1 (* Title: Pure/axclass.ML
2 Author: Markus Wenzel, TU Muenchen
4 Type classes defined as predicates, associated with a record of
10 val define_class: binding * class list -> string list ->
11 (Thm.binding * term list) list -> theory -> class * theory
12 val add_classrel: thm -> theory -> theory
13 val add_arity: thm -> theory -> theory
14 val prove_classrel: class * class -> tactic -> theory -> theory
15 val prove_arity: string * sort list * sort -> tactic -> theory -> theory
16 val get_info: theory -> class ->
17 {def: thm, intro: thm, axioms: thm list, params: (string * typ) list}
18 val class_intros: theory -> thm list
19 val class_of_param: theory -> string -> class option
20 val cert_classrel: theory -> class * class -> class * class
21 val read_classrel: theory -> xstring * xstring -> class * class
22 val axiomatize_class: binding * class list -> theory -> theory
23 val axiomatize_class_cmd: binding * xstring list -> theory -> theory
24 val axiomatize_classrel: (class * class) list -> theory -> theory
25 val axiomatize_classrel_cmd: (xstring * xstring) list -> theory -> theory
26 val axiomatize_arity: arity -> theory -> theory
27 val axiomatize_arity_cmd: xstring * string list * string -> theory -> theory
28 val instance_name: string * class -> string
29 val declare_overloaded: string * typ -> theory -> term * theory
30 val define_overloaded: binding -> string * term -> theory -> thm * theory
31 val unoverload: theory -> thm -> thm
32 val overload: theory -> thm -> thm
33 val unoverload_conv: theory -> conv
34 val overload_conv: theory -> conv
35 val unoverload_const: theory -> string * typ -> string
36 val lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option
37 val param_of_inst: theory -> string * string -> string
38 val inst_of_param: theory -> string -> (string * string) option
39 val thynames_of_arity: theory -> class * string -> string list
44 structure AxClass: AX_CLASS =
49 (* class parameters (canonical order) *)
51 type param = string * class;
53 fun add_param pp ((x, c): param) params =
54 (case AList.lookup (op =) params x of
55 NONE => (x, c) :: params
56 | SOME c' => error ("Duplicate class parameter " ^ quote x ^
57 " for " ^ Pretty.string_of_sort pp [c] ^
58 (if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));
60 fun merge_params _ ([], qs) = qs
61 | merge_params pp (ps, qs) =
62 fold_rev (fn q => if member (op =) ps q then I else add_param pp q) qs ps;
69 val axiomsN = "axioms";
71 datatype axclass = AxClass of
75 params: (string * typ) list};
77 type axclasses = axclass Symtab.table * param list;
79 fun make_axclass ((def, intro, axioms), params) = AxClass
80 {def = def, intro = intro, axioms = axioms, params = params};
82 fun merge_axclasses pp ((tab1, params1), (tab2, params2)) : axclasses =
83 (Symtab.merge (K true) (tab1, tab2), merge_params pp (params1, params2));
88 val classrel_prefix = "classrel_";
89 val arity_prefix = "arity_";
92 (thm * proof) Symreltab.table * (*classrel theorems*)
93 ((class * sort list) * ((thm * string) * proof)) list Symtab.table; (*arity theorems with theory name*)
95 (*transitive closure of classrels and arity completion is done in Theory.at_begin hook*)
96 fun merge_instances ((classrel1, arities1): instances, (classrel2, arities2)) =
97 (Symreltab.join (K fst) (classrel1, classrel2),
98 Symtab.join (K (merge (eq_fst op =))) (arities1, arities2));
101 (* instance parameters *)
104 (string * thm) Symtab.table Symtab.table
