Added
goal Set.thy "(Union M = {}) = (! A : M. A = {})";
AddIffs [Union_empty_conv];
Good idea??
(* Title: Pure/axclass.ML
ID: $Id$
Author: Markus Wenzel, TU Muenchen
User interfaces for axiomatic type classes.
*)
signature AX_CLASS =
sig
val add_classrel_thms: thm list -> theory -> theory
val add_arity_thms: thm list -> theory -> theory
val add_axclass: bclass * xclass list -> (string * string) list
-> theory -> theory
val add_axclass_i: bclass * class list -> (string * term) list
-> theory -> theory
val add_inst_subclass: xclass * xclass -> string list -> thm list
-> tactic option -> theory -> theory
val add_inst_subclass_i: class * class -> string list -> thm list
-> tactic option -> theory -> theory
val add_inst_arity: xstring * xsort list * xclass list -> string list
-> thm list -> tactic option -> theory -> theory
val add_inst_arity_i: string * sort list * class list -> string list
-> thm list -> tactic option -> theory -> theory
val axclass_tac: theory -> thm list -> tactic
val prove_subclass: theory -> class * class -> thm list
-> tactic option -> thm
val prove_arity: theory -> string * sort list * class -> thm list
-> tactic option -> thm
val goal_subclass: theory -> xclass * xclass -> thm list
val goal_arity: theory -> xstring * xsort list * xclass -> thm list
end;
structure AxClass : AX_CLASS =
struct
(** utilities **)
(* 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], []);
(* get axioms and theorems *)
fun get_ax thy name =
Some (get_axiom thy name) handle THEORY _ => None;
val get_axioms = mapfilter o get_ax;
val is_def = Logic.is_equals o #prop o rep_thm;
fun witnesses thy axms thms =
map (get_axiom thy) axms @ thms @ filter is_def (map snd (axioms_of thy));
(** abstract syntax operations **)
(* 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 names = tl (variantlist (replicate (length ss + 1) "'", []));
val tfrees = ListPair.map TFree (names, ss);
in
Logic.mk_inclass (Type (t, tfrees), c)
end;
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 tvars)
then map snd tvars else err ();
in
(t, ss, c)
end;
(** add theorems as axioms **)
fun prep_thm_axm thy thm =
let
fun err msg = raise THM ("prep_thm_axm: " ^ msg, 0, [thm]);
val {sign, hyps, prop, ...} = rep_thm thm;
in
if not (Sign.subsig (sign, sign_of thy)) then
err "theorem not of same theory"
else if not (null (extra_shyps thm)) orelse not (null hyps) then
err "theorem may not contain hypotheses"
else prop
end;
(*theorems expressing class relations*)
fun add_classrel_thms thms thy =
let
fun prep_thm thm =
let
val prop = prep_thm_axm thy thm;
val (c1, c2) = dest_classrel prop handle TERM _ =>
raise THM ("add_classrel_thms: theorem is not a class relation", 0, [thm]);
in (c1, c2) end;
in
Theory.add_classrel (map prep_thm thms) thy
end;
(*theorems expressing arities*)
fun add_arity_thms thms thy =
let
fun prep_thm thm =
let
val prop = prep_thm_axm thy thm;
val (t, ss, c) = dest_arity prop handle TERM _ =>
raise THM ("add_arity_thms: theorem is not an arity", 0, [thm]);
in (t, ss, [c]) end;
in
Theory.add_arities (map prep_thm thms) thy
end;
(** add axiomatic type classes **)
(* errors *)
fun err_not_logic c =
error ("Axiomatic class " ^ quote c ^ " not subclass of \"logic\"");
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);
(* ext_axclass *)
fun ext_axclass int prep_axm (raw_class, raw_super_classes) raw_axioms old_thy =
let
val old_sign = sign_of old_thy;
val axioms = map (prep_axm old_sign) raw_axioms;
val class = Sign.full_name old_sign raw_class;
val thy =
(if int then Theory.add_classes else Theory.add_classes_i)
[(raw_class, raw_super_classes)] old_thy;
