1 (* Title: Pure/axclass.ML
2 Author: Markus Wenzel, TU Muenchen
4 Type classes defined as predicates, associated with a record of
5 parameters. Proven class relations and type arities.
10 type info = {def: thm, intro: thm, axioms: thm list, params: (string * typ) list}
11 val get_info: theory -> class -> info
12 val class_of_param: theory -> string -> class option
13 val instance_name: string * class -> string
14 val thynames_of_arity: theory -> class * string -> string list
15 val param_of_inst: theory -> string * string -> string
16 val inst_of_param: theory -> string -> (string * string) option
17 val unoverload: theory -> thm -> thm
18 val overload: theory -> thm -> thm
19 val unoverload_conv: theory -> conv
20 val overload_conv: theory -> conv
21 val lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option
22 val unoverload_const: theory -> string * typ -> string
23 val cert_classrel: theory -> class * class -> class * class
24 val read_classrel: theory -> xstring * xstring -> class * class
25 val declare_overloaded: string * typ -> theory -> term * theory
26 val define_overloaded: binding -> string * term -> theory -> thm * theory
27 val add_classrel: thm -> theory -> theory
28 val add_arity: thm -> theory -> theory
29 val prove_classrel: class * class -> tactic -> theory -> theory
30 val prove_arity: string * sort list * sort -> tactic -> theory -> theory
31 val define_class: binding * class list -> string list ->
32 (Thm.binding * term list) list -> theory -> class * theory
33 val axiomatize_class: binding * class list -> theory -> theory
34 val axiomatize_class_cmd: binding * xstring list -> theory -> theory
35 val axiomatize_classrel: (class * class) list -> theory -> theory
36 val axiomatize_classrel_cmd: (xstring * xstring) list -> theory -> theory
37 val axiomatize_arity: arity -> theory -> theory
38 val axiomatize_arity_cmd: xstring * string list * string -> theory -> theory
41 structure AxClass: AX_CLASS =
52 params: (string * typ) list};
54 fun make_axclass (def, intro, axioms, params): info =
55 {def = def, intro = intro, axioms = axioms, params = params};
58 (* class parameters (canonical order) *)
60 type param = string * class;
62 fun add_param pp ((x, c): param) params =
63 (case AList.lookup (op =) params x of
64 NONE => (x, c) :: params
65 | SOME c' => error ("Duplicate class parameter " ^ quote x ^
66 " for " ^ Pretty.string_of_sort pp [c] ^
67 (if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));
72 datatype data = Data of
73 {axclasses: info Symtab.table,
75 proven_classrels: thm Symreltab.table,
76 proven_arities: ((class * sort list) * (thm * string)) list Symtab.table,
77 (*arity theorems with theory name*)
79 (string * thm) Symtab.table Symtab.table *
80 (*constant name ~> type constructor ~> (constant name, equation)*)
81 (string * string) Symtab.table (*constant name ~> (constant name, type constructor)*),
82 diff_classrels: (class * class) list};
85 (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =
86 Data {axclasses = axclasses, params = params, proven_classrels = proven_classrels,
87 proven_arities = proven_arities, inst_params = inst_params,
88 diff_classrels = diff_classrels};
90 fun diff_table tab1 tab2 =
91 Symreltab.fold (fn (x, _) => if Symreltab.defined tab2 x then I else cons x) tab1 [];
93 structure Data = Theory_Data_PP
97 make_data (Symtab.empty, [], Symreltab.empty, Symtab.empty, (Symtab.empty, Symtab.empty), []);
100 (Data {axclasses = axclasses1, params = params1, proven_classrels = proven_classrels1,
101 proven_arities = proven_arities1, inst_params = inst_params1,
102 diff_classrels = diff_classrels1},
103 Data {axclasses = axclasses2, params = params2, proven_classrels = proven_classrels2,
104 proven_arities = proven_arities2, inst_params = inst_params2,
105 diff_classrels = diff_classrels2}) =
107 val axclasses' = Symtab.merge (K true) (axclasses1, axclasses2);
109 if null params1 then params2
110 else fold_rev (fn p => if member (op =) params1 p then I else add_param pp p) params2 params1;
112 (*transitive closure of classrels and arity completion is done in Theory.at_begin hook*)
