(* Title: Pure/axclass.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Type classes defined as predicates, associated with a record of
parameters.
*)
signature AX_CLASS =
sig
val define_class: bstring * class list -> string list ->
((bstring * attribute list) * term list) list -> theory -> class * 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 get_info: theory -> class ->
{def: thm, intro: thm, axioms: thm list, params: (string * typ) list}
val class_intros: theory -> thm list
val class_of_param: theory -> string -> class option
val cert_classrel: theory -> class * class -> class * class
val read_classrel: theory -> xstring * xstring -> class * class
val axiomatize_class: bstring * class list -> theory -> theory
val axiomatize_class_cmd: bstring * 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
val instance_name: string * class -> string
val declare_overloaded: string * typ -> theory -> term * theory
val define_overloaded: string -> string * term -> theory -> thm * theory
val inst_tyco_of: theory -> string * typ -> string option
val unoverload: theory -> thm -> thm
val overload: theory -> thm -> thm
val unoverload_conv: theory -> conv
val overload_conv: theory -> conv
val unoverload_const: theory -> string * typ -> string
val param_of_inst: theory -> string * string -> string
val inst_of_param: theory -> string -> (string * string) option
type cache
val of_sort: theory -> typ * sort -> cache -> thm list * cache (*exception Sorts.CLASS_ERROR*)
val cache: cache
val axiomsN: string
end;
structure AxClass: AX_CLASS =
struct
(** theory data **)
(* class parameters (canonical order) *)
type param = string * class;
fun add_param pp ((x, c): param) params =
(case AList.lookup (op =) params x of
NONE => (x, c) :: params
| SOME c' => error ("Duplicate class parameter " ^ quote x ^
" for " ^ Pretty.string_of_sort pp [c] ^
(if c = c' then "" else " and " ^ Pretty.string_of_sort pp [c'])));
fun merge_params _ ([], qs) = qs
| merge_params pp (ps, qs) =
fold_rev (fn q => if member (op =) ps q then I else add_param pp q) qs ps;
(* axclasses *)
val introN = "intro";
val superN = "super";
val axiomsN = "axioms";
datatype axclass = AxClass of
{def: thm,
intro: thm,
axioms: thm list,
params: (string * typ) list};
type axclasses = axclass Symtab.table * param list;
fun make_axclass ((def, intro, axioms), params) =
AxClass {def = def, intro = intro, axioms = axioms, params = params};
fun merge_axclasses pp ((tab1, params1), (tab2, params2)) : axclasses =
(Symtab.merge (K true) (tab1, tab2), merge_params pp (params1, params2));
(* instances *)
val classrel_prefix = "classrel_";
val arity_prefix = "arity_";
type instances =
((class * class) * thm) list *
((class * sort list) * thm) list Symtab.table;
fun merge_instances ((classrel1, arities1): instances, (classrel2, arities2)) =
(merge (eq_fst op =) (classrel1, classrel2),
Symtab.join (K (merge (eq_fst op =))) (arities1, arities2));
(* instance parameters *)
type 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 merge_inst_params ((const_param1, param_const1), (const_param2, param_const2)) =
(Symtab.join (K (Symtab.merge (K true))) (const_param1, const_param2),
Symtab.merge (K true) (param_const1, param_const2));
(* setup data *)
structure AxClassData = TheoryDataFun
(
type T = axclasses * (instances * inst_params);
val empty = ((Symtab.empty, []), (([], Symtab.empty), (Symtab.empty, Symtab.empty)));
val copy = I;
val extend = I;
fun merge pp ((axclasses1, (instances1, inst_params1)), (axclasses2, (instances2, inst_params2))) =