val sign = sign_of thy;
val super_classes =
if int then map (Sign.intern_class sign) raw_super_classes
else raw_super_classes;
(* prepare abstract axioms *)
fun abs_axm ax =
if null (term_tfrees ax) then
Logic.mk_implies (Logic.mk_inclass (aT logicS, class), ax)
else map_term_tfrees (K (aT [class])) ax;
val abs_axioms = map (apsnd abs_axm) axioms;
(* prepare introduction orule *)
val _ =
if Sign.subsort sign ([class], logicS) then ()
else err_not_logic class;
fun axm_sort (name, ax) =
(case term_tfrees ax of
[] => []
| [(_, S)] =>
if Sign.subsort sign ([class], S) then S
else err_bad_axsort name class
| _ => err_bad_tfrees name);
val axS = Sign.norm_sort sign (logicC :: flat (map axm_sort axioms))
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 snd) axioms, inclass class);
in
Theory.add_axioms_i ((raw_class ^ "I", intro_axm) :: abs_axioms) thy
end;
(* external interfaces *)
val add_axclass = ext_axclass true read_axm;
val add_axclass_i = ext_axclass false cert_axm;
(** prove class relations and type arities **)
(* class_axms *)
fun class_axms thy =
let
val classes = Sign.classes (sign_of thy);
val intros = map (fn c => c ^ "I") classes;
in
map (class_triv thy) classes @
get_axioms thy intros
end;
(* 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*)
fun axclass_tac thy thms =
let
val defs = filter is_def thms;
val non_defs = filter_out is_def thms;
in
TRY (REPEAT_FIRST (resolve_tac (class_axms thy))) THEN
TRY (rewrite_goals_tac defs) THEN
TRY (REPEAT_FIRST (fn i => assume_tac i ORELSE resolve_tac non_defs i))
end;
(* provers *)
fun prove term_of str_of thy sig_prop thms usr_tac =
let
val sign = sign_of thy;
val goal = cterm_of sign (term_of sig_prop);
val tac = axclass_tac thy thms THEN (if_none usr_tac all_tac);
in
prove_goalw_cterm [] goal (K [tac])
end
handle ERROR => error ("The error(s) above occurred while trying to prove "
^ quote (str_of (sign_of thy, sig_prop)));
val prove_subclass =
prove mk_classrel (fn (sg, c1_c2) => Sign.str_of_classrel sg c1_c2);
val prove_arity =
prove mk_arity (fn (sg, (t, Ss, c)) => Sign.str_of_arity sg (t, Ss, [c]));
(** add proved subclass relations and arities **)
fun intrn_classrel sg c1_c2 =
pairself (Sign.intern_class sg) c1_c2;
fun ext_inst_subclass int raw_c1_c2 axms thms usr_tac thy =
let
val c1_c2 =
if int then intrn_classrel (sign_of thy) raw_c1_c2
else raw_c1_c2;
in
writeln ("Proving class inclusion " ^
quote (Sign.str_of_classrel (sign_of thy) c1_c2) ^ " ...");
add_classrel_thms
[prove_subclass thy c1_c2 (witnesses thy axms thms) usr_tac] thy
end;
fun intrn_arity sg intrn (t, Ss, x) =
(Sign.intern_tycon sg t, map (Sign.intern_sort sg) Ss, intrn sg x);
fun ext_inst_arity int (raw_t, raw_Ss, raw_cs) axms thms usr_tac thy =
let
val sign = sign_of thy;
val (t, Ss, cs) =
if int then intrn_arity sign Sign.intern_sort (raw_t, raw_Ss, raw_cs)
else (raw_t, raw_Ss, raw_cs);
val wthms = witnesses thy axms thms;
fun prove c =
(writeln ("Proving type arity " ^
quote (Sign.str_of_arity sign (t, Ss, [c])) ^ " ...");
prove_arity thy (t, Ss, c) wthms usr_tac);
in
add_arity_thms (map prove cs) thy
end;
val add_inst_subclass = ext_inst_subclass true;
val add_inst_subclass_i = ext_inst_subclass false;
val add_inst_arity = ext_inst_arity true;
val add_inst_arity_i = ext_inst_arity false;
(* make goals (for interactive use) *)
fun mk_goal term_of thy sig_prop =
goalw_cterm [] (cterm_of (sign_of thy) (term_of sig_prop));
fun goal_subclass thy =
mk_goal (mk_classrel o intrn_classrel (sign_of thy)) thy;
fun goal_arity thy =
mk_goal (mk_arity o intrn_arity (sign_of thy) Sign.intern_class) thy;
end;