(* Title: Pure/axclass.ML ID: $Id: axclass.ML,v 1.98 2005/09/28 22:58:56 wenzelm Exp $ 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.$$$ "\\" || P.$$$ "<") |-- 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.$$$ "\\" || P.$$$ "<") |-- 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;