(* Title: HOL/Tools/enriched_type.ML
Author: Florian Haftmann, TU Muenchen
Functorial structure of types.
*)
signature ENRICHED_TYPE =
sig
val find_atomic: Proof.context -> typ -> (typ * (bool * bool)) list
val construct_mapper: Proof.context -> (string * bool -> term)
-> bool -> typ -> typ -> term
val enriched_type: string option -> term -> local_theory -> Proof.state
type entry
val entries: Proof.context -> entry list Symtab.table
end;
structure Enriched_Type : ENRICHED_TYPE =
struct
(* bookkeeping *)
val compN = "comp";
val idN = "id";
val compositionalityN = "compositionality";
val identityN = "identity";
type entry = { mapper: term, variances: (sort * (bool * bool)) list,
comp: thm, id: thm };
structure Data = Generic_Data
(
type T = entry list Symtab.table
val empty = Symtab.empty
val extend = I
fun merge data = Symtab.merge (K true) data
);
val entries = Data.get o Context.Proof;
(* type analysis *)
fun term_with_typ ctxt T t = Envir.subst_term_types
(Type.typ_match (Proof_Context.tsig_of ctxt) (fastype_of t, T) Vartab.empty) t;
fun find_atomic ctxt T =
let
val variances_of = Option.map #variances o try hd o Symtab.lookup_list (entries ctxt);
fun add_variance is_contra T =
AList.map_default (op =) (T, (false, false))
((if is_contra then apsnd else apfst) (K true));
fun analyze' is_contra (_, (co, contra)) T =
(if co then analyze is_contra T else I)
#> (if contra then analyze (not is_contra) T else I)
and analyze is_contra (T as Type (tyco, Ts)) = (case variances_of tyco
of NONE => add_variance is_contra T
| SOME variances => fold2 (analyze' is_contra) variances Ts)
| analyze is_contra T = add_variance is_contra T;
in analyze false T [] end;
fun construct_mapper ctxt atomic =
let
val lookup = hd o Symtab.lookup_list (entries ctxt);
fun constructs is_contra (_, (co, contra)) T T' =
(if co then [construct is_contra T T'] else [])
@ (if contra then [construct (not is_contra) T T'] else [])
and construct is_contra (T as Type (tyco, Ts)) (T' as Type (_, Ts')) =
let
val { mapper = raw_mapper, variances, ... } = lookup tyco;
val args = maps (fn (arg_pattern, (T, T')) =>
constructs is_contra arg_pattern T T')
(variances ~~ (Ts ~~ Ts'));
val (U, U') = if is_contra then (T', T) else (T, T');
val mapper = term_with_typ ctxt (map fastype_of args ---> U --> U') raw_mapper;
in list_comb (mapper, args) end
| construct is_contra (TFree (v, _)) (TFree _) = atomic (v, is_contra);
in construct end;
(* mapper properties *)
val compositionality_ss = Simplifier.add_simp (Simpdata.mk_eq @{thm comp_def}) HOL_basic_ss;
fun make_comp_prop ctxt variances (tyco, mapper) =
let
val sorts = map fst variances
val (((vs3, vs2), vs1), _) = ctxt
|> Variable.invent_types sorts
||>> Variable.invent_types sorts
||>> Variable.invent_types sorts
val (Ts1, Ts2, Ts3) = (map TFree vs1, map TFree vs2, map TFree vs3);
fun mk_argT ((T, T'), (_, (co, contra))) =
(if co then [(T --> T')] else [])
@ (if contra then [(T' --> T)] else []);
val contras = maps (fn (_, (co, contra)) =>
(if co then [false] else []) @ (if contra then [true] else [])) variances;
val Ts21 = maps mk_argT ((Ts2 ~~ Ts1) ~~ variances);
val Ts32 = maps mk_argT ((Ts3 ~~ Ts2) ~~ variances);