105 (*constant name ~> type constructor ~> (constant name, equation)*)
106 * (string * string) Symtab.table; (*constant name ~> (constant name, type constructor)*)
108 fun merge_inst_params ((const_param1, param_const1), (const_param2, param_const2)) =
109 (Symtab.join (K (Symtab.merge (K true))) (const_param1, const_param2),
110 Symtab.merge (K true) (param_const1, param_const2));
115 structure AxClassData = Theory_Data_PP
117 type T = axclasses * ((instances * inst_params) * (class * class) list);
118 val empty = ((Symtab.empty, []), (((Symreltab.empty, Symtab.empty), (Symtab.empty, Symtab.empty)), []));
120 fun merge pp ((axclasses1, ((instances1, inst_params1), diff_merge_classrels1)),
121 (axclasses2, ((instances2, inst_params2), diff_merge_classrels2))) =
123 val (classrels1, classrels2) = pairself (Symreltab.keys o fst) (instances1, instances2)
124 val diff_merge_classrels = subtract (op =) classrels1 classrels2
125 @ subtract (op =) classrels2 classrels1
126 @ diff_merge_classrels1 @ diff_merge_classrels2
128 (merge_axclasses pp (axclasses1, axclasses2),
129 ((merge_instances (instances1, instances2), merge_inst_params (inst_params1, inst_params2)),
130 diff_merge_classrels))
135 (* maintain axclasses *)
137 val get_axclasses = #1 o AxClassData.get;
138 val map_axclasses = AxClassData.map o apfst;
140 val lookup_def = Symtab.lookup o #1 o get_axclasses;
143 (case lookup_def thy c of
144 SOME (AxClass info) => info
145 | NONE => error ("No such axclass: " ^ quote c));
147 fun class_intros thy =
150 (case lookup_def thy c of SOME (AxClass {intro, ...}) => cons intro | _ => I);
151 val classes = Sign.all_classes thy;
152 in map (Thm.class_triv thy) classes @ fold add_intro classes [] end;
155 fun get_params thy pred =
156 let val params = #2 (get_axclasses thy);
157 in fold (fn (x, c) => if pred c then cons x else I) params [] end;
159 fun all_params_of thy S = get_params thy (fn c => Sign.subsort thy (S, [c]));
161 fun class_of_param thy = AList.lookup (op =) (#2 (get_axclasses thy));
164 (* maintain instances *)
166 fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;
168 val get_instances = #1 o #1 o #2 o AxClassData.get;
169 val map_instances = AxClassData.map o apsnd o apfst o apfst;
171 val get_diff_merge_classrels = #2 o #2 o AxClassData.get;
172 val clear_diff_merge_classrels = AxClassData.map (apsnd (apsnd (K [])));
175 fun the_classrel thy (c1, c2) =
176 (case Symreltab.lookup (#1 (get_instances thy)) (c1, c2) of
177 SOME classrel => classrel
178 | NONE => error ("Unproven class relation " ^
179 Syntax.string_of_classrel (ProofContext.init thy) [c1, c2]));
181 fun the_classrel_thm thy = Thm.transfer thy o fst o the_classrel thy;
182 fun the_classrel_prf thy = snd o the_classrel thy;
184 fun put_trancl_classrel ((c1, c2), th) thy =
186 val classrels = fst (get_instances thy)
187 val alg = Sign.classes_of thy
188 val {classes, ...} = alg |> Sorts.rep_algebra
190 fun reflcl_classrel (c1', c2') =
191 if c1' = c2' then Thm.trivial (Logic.mk_of_class (TVar(("'a",0),[]), c1') |> cterm_of thy)
192 else the_classrel_thm thy (c1', c2')
193 fun gen_classrel (c1_pred, c2_succ) =
195 val th' = ((reflcl_classrel (c1_pred, c1) RS th) RS reflcl_classrel (c2, c2_succ))
196 |> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [])))] []
197 |> Thm.close_derivation
198 val prf' = th' |> Thm.proof_of
199 in ((c1_pred, c2_succ), (th',prf')) end
201 val new_classrels = Library.map_product pair
