(* Title: Pure/axclass.ML
Author: Markus Wenzel, TU Muenchen
Type classes defined as predicates, associated with a record of
parameters.
*)
signature AX_CLASS =
sig
val define_class: binding * class list -> string list ->
(Thm.binding * 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: binding * class list -> theory -> theory
val axiomatize_class_cmd: binding * 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: binding -> string * term -> theory -> thm * theory
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 lookup_inst_param: Consts.T -> ((string * string) * 'a) list -> string * typ -> 'a option
val param_of_inst: theory -> string * string -> string
val inst_of_param: theory -> string -> (string * string) option
val thynames_of_arity: theory -> class * string -> string list
val introN: string
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 =
(thm * proof) Symreltab.table * (*classrel theorems*)
((class * sort list) * ((thm * string) * proof)) list Symtab.table; (*arity theorems with theory name*)
(*transitive closure of classrels and arity completion is done in Theory.at_begin hook*)
fun merge_instances ((classrel1, arities1): instances, (classrel2, arities2)) =
(Symreltab.join (K fst) (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 = Theory_Data_PP
(
type T = axclasses * ((instances * inst_params) * (class * class) list);
val empty = ((Symtab.empty, []), (((Symreltab.empty, Symtab.empty), (Symtab.empty, Symtab.empty)), []));
val extend = I;
fun merge pp ((axclasses1, ((instances1, inst_params1), diff_merge_classrels1)),
(axclasses2, ((instances2, inst_params2), diff_merge_classrels2))) =
let
val (classrels1, classrels2) = pairself (Symreltab.keys o fst) (instances1, instances2)
val diff_merge_classrels = subtract (op =) classrels1 classrels2
@ subtract (op =) classrels2 classrels1
@ diff_merge_classrels1 @ diff_merge_classrels2
in
(merge_axclasses pp (axclasses1, axclasses2),
((merge_instances (instances1, instances2), merge_inst_params (inst_params1, inst_params2)),
diff_merge_classrels))
end;
);
(* 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 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) = Long_Name.base_name c ^ "_" ^ Long_Name.base_name a;
val get_instances = #1 o #1 o #2 o AxClassData.get;
val map_instances = AxClassData.map o apsnd o apfst o apfst;
val get_diff_merge_classrels = #2 o #2 o AxClassData.get;
val clear_diff_merge_classrels = AxClassData.map (apsnd (apsnd (K [])));
fun the_classrel thy (c1, c2) =
(case Symreltab.lookup (#1 (get_instances thy)) (c1, c2) of
SOME classrel => classrel
| NONE => error ("Unproven class relation " ^
Syntax.string_of_classrel (ProofContext.init thy) [c1, c2]));
fun the_classrel_thm thy = Thm.transfer thy o fst o the_classrel thy;
fun the_classrel_prf thy = snd o the_classrel thy;
fun put_trancl_classrel ((c1, c2), th) thy =
let
val classrels = fst (get_instances thy)
val alg = Sign.classes_of thy
val {classes, ...} = alg |> Sorts.rep_algebra
fun reflcl_classrel (c1', c2') =
if c1' = c2' then Thm.trivial (Logic.mk_of_class (TVar(("'a",0),[]), c1') |> cterm_of thy)
else the_classrel_thm thy (c1', c2')
fun gen_classrel (c1_pred, c2_succ) =
let
val th' = ((reflcl_classrel (c1_pred, c1) RS th) RS reflcl_classrel (c2, c2_succ))
|> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [])))] []
|> Thm.close_derivation
val prf' = th' |> Thm.proof_of
in ((c1_pred, c2_succ), (th',prf')) end
val new_classrels = Library.map_product pair
(c1 :: Graph.imm_preds classes c1) (c2 :: Graph.imm_succs classes c2)
|> filter_out (Symreltab.defined classrels)
|> map gen_classrel
val needed = length new_classrels > 0
in
(needed,
if needed then
thy |> map_instances (fn (classrels, arities) =>
(classrels |> fold Symreltab.update new_classrels, arities))
else thy)
end;
fun complete_classrels thy =
let
val diff_merge_classrels = get_diff_merge_classrels thy
val classrels = fst (get_instances thy)
val (needed, thy') = (false, thy) |>
fold (fn c12 => fn (needed, thy) =>
put_trancl_classrel (c12, Symreltab.lookup classrels c12 |> the |> fst) thy
|>> (fn b => needed orelse b))
diff_merge_classrels
in
if null diff_merge_classrels then NONE
else thy' |> clear_diff_merge_classrels |> SOME
end;
fun the_arity thy a (c, Ss) =
(case AList.lookup (op =) (Symtab.lookup_list (#2 (get_instances thy)) a) (c, Ss) of
