(* 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 AX_CLASS =
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 thynames_of_arity: theory -> class * string -> string list
val param_of_inst: theory -> string * string -> string
val inst_of_param: theory -> string -> (string * string) option
val unoverload: theory -> thm -> thm
val overload: theory -> thm -> thm
val unoverload_conv: theory -> conv
val overload_conv: theory -> 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 -> tactic -> theory -> theory
val prove_arity: string * sort list * sort -> tactic -> theory -> theory
val define_class: binding * class list -> string list ->
(Thm.binding * term list) list -> theory -> class * theory
val axiomatize_class: binding * class list -> theory -> theory
val axiomatize_class_cmd: binding * xstring list -> theory -> theory
val axiomatize_classrel: (class * class) list -> theory -> theory
val axiomatize_classrel_cmd: (xstring * xstring) list -> theory -> theory
val axiomatize_arity: arity -> theory -> theory
val axiomatize_arity_cmd: xstring * string list * string -> theory -> theory
end;
structure AxClass: AX_CLASS =
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,
proven_classrels: thm Symreltab.table,
proven_arities: ((class * sort list) * (thm * string)) list Symtab.table,
(*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)*),
diff_classrels: (class * class) list};
fun make_data
(axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =
Data {axclasses = axclasses, params = params, proven_classrels = proven_classrels,
proven_arities = proven_arities, inst_params = inst_params,
diff_classrels = diff_classrels};
fun diff_table tab1 tab2 =
Symreltab.fold (fn (x, _) => if Symreltab.defined tab2 x then I else cons x) tab1 [];
structure Data = Theory_Data_PP
(
type T = data;
val empty =
make_data (Symtab.empty, [], Symreltab.empty, Symtab.empty, (Symtab.empty, Symtab.empty), []);
val extend = I;
fun merge pp
(Data {axclasses = axclasses1, params = params1, proven_classrels = proven_classrels1,
proven_arities = proven_arities1, inst_params = inst_params1,
diff_classrels = diff_classrels1},
Data {axclasses = axclasses2, params = params2, proven_classrels = proven_classrels2,
proven_arities = proven_arities2, inst_params = inst_params2,
diff_classrels = diff_classrels2}) =
let
val ctxt = Syntax.init_pretty pp;
val axclasses' = Symtab.merge (K true) (axclasses1, axclasses2);
val params' =
if null params1 then params2
else
fold_rev (fn p => if member (op =) params1 p then I else add_param ctxt p)
params2 params1;
(*transitive closure of classrels and arity completion is done in Theory.at_begin hook*)
val proven_classrels' = Symreltab.join (K #1) (proven_classrels1, proven_classrels2);
val proven_arities' =
Symtab.join (K (Library.merge (eq_fst op =))) (proven_arities1, proven_arities2);
val diff_classrels' =
diff_table proven_classrels1 proven_classrels2 @
diff_table proven_classrels2 proven_classrels1 @
diff_classrels1 @ diff_classrels2;
val inst_params' =
(Symtab.join (K (Symtab.merge (K true))) (#1 inst_params1, #1 inst_params2),
Symtab.merge (K true) (#2 inst_params1, #2 inst_params2));
in
make_data
(axclasses', params', proven_classrels', proven_arities', inst_params', diff_classrels')
end;
);
fun map_data f =
Data.map (fn Data {axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels} =>
make_data (f (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels)));
fun map_axclasses f =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
(f axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels));
fun map_params f =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
(axclasses, f params, proven_classrels, proven_arities, inst_params, diff_classrels));
fun map_proven_classrels f =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