202 (c1 :: Graph.imm_preds classes c1) (c2 :: Graph.imm_succs classes c2)
203 |> filter_out (Symreltab.defined classrels)
205 val needed = length new_classrels > 0
209 thy |> map_instances (fn (classrels, arities) =>
210 (classrels |> fold Symreltab.update new_classrels, arities))
214 fun complete_classrels thy =
216 val diff_merge_classrels = get_diff_merge_classrels thy
217 val classrels = fst (get_instances thy)
218 val (needed, thy') = (false, thy) |>
219 fold (fn c12 => fn (needed, thy) =>
220 put_trancl_classrel (c12, Symreltab.lookup classrels c12 |> the |> fst) thy
221 |>> (fn b => needed orelse b))
224 if null diff_merge_classrels then NONE
225 else thy' |> clear_diff_merge_classrels |> SOME
229 fun the_arity thy a (c, Ss) =
230 (case AList.lookup (op =) (Symtab.lookup_list (#2 (get_instances thy)) a) (c, Ss) of
232 | NONE => error ("Unproven type arity " ^
233 Syntax.string_of_arity (ProofContext.init thy) (a, Ss, [c])));
235 fun the_arity_thm thy a c_Ss = the_arity thy a c_Ss |> fst |> fst |> Thm.transfer thy;
236 fun the_arity_prf thy a c_Ss = the_arity thy a c_Ss |> snd;
238 fun thynames_of_arity thy (c, a) =
239 Symtab.lookup_list (#2 (get_instances thy)) a
240 |> map_filter (fn ((c', _), ((_, name),_)) => if c = c' then SOME name else NONE)
243 fun insert_arity_completions thy (t, ((c, Ss), ((th, thy_name), _))) arities =
245 val algebra = Sign.classes_of thy;
246 val super_class_completions =
247 Sign.super_classes thy c
248 |> filter_out (fn c1 => exists (fn ((c2, Ss2), _) => c1 = c2
249 andalso Sorts.sorts_le algebra (Ss2, Ss)) (Symtab.lookup_list arities t));
250 val names_and_Ss = Name.names Name.context Name.aT (map (K []) Ss);
251 val completions = super_class_completions |> map (fn c1 =>
253 val th1 = (th RS the_classrel_thm thy (c, c1))
254 |> Drule.instantiate' (map (SOME o ctyp_of thy o TVar o apfst (rpair 0)) names_and_Ss) []
255 |> Thm.close_derivation
256 val prf1 = Thm.proof_of th1
257 in (((th1,thy_name), prf1), c1) end)
258 val arities' = fold (fn (th_thy_prf1, c1) => Symtab.cons_list (t, ((c1, Ss), th_thy_prf1)))
260 in (null completions, arities') end;
262 fun put_arity ((t, Ss, c), th) thy =
264 val arity' = (t, ((c, Ss), ((th, Context.theory_name thy), Thm.proof_of th)));
267 |> map_instances (fn (classrel, arities) => (classrel,
269 |> Symtab.insert_list (eq_fst op =) arity'
270 |> insert_arity_completions thy arity'
274 fun complete_arities thy =
276 val arities = snd (get_instances thy);
277 val (finished, arities') = arities
278 |> fold_map (insert_arity_completions thy) (Symtab.dest_list arities);
280 if forall I finished then NONE
281 else SOME (thy |> map_instances (fn (classrel, _) => (classrel, arities')))
284 val _ = Context.>> (Context.map_theory
285 (Theory.at_begin complete_classrels #> Theory.at_begin complete_arities))
288 (* maintain instance parameters *)
290 val get_inst_params = #2 o #1 o #2 o AxClassData.get;
291 val map_inst_params = AxClassData.map o apsnd o apfst o apsnd;
293 fun get_inst_param thy (c, tyco) =
294 case Symtab.lookup ((the_default Symtab.empty o Symtab.lookup (fst (get_inst_params thy))) c) tyco
296 | NONE => error ("No instance parameter for constant " ^ quote c
297 ^ " on type constructor " ^ quote tyco);
299 fun add_inst_param (c, tyco) inst = (map_inst_params o apfst
300 o Symtab.map_default (c, Symtab.empty)) (Symtab.update_new (tyco, inst))
301 #> (map_inst_params o apsnd) (Symtab.update_new (fst inst, (c, tyco)));
303 val inst_of_param = Symtab.lookup o snd o get_inst_params;
