(* Title: Pure/axclass.ML Author: Markus Wenzel, TU Muenchen
Type classes defined as predicates, associated with a record of parameters. Proven class relations and type arities.
*)
signature AXCLASS = sig type info = {def: thm, intro: thm, axioms: thm list, params: (string * typ) list} val get_info: theory -> class -> info val class_of_param: theory -> string -> class option val instance_name: string * class -> string val param_of_inst: theory -> string * string -> string val inst_of_param: theory -> string -> (string * string) option val unoverload: Proof.context -> thm -> thm val overload: Proof.context -> thm -> thm val unoverload_conv: Proof.context -> conv val overload_conv: Proof.context -> conv val lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option val unoverload_const: theory -> string * typ -> string val cert_classrel: theory -> class * class -> class * class val read_classrel: theory -> xstring * xstring -> class * class val declare_overloaded: string * typ -> theory -> term * theory val define_overloaded: binding -> string * term -> theory -> thm * theory val add_classrel: thm -> theory -> theory val add_arity: thm -> theory -> theory val prove_classrel: class * class -> (Proof.context -> tactic) -> theory -> theory val prove_arity: string * sort list * sort -> (Proof.context -> tactic) -> theory -> theory val define_class: binding * class list -> stringlist ->
(Thm.binding * term list) list -> theory -> class * theory val classrel_axiomatization: (class * class) list -> theory -> theory val arity_axiomatization: arity -> theory -> theory val class_axiomatization: binding * class list -> theory -> theory end;
structure Axclass: AXCLASS = struct
(** theory data **)
(* axclass info *)
type info =
{def: thm,
intro: thm,
axioms: thm list,
params: (string * typ) list};
fun make_axclass (def, intro, axioms, params): info =
{def = def, intro = intro, axioms = axioms, params = params};
(* class parameters (canonical order) *)
type param = string * class;
fun add_param ctxt ((x, c): param) params =
(case AList.lookup (op =) params x of
NONE => (x, c) :: params
| SOME c' =>
error ("Duplicate class parameter " ^ quote x ^ " for " ^ Syntax.string_of_sort ctxt [c] ^
(if c = c' then "" else " and " ^ Syntax.string_of_sort ctxt [c'])));
(* setup data *)
datatype data = Data of
{axclasses: info Symtab.table,
params: param list, (*arity theorems with theory name*)
inst_params:
(string * thm) Symtab.table Symtab.table * (*constant name ~> type constructor ~> (constant name, equation)*)
(string * string) Symtab.table (*constant name ~> (constant name, type constructor)*)};
fun rep_data (Data args) = args;
fun make_data (axclasses, params, inst_params) =
Data {axclasses = axclasses, params = params, inst_params = inst_params};
structure Data = Theory_Data'
( type T = data; val empty = make_data (Symtab.empty, [], (Symtab.empty, Symtab.empty)); fun merge args = let val ctxt0 = Syntax.init_pretty_global (#1 (hd args));
fun merge_params (params1, params2) = if null params1 then params2 else
fold_rev (fn p => if member (op =) params1 p then I else add_param ctxt0 p)
params2 params1;
val axclasses' = Library.foldl1 (Symtab.merge (K true)) (map (#axclasses o rep_data o #2) args); val params' = Library.foldl1 merge_params (map (#params o rep_data o #2) args); val inst_params' = Library.foldl1 merge_inst_params (map (#inst_params o rep_data o #2) args); in make_data (axclasses', params', inst_params') end;
);
fun map_data f =
Data.map (fn Data {axclasses, params, inst_params} =>
make_data (f (axclasses, params, inst_params)));
fun map_axclasses f =
map_data (fn (axclasses, params, inst_params) =>
(f axclasses, params, inst_params));
fun map_params f =
map_data (fn (axclasses, params, inst_params) =>
(axclasses, f params, inst_params));
fun map_inst_params f =
map_data (fn (axclasses, params, inst_params) =>
(axclasses, params, f inst_params));
val rep_theory_data = Data.get #> rep_data;
val axclasses_of = #axclasses o rep_theory_data; val params_of = #params o rep_theory_data; val inst_params_of = #inst_params o rep_theory_data;
(* axclasses with parameters *)
fun get_info thy c =
(case Symtab.lookup (axclasses_of thy) c of
SOME {def, intro, axioms, params} =>
{def = Thm.transfer thy def,
intro = Thm.transfer thy intro,
axioms = map (Thm.transfer thy) axioms,
params = params}
| NONE => error ("No such axclass: " ^ quote c));
fun all_params_of thy S = letval params = params_of thy; in fold (fn (x, c) => if Sign.subsort thy (S, [c]) then cons x else I) params [] end;