(axclasses, params, f proven_classrels, proven_arities, inst_params, diff_classrels));
fun map_proven_arities f =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
(axclasses, params, proven_classrels, f proven_arities, inst_params, diff_classrels));
fun map_inst_params f =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, diff_classrels) =>
(axclasses, params, proven_classrels, proven_arities, f inst_params, diff_classrels));
val clear_diff_classrels =
map_data (fn (axclasses, params, proven_classrels, proven_arities, inst_params, _) =>
(axclasses, params, proven_classrels, proven_arities, inst_params, []));
val rep_data = Data.get #> (fn Data args => args);
val axclasses_of = #axclasses o rep_data;
val params_of = #params o rep_data;
val proven_classrels_of = #proven_classrels o rep_data;
val proven_arities_of = #proven_arities o rep_data;
val inst_params_of = #inst_params o rep_data;
val diff_classrels_of = #diff_classrels o rep_data;
(* axclasses with parameters *)
fun get_info thy c =
(case Symtab.lookup (axclasses_of thy) c of
SOME info => info
| NONE => error ("No such axclass: " ^ quote c));
fun all_params_of thy S =
let val 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 instances *)
val classrel_prefix = "classrel_";
val arity_prefix = "arity_";
fun instance_name (a, c) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;
infix 0 RSO;
fun (SOME a) RSO (SOME b) = SOME (a RS b)
| x RSO NONE = x
| NONE RSO y = y;
fun the_classrel thy (c1, c2) =
(case Symreltab.lookup (proven_classrels_of thy) (c1, c2) of
SOME thm => thm
| NONE => error ("Unproven class relation " ^
Syntax.string_of_classrel (Proof_Context.init_global thy) [c1, c2])); (* FIXME stale thy (!?) *)
fun put_trancl_classrel ((c1, c2), th) thy =
let
val classes = Sorts.classes_of (Sign.classes_of thy);
val classrels = proven_classrels_of thy;
fun reflcl_classrel (c1', c2') =
if c1' = c2' then NONE else SOME (Thm.transfer thy (the_classrel thy (c1', c2')));
fun gen_classrel (c1_pred, c2_succ) =
let
val th' =
the ((reflcl_classrel (c1_pred, c1) RSO SOME th) RSO reflcl_classrel (c2, c2_succ))
|> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [])))] []
|> Thm.close_derivation;
in ((c1_pred, c2_succ), th') end;
val new_classrels =
Library.map_product pair
(c1 :: Graph.immediate_preds classes c1)
(c2 :: Graph.immediate_succs classes c2)
|> filter_out ((op =) orf Symreltab.defined classrels)
|> map gen_classrel;
val needed = not (null new_classrels);
in
(needed,
if needed then map_proven_classrels (fold Symreltab.update new_classrels) thy
else thy)
end;
fun complete_classrels thy =
let
val classrels = proven_classrels_of thy;
val diff_classrels = diff_classrels_of thy;
val (needed, thy') = (false, thy) |>
fold (fn rel => fn (needed, thy) =>
put_trancl_classrel (rel, Symreltab.lookup classrels rel |> the) thy
|>> (fn b => needed orelse b))
diff_classrels;
in
if null diff_classrels then NONE
else SOME (clear_diff_classrels thy')
end;
fun the_arity thy (a, Ss, c) =
(case AList.lookup (op =) (Symtab.lookup_list (proven_arities_of thy) a) (c, Ss) of
SOME (thm, _) => thm
| NONE => error ("Unproven type arity " ^
Syntax.string_of_arity (Proof_Context.init_global thy) (a, Ss, [c]))); (* FIXME stale thy (!?) *)
fun thynames_of_arity thy (c, a) =
Symtab.lookup_list (proven_arities_of thy) a
|> map_filter (fn ((c', _), (_, name)) => if c = c' then SOME name else NONE)
|> rev;
fun insert_arity_completions thy t ((c, Ss), ((th, thy_name))) (finished, arities) =
let
val algebra = Sign.classes_of thy;
val ars = Symtab.lookup_list arities t;
val super_class_completions =
Sign.super_classes thy c
|> filter_out (fn c1 => exists (fn ((c2, Ss2), _) =>
c1 = c2 andalso Sorts.sorts_le algebra (Ss2, Ss)) ars);
val names = Name.invent Name.context Name.aT (length Ss);