304 val param_of_inst = fst oo get_inst_param;
306 fun inst_thms thy = (Symtab.fold (Symtab.fold (cons o snd o snd) o snd) o fst)
307 (get_inst_params thy) [];
309 fun get_inst_tyco consts = try (fst o dest_Type o the_single o Consts.typargs consts);
311 fun unoverload thy = MetaSimplifier.simplify true (inst_thms thy);
312 fun overload thy = MetaSimplifier.simplify true (map Thm.symmetric (inst_thms thy));
314 fun unoverload_conv thy = MetaSimplifier.rewrite true (inst_thms thy);
315 fun overload_conv thy = MetaSimplifier.rewrite true (map Thm.symmetric (inst_thms thy));
317 fun lookup_inst_param consts params (c, T) = case get_inst_tyco consts (c, T)
318 of SOME tyco => AList.lookup (op =) params (c, tyco)
321 fun unoverload_const thy (c_ty as (c, _)) =
322 if is_some (class_of_param thy c)
323 then case get_inst_tyco (Sign.consts_of thy) c_ty
324 of SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c
331 (* class relations *)
333 fun cert_classrel thy raw_rel =
335 val string_of_sort = Syntax.string_of_sort_global thy;
336 val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;
337 val _ = Sign.primitive_classrel (c1, c2) (Theory.copy thy);
339 (case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of
341 | xs => raise TYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^
342 commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));
345 fun read_classrel thy raw_rel =
346 cert_classrel thy (pairself (ProofContext.read_class (ProofContext.init thy)) raw_rel)
347 handle TYPE (msg, _, _) => error msg;
349 fun check_shyps_topped th errmsg =
350 let val {shyps, ...} = Thm.rep_thm th
352 forall null shyps orelse raise Fail errmsg
355 (* declaration and definition of instances of overloaded constants *)
357 fun inst_tyco_of thy (c, T) =
358 (case get_inst_tyco (Sign.consts_of thy) (c, T) of
360 | NONE => error ("Illegal type for instantiation of class parameter: " ^
361 quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T)));
363 fun declare_overloaded (c, T) thy =
366 (case class_of_param thy c of
368 | NONE => error ("Not a class parameter: " ^ quote c));
369 val tyco = inst_tyco_of thy (c, T);
370 val name_inst = instance_name (tyco, class) ^ "_inst";
371 val c' = Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco;
372 val T' = Type.strip_sorts T;
375 |> Sign.qualified_path true (Binding.name name_inst)
376 |> Sign.declare_const ((Binding.name c', T'), NoSyn)
377 |-> (fn const' as Const (c'', _) =>
378 Thm.add_def false true
379 (Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))
380 #>> apsnd Thm.varifyT_global
381 #-> (fn (_, thm) => add_inst_param (c, tyco) (c'', thm)
382 #> PureThy.add_thm ((Binding.conceal (Binding.name c'), thm), [])
384 #> pair (Const (c, T))))
385 ||> Sign.restore_naming thy
388 fun define_overloaded b (c, t) thy =
390 val T = Term.fastype_of t;
391 val tyco = inst_tyco_of thy (c, T);
392 val (c', eq) = get_inst_param thy (c, tyco);
393 val prop = Logic.mk_equals (Const (c', T), t);
394 val b' = Thm.def_binding_optional
395 (Binding.name (Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco)) b;
398 |> Thm.add_def false false (b', prop)
399 |>> (fn (_, thm) => Drule.transitive_thm OF [eq, thm])
403 (* primitive rules *)
405 fun add_classrel raw_th thy =
407 val th = Thm.strip_shyps (Thm.transfer thy raw_th);
408 val prop = Thm.plain_prop_of th;
409 fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