fun invents n k nctxt =
let
val names = Name.invents nctxt n k;
in (names, fold Name.declare names nctxt) end;
val ((names21, names32), nctxt) = Variable.names_of ctxt
|> invents "f" (length Ts21)
||>> invents "f" (length Ts32);
val T1 = Type (tyco, Ts1);
val T2 = Type (tyco, Ts2);
val T3 = Type (tyco, Ts3);
val (args21, args32) = (names21 ~~ Ts21, names32 ~~ Ts32);
val args31 = map2 (fn is_contra => fn ((f21, T21), (f32, T32)) =>
if not is_contra then
HOLogic.mk_comp (Free (f21, T21), Free (f32, T32))
else
HOLogic.mk_comp (Free (f32, T32), Free (f21, T21))
) contras (args21 ~~ args32)
fun mk_mapper T T' args = list_comb
(term_with_typ ctxt (map fastype_of args ---> T --> T') mapper, args);
val mapper21 = mk_mapper T2 T1 (map Free args21);
val mapper32 = mk_mapper T3 T2 (map Free args32);
val mapper31 = mk_mapper T3 T1 args31;
val eq1 = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(HOLogic.mk_comp (mapper21, mapper32), mapper31);
val x = Free (the_single (Name.invents nctxt (Long_Name.base_name tyco) 1), T3)
val eq2 = (HOLogic.mk_Trueprop o HOLogic.mk_eq)
(mapper21 $ (mapper32 $ x), mapper31 $ x);
val comp_prop = fold_rev Logic.all (map Free (args21 @ args32)) eq1;
val compositionality_prop = fold_rev Logic.all (map Free (args21 @ args32) @ [x]) eq2;
fun prove_compositionality ctxt comp_thm = Skip_Proof.prove ctxt [] [] compositionality_prop
(K (ALLGOALS (Method.insert_tac [@{thm fun_cong} OF [comp_thm]]
THEN' Simplifier.asm_lr_simp_tac compositionality_ss
THEN_ALL_NEW (Goal.assume_rule_tac ctxt))));
in (comp_prop, prove_compositionality) end;
val identity_ss = Simplifier.add_simp (Simpdata.mk_eq @{thm id_def}) HOL_basic_ss;
fun make_id_prop ctxt variances (tyco, mapper) =
let
val (vs, ctxt') = Variable.invent_types (map fst variances) ctxt;
val Ts = map TFree vs;
fun bool_num b = if b then 1 else 0;
fun mk_argT (T, (_, (co, contra))) =
replicate (bool_num co + bool_num contra) T
val arg_Ts = maps mk_argT (Ts ~~ variances)
val T = Type (tyco, Ts);
val head = term_with_typ ctxt (map (fn T => T --> T) arg_Ts ---> T --> T) mapper;
val lhs1 = list_comb (head, map (HOLogic.id_const) arg_Ts);
val lhs2 = list_comb (head, map (fn arg_T => Abs ("x", arg_T, Bound 0)) arg_Ts);
val rhs = HOLogic.id_const T;
val (id_prop, identity_prop) = pairself
(HOLogic.mk_Trueprop o HOLogic.mk_eq o rpair rhs) (lhs1, lhs2);
fun prove_identity ctxt id_thm = Skip_Proof.prove ctxt [] [] identity_prop
(K (ALLGOALS (Method.insert_tac [id_thm] THEN' Simplifier.asm_lr_simp_tac identity_ss)));
in (id_prop, prove_identity) end;
(* analyzing and registering mappers *)
fun consume eq x [] = (false, [])
| consume eq x (ys as z :: zs) = if eq (x, z) then (true, zs) else (false, ys);
fun split_mapper_typ "fun" T =
let
val (Ts', T') = strip_type T;
val (Ts'', T'') = split_last Ts';
val (Ts''', T''') = split_last Ts'';
in (Ts''', T''', T'' --> T') end
| split_mapper_typ tyco T =
let
val (Ts', T') = strip_type T;
val (Ts'', T'') = split_last Ts';
in (Ts'', T'', T') end;
fun analyze_variances ctxt tyco T =
let
fun bad_typ () = error ("Bad mapper type: " ^ Syntax.string_of_typ ctxt T);
val (Ts, T1, T2) = split_mapper_typ tyco T
handle List.Empty => bad_typ ();
val _ = pairself