val std_vars = map (fn a => SOME (ctyp_of thy (TVar ((a, 0), [])))) names;
val completions = super_class_completions |> map (fn c1 =>
let
val th1 =
(th RS Thm.transfer thy (the_classrel thy (c, c1)))
|> Drule.instantiate' std_vars []
|> Thm.close_derivation;
in ((th1, thy_name), c1) end);
val finished' = finished andalso null completions;
val arities' = fold (fn (th, c1) => Symtab.cons_list (t, ((c1, Ss), th))) completions arities;
in (finished', arities') end;
fun put_arity ((t, Ss, c), th) thy =
let val ar = ((c, Ss), (th, Context.theory_name thy)) in
thy
|> map_proven_arities
(Symtab.insert_list (eq_fst op =) (t, ar) #>
curry (insert_arity_completions thy t ar) true #> #2)
end;
fun complete_arities thy =
let
val arities = proven_arities_of thy;
val (finished, arities') =
Symtab.fold (fn (t, ars) => fold (insert_arity_completions thy t) ars) arities (true, arities);
in
if finished then NONE
else SOME (map_proven_arities (K arities') thy)
end;
val _ = Context.>> (Context.map_theory
(Theory.at_begin complete_classrels #>
Theory.at_begin complete_arities));
val _ = Proofterm.install_axclass_proofs
{classrel_proof = Thm.proof_of oo the_classrel,
arity_proof = Thm.proof_of oo the_arity};
(* 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 c' => c'
| NONE => error ("No instance parameter for constant " ^ quote c ^ " on type " ^ quote tyco));
fun add_inst_param (c, tyco) inst =
(map_inst_params o apfst o Symtab.map_default (c, Symtab.empty)) (Symtab.update_new (tyco, inst))
#> (map_inst_params o apsnd) (Symtab.update_new (#1 inst, (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 thy =
Symtab.fold (Symtab.fold (cons o #2 o #2) o #2) (#1 (inst_params_of thy)) [];
fun get_inst_tyco consts = try (#1 o dest_Type o the_single o Consts.typargs consts);
fun unoverload thy = Raw_Simplifier.simplify true (inst_thms thy);
fun overload thy = Raw_Simplifier.simplify true (map Thm.symmetric (inst_thms thy));
fun unoverload_conv thy = Raw_Simplifier.rewrite true (inst_thms thy);
fun overload_conv thy = Raw_Simplifier.rewrite true (map Thm.symmetric (inst_thms thy));
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 **)
(* class relations *)
fun cert_classrel thy raw_rel =
let
val string_of_sort = Syntax.string_of_sort_global thy;
val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;
val _ = Sign.primitive_classrel (c1, c2) (Theory.copy thy);
val _ =
(case subtract (op =) (all_params_of thy [c1]) (all_params_of thy [c2]) of
[] => ()
| xs => raise TYPE ("Class " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^
commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));
in (c1, c2) end;
fun read_classrel thy raw_rel =
cert_classrel thy (pairself (Proof_Context.read_class (Proof_Context.init_global thy)) raw_rel)
handle TYPE (msg, _, _) => error msg;
(* declaration and definition of instances of overloaded constants *)
fun inst_tyco_of thy (c, T) =
(case get_inst_tyco (Sign.consts_of thy) (c, T) of
SOME tyco => tyco
| NONE => error ("Illegal type for instantiation of class parameter: " ^
quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T)));
fun declare_overloaded (c, T) thy =
let
val class =
(case class_of_param thy c of
SOME class => class
| NONE => error ("Not a class parameter: " ^ quote c));
val tyco = inst_tyco_of thy (c, T);
val name_inst = instance_name (tyco, class) ^ "_inst";
val c' = instance_name (tyco, c);
val T' = Type.strip_sorts 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 false true
(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.conceal (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 false false (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 handle TYPE _ => err ();
val (th', thy') = Global_Theory.store_thm
(Binding.name (prefix classrel_prefix (Logic.name_classrel (c1, c2))), th) thy;
