(* Title: Pure/axclass.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Type classes as parameter records and predicates, with explicit
definitions and proofs.
*)
signature AX_CLASS =
sig
val get_definition: 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 params_of_class: theory -> class -> string * (string * typ) list
val param_tyvarname: string
val print_axclasses: theory -> unit
val cert_classrel: theory -> class * class -> class * class
val read_classrel: theory -> xstring * xstring -> class * class
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: bstring * class list -> string list ->
((bstring * attribute list) * term list) list -> theory -> class * theory
val axiomatize_class: bstring * xstring list -> theory -> theory
val axiomatize_class_i: bstring * class list -> theory -> theory
val axiomatize_classrel: (xstring * xstring) list -> theory -> theory
val axiomatize_classrel_i: (class * class) list -> theory -> theory
val axiomatize_arity: xstring * string list * string -> theory -> theory
val axiomatize_arity_i: arity -> theory -> theory
type cache
val cache: cache
val of_sort: theory -> typ * sort -> cache -> thm list * cache (*exception Sorts.CLASS_ERROR*)
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";
val param_tyvarname = "'a";
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));
(* setup data *)
structure AxClassData = TheoryDataFun
(
type T = axclasses * instances;
val empty = ((Symtab.empty, []), ([], Symtab.empty));
val copy = I;
val extend = I;
fun merge pp ((axclasses1, instances1), (axclasses2, instances2)) =
(merge_axclasses pp (axclasses1, axclasses2), (merge_instances (instances1, instances2)));
);
(* maintain axclasses *)
val get_axclasses = #1 o AxClassData.get;
fun map_axclasses f = AxClassData.map (apfst f);
val lookup_def = Symtab.lookup o #1 o get_axclasses;
fun get_definition 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));
fun params_of_class thy class =
(param_tyvarname, (#params o get_definition thy) class);
(* maintain instances *)
val get_instances = #2 o AxClassData.get;
fun map_instances f = AxClassData.map (apsnd f);
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 " ^ Sign.string_of_classrel 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 " ^ Sign.string_of_arity 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))));
(* print data *)
fun print_axclasses thy =
let
val axclasses = #1 (get_axclasses thy);
val ctxt = ProofContext.init thy;
fun pretty_axclass (class, AxClass {def, intro, axioms, params}) =
Pretty.block (Pretty.fbreaks
[Pretty.block
[Pretty.str "class ", ProofContext.pretty_sort ctxt [class], Pretty.str ":"],
Pretty.strs ("parameters:" :: params_of thy class),
ProofContext.pretty_fact ctxt ("def", [def]),
ProofContext.pretty_fact ctxt (introN, [intro]),
ProofContext.pretty_fact ctxt (axiomsN, axioms)]);
in Pretty.writeln (Pretty.chunks (map pretty_axclass (Symtab.dest axclasses))) end;
(** instances **)
(* class relations *)
fun cert_classrel thy raw_rel =
let
val (c1, c2) = pairself (Sign.certify_class thy) raw_rel;
val _ = Type.add_classrel (Sign.pp thy) (c1, c2) (Sign.tsig_of 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 _ = if map (Sign.certify_sort thy) Ss = Ss then () else err ();
in
thy
|> Sign.primitive_arity (t, Ss, [c])
|> put_arity ((t, Ss, c), Drule.unconstrainTs th)
end;
(* tactical proofs *)
fun prove_classrel raw_rel tac thy =
let
val (c1, c2) = cert_classrel thy raw_rel;
val th = Goal.prove (ProofContext.init thy) [] []
(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 (Sign.string_of_classrel thy [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 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 (ProofContext.init thy) [] [] 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 (Sign.string_of_arity thy arity));
in
thy
|> PureThy.add_thms (map (rpair []) (names ~~ ths))
|-> fold add_arity
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) params raw_specs thy =
let
val ctxt = ProofContext.init thy;
val pp = ProofContext.pp ctxt;
(* prepare specification *)
val bconst = Logic.const_of_class bclass;
val class = Sign.full_name thy bclass;
val super = Sign.certify_sort thy raw_super;
fun prep_axiom t =
(case Term.add_tfrees t [] of
[(a, S)] =>
if Sign.subsort thy (super, S) then t
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)
| [] => t
| _ => error ("Multiple type variables in class axiom:\n" ^ Pretty.string_of_term pp t))
|> map_types (Term.map_atyps (fn TFree _ => Term.aT [] | U => U))
|> Logic.close_form;
(*FIXME is ProofContext.cert_propp really neccessary?*)
val axiomss = ProofContext.cert_propp (ctxt, map (map (rpair []) o snd) raw_specs)
|> snd |> map (map (prep_axiom o fst));
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, [])])];
(* params *)
val params_typs = map (fn param =>
let
val ty = Sign.the_const_type thy param;
val _ = case Term.typ_tvars ty
of [_] => ()
| _ => error ("Exactly one type variable required in parameter " ^ quote param);
val ty' = Term.map_type_tvar (fn _ => TFree (param_tyvarname, [class])) ty;
in (param, ty') end) params;
(* result *)
val axclass = make_axclass ((def, intro, axioms), params_typs);
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.certify_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.read_class read_classrel;
val axiomatize_class_i = ax_class Sign.certify_class cert_classrel;
val axiomatize_classrel = ax_classrel read_classrel;
val axiomatize_classrel_i = ax_classrel cert_classrel;
val axiomatize_arity = ax_arity Sign.read_arity;
val axiomatize_arity_i = ax_arity Sign.cert_arity;
end;
(** explicit derivations -- cached **)
datatype cache = Types of (class * thm) list Typtab.table;
val cache = Types Typtab.empty;
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;
end;