(merge_axclasses pp (axclasses1, axclasses2),
(merge_instances (instances1, instances2), merge_inst_params (inst_params1, inst_params2)));
);
(* maintain axclasses *)
val get_axclasses = #1 o AxClassData.get;
val map_axclasses = AxClassData.map o apfst;
val lookup_def = Symtab.lookup o #1 o get_axclasses;
fun get_info thy c =
(case lookup_def thy c of
SOME (AxClass info) => info
| NONE => error ("No such axclass: " ^ quote c));
fun class_intros thy =
let
fun add_intro c =
(case lookup_def thy c of SOME (AxClass {intro, ...}) => cons intro | _ => I);
val classes = Sign.all_classes thy;
in map (Thm.class_triv thy) classes @ fold add_intro classes [] end;
fun get_params thy pred =
let val params = #2 (get_axclasses thy);
in fold (fn (x, c) => if pred c then cons x else I) params [] end;
fun params_of thy c = get_params thy (fn c' => c' = c);
fun all_params_of thy S = get_params thy (fn c => Sign.subsort thy (S, [c]));
fun class_of_param thy = AList.lookup (op =) (#2 (get_axclasses thy));
(* maintain instances *)
fun instance_name (a, c) = NameSpace.base c ^ "_" ^ NameSpace.base a;
val get_instances = #1 o #2 o AxClassData.get;
val map_instances = AxClassData.map o apsnd o apfst;
fun the_classrel thy (c1, c2) =
(case AList.lookup (op =) (#1 (get_instances thy)) (c1, c2) of
SOME th => Thm.transfer thy th
| NONE => error ("Unproven class relation " ^
Syntax.string_of_classrel (ProofContext.init thy) [c1, c2]));
fun put_classrel arg = map_instances (fn (classrel, arities) =>
(insert (eq_fst op =) arg classrel, arities));
fun the_arity thy a (c, Ss) =
(case AList.lookup (op =) (Symtab.lookup_list (#2 (get_instances thy)) a) (c, Ss) of
SOME th => Thm.transfer thy th
| NONE => error ("Unproven type arity " ^
Syntax.string_of_arity (ProofContext.init thy) (a, Ss, [c])));
fun put_arity ((t, Ss, c), th) = map_instances (fn (classrel, arities) =>
(classrel, arities |> Symtab.insert_list (eq_fst op =) (t, ((c, Ss), th))));
(* maintain instance parameters *)
val get_inst_params = #2 o #2 o AxClassData.get;
val map_inst_params = AxClassData.map o apsnd o apsnd;
fun get_inst_param thy (c, tyco) =
(the o Symtab.lookup ((the o Symtab.lookup (fst (get_inst_params thy))) c)) 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 (fst inst, (c, tyco)));
val inst_of_param = Symtab.lookup o snd o get_inst_params;
val param_of_inst = fst oo get_inst_param;
fun inst_thms thy = (Symtab.fold (Symtab.fold (cons o snd o snd) o snd) o fst)
(get_inst_params thy) [];
fun inst_tyco_of thy = try (fst o dest_Type o the_single o Sign.const_typargs thy);
fun unoverload thy = MetaSimplifier.simplify true (inst_thms thy);
fun overload thy = MetaSimplifier.simplify true (map Thm.symmetric (inst_thms thy));
fun unoverload_conv thy = MetaSimplifier.rewrite true (inst_thms thy);
fun overload_conv thy = MetaSimplifier.rewrite true (map Thm.symmetric (inst_thms thy));
fun unoverload_const thy (c_ty as (c, _)) =
case class_of_param thy c
of SOME class => (case inst_tyco_of thy c_ty
of SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c
| NONE => c)
| NONE => c;
(** instances **)
(* class relations *)
fun cert_classrel thy raw_rel =
let
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 " ^ Sign.string_of_sort thy [c1] ^ " lacks parameter(s) " ^
commas_quote xs ^ " of " ^ Sign.string_of_sort thy [c2], [], []));
in (c1, c2) end;
fun read_classrel thy raw_rel =
cert_classrel thy (pairself (Sign.read_class thy) raw_rel)
handle TYPE (msg, _, _) => error msg;
(* primitive rules *)
fun add_classrel th thy =
let
fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
val prop = Thm.plain_prop_of (Thm.transfer thy th);
val rel = Logic.dest_classrel prop handle TERM _ => err ();
val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
in
thy
|> Sign.primitive_classrel (c1, c2)
|> put_classrel ((c1, c2), Drule.unconstrainTs th)
end;
fun add_arity th thy =
let
fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
val prop = Thm.plain_prop_of (Thm.transfer thy th);
val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();
(*FIXME turn this into a mere check as soon as "attach" has disappeared*)
val missing_params = Sign.complete_sort thy [c]
|> maps (these o Option.map (fn AxClass { params, ... } => params) o lookup_def thy)
|> filter_out (fn (p, _) => can (get_inst_param thy) (p, t));
fun declare_missing (p, T0) thy =
let
val name_inst = instance_name (t, c) ^ "_inst";
val p' = NameSpace.base p ^ "_" ^ NameSpace.base t;
val vs = Name.names Name.context Name.aT (replicate (Sign.arity_number thy t) []);
val T = map_atyps (fn _ => Type (t, map TFree vs)) T0;
in
thy
|> Sign.sticky_prefix name_inst
|> Sign.no_base_names
|> Sign.declare_const [] (p', T, NoSyn)
|-> (fn const' as Const (p'', _) => Thm.add_def false true
(Thm.def_name p', Logic.mk_equals (const', Const (p, T)))
#>> Thm.varifyT
#-> (fn thm => add_inst_param (p, t) (p'', Thm.symmetric thm)
#> PureThy.note Thm.internalK (p', thm)
#> snd
#> Sign.restore_naming thy))
end;
in
thy
|> Sign.primitive_arity (t, Ss, [c])
|> put_arity ((t, Ss, c), Drule.unconstrainTs th)
|> fold declare_missing missing_params
end;
(* tactical proofs *)
fun prove_classrel raw_rel tac thy =
let
val ctxt = ProofContext.init thy;
val (c1, c2) = cert_classrel thy raw_rel;
val th = Goal.prove ctxt [] []
(Logic.mk_classrel (c1, c2)) (fn _ => 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
|> PureThy.add_thms [((prefix classrel_prefix (Logic.name_classrel (c1, c2)), th), [])]
|-> (fn [th'] => add_classrel th')
end;
fun prove_arity raw_arity tac thy =
let
val ctxt = ProofContext.init thy;
val arity = Sign.cert_arity thy 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
|> PureThy.add_thms (map (rpair []) (names ~~ ths))
|-> fold add_arity
end;
(* instance parameters and overloaded definitions *)
(* declaration and definition of instances of overloaded constants *)
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 = case inst_tyco_of thy (c, T)
of SOME tyco => tyco
| NONE => error ("Illegal type for instantiation of class parameter: "
^ quote (c ^ " :: " ^ Sign.string_of_typ thy T));
val name_inst = instance_name (tyco, class) ^ "_inst";
val c' = NameSpace.base c ^ "_" ^ NameSpace.base tyco;
val T' = Type.strip_sorts T;
in
thy
|> Sign.sticky_prefix name_inst
|> Sign.no_base_names
|> Sign.declare_const [] (c', T', NoSyn)
|-> (fn const' as Const (c'', _) => Thm.add_def false true
(Thm.def_name c', Logic.mk_equals (Const (c, T'), const'))
#>> Thm.varifyT
#-> (fn thm => add_inst_param (c, tyco) (c'', thm)
#> PureThy.note Thm.internalK (c', thm)
#> snd
#> Sign.restore_naming thy
#> pair (Const (c, T))))
end;
fun define_overloaded name (c, t) thy =
let
val T = Term.fastype_of t;
val SOME 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 name' = Thm.def_name_optional
(NameSpace.base c ^ "_" ^ NameSpace.base tyco) name;
in
thy
|> Thm.add_def false false (name', prop)
|>> (fn thm => Drule.transitive_thm OF [eq, thm])
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)
|> BalancedTree.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 = ProofContext.init thy;
