(* Title: Pure/axclass.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Axiomatic type class package.
*)
signature AX_CLASS =
sig
val quiet_mode: bool ref
val print_axclasses: theory -> unit
val get_info: theory -> string -> {super_classes: class list, intro: thm, axioms: thm list}
val add_axclass: bstring * xstring list -> ((bstring * string) * Attrib.src list) list ->
theory -> theory * {intro: thm, axioms: thm list}
val add_axclass_i: bstring * class list -> ((bstring * term) * theory attribute list) list ->
theory -> theory * {intro: thm, axioms: thm list}
val add_classrel_thms: thm list -> theory -> theory
val add_arity_thms: thm list -> theory -> theory
val add_inst_subclass: xstring * xstring -> tactic -> theory -> theory
val add_inst_subclass_i: class * class -> tactic -> theory -> theory
val add_inst_arity: xstring * string list * string -> tactic -> theory -> theory
val add_inst_arity_i: string * sort list * sort -> tactic -> theory -> theory
val instance_subclass: xstring * xstring -> theory -> Proof.state
val instance_subclass_i: class * class -> theory -> Proof.state
val instance_arity: xstring * string list * string -> theory -> Proof.state
val instance_arity_i: string * sort list * sort -> theory -> Proof.state
val intro_classes_tac: thm list -> tactic
val default_intro_classes_tac: thm list -> tactic
(*legacy interfaces*)
val axclass_tac: thm list -> tactic
val add_inst_subclass_x: xstring * xstring -> string list -> thm list
-> tactic option -> theory -> theory
val add_inst_arity_x: xstring * string list * string -> string list
-> thm list -> tactic option -> theory -> theory
end;
structure AxClass: AX_CLASS =
struct
(** utilities **)
(* messages *)
val quiet_mode = ref false;
fun message s = if ! quiet_mode then () else writeln s;
(* type vars *)
fun map_typ_frees f (Type (t, tys)) = Type (t, map (map_typ_frees f) tys)
| map_typ_frees f (TFree a) = f a
| map_typ_frees _ a = a;
val map_term_tfrees = map_term_types o map_typ_frees;
fun aT S = TFree ("'a", S);
fun dest_varT (TFree (x, S)) = ((x, ~1), S)
| dest_varT (TVar xi_S) = xi_S
| dest_varT T = raise TYPE ("dest_varT", [T], []);
(** abstract syntax operations **)
(* names *)
val introN = "intro";
val axiomsN = "axioms";
(* subclass relations as terms *)
fun mk_classrel (c1, c2) = Logic.mk_inclass (aT [c1], c2);
fun dest_classrel tm =
let
fun err () = raise TERM ("dest_classrel", [tm]);
val (ty, c2) = Logic.dest_inclass tm handle TERM _ => err ();
val c1 = (case dest_varT ty of (_, [c]) => c | _ => err ())
handle TYPE _ => err ();
in (c1, c2) end;
(* arities as terms *)
fun mk_arity (t, Ss, c) =
let
val tfrees = ListPair.map TFree (Term.invent_names [] "'a" (length Ss), Ss);
in Logic.mk_inclass (Type (t, tfrees), c) end;
fun mk_arities (t, Ss, S) = map (fn c => mk_arity (t, Ss, c)) S;
fun dest_arity tm =
let
fun err () = raise TERM ("dest_arity", [tm]);
val (ty, c) = Logic.dest_inclass tm handle TERM _ => err ();
val (t, tvars) =
(case ty of
Type (t, tys) => (t, map dest_varT tys handle TYPE _ => err ())
| _ => err ());
val ss =
if null (gen_duplicates (eq_fst (op =)) tvars)
then map snd tvars else err ();
in (t, ss, c) end;
(** axclass info **)
(* data kind 'Pure/axclasses' *)
type axclass_info =
{super_classes: class list,
intro: thm,