113 val proven_classrels' = Symreltab.join (K #1) (proven_classrels1, proven_classrels2);
114 val proven_arities' =
115 Symtab.join (K (Library.merge (eq_fst op =))) (proven_arities1, proven_arities2);
117 val diff_classrels' =
118 diff_table proven_classrels1 proven_classrels2 @
119 diff_table proven_classrels2 proven_classrels1 @
120 diff_classrels1 @ diff_classrels2;
123 (Symtab.join (K (Symtab.merge (K true))) (#1 inst_params1, #1 inst_params2),
124 Symtab.merge (K true) (#2 inst_params1, #2 inst_params2));
127 (axclasses', params', proven_classrels', proven_arities', inst_params', diff_classrels')
132 Data.map (fn Data {axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels} =>
133 make_data (f (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels)));
135 fun map_axclasses f =
136 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
137 (f axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels));
140 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
141 (axclasses, f params, proven_classrels, proven_arities, inst_params, diff_classrels));
143 fun map_proven_classrels f =
144 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
145 (axclasses, params, f proven_classrels, proven_arities, inst_params, diff_classrels));
147 fun map_proven_arities f =
148 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
149 (axclasses, params, proven_classrels, f proven_arities, inst_params, diff_classrels));
151 fun map_inst_params f =
152 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
153 (axclasses, params, proven_classrels, proven_arities, f inst_params, diff_classrels));
155 val clear_diff_classrels =
156 map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, _) =>
157 (axclasses, params, proven_classrels, proven_arities, inst_params, []));
159 val rep_data = Data.get #> (fn Data args => args);
161 val axclasses_of = #axclasses o rep_data;
162 val params_of = #params o rep_data;
163 val proven_classrels_of = #proven_classrels o rep_data;
164 val proven_arities_of = #proven_arities o rep_data;
165 val inst_params_of = #inst_params o rep_data;
166 val diff_classrels_of = #diff_classrels o rep_data;
169 (* axclasses with parameters *)
172 (case Symtab.lookup (axclasses_of thy) c of
174 | NONE => error ("No such axclass: " ^ quote c));
176 fun all_params_of thy S =
177 let val params = params_of thy;
178 in fold (fn (x, c) => if Sign.subsort thy (S, [c]) then cons x else I) params [] end;
180 fun class_of_param thy = AList.lookup (op =) (params_of thy);
183 (* maintain instances *)
185 val classrel_prefix = "classrel_";
186 val arity_prefix = "arity_";
188 fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;
193 fun (SOME a) RSO (SOME b) = SOME (a RS b)
197 fun the_classrel thy (c1, c2) =
198 (case Symreltab.lookup (proven_classrels_of thy) (c1, c2) of
200 | NONE => error ("Unproven class relation " ^
201 Syntax.string_of_classrel (ProofContext.init_global thy) [c1, c2])); (* FIXME stale thy (!?) *)
203 fun put_trancl_classrel ((c1, c2), th) thy =
205 val classes = Sorts.classes_of (Sign.classes_of thy);
206 val classrels = proven_classrels_of thy;
208 fun reflcl_classrel (c1', c2') =
209 if c1' = c2' then NONE else SOME (Thm.transfer thy (the_classrel thy (c1', c2')));
210 fun gen_classrel (c1_pred, c2_succ) =
213 the ((reflcl_classrel (c1_pred, c1) RSO SOME th) RSO reflcl_classrel (c2, c2_succ))
214 |> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [])))] []
215 |> Thm.close_derivation;
216 in ((c1_pred, c2_succ), th') end;
219 Library.map_product pair (c1 :: Graph.imm_preds classes c1) (c2 :: Graph.imm_succs classes c2)
220 |> filter_out ((op =) orf Symreltab.defined classrels)
222 val needed = not (null new_classrels);
225 if needed then map_proven_classrels (fold Symreltab.update new_classrels) thy
229 fun complete_classrels thy =
231 val classrels = proven_classrels_of thy;
232 val diff_classrels = diff_classrels_of thy;
233 val (needed, thy') = (false, thy) |>
234 fold (fn rel => fn (needed, thy) =>
235 put_trancl_classrel (rel, Symreltab.lookup classrels rel |> the) thy
236 |>> (fn b => needed orelse b))
239 if null diff_classrels then NONE
240 else SOME (clear_diff_classrels thy')
244 fun the_arity thy (a, Ss, c) =