((fn tyco' => if tyco' = tyco then () else bad_typ ()) o fst o dest_Type) (T1, T2)
handle TYPE _ => bad_typ ();
val (vs1, vs2) = pairself (map dest_TFree o snd o dest_Type) (T1, T2)
handle TYPE _ => bad_typ ();
val _ = if has_duplicates (eq_fst (op =)) (vs1 @ vs2)
then bad_typ () else ();
fun check_variance_pair (var1 as (v1, sort1), var2 as (v2, sort2)) =
let
val coT = TFree var1 --> TFree var2;
val contraT = TFree var2 --> TFree var1;
val sort = Sign.inter_sort (Proof_Context.theory_of ctxt) (sort1, sort2);
in
consume (op =) coT
##>> consume (op =) contraT
#>> pair sort
end;
val (variances, left_variances) = fold_map check_variance_pair (vs1 ~~ vs2) Ts;
val _ = if null left_variances then () else bad_typ ();
in variances end;
fun gen_enriched_type prep_term some_prfx raw_mapper lthy =
let
val input_mapper = prep_term lthy raw_mapper;
val T = fastype_of input_mapper;
val _ = Type.no_tvars T;
val mapper = singleton (Variable.polymorphic lthy) input_mapper;
val _ = if null (Term.add_tfreesT (fastype_of mapper) []) then ()
else error ("Illegal locally fixed variables in type: " ^ Syntax.string_of_typ lthy T);
fun add_tycos (Type (tyco, Ts)) = insert (op =) tyco #> fold add_tycos Ts
| add_tycos _ = I;
val tycos = add_tycos T [];
val tyco = if tycos = ["fun"] then "fun"
else case remove (op =) "fun" tycos
of [tyco] => tyco
| _ => error ("Bad number of type constructors: " ^ Syntax.string_of_typ lthy T);
val prfx = the_default (Long_Name.base_name tyco) some_prfx;
val variances = analyze_variances lthy tyco T;
val (comp_prop, prove_compositionality) = make_comp_prop lthy variances (tyco, mapper);
val (id_prop, prove_identity) = make_id_prop lthy variances (tyco, mapper);
val qualify = Binding.qualify true prfx o Binding.name;
fun mapper_declaration comp_thm id_thm phi context =
let
val typ_instance = Type.typ_instance (Proof_Context.tsig_of (Context.proof_of context));
val mapper' = Morphism.term phi mapper;
val T_T' = pairself fastype_of (mapper, mapper');
in if typ_instance T_T' andalso typ_instance (swap T_T')
then (Data.map o Symtab.cons_list) (tyco,
{ mapper = mapper', variances = variances,
comp = Morphism.thm phi comp_thm, id = Morphism.thm phi id_thm }) context
else context
end;
fun after_qed [single_comp_thm, single_id_thm] lthy =
lthy
|> Local_Theory.note ((qualify compN, []), single_comp_thm)
||>> Local_Theory.note ((qualify idN, []), single_id_thm)
|-> (fn ((_, [comp_thm]), (_, [id_thm])) => fn lthy =>
lthy
|> Local_Theory.note ((qualify compositionalityN, []),
[prove_compositionality lthy comp_thm])
|> snd
|> Local_Theory.note ((qualify identityN, []),
[prove_identity lthy id_thm])
|> snd
|> Local_Theory.declaration false (mapper_declaration comp_thm id_thm))
in
lthy
|> Proof.theorem NONE after_qed (map (fn t => [(t, [])]) [comp_prop, id_prop])
end
val enriched_type = gen_enriched_type Syntax.check_term;
val enriched_type_cmd = gen_enriched_type Syntax.read_term;
val _ = Outer_Syntax.local_theory_to_proof "enriched_type"
"register operations managing the functorial structure of a type"
Keyword.thy_goal (Scan.option (Parse.name --| Parse.$$$ ":") -- Parse.term
>> (fn (prfx, t) => enriched_type_cmd prfx t));
end;