axioms: thm list};
structure AxclassesData = TheoryDataFun
(struct
val name = "Pure/axclasses";
type T = axclass_info Symtab.table;
val empty = Symtab.empty;
val copy = I;
val extend = I;
fun merge _ = Symtab.merge (K true);
fun print thy tab =
let
fun pretty_class c cs = Pretty.block
(Pretty.str (Sign.extern_class thy c) :: Pretty.str " <" :: Pretty.brk 1 ::
Pretty.breaks (map (Pretty.str o Sign.extern_class thy) cs));
fun pretty_thms name thms = Pretty.big_list (name ^ ":")
(map (Display.pretty_thm_sg thy) thms);
fun pretty_axclass (name, {super_classes, intro, axioms}) = Pretty.block (Pretty.fbreaks
[pretty_class name super_classes, pretty_thms introN [intro], pretty_thms axiomsN axioms]);
in Pretty.writeln (Pretty.chunks (map pretty_axclass (Symtab.dest tab))) end;
end);
val _ = Context.add_setup [AxclassesData.init];
val print_axclasses = AxclassesData.print;
(* get and put data *)
val lookup_info = Symtab.lookup o AxclassesData.get;
fun get_info thy c =
(case lookup_info thy c of
NONE => error ("Unknown axclass " ^ quote c)
| SOME info => info);
(* class_axms *)
fun class_axms thy =
let val classes = Sign.classes thy in
map (Thm.class_triv thy) classes @
List.mapPartial (Option.map #intro o lookup_info thy) classes
end;
(** add axiomatic type classes **)
local
fun err_bad_axsort ax c =
error ("Sort constraint in axiom " ^ quote ax ^ " not supersort of " ^ quote c);
fun err_bad_tfrees ax =
error ("More than one type variable in axiom " ^ quote ax);
fun ext_axclass prep_class prep_axm prep_att (bclass, raw_super_classes) raw_axioms_atts thy =
let
val class = Sign.full_name thy bclass;
val super_classes = map (prep_class thy) raw_super_classes;
val axms = map (prep_axm thy o fst) raw_axioms_atts;
val atts = map (map (prep_att thy) o snd) raw_axioms_atts;
(*declare class*)
val class_thy =
thy |> Theory.add_classes_i [(bclass, super_classes)];
(*prepare abstract axioms*)
fun abs_axm ax =
if null (term_tfrees ax) then
Logic.mk_implies (Logic.mk_inclass (aT [], class), ax)
else map_term_tfrees (K (aT [class])) ax;
val abs_axms = map (abs_axm o #2) axms;
fun axm_sort (name, ax) =
(case term_tfrees ax of
[] => []
| [(_, S)] => if Sign.subsort class_thy ([class], S) then S else err_bad_axsort name class
| _ => err_bad_tfrees name);
val axS = Sign.certify_sort class_thy (List.concat (map axm_sort axms));
val int_axm = Logic.close_form o map_term_tfrees (K (aT axS));
fun inclass c = Logic.mk_inclass (aT axS, c);
val intro_axm = Logic.list_implies
(map inclass super_classes @ map (int_axm o #2) axms, inclass class);
(*declare axioms and rule*)
val (axms_thy, ([intro], [axioms])) =
class_thy
|> Theory.add_path (Sign.const_of_class bclass)
|> PureThy.add_axioms_i [Thm.no_attributes (introN, intro_axm)]
|>>> PureThy.add_axiomss_i [Thm.no_attributes (axiomsN, abs_axms)];
val info = {super_classes = super_classes, intro = intro, axioms = axioms};
(*store info*)
val final_thy =
axms_thy
|> Theory.add_finals_i false [Const (Sign.const_of_class class, a_itselfT --> propT)]
|> (#1 o PureThy.add_thms ((map #1 axms ~~ axioms) ~~ atts))
|> Theory.restore_naming class_thy
|> AxclassesData.map (Symtab.update (class, info));
in (final_thy, {intro = intro, axioms = axioms}) end;
in
val add_axclass = ext_axclass Sign.intern_class Theory.read_axm Attrib.global_attribute;