245 (case AList.lookup (op =) (Symtab.lookup_list (proven_arities_of thy) a) (c, Ss) of
247 | NONE => error ("Unproven type arity " ^
248 Syntax.string_of_arity (ProofContext.init_global thy) (a, Ss, [c]))); (* FIXME stale thy (!?) *)
250 fun thynames_of_arity thy (c, a) =
251 Symtab.lookup_list (proven_arities_of thy) a
252 |> map_filter (fn ((c', _), (_, name)) => if c = c' then SOME name else NONE)
255 fun insert_arity_completions thy t ((c, Ss), ((th, thy_name))) (finished, arities) =
257 val algebra = Sign.classes_of thy;
258 val ars = Symtab.lookup_list arities t;
259 val super_class_completions =
260 Sign.super_classes thy c
261 |> filter_out (fn c1 => exists (fn ((c2, Ss2), _) =>
262 c1 = c2 andalso Sorts.sorts_le algebra (Ss2, Ss)) ars);
264 val names = Name.invents Name.context Name.aT (length Ss);
265 val std_vars = map (fn a => SOME (ctyp_of thy (TVar ((a, 0), [])))) names;
267 val completions = super_class_completions |> map (fn c1 =>
270 (th RS Thm.transfer thy (the_classrel thy (c, c1)))
271 |> Drule.instantiate' std_vars []
272 |> Thm.close_derivation;
273 in ((th1, thy_name), c1) end);
275 val finished' = finished andalso null completions;
276 val arities' = fold (fn (th, c1) => Symtab.cons_list (t, ((c1, Ss), th))) completions arities;
277 in (finished', arities') end;
279 fun put_arity ((t, Ss, c), th) thy =
280 let val ar = ((c, Ss), (th, Context.theory_name thy)) in
282 |> map_proven_arities
283 (Symtab.insert_list (eq_fst op =) (t, ar) #>
284 curry (insert_arity_completions thy t ar) true #> #2)
287 fun complete_arities thy =
289 val arities = proven_arities_of thy;
290 val (finished, arities') =
291 Symtab.fold (fn (t, ars) => fold (insert_arity_completions thy t) ars) arities (true, arities);
293 if finished then NONE
294 else SOME (map_proven_arities (K arities') thy)
297 val _ = Context.>> (Context.map_theory
298 (Theory.at_begin complete_classrels #>
299 Theory.at_begin complete_arities));
301 val _ = Proofterm.install_axclass_proofs
302 {classrel_proof = Thm.proof_of oo the_classrel,
303 arity_proof = Thm.proof_of oo the_arity};
306 (* maintain instance parameters *)
308 fun get_inst_param thy (c, tyco) =
309 (case Symtab.lookup (the_default Symtab.empty (Symtab.lookup (#1 (inst_params_of thy)) c)) tyco of
311 | NONE => error ("No instance parameter for constant " ^ quote c ^ " on type " ^ quote tyco));
313 fun add_inst_param (c, tyco) inst =
314 (map_inst_params o apfst o Symtab.map_default (c, Symtab.empty)) (Symtab.update_new (tyco, inst))
315 #> (map_inst_params o apsnd) (Symtab.update_new (#1 inst, (c, tyco)));
317 val inst_of_param = Symtab.lookup o #2 o inst_params_of;
318 val param_of_inst = #1 oo get_inst_param;
321 Symtab.fold (Symtab.fold (cons o #2 o #2) o #2) (#1 (inst_params_of thy)) [];
323 fun get_inst_tyco consts = try (#1 o dest_Type o the_single o Consts.typargs consts);
325 fun unoverload thy = MetaSimplifier.simplify true (inst_thms thy);
326 fun overload thy = MetaSimplifier.simplify true (map Thm.symmetric (inst_thms thy));
328 fun unoverload_conv thy = MetaSimplifier.rewrite true (inst_thms thy);
329 fun overload_conv thy = MetaSimplifier.rewrite true (map Thm.symmetric (inst_thms thy));
331 fun lookup_inst_param consts params (c, T) =
332 (case get_inst_tyco consts (c, T) of
333 SOME tyco => AList.lookup (op =) params (c, tyco)
336 fun unoverload_const thy (c_ty as (c, _)) =
337 if is_some (class_of_param thy c) then
338 (case get_inst_tyco (Sign.consts_of thy) c_ty of
339 SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c
347 (* class relations *)
349 fun cert_classrel thy raw_rel =
351 val string_of_sort = Syntax.string_of_sort_global thy;
352 val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;
353 val _ = Sign.primitive_classrel (c1, c2) (Theory.copy thy);
355 (case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of
357 | xs => raise TYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^
358 commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));
361 fun read_classrel thy raw_rel =
362 cert_classrel thy (pairself (ProofContext.read_class (ProofContext.init_global thy)) raw_rel)
363 handle TYPE (msg, _, _) => error msg;