SOME arity => arity
| NONE => error ("Unproven type arity " ^
Syntax.string_of_arity (ProofContext.init thy) (a, Ss, [c])));
fun the_arity_thm thy a c_Ss = the_arity thy a c_Ss |> fst |> fst |> Thm.transfer thy;
fun the_arity_prf thy a c_Ss = the_arity thy a c_Ss |> snd;
fun thynames_of_arity thy (c, a) =
Symtab.lookup_list (#2 (get_instances thy)) a
|> map_filter (fn ((c', _), ((_, name),_)) => if c = c' then SOME name else NONE)
|> rev;
fun insert_arity_completions thy (t, ((c, Ss), ((th, thy_name), _))) arities =
let
val algebra = Sign.classes_of thy;
val super_class_completions =
Sign.super_classes thy c
|> filter_out (fn c1 => exists (fn ((c2, Ss2), _) => c1 = c2
andalso Sorts.sorts_le algebra (Ss2, Ss)) (Symtab.lookup_list arities t));
val names_and_Ss = Name.names Name.context Name.aT (map (K []) Ss);
val completions = super_class_completions |> map (fn c1 =>
let
val th1 = (th RS the_classrel_thm thy (c, c1))
|> Drule.instantiate' (map (SOME o ctyp_of thy o TVar o apfst (rpair 0)) names_and_Ss) []
|> Thm.close_derivation
val prf1 = Thm.proof_of th1
in (((th1,thy_name), prf1), c1) end)
val arities' = fold (fn (th_thy_prf1, c1) => Symtab.cons_list (t, ((c1, Ss), th_thy_prf1)))
completions arities;
in (null completions, arities') end;
fun put_arity ((t, Ss, c), th) thy =
let
val arity' = (t, ((c, Ss), ((th, Context.theory_name thy), Thm.proof_of th)));
in
thy
|> map_instances (fn (classrel, arities) => (classrel,
arities
|> Symtab.insert_list (eq_fst op =) arity'
|> insert_arity_completions thy arity'
|> snd))
end;
fun complete_arities thy =
let
val arities = snd (get_instances thy);
val (finished, arities') = arities
|> fold_map (insert_arity_completions thy) (Symtab.dest_list arities);
in
if forall I finished then NONE
else SOME (thy |> map_instances (fn (classrel, _) => (classrel, arities')))
end;
val _ = Context.>> (Context.map_theory
(Theory.at_begin complete_classrels #> Theory.at_begin complete_arities))
(* maintain instance parameters *)
val get_inst_params = #2 o #1 o #2 o AxClassData.get;
val map_inst_params = AxClassData.map o apsnd o apfst o apsnd;
fun get_inst_param thy (c, tyco) =
case Symtab.lookup ((the_default Symtab.empty o Symtab.lookup (fst (get_inst_params thy))) c) tyco
of SOME c' => c'
| NONE => error ("No instance parameter for constant " ^ quote c
^ " on type constructor " ^ quote 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 get_inst_tyco consts = try (fst o dest_Type o the_single o Consts.typargs consts);
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 lookup_inst_param consts params (c, T) = case get_inst_tyco consts (c, T)
of SOME tyco => AList.lookup (op =) params (c, tyco)
| NONE => NONE;
fun unoverload_const thy (c_ty as (c, _)) =
if is_some (class_of_param thy c)
then case get_inst_tyco (Sign.consts_of thy) c_ty
of SOME tyco => try (param_of_inst thy) (c, tyco) |> the_default c
| NONE => c
else c;
(** instances **)
(* class relations *)
fun cert_classrel thy raw_rel =
let
val string_of_sort = Syntax.string_of_sort_global thy;
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 " ^ string_of_sort [c1] ^ " lacks parameter(s) " ^
commas_quote xs ^ " of " ^ string_of_sort [c2], [], []));
in (c1, c2) end;
fun read_classrel thy raw_rel =
cert_classrel thy (pairself (ProofContext.read_class (ProofContext.init thy)) raw_rel)
handle TYPE (msg, _, _) => error msg;
fun check_shyps_topped th errmsg =
let val {shyps, ...} = Thm.rep_thm th
in
forall null shyps orelse raise Fail errmsg
end;
(* declaration and definition of instances of overloaded constants *)
fun inst_tyco_of thy (c, T) =
(case get_inst_tyco (Sign.consts_of thy) (c, T) of
SOME tyco => tyco
| NONE => error ("Illegal type for instantiation of class parameter: " ^
quote (c ^ " :: " ^ Syntax.string_of_typ_global thy T)));
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 = inst_tyco_of thy (c, T);
val name_inst = instance_name (tyco, class) ^ "_inst";
val c' = Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco;
val T' = Type.strip_sorts T;
in
thy
|> Sign.qualified_path true (Binding.name name_inst)
|> Sign.declare_const ((Binding.name c', T'), NoSyn)
|-> (fn const' as Const (c'', _) =>
Thm.add_def false true
(Binding.name (Thm.def_name c'), Logic.mk_equals (Const (c, T'), const'))
#>> apsnd Thm.varifyT_global