val add_axclass_i = ext_axclass (K I) Theory.cert_axm (K I);
end;
(** proven class instantiation **)
(* add thms to type signature *)
fun add_classrel_thms thms thy =
let
fun prep_thm thm =
let
val prop = Drule.plain_prop_of (Thm.transfer thy thm);
val (c1, c2) = dest_classrel prop handle TERM _ =>
raise THM ("add_classrel_thms: not a class relation", 0, [thm]);
in (c1, c2) end;
in Theory.add_classrel_i (map prep_thm thms) thy end;
fun add_arity_thms thms thy =
let
fun prep_thm thm =
let
val prop = Drule.plain_prop_of (Thm.transfer thy thm);
val (t, ss, c) = dest_arity prop handle TERM _ =>
raise THM ("add_arity_thms: not an arity", 0, [thm]);
in (t, ss, [c]) end;
in Theory.add_arities_i (map prep_thm thms) thy end;
(* prepare classes and arities *)
fun read_class thy c = Sign.certify_class thy (Sign.intern_class thy c);
fun test_classrel thy cc = (Type.add_classrel (Sign.pp thy) [cc] (Sign.tsig_of thy); cc);
fun cert_classrel thy = test_classrel thy o Library.pairself (Sign.certify_class thy);
fun read_classrel thy = test_classrel thy o Library.pairself (read_class thy);
fun test_arity thy ar = (Type.add_arities (Sign.pp thy) [ar] (Sign.tsig_of thy); ar);
fun prep_arity prep_tycon prep_sort prep thy (t, Ss, x) =
test_arity thy (prep_tycon thy t, map (prep_sort thy) Ss, prep thy x);
val read_arity = prep_arity Sign.intern_type Sign.read_sort Sign.read_sort;
val cert_arity = prep_arity (K I) Sign.certify_sort Sign.certify_sort;
(* instance declarations -- tactical proof *)
local
fun ext_inst_subclass prep_classrel raw_cc tac thy =
let
val (c1, c2) = prep_classrel thy raw_cc;
val txt = quote (Sign.string_of_classrel thy [c1, c2]);
val _ = message ("Proving class inclusion " ^ txt ^ " ...");
val th = Tactic.prove thy [] [] (mk_classrel (c1, c2)) (K tac) handle ERROR =>
error ("The error(s) above occurred while trying to prove " ^ txt);
in add_classrel_thms [th] thy end;
fun ext_inst_arity prep_arity raw_arity tac thy =
let
val arity = prep_arity thy raw_arity;
val txt = quote (Sign.string_of_arity thy arity);
val _ = message ("Proving type arity " ^ txt ^ " ...");
val props = (mk_arities arity);
val ths = Tactic.prove_multi thy [] [] props
(fn _ => Tactic.smart_conjunction_tac (length props) THEN tac)
handle ERROR => error ("The error(s) above occurred while trying to prove " ^ txt);
in add_arity_thms ths thy end;
in
val add_inst_subclass = ext_inst_subclass read_classrel;
val add_inst_subclass_i = ext_inst_subclass cert_classrel;
val add_inst_arity = ext_inst_arity read_arity;
val add_inst_arity_i = ext_inst_arity cert_arity;
end;
(* instance declarations -- Isar proof *)
local
fun gen_instance mk_prop add_thms inst thy = thy
|> ProofContext.init
|> Proof.theorem_i Drule.internalK (K (fold add_thms)) NONE ("", [])
(map (fn t => (("", []), [(t, ([], []))])) (mk_prop thy inst));
in
val instance_subclass =
gen_instance (single oo (mk_classrel oo read_classrel)) add_classrel_thms;
val instance_subclass_i =
gen_instance (single oo (mk_classrel oo cert_classrel)) add_classrel_thms;
val instance_arity = gen_instance (mk_arities oo read_arity) add_arity_thms;
val instance_arity_i = gen_instance (mk_arities oo cert_arity) add_arity_thms;
end;
(* tactics and methods *)