410 val rel = Logic.dest_classrel prop handle TERM _ => err ();
411 val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
413 |> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [c1])))] []
414 |> Drule.unconstrainTs;
415 val _ = check_shyps_topped th' "add_classrel: nontop shyps after unconstrain"
418 |> Sign.primitive_classrel (c1, c2)
419 |> (snd oo put_trancl_classrel) ((c1, c2), th')
420 |> perhaps complete_arities
423 fun add_arity raw_th thy =
425 val th = Thm.strip_shyps (Thm.transfer thy raw_th);
426 val prop = Thm.plain_prop_of th;
427 fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
428 val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
429 val names = Name.names Name.context Name.aT Ss;
430 val T = Type (t, map TFree names);
431 val missing_params = Sign.complete_sort thy [c]
432 |> maps (these o Option.map #params o try (get_info thy))
433 |> filter_out (fn (const, _) => can (get_inst_param thy) (const, t))
434 |> (map o apsnd o map_atyps) (K T);
435 val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();
437 |> Drule.instantiate' (map (SOME o ctyp_of thy o TVar o apfst (rpair 0)) names) []
438 |> Drule.unconstrainTs;
439 val _ = check_shyps_topped th' "add_arity: nontop shyps after unconstrain"
442 |> fold (snd oo declare_overloaded) missing_params
443 |> Sign.primitive_arity (t, Ss, [c])
444 |> put_arity ((t, Ss, c), th')
448 (* tactical proofs *)
450 fun prove_classrel raw_rel tac thy =
452 val ctxt = ProofContext.init thy;
453 val (c1, c2) = cert_classrel thy raw_rel;
454 val th = Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (K tac) handle ERROR msg =>
455 cat_error msg ("The error(s) above occurred while trying to prove class relation " ^
456 quote (Syntax.string_of_classrel ctxt [c1, c2]));
459 |> PureThy.add_thms [((Binding.name
460 (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th), [])]
461 |-> (fn [th'] => add_classrel th')
464 fun prove_arity raw_arity tac thy =
466 val ctxt = ProofContext.init thy;
467 val arity = ProofContext.cert_arity ctxt raw_arity;
468 val names = map (prefix arity_prefix) (Logic.name_arities arity);
469 val props = Logic.mk_arities arity;
470 val ths = Goal.prove_multi ctxt [] [] props
471 (fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) handle ERROR msg =>
472 cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
473 quote (Syntax.string_of_arity ctxt arity));
476 |> PureThy.add_thms (map (rpair []) (map Binding.name names ~~ ths))
482 (** class definitions **)
484 fun split_defined n eq =
487 (eq RS Drule.equal_elim_rule2)
488 |> Conjunction.curry_balanced n
489 |> n = 0 ? Thm.eq_assumption 1;
493 (eq RS Drule.equal_elim_rule1)
494 |> Balanced_Tree.dest (fn th =>
495 (th RS Conjunction.conjunctionD1, th RS Conjunction.conjunctionD2)) n;
496 in (intro, dests) end;
498 fun define_class (bclass, raw_super) raw_params raw_specs thy =
500 val ctxt = ProofContext.init thy;
501 val pp = Syntax.pp ctxt;
506 val bconst = Binding.map_name Logic.const_of_class bclass;
507 val class = Sign.full_name thy bclass;
508 val super = Sign.minimize_sort thy (Sign.certify_sort thy raw_super);
510 fun check_constraint (a, S) =
511 if Sign.subsort thy (super, S) then ()
512 else error ("Sort constraint of type variable " ^
513 setmp_CRITICAL show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^
514 " needs to be weaker than " ^ Pretty.string_of_sort pp super);
519 val params = raw_params |> map (fn p =>
521 val T = Sign.the_const_type thy p;