#-> (fn (_, thm) => add_inst_param (c, tyco) (c'', thm)
#> PureThy.add_thm ((Binding.conceal (Binding.name c'), thm), [])
#> snd
#> pair (Const (c, T))))
||> Sign.restore_naming thy
end;
fun define_overloaded b (c, t) thy =
let
val T = Term.fastype_of t;
val 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 b' = Thm.def_binding_optional
(Binding.name (Long_Name.base_name c ^ "_" ^ Long_Name.base_name tyco)) b;
in
thy
|> Thm.add_def false false (b', prop)
|>> (fn (_, thm) => Drule.transitive_thm OF [eq, thm])
end;
(* primitive rules *)
fun add_classrel raw_th thy =
let
val th = Thm.strip_shyps (Thm.transfer thy raw_th);
val prop = Thm.plain_prop_of th;
fun err () = raise THM ("add_classrel: malformed class relation", 0, [th]);
val rel = Logic.dest_classrel prop handle TERM _ => err ();
val (c1, c2) = cert_classrel thy rel handle TYPE _ => err ();
val th' = th
|> Drule.instantiate' [SOME (ctyp_of thy (TVar ((Name.aT, 0), [c1])))] []
|> Drule.unconstrainTs;
val _ = check_shyps_topped th' "add_classrel: nontop shyps after unconstrain"
in
thy
|> Sign.primitive_classrel (c1, c2)
|> (snd oo put_trancl_classrel) ((c1, c2), th')
|> perhaps complete_arities
end;
fun add_arity raw_th thy =
let
val th = Thm.strip_shyps (Thm.transfer thy raw_th);
val prop = Thm.plain_prop_of th;
fun err () = raise THM ("add_arity: malformed type arity", 0, [th]);
val (t, Ss, c) = Logic.dest_arity prop handle TERM _ => err ();
val names = Name.names Name.context Name.aT Ss;
val T = Type (t, map TFree names);
val missing_params = Sign.complete_sort thy [c]
|> maps (these o Option.map #params o try (get_info thy))
|> filter_out (fn (const, _) => can (get_inst_param thy) (const, t))
|> (map o apsnd o map_atyps) (K T);
val _ = map (Sign.certify_sort thy) Ss = Ss orelse err ();
val th' = th
|> Drule.instantiate' (map (SOME o ctyp_of thy o TVar o apfst (rpair 0)) names) []
|> Drule.unconstrainTs;
val _ = check_shyps_topped th' "add_arity: nontop shyps after unconstrain"
in
thy
|> fold (snd oo declare_overloaded) missing_params
|> Sign.primitive_arity (t, Ss, [c])
|> put_arity ((t, Ss, c), th')
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)) (K 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 [((Binding.name
(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 = ProofContext.cert_arity ctxt 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 []) (map Binding.name 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)
|> Balanced_Tree.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 = Binding.map_name 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_CRITICAL 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 = Logic.mk_of_sort (Term.aT [], super) @ flat axiomss;
val class_eq =
Logic.mk_equals (Logic.mk_of_class (Term.aT [], class), Logic.mk_conjunction_balanced conjs);
val ([def], def_thy) =
thy
|> Sign.primitive_class (bclass, super)
|> PureThy.add_defs false [((Thm.def_binding 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
|> Sign.qualified_path true bconst
|> PureThy.note_thmss ""
[((Binding.name introN, []), [([Drule.export_without_context raw_intro], [])]),
((Binding.name superN, []), [(map Drule.export_without_context raw_classrel, [])]),
((Binding.name axiomsN, []),
[(map (fn th => Drule.export_without_context (class_triv RS th)) raw_axioms, [])])]
||> Sign.restore_naming def_thy;
(* result *)
val axclass = make_axclass ((def, intro, axioms), params);
val result_thy =
facts_thy
|> fold (snd oo put_trancl_classrel) (map (pair class) super ~~ classrel)
|> Sign.qualified_path false bconst
|> PureThy.note_thmss "" (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
(* old-style axioms *)
fun add_axiom (b, prop) =
Thm.add_axiom (b, prop) #->
(fn (_, thm) => PureThy.add_thm ((b, Drule.export_without_context thm), []));
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
|> fold_map add_axiom (map Binding.name 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 o ProofContext.init) 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 (ProofContext.read_class o ProofContext.init) read_classrel;
val axiomatize_classrel = ax_classrel cert_classrel;
val axiomatize_classrel_cmd = ax_classrel read_classrel;
val axiomatize_arity = ax_arity ProofContext.cert_arity;
val axiomatize_arity_cmd = ax_arity ProofContext.read_arity;
end;
end;