minimal)) 1926 |> Thm.close_derivation \<^here>) 1927 goals col_minimal_thms 1928 end; 1929 1930 val (dtor_Jset_incl_thmss, dtor_set_Jset_incl_thmsss) = 1931 let 1932 fun mk_set_incl_Jset dtor x set Jset = 1933 HOLogic.mk_Trueprop (mk_leq (set $ (dtor $ x)) (Jset $ x)); 1934 1935 fun mk_set_Jset_incl_Jset dtor x y set Jset1 Jset2 = 1936 Logic.mk_implies (mk_Trueprop_mem (x, set $ (dtor $ y)), 1937 HOLogic.mk_Trueprop (mk_leq (Jset1 $ x) (Jset2 $ y))); 1938 1939 val set_incl_Jset_goalss = 1940 @{map 4} (fn dtor => fn x => fn sets => fn Jsets => 1941 map2 (mk_set_incl_Jset dtor x) (take m sets) Jsets) 1942 dtors Jzs FTs_setss Jsetss_by_bnf; 1943 1944 (*x(k) : F(i)set(m+k) (dtor(i) y(i)) ==> J(k)set(j) x(k) <= J(i)set(j) y(i)*) 1945 val set_Jset_incl_Jset_goalsss = 1946 @{map 4} (fn dtori => fn yi => fn sets => fn Jsetsi => 1947 @{map 3} (fn xk => fn set => fn Jsetsk => 1948 map2 (mk_set_Jset_incl_Jset dtori xk yi set) Jsetsk Jsetsi) 1949 Jzs_copy (drop m sets) Jsetss_by_bnf) 1950 dtors Jzs FTs_setss Jsetss_by_bnf; 1951 in 1952 (map2 (fn goals => fn rec_Sucs => 1953 map2 (fn goal => fn rec_Suc => 1954 Variable.add_free_names lthy goal [] 1955 |> (fn vars => Goal.prove_sorry lthy vars [] goal 1956 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN 1957 mk_set_incl_Jset_tac ctxt rec_Suc)) 1958 |> Thm.close_derivation \<^here>) 1959 goals rec_Sucs) 1960 set_incl_Jset_goalss col_Sucss, 1961 map2 (fn goalss => fn rec_Sucs => 1962 map2 (fn k => fn goals => 1963 map2 (fn goal => fn rec_Suc => 1964 Variable.add_free_names lthy goal [] 1965 |> (fn vars => Goal.prove_sorry lthy vars [] goal 1966 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jset_defs THEN 1967 mk_set_Jset_incl_Jset_tac ctxt n rec_Suc k)) 1968 |> Thm.close_derivation \<^here>) 1969 goals rec_Sucs) 1970 ks goalss) 1971 set_Jset_incl_Jset_goalsss col_Sucss) 1972 end; 1973 1974 val set_incl_Jset_thmss' = transpose dtor_Jset_incl_thmss; 1975 val set_Jset_incl_Jset_thmsss' = transpose (map transpose dtor_set_Jset_incl_thmsss); 1976 val set_Jset_thmss = map (map (fn thm => thm RS @{thm set_mp})) dtor_Jset_incl_thmss; 1977 val set_Jset_Jset_thmsss = map (map (map (fn thm => thm RS @{thm set_mp}))) 1978 dtor_set_Jset_incl_thmsss; 1979 val set_Jset_thmss' = transpose set_Jset_thmss; 1980 val set_Jset_Jset_thmsss' = transpose (map transpose set_Jset_Jset_thmsss); 1981 1982 val dtor_Jset_induct_thms = 1983 let 1984 val incls = 1985 maps (map (fn thm => thm RS @{thm subset_Collect_iff})) dtor_Jset_incl_thmss @ 1986 @{thms subset_Collect_iff[OF subset_refl]}; 1987 1988 val cTs = map (SOME o Thm.ctyp_of lthy) params'; 1989 fun mk_induct_tinst phis jsets y y' = 1990 @{map 4} (fn phi => fn jset => fn Jz => fn Jz' => 1991 SOME (Thm.cterm_of lthy (Term.absfree Jz' (HOLogic.mk_Collect (fst y', snd y', 1992 HOLogic.mk_conj (HOLogic.mk_mem (y, jset $ Jz), phi $ y $ Jz)))))) 1993 phis jsets Jzs Jzs'; 1994 in 1995 @{map 6} (fn set_minimal => fn set_set_inclss => fn jsets => fn y => fn y' => fn phis => 1996 ((set_minimal 1997 |> Thm.instantiate' cTs (mk_induct_tinst phis jsets y y') 1998 |> unfold_thms lthy incls) OF 1999 (replicate n ballI @ 2000 maps (map (fn thm => thm RS @{thm subset_CollectI})) set_set_inclss)) 2001 |> singleton (Proof_Context.export names_lthy lthy) 2002 |> rule_by_tactic lthy (ALLGOALS (TRY o etac lthy asm_rl))) 2003 Jset_minimal_thms set_Jset_incl_Jset_thmsss' Jsetss_by_range ys ys' dtor_set_induct_phiss 2004 end; 2005 2006 val (dtor_Jset_thmss', dtor_Jset_thmss) = 2007 let 2008 fun mk_simp_goal relate pas_set act_sets sets dtor z set = 2009 relate (set $ z, mk_union (pas_set $ (dtor $ z), 2010 