val th'' = th'
|> Thm.unconstrainT
|> Drule.instantiate' [SOME (ctyp_of thy' (TVar ((Name.aT, 0), [])))] [];
in
thy'
|> Sign.primitive_classrel (c1, c2)
|> (#2 oo put_trancl_classrel) ((c1, c2), th'')
|> perhaps complete_arities
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 (th', thy') = Global_Theory.store_thm
(Binding.name (prefix arity_prefix (Logic.name_arity arity)), th) thy;
val args = Name.invent_names Name.context Name.aT Ss;
val T = Type (t, map TFree args);
val std_vars = map (fn (a, S) => SOME (ctyp_of thy' (TVar ((a, 0), [])))) args;
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 T);
val th'' = th'
|> Thm.unconstrainT
|> Drule.instantiate' std_vars [];
in
thy'
|> fold (#2 oo declare_overloaded) missing_params
|> Sign.primitive_arity (t, Ss, [c])
|> put_arity ((t, Ss, c), th'')
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)) (K tac) 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 names = map (prefix arity_prefix) (Logic.name_arities arity);
val props = Logic.mk_arities arity;
val ths = Goal.prove_multi ctxt [] [] props
(fn _ => Goal.precise_conjunction_tac (length props) 1 THEN tac) 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 (fn _ => TFree (Name.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 (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))
|> 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_defs false [((Thm.def_binding bconst, class_eq), [])];
val (raw_intro, (raw_classrel, raw_axioms)) =
split_defined (length conjs) def ||> chop (length super);
(* facts *)
val class_triv = Thm.class_triv def_thy class;
val ([(_, [intro]), (_, classrel), (_, axioms)], facts_thy) =
def_thy
|> Sign.qualified_path true bconst
|> Global_Theory.note_thmss ""
[((Binding.name "intro", []), [([Drule.export_without_context raw_intro], [])]),
((Binding.name "super", []), [(map Drule.export_without_context raw_classrel, [])]),
((Binding.name "axioms", []),
[(map (fn th => Drule.export_without_context (class_triv RS th)) raw_axioms, [])])]
||> Sign.restore_naming def_thy;
(* result *)
val axclass = make_axclass (def, intro, axioms, params);
val result_thy =
facts_thy
|> fold (#2 oo put_trancl_classrel) (map (pair class) super ~~ classrel)
|> Sign.qualified_path false bconst
|> Global_Theory.note_thmss "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms))
|> #2
|> Sign.restore_naming facts_thy
|> map_axclasses (Symtab.update (class, axclass))
|> map_params (fold (fn (x, _) => add_param ctxt (x, class)) params);
in (class, result_thy) end;
(** axiomatizations **)
local
(*old-style axioms*)
fun axiomatize 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 ax_classrel prep =
axiomatize (map o prep) (map Logic.mk_classrel)
(map (prefix classrel_prefix o Logic.name_classrel)) add_classrel;
fun ax_arity prep =
axiomatize (prep o Proof_Context.init_global) Logic.mk_arities
(map (prefix arity_prefix) o Logic.name_arities) add_arity;
fun class_const c =
(Logic.const_of_class c, Term.itselfT (Term.aT []) --> propT);
fun ax_class prep_class prep_classrel (bclass, raw_super) thy =
let
val class = Sign.full_name thy bclass;
val super = map (prep_class thy) raw_super |> Sign.minimize_sort thy;
in
thy
|> Sign.primitive_class (bclass, super)
|> ax_classrel prep_classrel (map (fn c => (class, c)) super)
|> Theory.add_deps_global "" (class_const class) (map class_const super)
end;
in
val axiomatize_class = ax_class Sign.certify_class cert_classrel;
val axiomatize_class_cmd =
ax_class (Proof_Context.read_class o Proof_Context.init_global) read_classrel;
val axiomatize_classrel = ax_classrel cert_classrel;
val axiomatize_classrel_cmd = ax_classrel read_classrel;
val axiomatize_arity = ax_arity Proof_Context.cert_arity;
val axiomatize_arity_cmd = ax_arity Proof_Context.read_arity;
end;
end;