fun intro_classes_tac facts st =
(ALLGOALS (Method.insert_tac facts THEN'
REPEAT_ALL_NEW (resolve_tac (class_axms (Thm.theory_of_thm st))))
THEN Tactic.distinct_subgoals_tac) st;
fun default_intro_classes_tac [] = intro_classes_tac []
| default_intro_classes_tac _ = Tactical.no_tac; (*no error message!*)
fun default_tac rules ctxt facts =
HEADGOAL (Method.some_rule_tac rules ctxt facts) ORELSE
default_intro_classes_tac facts;
val _ = Context.add_setup [Method.add_methods
[("intro_classes", Method.no_args (Method.METHOD intro_classes_tac),
"back-chain introduction rules of axiomatic type classes"),
("default", Method.thms_ctxt_args (Method.METHOD oo default_tac), "apply some intro/elim rule")]];
(** outer syntax **)
local structure P = OuterParse and K = OuterKeyword in
val axclassP =
OuterSyntax.command "axclass" "define axiomatic type class" K.thy_decl
((P.name -- Scan.optional ((P.$$$ "\\<subseteq>" || P.$$$ "<") |--
P.!!! (P.list1 P.xname)) []) -- Scan.repeat P.spec_name
>> (Toplevel.theory o (#1 oo uncurry add_axclass)));
val instanceP =
OuterSyntax.command "instance" "prove type arity or subclass relation" K.thy_goal
((P.xname -- ((P.$$$ "\\<subseteq>" || P.$$$ "<") |-- P.!!! P.xname) >> instance_subclass ||
P.xname -- (P.$$$ "::" |-- P.!!! P.arity) >> (instance_arity o P.triple2))
>> (Toplevel.print oo Toplevel.theory_to_proof));
val _ = OuterSyntax.add_parsers [axclassP, instanceP];
end;
(** old-style instantiation proofs **)
(* axclass_tac *)
(*(1) repeatedly resolve goals of form "OFCLASS(ty, c_class)",
try class_trivs first, then "cI" axioms
(2) rewrite goals using user supplied definitions
(3) repeatedly resolve goals with user supplied non-definitions*)
val is_def = Logic.is_equals o #prop o rep_thm;
fun axclass_tac thms =
let
val defs = List.filter is_def thms;
val non_defs = filter_out is_def thms;
in
intro_classes_tac [] THEN
TRY (rewrite_goals_tac defs) THEN
TRY (REPEAT_FIRST (fn i => assume_tac i ORELSE resolve_tac non_defs i))
end;
(* instance declarations *)
local
fun prove mk_prop str_of thy prop thms usr_tac =
(Tactic.prove thy [] [] (mk_prop prop)
(K (axclass_tac thms THEN (if_none usr_tac all_tac)))
handle ERROR => error ("The error(s) above occurred while trying to prove " ^
quote (str_of thy prop))) |> Drule.standard;
val prove_subclass = prove mk_classrel (fn thy => fn (c1, c2) =>
Pretty.string_of_classrel (Sign.pp thy) [c1, c2]);
val prove_arity = prove mk_arity (fn thy => fn (t, Ss, c) =>
Pretty.string_of_arity (Sign.pp thy) (t, Ss, [c]));
fun witnesses thy names thms =
PureThy.get_thmss thy (map Name names) @
thms @
List.filter is_def (map snd (axioms_of thy));
in
fun add_inst_subclass_x raw_c1_c2 names thms usr_tac thy =
let val (c1, c2) = read_classrel thy raw_c1_c2 in
message ("Proving class inclusion " ^ quote (Sign.string_of_classrel thy [c1, c2]) ^ " ...");
thy |> add_classrel_thms [prove_subclass thy (c1, c2) (witnesses thy names thms) usr_tac]
end;
fun add_inst_arity_x raw_arity names thms usr_tac thy =
let
val (t, Ss, cs) = read_arity thy raw_arity;
val wthms = witnesses thy names thms;
fun prove c =
(message ("Proving type arity " ^
quote (Sign.string_of_arity thy (t, Ss, [c])) ^ " ...");
prove_arity thy (t, Ss, c) wthms usr_tac);
in add_arity_thms (map prove cs) thy end;
end;
end;