Library.foldl1 mk_union 2011 (map2 (fn X => mk_UNION (X $ (dtor $ z))) act_sets sets))); 2012 fun mk_goals eq = 2013 map2 (fn i => fn sets => 2014 @{map 4} (fn Fsets => 2015 mk_simp_goal eq (nth Fsets (i - 1)) (drop m Fsets) sets) 2016 FTs_setss dtors Jzs sets) 2017 ls Jsetss_by_range; 2018 2019 val le_goals = map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj) 2020 (mk_goals (uncurry mk_leq)); 2021 val set_le_thmss = map split_conj_thm 2022 (@{map 4} (fn goal => fn Jset_minimal => fn set_Jsets => fn set_Jset_Jsetss => 2023 Variable.add_free_names lthy goal [] 2024 |> (fn vars => Goal.prove_sorry lthy vars [] goal 2025 (fn {context = ctxt, prems = _} => 2026 mk_set_le_tac ctxt n Jset_minimal set_Jsets set_Jset_Jsetss)) 2027 |> Thm.close_derivation \<^here>) 2028 le_goals Jset_minimal_thms set_Jset_thmss' set_Jset_Jset_thmsss'); 2029 2030 val ge_goalss = map (map HOLogic.mk_Trueprop) (mk_goals (uncurry mk_leq o swap)); 2031 val set_ge_thmss = 2032 @{map 3} (@{map 3} (fn goal => fn set_incl_Jset => fn set_Jset_incl_Jsets => 2033 Variable.add_free_names lthy goal [] 2034 |> (fn vars => Goal.prove_sorry lthy vars [] goal 2035 (fn {context = ctxt, prems = _} => 2036 mk_set_ge_tac ctxt n set_incl_Jset set_Jset_incl_Jsets)) 2037 |> Thm.close_derivation \<^here>)) 2038 ge_goalss set_incl_Jset_thmss' set_Jset_incl_Jset_thmsss' 2039 in 2040 map2 (map2 (fn le => fn ge => equalityI OF [le, ge])) set_le_thmss set_ge_thmss 2041 |> `transpose 2042 end; 2043 2044 val timer = time (timer "set functions for the new codatatypes"); 2045 2046 val colss = map2 (fn j => fn T => 2047 map (fn i => mk_col Ts nat i j T) ks) ls passiveAs; 2048 val colss' = map2 (fn j => fn T => 2049 map (fn i => mk_col Ts' nat i j T) ks) ls passiveBs; 2050 2051 val col_natural_thmss = 2052 let 2053 fun mk_col_natural f map z col col' = 2054 HOLogic.mk_eq (mk_image f $ (col $ z), col' $ (map $ z)); 2055 2056 fun mk_goal f cols cols' = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj 2057 (@{map 4} (mk_col_natural f) fs_Jmaps Jzs cols cols')); 2058 2059 val goals = @{map 3} mk_goal fs colss colss'; 2060 2061 val ctss = 2062 map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat]) goals; 2063 2064 val thms = 2065 @{map 4} (fn goal => fn cts => fn rec_0s => fn rec_Sucs => 2066 Variable.add_free_names lthy goal [] 2067 |> (fn vars => Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 2068 (fn {context = ctxt, prems = _} => mk_col_natural_tac ctxt cts rec_0s rec_Sucs 2069 dtor_Jmap_thms set_mapss)) 2070 |> Thm.close_derivation \<^here>) 2071 goals ctss col_0ss' col_Sucss'; 2072 in 2073 map (split_conj_thm o mk_specN n) thms 2074 end; 2075 2076 val col_bd_thmss = 2077 let 2078 fun mk_col_bd z col bd = mk_ordLeq (mk_card_of (col $ z)) bd; 2079 2080 fun mk_goal bds cols = list_all_free Jzs (Library.foldr1 HOLogic.mk_conj 2081 (@{map 3} mk_col_bd Jzs cols bds)); 2082 2083 val goals = map (mk_goal Jbds) colss; 2084 2085 val ctss = map (fn phi => map (SOME o Thm.cterm_of lthy) [Term.absfree nat' phi, nat]) 2086 (map (mk_goal (replicate n sbd)) colss); 2087 2088 val thms = 2089 @{map 5} (fn j => fn goal => fn cts => fn rec_0s => fn rec_Sucs => 2090 Variable.add_free_names lthy goal [] 2091 |> (fn vars => Goal.prove_sorry lthy vars [] (HOLogic.mk_Trueprop goal) 2092 (fn {context = ctxt, prems = _} => unfold_thms_tac ctxt Jbd_defs THEN 2093 mk_col_bd_tac ctxt m j cts rec_0s rec_Sucs sbd_Card_order sbd_Cinfinite set_sbdss)) 2094 |> Thm.close_derivation \<^here>) 2095 ls goals ctss col_0ss' col_Sucss'; 2096 in 2097 map (split_conj_thm o mk_specN n) thms 2098 end; 2099 2100 val map_cong0_thms = 2101 let 2102 val