523 (case Term.add_tvarsT T [] of
524 [((a, _), S)] => check_constraint (a, S)
525 | _ => error ("Exactly one type variable expected in class parameter " ^ quote p));
526 val T' = Term.map_type_tvar (fn _ => TFree (Name.aT, [class])) T;
533 (case Term.add_tfrees t [] of
534 [(a, S)] => check_constraint (a, S)
536 | _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t);
538 |> Term.map_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))
539 |> Logic.close_form);
541 val axiomss = map (map (prep_axiom o Sign.cert_prop thy) o snd) raw_specs;
542 val name_atts = map fst raw_specs;
547 val conjs = Logic.mk_of_sort (Term.aT [], super) @ flat axiomss;
549 Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);
551 val ([def], def_thy) =
553 |> Sign.primitive_class (bclass, super)
554 |> PureThy.add_defs false [((Thm.def_binding bconst, class_eq), [])];
555 val (raw_intro, (raw_classrel, raw_axioms)) =
556 split_defined (length conjs) def ||> chop (length super);
561 val class_triv = Thm.class_triv def_thy class;
562 val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =
564 |> Sign.qualified_path true bconst
565 |> PureThy.note_thmss ""
566 [((Binding.name introN, []), [([Drule.export_without_context raw_intro], [])]),
567 ((Binding.name superN, []), [(map Drule.export_without_context raw_classrel, [])]),
568 ((Binding.name axiomsN, []),
569 [(map (fn th => Drule.export_without_context (class_triv RS th)) raw_axioms, [])])]
570 ||> Sign.restore_naming def_thy;
575 val axclass = make_axclass ((def, intro, axioms), params);
578 |> fold (snd oo put_trancl_classrel) (map (pair class) super ~~ classrel)
579 |> Sign.qualified_path false bconst
580 |> PureThy.note_thmss "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> snd
581 |> Sign.restore_naming facts_thy
582 |> map_axclasses (fn (axclasses, parameters) =>
583 (Symtab.update (class, axclass) axclasses,
584 fold (fn (x, _) => add_param pp (x, class)) params parameters));
586 in (class, result_thy) end;
590 (** axiomatizations **)
594 (* old-style axioms *)
596 fun add_axiom (b, prop) =
597 Thm.add_axiom (b, prop) #->
598 (fn (_, thm) => PureThy.add_thm ((b, Drule.export_without_context thm), []));
600 fun axiomatize prep mk name add raw_args thy =
602 val args = prep thy raw_args;
604 val names = name args;
607 |> fold_map add_axiom (map Binding.name names ~~ specs)
611 fun ax_classrel prep =
612 axiomatize (map o prep) (map Logic.mk_classrel)
613 (map (prefix classrel_prefix o Logic.name_classrel)) add_classrel;
616 axiomatize (prep o ProofContext.init) Logic.mk_arities
617 (map (prefix arity_prefix) o Logic.name_arities) add_arity;
620 (Logic.const_of_class c, Term.itselfT (Term.aT []) --> propT);
622 fun ax_class prep_class prep_classrel (bclass, raw_super) thy =
624 val class = Sign.full_name thy bclass;
625 val super = map (prep_class thy) raw_super |> Sign.minimize_sort thy;
628 |> Sign.primitive_class (bclass, super)
629 |> ax_classrel prep_classrel (map (fn c => (class, c)) super)
630 |> Theory.add_deps "" (class_const class) (map class_const super)
635 val axiomatize_class = ax_class Sign.certify_class cert_classrel;
636 val axiomatize_class_cmd = ax_class (ProofContext.read_class o ProofContext.init) read_classrel;
637 val axiomatize_classrel = ax_classrel cert_classrel;
638 val axiomatize_classrel_cmd = ax_classrel read_classrel;
639 val axiomatize_arity = ax_arity ProofContext.cert_arity;
640 val axiomatize_arity_cmd = ax_arity ProofContext.read_arity;