--- a/src/HOL/Tools/type_lifting.ML Mon Jan 10 22:03:24 2011 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,256 +0,0 @@
-(* Title: HOL/Tools/type_lifting.ML
- Author: Florian Haftmann, TU Muenchen
-
-Functorial structure of types.
-*)
-
-signature TYPE_LIFTING =
-sig
- val find_atomic: Proof.context -> typ -> (typ * (bool * bool)) list
- val construct_mapper: Proof.context -> (string * bool -> term)
- -> bool -> typ -> typ -> term
- val type_lifting: string option -> term -> local_theory -> Proof.state
- type entry
- val entries: Proof.context -> entry list Symtab.table
-end;
-
-structure Type_Lifting : TYPE_LIFTING =
-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 (ProofContext.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 (ProofContext.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_type_lifting 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 (ProofContext.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 type_lifting = gen_type_lifting Syntax.check_term;
-val type_lifting_cmd = gen_type_lifting Syntax.read_term;
-
-val _ = Outer_Syntax.local_theory_to_proof "type_lifting"
- "register operations managing the functorial structure of a type"
- Keyword.thy_goal (Scan.option (Parse.name --| Parse.$$$ ":") -- Parse.term
- >> (fn (prfx, t) => type_lifting_cmd prfx t));
-
-end;