cTs = map (SOME o Thm.ctyp_of lthy o 2103 Term.typ_subst_atomic (passiveAs ~~ passiveBs) o TFree) coinduct_params; 2104 2105 fun mk_prem z set f g y y' = 2106 mk_Ball (set $ z) (Term.absfree y' (HOLogic.mk_eq (f $ y, g $ y))); 2107 2108 fun mk_prems sets z = 2109 Library.foldr1 HOLogic.mk_conj (@{map 5} (mk_prem z) sets fs fs_copy ys ys') 2110 2111 fun mk_map_cong0 sets z fmap gmap = 2112 HOLogic.mk_imp (mk_prems sets z, HOLogic.mk_eq (fmap $ z, gmap $ z)); 2113 2114 fun mk_coind_body sets (x, T) z fmap gmap y y_copy = 2115 HOLogic.mk_conj 2116 (HOLogic.mk_mem (z, HOLogic.mk_Collect (x, T, mk_prems sets z)), 2117 HOLogic.mk_conj (HOLogic.mk_eq (y, fmap $ z), 2118 HOLogic.mk_eq (y_copy, gmap $ z))) 2119 2120 fun mk_cphi sets (z' as (x, T)) z fmap gmap y' y y'_copy y_copy = 2121 HOLogic.mk_exists (x, T, mk_coind_body sets z' z fmap gmap y y_copy) 2122 |> Term.absfree y'_copy 2123 |> Term.absfree y' 2124 |> Thm.cterm_of lthy; 2125 2126 val cphis = @{map 9} mk_cphi 2127 Jsetss_by_bnf Jzs' Jzs fs_Jmaps fs_copy_Jmaps Jys' Jys Jys'_copy Jys_copy; 2128 2129 val coinduct = Thm.instantiate' cTs (map SOME cphis) dtor_coinduct_thm; 2130 2131 val goal = 2132 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 2133 (@{map 4} mk_map_cong0 Jsetss_by_bnf Jzs fs_Jmaps fs_copy_Jmaps)); 2134 val vars = Variable.add_free_names lthy goal []; 2135 2136 val thm = 2137 Goal.prove_sorry lthy vars [] goal 2138 (fn {context = ctxt, prems = _} => mk_mcong_tac ctxt m (rtac ctxt coinduct) map_comps 2139 dtor_Jmap_thms map_cong0s 2140 set_mapss set_Jset_thmss set_Jset_Jset_thmsss in_rels) 2141 |> Thm.close_derivation \<^here>; 2142 in 2143 split_conj_thm thm 2144 end; 2145 2146 val in_Jrels = map (fn def => trans OF [def, @{thm OO_Grp_alt}] RS @{thm predicate2_eqD}) 2147 Jrel_unabs_defs; 2148 2149 val Jrels = mk_Jrels passiveAs passiveBs; 2150 val Jpreds = mk_Jpreds passiveAs; 2151 val Jrelphis = map (fn rel => Term.list_comb (rel, Jphis)) Jrels; 2152 val relphis = map (fn rel => Term.list_comb (rel, Jphis @ Jrelphis)) rels; 2153 val Jrelpsi1s = map (fn rel => Term.list_comb (rel, Jpsi1s)) (mk_Jrels passiveAs passiveCs); 2154 val Jrelpsi2s = map (fn rel => Term.list_comb (rel, Jpsi2s)) (mk_Jrels passiveCs passiveBs); 2155 val Jrelpsi12s = map (fn rel => 2156 Term.list_comb (rel, map2 (curry mk_rel_compp) Jpsi1s Jpsi2s)) Jrels; 2157 2158 val dtor_Jrel_thms = 2159 let 2160 fun mk_goal Jz Jz' dtor dtor' Jrelphi relphi = 2161 mk_Trueprop_eq (Jrelphi $ Jz $ Jz', relphi $ (dtor $ Jz) $ (dtor' $ Jz')); 2162 val goals = @{map 6} mk_goal Jzs Jz's dtors dtor's Jrelphis relphis; 2163 in 2164 @{map 12} (fn i => fn goal => fn in_rel => fn map_comp0 => fn map_cong0 => 2165 fn dtor_map => fn dtor_sets => fn dtor_inject => fn dtor_ctor => 2166 fn set_map0s => fn dtor_set_incls => fn dtor_set_set_inclss => 2167 Variable.add_free_names lthy goal [] 2168 |> (fn vars => Goal.prove_sorry lthy vars [] goal 2169 (fn {context = ctxt, prems = _} => 2170 mk_dtor_rel_tac ctxt in_Jrels i in_rel map_comp0 map_cong0 dtor_map dtor_sets 2171 dtor_inject dtor_ctor set_map0s dtor_set_incls dtor_set_set_inclss)) 2172 |> Thm.close_derivation \<^here>) 2173 ks goals in_rels map_comps map_cong0s dtor_Jmap_thms dtor_Jset_thmss' 2174 dtor_inject_thms dtor_ctor_thms set_mapss dtor_Jset_incl_thmss 2175 dtor_set_Jset_incl_thmsss 2176 end; 2177 2178 val passiveABs = map2 (curry HOLogic.mk_prodT) passiveAs passiveBs; 2179 val zip_ranTs = passiveABs @ prodTsTs'; 2180 val allJphis = Jphis @ activeJphis; 2181 val zipFTs = mk_FTs zip_ranTs; 2182 val zipTs = @{map 3} (fn T => fn T' => fn FT => T --> T' --> FT) Ts Ts' zipFTs; 2183 val zip_zTs = mk_Ts passiveABs; 2184 val (((zips, (abs, abs')), (zip_zs, zip_zs')), _) = 2185 names_lthy 2186 |> mk_Frees "zip" zipTs 2187 ||>> mk_Frees' "ab" passiveABs 2188 ||>> mk_Frees' "z" zip_zTs; 2189 2190 val Iphi_sets = 2191 map2 (fn phi => fn T => HOLogic.Collect_const T $ HOLogic.mk_case_prod phi) allJphis zip_ranTs; 2192 val in_phis = map2 (mk_in Iphi_sets) (mk_setss zip_ranTs) zipFTs; 2193 val fstABs = map fst_const passiveABs; 2194 val all_fsts = fstABs @ fstsTsTs'; 2195 val map_all_fsts = map2 (fn Ds => fn bnf => 2196 Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveAs @ Ts) bnf, all_fsts)) Dss bnfs; 2197 val Jmap_fsts = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T 2198 else Term.list_comb (map, fstABs)) (mk_Jmaps passiveABs passiveAs) Ts; 2199 2200 val sndABs = map snd_const passiveABs; 2201 val all_snds = sndABs @ sndsTsTs'; 2202 val map_all_snds = map2 (fn Ds => fn bnf => 2203 Term.list_comb (mk_map_of_bnf Ds zip_ranTs (passiveBs @ Ts') bnf, all_snds)) Dss bnfs; 2204 val Jmap_snds = map2 (fn map => fn T => if m = 0 then HOLogic.id_const T 2205 else Term.list_comb (map, sndABs)) (mk_Jmaps passiveABs passiveBs) Ts; 2206 val zip_unfolds = map (mk_unfold zip_zTs (map HOLogic.mk_case_prod zips)) ks; 2207 val zip_setss = mk_Jsetss passiveABs |> transpose; 2208 2209 fun Jrel_coinduct_tac {context = ctxt, prems = CIHs} = 2210 let 2211 fun mk_helper_prem phi in_phi zip x y map map' dtor dtor' = 2212 let 2213 val zipxy = zip $ x $ y; 2214 in 2215 HOLogic.mk_Trueprop (list_all_free [x, y] 2216 (HOLogic.mk_imp (phi $ x $ y, HOLogic.mk_conj (HOLogic.mk_mem (zipxy, in_phi), 2217 HOLogic.mk_conj (HOLogic.mk_eq (map $ zipxy, dtor $ x), 2218 HOLogic.mk_eq (map' $ zipxy, dtor' $ y)))))) 2219 end; 2220 val helper_prems = @{map 9} mk_helper_prem 2221 activeJphis in_phis zips Jzs Jz's map_all_fsts map_all_snds dtors dtor's; 2222 fun mk_helper_coind_phi fst phi x alt y map zip_unfold = 2223 list_exists_free [if fst then y else x] (HOLogic.mk_conj (phi $ x $ y, 2224 HOLogic.mk_eq (alt, map $ (zip_unfold $ HOLogic.mk_prod (x, y))))) 2225 val coind1_phis = @{map 6} (mk_helper_coind_phi true) 2226 activeJphis Jzs Jzs_copy Jz's Jmap_fsts zip_unfolds; 2227 val coind2_phis = @{map 6} (mk_helper_coind_phi false) 2228 activeJphis Jzs Jz's_copy Jz's Jmap_snds zip_unfolds; 2229 fun mk_cts zs z's phis = 2230 @{map 3} (fn z => fn z' => fn phi => 2231 SOME (Thm.cterm_of lthy (fold_rev (Term.absfree o Term.dest_Free) [z', z] phi))) 2232 zs z's phis @ 2233 map (SOME o Thm.cterm_of lthy) (splice z's zs); 2234 val cts1 = mk_cts Jzs Jzs_copy coind1_phis; 2235 val cts2 = mk_cts Jz's Jz's_copy coind2_phis; 2236 2237 fun mk_helper_coind_concl z alt coind_phi = 2238 HOLogic.mk_imp (coind_phi, HOLogic.mk_eq (alt, z)); 2239 val helper_coind1_concl = 2240 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 2241 (@{map 3} mk_helper_coind_concl Jzs Jzs_copy coind1_phis)); 2242 val helper_coind2_concl = 2243 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj 2244 (@{map 3} mk_helper_coind_concl Jz's Jz's_copy coind2_phis)); 2245 2246 fun mk_helper_coind_thms fst concl cts = 2247 let 2248 val vars = fold (Variable.add_free_names lthy) (concl :: helper_prems) []; 2249 in 2250 Goal.prove_sorry lthy vars [] (Logic.list_implies (helper_prems, concl)) 2251 (fn {context = ctxt, prems = _} => 2252 mk_rel_coinduct_coind_tac ctxt fst m 2253 (infer_instantiate' ctxt cts dtor_coinduct_thm) ks map_comps map_cong0s 2254 map_arg_cong_thms set_mapss dtor_unfold_thms dtor_Jmap_thms in_rels) 2255 |> Thm.close_derivation \<^here> 2256 |> split_conj_thm 2257 end; 2258 2259 val helper_coind1_thms = mk_helper_coind_thms true helper_coind1_concl cts1; 2260 val helper_coind2_thms = mk_helper_coind_thms false helper_coind2_concl cts2; 2261 2262 fun mk_helper_ind_phi phi ab fst snd z active_phi x y zip_unfold = 2263 list_all_free [x, y] (HOLogic.mk_imp 2264 (HOLogic.mk_conj (active_phi $ x $ y, 2265 HOLogic.mk_eq (z, zip_unfold $ HOLogic.mk_prod (x, y))), 2266 phi $ (fst $ ab) $ (snd $ ab))); 2267 val helper_ind_phiss = 2268 @{map 4} (fn Jphi => fn ab => fn fst => fn snd => 2269 @{map 5} (mk_helper_ind_phi Jphi ab fst snd) 2270 zip_zs activeJphis Jzs Jz's zip_unfolds) 2271 Jphis abs fstABs sndABs; 2272 val ctss = map2 (fn ab' => fn phis => 2273 map2 (fn z' => fn phi => 2274 SOME (Thm.cterm_of lthy (Term.absfree ab' (Term.absfree z' phi)))) 2275 zip_zs' phis @ 2276 map (SOME o Thm.cterm_of lthy) zip_zs) 2277 abs' helper_ind_phiss; 2278 fun mk_helper_ind_concl ab' z ind_phi set = 2279 mk_Ball (set $ z) (Term.absfree ab' ind_phi); 2280 2281 val mk_helper_ind_concls = 2282 @{map 3} (fn ab' => fn ind_phis => fn zip_sets => 2283 @{map 3} (mk_helper_ind_concl ab') zip_zs ind_phis zip_sets) 2284 abs' helper_ind_phiss zip_setss 2285 |> map (HOLogic.mk_Trueprop o Library.foldr1 HOLogic.mk_conj); 2286 2287 val helper_ind_thmss = if m = 0 then replicate n [] else 2288 @{map 4} (fn concl => fn j => fn set_induct => fn cts => 2289 fold (Variable.add_free_names lthy) (concl :: helper_prems) [] 2290 |> (fn vars => Goal.prove_sorry lthy vars [] (Logic.list_implies (helper_prems, concl)) 2291 (fn {context = ctxt, prems = _} => 2292 mk_rel_coinduct_ind_tac ctxt m ks 2293 dtor_unfold_thms set_mapss j (infer_instantiate' ctxt cts set_induct))) 2294 |> Thm.close_derivation \<^here> 2295 |> split_conj_thm) 2296 mk_helper_ind_concls ls dtor_Jset_induct_thms ctss 2297 |> transpose; 2298 in 2299 mk_rel_coinduct_tac ctxt CIHs in_rels in_Jrels 2300 helper_ind_thmss helper_coind1_thms helper_coind2_thms 2301 end; 2302 2303 val Jrel_coinduct_thm = 2304 mk_xtor_rel_co_induct_thm Greatest_FP rels activeJphis Jrels Jphis Jzs Jz's dtors dtor's 2305 Jrel_coinduct_tac lthy; 2306 2307 val le_Jrel_OO_thm = 2308 let 2309 fun mk_le_Jrel_OO Jrelpsi1 Jrelpsi2 Jrelpsi12 = 2310 mk_leq (mk_rel_compp (Jrelpsi1, Jrelpsi2)) Jrelpsi12; 2311 val goals = @{map 3} mk_le_Jrel_OO Jrelpsi1s Jrelpsi2s Jrelpsi12s; 2312 2313 val goal = HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj goals); 2314 val vars = Variable.add_free_names lthy goal []; 2315 in 2316 Goal.prove_sorry lthy vars [] goal (fn {context = ctxt, prems = _} => 2317 mk_le_rel_OO_tac ctxt Jrel_coinduct_thm dtor_Jrel_thms le_rel_OOs) 2318 |> Thm.close_derivation \<^here> 2319 end; 2320 2321 val timer = time (timer "helpers for BNF properties"); 2322 2323 fun close_wit I wit = (I, fold_rev Term.absfree (map (nth ys') I) wit); 2324 2325 val all_unitTs = replicate live HOLogic.unitT; 2326 val unitTs = replicate n HOLogic.unitT; 2327 val unit_funs = replicate n (Term.absdummy HOLogic.unitT HOLogic.unit); 2328 fun mk_map_args I = 2329 map (fn i => 2330 if member (op =) I i then Term.absdummy HOLogic.unitT (nth ys i) 2331 else mk_undefined (HOLogic.unitT --> nth passiveAs i)) 2332 (0 upto (m - 1)); 2333 2334 fun mk_nat_wit Ds bnf (I, wit) () = 2335 let 2336 val passiveI = filter (fn i => i < m) I; 2337 val map_args = mk_map_args passiveI; 2338 in 2339 Term.absdummy HOLogic.unitT (Term.list_comb 2340 (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ wit) 2341 end; 2342 2343 fun mk_dummy_wit Ds bnf I = 2344 let 2345 val map_args = mk_map_args I; 2346 in 2347 Term.absdummy HOLogic.unitT (Term.list_comb 2348 (mk_map_of_bnf Ds all_unitTs (passiveAs @ unitTs) bnf, map_args @ unit_funs) $ 2349 mk_undefined (mk_T_of_bnf Ds all_unitTs bnf)) 2350 end; 2351 2352 val nat_witss = 2353 map2 (fn Ds => fn bnf => mk_wits_of_bnf (replicate (nwits_of_bnf bnf) Ds) 2354 (replicate (nwits_of_bnf bnf) (replicate live HOLogic.unitT)) bnf 2355 |> map (fn (I, wit) => 2356 (I, Lazy.lazy (mk_nat_wit Ds bnf (I, Term.list_comb (wit, map (K HOLogic.unit) I)))))) 2357 Dss bnfs; 2358 2359 val nat_wit_thmss = map2 (curry op ~~) nat_witss (map wit_thmss_of_bnf bnfs) 2360 2361 val Iss = map (map fst) nat_witss; 2362 2363 fun filter_wits (I, wit) = 2364 let val J = filter (fn i => i < m) I; 2365 in (J, (length J < length I, wit)) end; 2366 2367 val wit_treess = map_index (fn (i, Is) => 2368 map_index (finish Iss m [i+m] (i+m)) Is) Iss 2369 |> map (minimize_wits o map filter_wits o minimize_wits o flat); 2370 2371 val coind_wit_argsss = 2372 map (map (tree_to_coind_wits nat_wit_thmss o snd o snd) o filter (fst o snd)) wit_treess; 2373 2374 val nonredundant_coind_wit_argsss = 2375 fold (fn i => fn argsss => 2376 nth_map (i - 1) (filter_out (fn xs => 2377 exists (fn ys => 2378 let 2379 val xs' = (map (fst o fst) xs, snd (fst (hd xs))); 2380 val ys' = (map (fst o fst) ys, snd (fst (hd ys))); 2381 in 2382 eq_pair (subset (op =)) (eq_set (op =)) (xs', ys') andalso not (fst xs' = fst ys') 2383 end) 2384 (flat argsss))) 2385 argsss) 2386 ks coind_wit_argsss; 2387 2388 fun prepare_args args = 2389 let 2390 val I = snd (fst (hd args)); 2391 val (dummys, args') = 2392 map_split (fn i => 2393 (case find_first (fn arg => fst (fst arg) = i - 1) args of 2394 SOME (_, ((_, wit), thms)) => (NONE, (Lazy.force wit, thms)) 2395 | NONE => 2396 (SOME (i - 1), (mk_dummy_wit (nth Dss (i - 1)) (nth bnfs (i - 1)) I, [])))) 2397 ks; 2398 in 2399 ((I, dummys), apsnd flat (split_list args')) 2400 end; 2401 2402 fun mk_coind_wits ((I, dummys), (args, thms)) = 2403 ((I, dummys), (map (fn i => mk_unfold Ts args i $ HOLogic.unit) ks, thms)); 2404 2405 val coind_witss = 2406 maps (map (mk_coind_wits o prepare_args)) nonredundant_coind_wit_argsss; 2407 2408 val witss = map2 (fn Ds => fn bnf => mk_wits_of_bnf 2409 (replicate (nwits_of_bnf bnf) Ds) 2410 (replicate (nwits_of_bnf bnf) (passiveAs @ Ts)) bnf) Dss bnfs; 2411 2412 val ctor_witss = 2413 map (map (uncurry close_wit o tree_to_ctor_wit ys ctors witss o snd o snd) o 2414 filter_out (fst o snd)) wit_treess; 2415 2416 fun mk_coind_wit_thms ((I, dummys), (wits, wit_thms)) = 2417 let 2418 fun mk_goal sets y y_copy y'_copy j = 2419 let 2420 fun mk_conjunct set z dummy wit = 2421 mk_Ball (set $ z) (Term.absfree y'_copy 2422 (if dummy = NONE orelse member (op =) I (j - 1) then 2423 HOLogic.mk_imp (HOLogic.mk_eq (z, wit), 2424 if member (op =) I (j - 1) then HOLogic.mk_eq (y_copy, y) 2425 else \<^term>\<open>False\<close>) 2426 else \<^term>\<open>True\<close>)); 2427 in 2428 HOLogic.mk_Trueprop 2429 (Library.foldr1 HOLogic.mk_conj (@{map 4} mk_conjunct sets Jzs dummys wits)) 2430 end; 2431 val goals = @{map 5} mk_goal Jsetss_by_range ys ys_copy ys'_copy ls; 2432 in 2433 map2 (fn goal => fn induct => 2434 Variable.add_free_names lthy goal [] 2435 |> (fn vars => Goal.prove_sorry lthy vars [] goal 2436 (fn {context = ctxt, prems = _} => mk_coind_wit_tac ctxt induct dtor_unfold_thms 2437 (flat set_mapss) wit_thms)) 2438 |> Thm.close_derivation \<^here>) 2439 goals dtor_Jset_induct_thms 2440 |> map split_conj_thm 2441 |> transpose 2442 |> map (map_filter (try (fn thm => thm RS bspec RS mp))) 2443 |> curry op ~~ (map_index Library.I (map (close_wit I) wits)) 2444 |> filter (fn (_, thms) => length thms = m) 2445 end; 2446 2447 val coind_wit_thms = maps mk_coind_wit_thms coind_witss; 2448 2449 val (wit_thmss, all_witss) = 2450 fold (fn ((i, wit), thms) => fn witss => 2451 nth_map i (fn (thms', wits) => (thms @ thms', wit :: wits)) witss) 2452 coind_wit_thms (map (pair []) ctor_witss) 2453 |> map (apsnd (map snd o minimize_wits)) 2454 |> split_list; 2455 2456 val timer = time (timer "witnesses"); 2457 2458 val map_id0_tacs = 2459 map2 (fn thm => fn thm' => fn ctxt => 2460 mk_map_id0_tac ctxt Jmap_thms thm thm') 2461 dtor_unfold_unique_thms unfold_dtor_thms; 2462 val map_comp0_tacs = map (fn thm => fn ctxt => rtac ctxt (thm RS sym) 1) Jmap_comp0_thms; 2463 val map_cong0_tacs = map (fn thm => fn ctxt => mk_map_cong0_tac ctxt m thm) map_cong0_thms; 2464 val set_map0_tacss = 2465 map (map (fn col => fn ctxt => 2466 unfold_thms_tac ctxt Jset_defs THEN mk_set_map0_tac ctxt col)) 2467 (transpose col_natural_thmss); 2468 2469 val Jbd_card_orders = map (fn def => Local_Defs.fold lthy [def] sbd_card_order) Jbd_defs; 2470 val Jbd_Cinfinites = map (fn def => Local_Defs.fold lthy [def] sbd_Cinfinite) Jbd_defs; 2471 2472 val bd_co_tacs = map (fn thm => fn ctxt => rtac ctxt thm 1) Jbd_card_orders; 2473 val bd_cinf_tacs = map (fn thm => fn ctxt => rtac ctxt (thm RS conjunct1) 1) Jbd_Cinfinites; 2474 2475 val set_bd_tacss = 2476 map2 (fn Cinf => map (fn col => fn ctxt => 2477 unfold_thms_tac ctxt Jset_defs THEN mk_set_bd_tac ctxt Cinf col)) 2478 Jbd_Cinfinites (transpose col_bd_thmss); 2479 2480 val le_rel_OO_tacs = map (fn i => fn ctxt => 2481 rtac ctxt (le_Jrel_OO_thm RS mk_conjunctN n i) 1) ks; 2482 2483 val rel_OO_Grp_tacs = map (fn def => fn ctxt => rtac ctxt def 1) Jrel_unabs_defs; 2484 2485 val pred_set_tacs = map (fn def => fn ctxt => rtac ctxt def 1) Jpred_unabs_defs; 2486 2487 val tacss = @{map 10} zip_axioms map_id0_tacs map_comp0_tacs map_cong0_tacs set_map0_tacss 2488 bd_co_tacs bd_cinf_tacs set_bd_tacss le_rel_OO_tacs rel_OO_Grp_tacs pred_set_tacs; 2489 2490 fun wit_tac thms ctxt = 2491 mk_wit_tac ctxt n dtor_ctor_thms (flat dtor_Jset_thmss) (maps wit_thms_of_bnf bnfs) thms; 2492 2493 val (Jbnfs, lthy) = 2494 @{fold_map 7} (fn tacs => fn map_b => fn rel_b => fn pred_b => fn set_bs => fn wit_thms => 2495 fn consts => 2496 bnf_def Hardly_Inline (user_policy Note_Some) false I tacs (wit_tac wit_thms) 2497 (SOME deads) map_b rel_b pred_b set_bs consts) 2498 tacss map_bs rel_bs pred_bs set_bss wit_thmss 2499 (((((((replicate n Binding.empty ~~ Ts) ~~ Jmaps) ~~ Jsetss_by_bnf) ~~ Jbds) ~~ 2500 all_witss) ~~ map SOME Jrels) ~~ map SOME Jpreds) 2501 lthy; 2502 2503 val timer = time (timer "registered new codatatypes as BNFs"); 2504 2505 val ls' = if m = 1 then [0] else ls; 2506 2507 val Jbnf_common_notes = 2508 map2 (fn i => fn thm => (mk_dtor_set_inductN i, [thm])) ls' dtor_Jset_induct_thms 2509 |> map (fn (thmN, thms) => 2510 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])])); 2511 2512 val Jbnf_notes = 2513 [(dtor_mapN, map single dtor_Jmap_thms), 2514 (dtor_map_uniqueN, map single dtor_Jmap_unique_thms), 2515 (dtor_relN, map single dtor_Jrel_thms), 2516 (dtor_set_inclN, dtor_Jset_incl_thmss), 2517 (dtor_set_set_inclN, map flat dtor_set_Jset_incl_thmsss)] @ 2518 map2 (fn i => fn thms => (mk_dtor_setN i, map single thms)) ls' dtor_Jset_thmss 2519 |> maps (fn (thmN, thmss) => 2520 map2 (fn b => fn thms => 2521 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])])) 2522 bs thmss) 2523 in 2524 (timer, Jbnfs, (Jmap_thms, dtor_Jmap_thms), dtor_Jmap_unique_thm, dtor_Jset_thmss', 2525 dtor_Jrel_thms, Jrel_coinduct_thm, Jbnf_common_notes @ Jbnf_notes, dtor_Jset_induct_thms, 2526 lthy) 2527 end; 2528 2529 val ((Jphis, activephis), _) = 2530 lthy 2531 |> mk_Frees "R" JphiTs 2532 ||>> mk_Frees "S" activephiTs; 2533 2534 val dtor_unfold_o_Jmap_thms = mk_xtor_co_iter_o_map_thms Greatest_FP false m 2535 dtor_unfold_unique_thm dtor_Jmap_o_thms (map (mk_pointfree2 lthy) dtor_unfold_thms) 2536 sym_map_comps map_cong0s; 2537 2538 val rels = map2 (fn Ds => mk_rel_of_bnf Ds allAs allBs') Dss bnfs; 2539 val Jrels = if m = 0 then map HOLogic.eq_const Ts 2540 else map (mk_rel_of_bnf deads passiveAs passiveBs) Jbnfs; 2541 2542 val dtor_unfold_transfer_thms = 2543 mk_xtor_co_iter_transfer_thms Greatest_FP rels activephis activephis Jrels Jphis 2544 (mk_unfolds passiveAs activeAs) (mk_unfolds passiveBs activeBs) 2545 (fn {context = ctxt, prems = _} => mk_unfold_transfer_tac ctxt m Jrel_coinduct_thm 2546 (map map_transfer_of_bnf bnfs) dtor_unfold_thms) 2547 lthy; 2548 2549 val timer = time (timer "relator coinduction"); 2550 2551 fun mk_Ts As = map (typ_subst_atomic (passiveAs ~~ As)) Ts; 2552 val export = map (Morphism.term (Local_Theory.target_morphism lthy)) 2553 val ((corecs, (dtor_corec_thms, dtor_corec_unique_thm, dtor_corec_o_Jmap_thms, 2554 dtor_corec_transfer_thms)), lthy) = lthy 2555 |> derive_xtor_co_recs Greatest_FP external_bs mk_Ts (Dss, resDs) bnfs 2556 (export dtors) (export unfolds) 2557 dtor_unfold_unique_thm dtor_unfold_thms dtor_unfold_transfer_thms 2558 dtor_Jmap_thms dtor_Jrel_thms (replicate n NONE); 2559 2560 val timer = time (timer "recursor"); 2561 2562 val common_notes = 2563 [(dtor_coinductN, [dtor_coinduct_thm]), 2564 (dtor_rel_coinductN, [Jrel_coinduct_thm])] 2565 |> map (fn (thmN, thms) => 2566 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])])); 2567 2568 val notes = 2569 [(ctor_dtorN, ctor_dtor_thms), 2570 (ctor_exhaustN, ctor_exhaust_thms), 2571 (ctor_injectN, ctor_inject_thms), 2572 (dtor_ctorN, dtor_ctor_thms), 2573 (dtor_exhaustN, dtor_exhaust_thms), 2574 (dtor_injectN, dtor_inject_thms), 2575 (dtor_unfoldN, dtor_unfold_thms), 2576 (dtor_unfold_o_mapN, dtor_unfold_o_Jmap_thms), 2577 (dtor_unfold_transferN, dtor_unfold_transfer_thms), 2578 (dtor_unfold_uniqueN, dtor_unfold_unique_thms)] 2579 |> map (apsnd (map single)) 2580 |> maps (fn (thmN, thmss) => 2581 map2 (fn b => fn thms => 2582 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])])) 2583 bs thmss); 2584 2585 val lthy' = lthy |> internals ? snd o Local_Theory.notes (common_notes @ notes @ Jbnf_notes); 2586 2587 val fp_res = 2588 {Ts = Ts, bnfs = Jbnfs, pre_bnfs = bnfs, absT_infos = absT_infos, 2589 ctors = ctors, dtors = dtors, xtor_un_folds = unfolds, xtor_co_recs = export corecs, 2590 xtor_co_induct = dtor_coinduct_thm, dtor_ctors = dtor_ctor_thms, ctor_dtors = ctor_dtor_thms, 2591 ctor_injects = ctor_inject_thms, dtor_injects = dtor_inject_thms, xtor_maps = dtor_Jmap_thms, 2592 xtor_map_unique = dtor_Jmap_unique_thm, xtor_setss = dtor_Jset_thmss', 2593 xtor_rels = dtor_Jrel_thms, xtor_un_fold_thms = dtor_unfold_thms, 2594 xtor_co_rec_thms = dtor_corec_thms, xtor_un_fold_unique = dtor_unfold_unique_thm, 2595 xtor_co_rec_unique = dtor_corec_unique_thm, 2596 xtor_un_fold_o_maps = dtor_unfold_o_Jmap_thms, 2597 xtor_co_rec_o_maps = dtor_corec_o_Jmap_thms, 2598 xtor_un_fold_transfers = dtor_unfold_transfer_thms, 2599 xtor_co_rec_transfers = dtor_corec_transfer_thms, xtor_rel_co_induct = Jrel_coinduct_thm, 2600 dtor_set_inducts = dtor_Jset_induct_thms}; 2601 in 2602 timer; (fp_res, lthy') 2603 end; 2604 2605val _ = 2606 Outer_Syntax.local_theory \<^command_keyword>\<open>codatatype\<close> "define coinductive datatypes" 2607 (parse_co_datatype_cmd Greatest_FP construct_gfp); 2608 2609end; 2610