val pp = Syntax.pp ctxt;
(* class *)
val bconst = 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 " ^
setmp show_sorts true (Pretty.string_of_typ pp) (TFree (a, S)) ^
" needs to be weaker than " ^ Pretty.string_of_sort pp 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" ^ Pretty.string_of_term pp 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 = map (curry Logic.mk_inclass (Term.aT [])) super @ flat axiomss;
val class_eq =
Logic.mk_equals (Logic.mk_inclass (Term.aT [], class), Logic.mk_conjunction_balanced conjs);
val ([def], def_thy) =
thy
|> Sign.primitive_class (bclass, super)
|> PureThy.add_defs_i false [((Thm.def_name 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
|> PureThy.note_thmss_qualified "" bconst
[((introN, []), [([Drule.standard raw_intro], [])]),
((superN, []), [(map Drule.standard raw_classrel, [])]),
((axiomsN, []), [(map (fn th => Drule.standard (class_triv RS th)) raw_axioms, [])])];
(* result *)
val axclass = make_axclass ((def, intro, axioms), params);
val result_thy =
facts_thy
|> fold put_classrel (map (pair class) super ~~ classrel)
|> Sign.add_path bconst
|> PureThy.note_thmss_i "" (name_atts ~~ map Thm.simple_fact (unflat axiomss axioms)) |> snd
|> Sign.restore_naming facts_thy
|> map_axclasses (fn (axclasses, parameters) =>
(Symtab.update (class, axclass) axclasses,
fold (fn (x, _) => add_param pp (x, class)) params parameters));
in (class, result_thy) end;
(** axiomatizations **)
local
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 |> PureThy.add_axioms_i (map (rpair []) (names ~~ specs)) |-> fold add 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 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 "" (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 Sign.read_class read_classrel;
val axiomatize_classrel = ax_classrel cert_classrel;
val axiomatize_classrel_cmd = ax_classrel read_classrel;
val axiomatize_arity = ax_arity Sign.cert_arity;
val axiomatize_arity_cmd = ax_arity Sign.read_arity;
end;
(** explicit derivations -- cached **)
datatype cache = Types of (class * thm) list Typtab.table;
local
fun lookup_type (Types cache) = AList.lookup (op =) o Typtab.lookup_list cache;
fun insert_type T der (Types cache) = Types (Typtab.insert_list (eq_fst op =) (T, der) cache);
fun derive_type _ (_, []) = []
| derive_type thy (typ, sort) =
let
val vars = Term.fold_atyps
(fn T as TFree (_, S) => insert (eq_fst op =) (T, S)
| T as TVar (_, S) => insert (eq_fst op =) (T, S)
| _ => I) typ [];
val hyps = vars
|> map (fn (T, S) => (T, Drule.sort_triv thy (T, S) ~~ S));
val ths = (typ, sort) |> Sorts.of_sort_derivation (Sign.pp thy) (Sign.classes_of thy)
{class_relation =
fn (th, c1) => fn c2 => th RS the_classrel thy (c1, c2),
type_constructor =
fn a => fn dom => fn c =>
let val Ss = map (map snd) dom and ths = maps (map fst) dom
in ths MRS the_arity thy a (c, Ss) end,
type_variable =
the_default [] o AList.lookup (op =) hyps};
in ths end;
in
fun of_sort thy (typ, sort) cache =
let
val sort' = filter (is_none o lookup_type cache typ) sort;
val ths' = derive_type thy (typ, sort')
handle ERROR msg => cat_error msg ("The error(s) above occurred for sort derivation: " ^
Sign.string_of_typ thy typ ^ " :: " ^ Sign.string_of_sort thy sort');
val cache' = cache |> fold (insert_type typ) (sort' ~~ ths');
val ths =
sort |> map (fn c =>
Goal.finish (the (lookup_type cache' typ c) RS
Goal.init (Thm.cterm_of thy (Logic.mk_inclass (typ, c))))
|> Thm.adjust_maxidx_thm ~1);
in (ths, cache') end;
end;
val cache = Types Typtab.empty;
end;