fun class_of_param thy = AList.lookup (op =) (params_of thy);
(* maintain instance parameters *)
fun get_inst_param thy (c, tyco) =
(case Symtab.lookup (the_default Symtab.empty (Symtab.lookup (#1 (inst_params_of thy)) c)) tyco of
SOME (a, th) => (a, Thm.transfer thy th)
| NONE => error ("No instance parameter for constant " ^ quote c ^ " on type " ^ quote tyco));
fun add_inst_param (c, tyco) (a, th) =
(map_inst_params o apfst o Symtab.map_default (c, Symtab.empty))
(Symtab.update_new (tyco, (a, Thm.trim_context th)))
#> (map_inst_params o apsnd) (Symtab.update_new (a, (c, tyco)));
val inst_of_param = Symtab.lookup o #2 o inst_params_of; val param_of_inst = #1 oo get_inst_param;
fun inst_thms ctxt =
Symtab.fold
(Symtab.fold (cons o #2 o #2) o #2) (#1 (inst_params_of (Proof_Context.theory_of ctxt))) []
|> map (Thm.transfer' ctxt);
fun get_inst_tyco consts = try (dest_Type_name o the_single o Consts.typargs consts);
fun unoverload ctxt = rewrite_rule ctxt (inst_thms ctxt); fun overload ctxt = rewrite_rule ctxt (map Thm.symmetric (inst_thms ctxt));
fun unoverload_conv ctxt = Raw_Simplifier.rewrite_wrt ctxt true (inst_thms ctxt); fun overload_conv ctxt = Raw_Simplifier.rewrite_wrt ctxt true (map Thm.symmetric (inst_thms ctxt));
fun lookup_inst_param consts params (c, T) =
(case get_inst_tyco consts (c, T) of
SOME tyco => AList.lookup (op =) params (c, tyco)
| NONE => NONE);
fun unoverload_const thy (c_ty as (c, _)) = if is_some (class_of_param thy c) then
(case get_inst_tyco (Sign.consts_of thy) c_ty of
SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c
| NONE => c) else c;
(** instances **)
val classrel_prefix = "classrel_"; val arity_prefix = "arity_";
fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;
(* class relations *)
fun cert_classrel thy raw_rel = let val string_of_sort = Syntax.string_of_sort_global thy; val (c1, c2) = apply2 (Sign.certify_class thy) raw_rel; val _ = Sign.primitive_classrel (c1, c2) thy; val _ =
(case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of
[] => ()
| xs => raiseTYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^
commas_quote xs ^ " of " ^ string_of_sort [c2], [], [])); in (c1, c2) end;
(* declaration and definition of instances of overloaded constants *)
fun inst_tyco_of thy (c, T) =
get_inst_tyco (Sign.consts_of thy) (c, T)
|> \<^if_none>\<open>
error ("Illegal type for instantiation of class parameter: " ^
quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T))\<close>;
fun declare_overloaded (c, T) thy = let val class = class_of_param thy c |> \<^if_none>\<open>error ("Not a class parameter: " ^ quote c)\<close>; val tyco = inst_tyco_of thy (c, T); val name_inst = instance_name (tyco, class) ^ "_inst"; val c' = instance_name (tyco, c); val T' = Term.strip_sortsT T; in
thy
|> Sign.qualified_path true (Binding.name name_inst)
|> Sign.declare_const_global ((Binding.name c', T'), NoSyn)
|-> (fn const' as Const (c'', _) =>
Thm.add_def_global falsetrue
(Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))
#>> apsnd Thm.varifyT_global
#-> (fn (_, thm) => add_inst_param (c, tyco) (c'', thm)
#> Global_Theory.add_thm ((Binding.concealed (Binding.name c'), thm), [])
#> #2
#> pair (Const (c, T))))
||> Sign.restore_naming thy end;
fun define_overloaded b (c, t) thy = let val T = Term.fastype_of t; val tyco = inst_tyco_of thy (c, T); val (c', eq) = get_inst_param thy (c, tyco); val prop = Logic.mk_equals (Const (c', T), t); val b' = Thm.def_binding_optional (Binding.name (instance_name (tyco, c))) b; in
thy
|> Thm.add_def_global falsefalse (b', prop)
|>> (fn (_, thm) => Drule.transitive_thm OF [eq, thm]) end;
(* primitive rules *)
fun add_classrel raw_th thy = let val th = Thm.strip_shyps (Thm.transfer thy raw_th); val prop = Thm.plain_prop_of th; fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]); val rel = Logic.dest_classrel prop handle TERM _ => err (); val (c1, c2) = cert_classrel thy rel handleTYPE _ => err (); val binding =
Binding.concealed (Binding.name (prefix classrel_prefix (Logic.name_classrel (c1, c2)))); in thy |> Global_Theory.store_thm (binding, th) |-> Thm.add_classrel end;
fun add_arity raw_th thy = let val th = Thm.strip_shyps (Thm.transfer thy raw_th); val prop = Thm.plain_prop_of th; fun err () = raise THM ("add_arity: malformed type arity", 0, [th]); val arity as (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
val binding =
Binding.concealed (Binding.name (prefix arity_prefix (Logic.name_arity arity)));
val args = map TFree (Name.invent_types_global Ss); val missing_params =