366 (* declaration and definition of instances of overloaded constants *)
368 fun inst_tyco_of thy (c, T) =
369 (case get_inst_tyco (Sign.consts_of thy) (c, T) of
371 | NONE => error ("Illegal type for instantiation of class parameter: " ^
372 quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T)));
374 fun declare_overloaded (c, T) thy =
377 (case class_of_param thy c of
379 | NONE => error ("Not a class parameter: " ^ quote c));
380 val tyco = inst_tyco_of thy (c, T);
381 val name_inst = instance_name (tyco, class) ^ "_inst";
382 val c' = instance_name (tyco, c);
383 val T' = Type.strip_sorts T;
386 |> Sign.qualified_path true (Binding.name name_inst)
387 |> Sign.declare_const ((Binding.name c', T'), NoSyn)
388 |-> (fn const' as Const (c'', _) =>
389 Thm.add_def false true
390 (Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))
391 #>> apsnd Thm.varifyT_global
392 #-> (fn (_, thm) => add_inst_param (c, tyco) (c'', thm)
393 #> PureThy.add_thm ((Binding.conceal (Binding.name c'), thm), [])
395 #> pair (Const (c, T))))
396 ||> Sign.restore_naming thy
399 fun define_overloaded b (c, t) thy =
401 val T = Term.fastype_of t;
402 val tyco = inst_tyco_of thy (c, T);
403 val (c', eq) = get_inst_param thy (c, tyco);
404 val prop = Logic.mk_equals (Const (c', T), t);
405 val b' = Thm.def_binding_optional (Binding.name (instance_name (tyco, c))) b;
408 |> Thm.add_def false false (b', prop)
409 |>> (fn (_, thm) => Drule.transitive_thm OF [eq, thm])
413 (* primitive rules *)
415 fun add_classrel raw_th thy =
417 val th = Thm.strip_shyps (Thm.transfer thy raw_th);
418 val prop = Thm.plain_prop_of th;
419 fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
420 val rel = Logic.dest_classrel prop handle TERM _ => err ();
421 val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
422 val (th', thy') = PureThy.store_thm
423 (Binding.name (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th) thy;
426 |> Drule.instantiate' [SOME (ctyp_of thy' (TVar ((Name.aT, 0), [])))] [];
429 |> Sign.primitive_classrel (c1, c2)
430 |> (#2 oo put_trancl_classrel) ((c1, c2), th'')
431 |> perhaps complete_arities
434 fun add_arity raw_th thy =
436 val th = Thm.strip_shyps (Thm.transfer thy raw_th);
437 val prop = Thm.plain_prop_of th;
438 fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
439 val arity as (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
441 val (th', thy') = PureThy.store_thm
442 (Binding.name (prefix arity_prefix (Logic.name_arity arity)), th) thy;
444 val args = Name.names Name.context Name.aT Ss;
445 val T = Type (t, map TFree args);
446 val std_vars = map (fn (a, S) => SOME (ctyp_of thy' (TVar ((a, 0), [])))) args;
448 val missing_params = Sign.complete_sort thy' [c]
449 |> maps (these o Option.map #params o try (get_info thy'))
450 |> filter_out (fn (const, _) => can (get_inst_param thy') (const, t))
451 |> (map o apsnd o map_atyps) (K T);
454 |> Drule.instantiate' std_vars [];
457 |> fold (#2 oo declare_overloaded) missing_params
458 |> Sign.primitive_arity (t, Ss, [c])
459 |> put_arity ((t, Ss, c), th'')
463 (* tactical proofs *)
465 fun prove_classrel raw_rel tac thy =
467 val ctxt = ProofContext.init_global thy;
468 val (c1, c2) = cert_classrel thy raw_rel;
469 val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>
470 cat_error msg ("The error(s) above occurred while trying to prove class relation " ^
471 quote (Syntax.string_of_classrel ctxt [c1, c2]));
473 thy |> add_classrel th
476 fun prove_arity raw_arity tac thy =
478 val ctxt = ProofContext.init_global thy;
479 val arity = ProofContext.cert_arity ctxt raw_arity;
480 val names = map (prefix arity_prefix) (Logic.name_arities arity);
481 val props = Logic.mk_arities arity;
482 val ths = Goal.prove_multi ctxt [] [] props
483 (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>
484 cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
485 quote (Syntax.string_of_arity ctxt arity));
487 thy |> fold add_arity ths
492 (** class definitions **)
494 fun split_defined n eq =
497 (eq RS Drule.equal_elim_rule2)
498 |> Conjunction.curry_balanced n
499 |> n = 0 ? Thm.eq_assumption 1;
503 (eq RS Drule.equal_elim_rule1)