Sign.complete_sort thy [c]
|> maps (these o Option.map #params o try (get_info thy))
|> filter_out (fn (const, _) => can (get_inst_param thy) (const, t))
|> (map o apsnd o map_atyps) (K (Type (t, args))); in
thy
|> Global_Theory.store_thm (binding, th)
|-> Thm.add_arity
|> fold (#2 oo declare_overloaded) missing_params end;
(* tactical proofs *)
fun prove_classrel raw_rel tac thy = let val ctxt = Proof_Context.init_global thy; val (c1, c2) = cert_classrel thy raw_rel; val th =
Goal.prove ctxt [] [] (Logic.mk_classrel (c1, c2)) (fn {context, ...} => tac context) handle ERROR msg =>
cat_error msg ("The error(s) above occurred while trying to prove class relation "^
quote (Syntax.string_of_classrel ctxt [c1, c2])); in
thy |> add_classrel th end;
fun prove_arity raw_arity tac thy = let val ctxt = Proof_Context.init_global thy; val arity = Proof_Context.cert_arity ctxt raw_arity; val props = Logic.mk_arities arity; val ths =
Goal.prove_common ctxt NONE [] [] props
(fn {context, ...} => Goal.precise_conjunction_tac (length props) 1 THEN tac context) handle ERROR msg =>
cat_error msg ("The error(s) above occurred while trying to prove type arity " ^
quote (Syntax.string_of_arity ctxt arity)); in
thy |> fold add_arity ths end;
(** class definitions **)
fun split_defined n eq = let val intro =
(eq RS Drule.equal_elim_rule2)
|> Conjunction.curry_balanced n
|> n = 0 ? Thm.eq_assumption 1; val dests = if n = 0 then [] else
(eq RS Drule.equal_elim_rule1)
|> Balanced_Tree.dest (fn th =>
(th RS Conjunction.conjunctionD1, th RS Conjunction.conjunctionD2)) n; in (intro, dests) end;
fun define_class (bclass, raw_super) raw_params raw_specs thy = let val ctxt = Syntax.init_pretty_global thy;
(* class *)
val bconst = Binding.map_name Logic.const_of_class bclass; val class = Sign.full_name thy bclass; val super = Sign.minimize_sort thy (Sign.certify_sort thy raw_super);
fun check_constraint (a, S) = if Sign.subsort thy (super, S) then () else error ("Sort constraint of type variable " ^
Syntax.string_of_typ (Config.put show_sorts true ctxt) (TFree (a, S)) ^ " needs to be weaker than " ^ Syntax.string_of_sort ctxt super);
(* params *)
val params = raw_params |> map (fn p => let val T = Sign.the_const_type thy p; val _ =
(case Term.add_tvarsT T [] of
[((a, _), S)] => check_constraint (a, S)
| _ => error ("Exactly one type variable expected in class parameter " ^ quote p)); val T' = Term.map_type_tvar (K (Term.aT [class])) T; in (p, T') end);
(* axioms *)
fun prep_axiom t =
(case Term.add_tfrees t [] of
[(a, S)] => check_constraint (a, S)
| [] => ()
| _ => error ("Multiple type variables in class axiom:\n" ^ Syntax.string_of_term ctxt t);
t
|> (Term.map_types o Term.map_atyps_same) (fn TFree _ => Term.aT [] | _ => raise Same.SAME)
|> Logic.close_form);
val axiomss = map (map (prep_axiom o Sign.cert_prop thy) o snd) raw_specs; val name_atts = map fst raw_specs;
(* definition *)
val conjs = Logic.mk_of_sort (Term.aT [], super) @ flat axiomss; val class_eq =
Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);
val (def, def_thy) =
thy
|> Sign.primitive_class (bclass, super)
|> Global_Theory.add_def (Thm.def_binding bconst, class_eq); val (raw_intro, (raw_classrel, raw_axioms)) =
split_defined (length conjs) def ||> chop (length super);
(*old-style axioms*) fun add_axioms prep mk name add raw_args thy = let val args = prep thy raw_args; val specs = mk args; val names = name args; in
thy
|> fold_map Thm.add_axiom_global (map Binding.name names ~~ specs)
|-> fold (add o Drule.export_without_context o snd) end;
fun class_const_dep c =
((Defs.Const, Logic.const_of_class c), [Term.aT []]);
in
val classrel_axiomatization =
add_axioms (map o cert_classrel) (map Logic.mk_classrel)
(map (prefix classrel_prefix o Logic.name_classrel)) add_classrel;
val arity_axiomatization =
add_axioms (Proof_Context.cert_arity o Proof_Context.init_global) Logic.mk_arities
(map (prefix arity_prefix) o Logic.name_arities) add_arity;
fun class_axiomatization (bclass, raw_super) thy = let val class = Sign.full_name thy bclass; val super = map (Sign.certify_class thy) raw_super |> Sign.minimize_sort thy; in
thy
|> Sign.primitive_class (bclass, super)
|> classrel_axiomatization (map (fn c => (class, c)) super)
|> Theory.add_deps_global "" (class_const_dep class) (map class_const_dep super) end;
end;
end;
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Die Informationen auf dieser Webseite wurden
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noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