504 |> Balanced_Tree.dest (fn th =>
505 (th RS Conjunction.conjunctionD1, th RS Conjunction.conjunctionD2)) n;
506 in (intro, dests) end;
508 fun define_class (bclass, raw_super) raw_params raw_specs thy =
510 val ctxt = ProofContext.init_global thy;
511 val pp = Syntax.pp ctxt;
516 val bconst = Binding.map_name Logic.const_of_class bclass;
517 val class = Sign.full_name thy bclass;
518 val super = Sign.minimize_sort thy (Sign.certify_sort thy raw_super);
520 fun check_constraint (a, S) =
521 if Sign.subsort thy (super, S) then ()
522 else error ("Sort constraint of type variable " ^
523 setmp_CRITICAL show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^
524 " needs to be weaker than " ^ Pretty.string_of_sort pp super);
529 val params = raw_params |> map (fn p =>
531 val T = Sign.the_const_type thy p;
533 (case Term.add_tvarsT T [] of
534 [((a, _), S)] => check_constraint (a, S)
535 | _ => error ("Exactly one type variable expected in class parameter " ^ quote p));
536 val T' = Term.map_type_tvar (fn _ => TFree (Name.aT, [class])) T;
543 (case Term.add_tfrees t [] of
544 [(a, S)] => check_constraint (a, S)
546 | _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t);
548 |> Term.map_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))
549 |> Logic.close_form);
551 val axiomss = map (map (prep_axiom o Sign.cert_prop thy) o snd) raw_specs;
552 val name_atts = map fst raw_specs;
557 val conjs = Logic.mk_of_sort (Term.aT [], super) @ flat axiomss;
559 Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);
561 val ([def], def_thy) =
563 |> Sign.primitive_class (bclass, super)
564 |> PureThy.add_defs false [((Thm.def_binding bconst, class_eq), [])];
565 val (raw_intro, (raw_classrel, raw_axioms)) =
566 split_defined (length conjs) def ||> chop (length super);
571 val class_triv = Thm.class_triv def_thy class;
572 val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =
574 |> Sign.qualified_path true bconst
575 |> PureThy.note_thmss ""
576 [((Binding.name "intro", []), [([Drule.export_without_context raw_intro], [])]),
577 ((Binding.name "super", []), [(map Drule.export_without_context raw_classrel, [])]),
578 ((Binding.name "axioms", []),
579 [(map (fn th => Drule.export_without_context (class_triv RS th)) raw_axioms, [])])]
580 ||> Sign.restore_naming def_thy;
585 val axclass = make_axclass (def, intro, axioms, params);
588 |> fold (#2 oo put_trancl_classrel) (map (pair class) super ~~ classrel)
589 |> Sign.qualified_path false bconst
590 |> PureThy.note_thmss "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> #2
591 |> Sign.restore_naming facts_thy
592 |> map_axclasses (Symtab.update (class, axclass))
593 |> map_params (fold (fn (x, _) => add_param pp (x, class)) params);
595 in (class, result_thy) end;
599 (** axiomatizations **)
604 fun axiomatize prep mk name add raw_args thy =
606 val args = prep thy raw_args;
608 val names = name args;
611 |> fold_map Thm.add_axiom (map Binding.name names ~~ specs)
612 |-> fold (add o Drule.export_without_context o snd)
615 fun ax_classrel prep =
616 axiomatize (map o prep) (map Logic.mk_classrel)
617 (map (prefix classrel_prefix o Logic.name_classrel)) add_classrel;
620 axiomatize (prep o ProofContext.init_global) Logic.mk_arities
621 (map (prefix arity_prefix) o Logic.name_arities) add_arity;
624 (Logic.const_of_class c, Term.itselfT (Term.aT []) --> propT);
626 fun ax_class prep_class prep_classrel (bclass, raw_super) thy =
628 val class = Sign.full_name thy bclass;
629 val super = map (prep_class thy) raw_super |> Sign.minimize_sort thy;
632 |> Sign.primitive_class (bclass, super)
633 |> ax_classrel prep_classrel (map (fn c => (class, c)) super)
634 |> Theory.add_deps "" (class_const class) (map class_const super)
639 val axiomatize_class = ax_class Sign.certify_class cert_classrel;
640 val axiomatize_class_cmd = ax_class (ProofContext.read_class o ProofContext.init_global) read_classrel;
641 val axiomatize_classrel = ax_classrel cert_classrel;
642 val axiomatize_classrel_cmd = ax_classrel read_classrel;
643 val axiomatize_arity = ax_arity ProofContext.cert_arity;
644 val axiomatize_arity_cmd = ax_arity ProofContext.read_arity;