diff -r 3d54fcc5f41a -r 6f3b75135638 Quot/Nominal/Fv.thy --- a/Quot/Nominal/Fv.thy Thu Feb 18 23:07:28 2010 +0100 +++ b/Quot/Nominal/Fv.thy Thu Feb 18 23:07:52 2010 +0100 @@ -1,5 +1,5 @@ theory Fv -imports "Nominal2_Atoms" +imports "Nominal2_Atoms" "Abs" begin (* Bindings are given as a list which has a length being equal @@ -23,6 +23,35 @@ [], [[], [], [(NONE, 0)]], [[], [], [(SOME (Const f), 0), (Some (Const g), 1)]]] + +A SOME binding has to have a function returning an atom set, +and a NONE binding has to be on an argument that is an atom +or an atom set. + +How the procedure works: + For each of the defined datatypes, + For each of the constructors, + It creates a union of free variables for each argument. + + For an argument the free variables are the variables minus + bound variables. + + The variables are: + For an atom, a singleton set with the atom itself. + For an atom set, the atom set itself. + For a recursive argument, the appropriate fv function applied to it. + (* TODO: This one is not implemented *) + For other arguments it should be an appropriate fv function stored + in the database. + The bound variables are a union of results of all bindings that + involve the given argument. For a paricular binding the result is: + For a function applied to an argument this function with the argument. + For an atom, a singleton set with the atom itself. + For an atom set, the atom set itself. + For a recursive argument, the appropriate fv function applied to it. + (* TODO: This one is not implemented *) + For other arguments it should be an appropriate fv function stored + in the database. *) ML {* @@ -40,13 +69,32 @@ if b = noatoms then a else if b = a then noatoms else HOLogic.mk_binop @{const_name minus} (a, b); + fun mk_atoms t = + let + val ty = fastype_of t; + val atom_ty = HOLogic.dest_setT ty --> @{typ atom}; + val img_ty = atom_ty --> ty --> @{typ "atom set"}; + in + (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t) + end; + (* Copy from Term *) + fun is_funtype (Type ("fun", [_, _])) = true + | is_funtype _ = false; + (* Similar to one in USyntax *) + fun mk_pair (fst, snd) = + let val ty1 = fastype_of fst + val ty2 = fastype_of snd + val c = HOLogic.pair_const ty1 ty2 + in c $ fst $ snd + end; + *} ML {* (* Currently needs just one full_tname to access Datatype *) -fun define_raw_fv full_tname bindsall lthy = +fun define_fv_alpha full_tname bindsall lthy = let - val thy = ProofContext.theory_of lthy + val thy = ProofContext.theory_of lthy; val {descr, ...} = Datatype.the_info thy full_tname; val sorts = []; (* TODO *) fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i); @@ -54,46 +102,92 @@ "fv_" ^ name_of_typ (nth_dtyp i)) descr); val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr; val fv_frees = map Free (fv_names ~~ fv_types); - fun fv_eq_constr i (cname, dts) bindcs = + val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) => + "alpha_" ^ name_of_typ (nth_dtyp i)) descr); + val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr; + val alpha_frees = map Free (alpha_names ~~ alpha_types); + fun fv_alpha_constr i (cname, dts) bindcs = let val Ts = map (typ_of_dtyp descr sorts) dts; val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts); val args = map Free (names ~~ Ts); + val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts); + val args2 = map Free (names2 ~~ Ts); val c = Const (cname, Ts ---> (nth_dtyp i)); - val fv_c = Free (nth fv_names i, (nth_dtyp i) --> @{typ "atom set"}); - fun fv_bind (NONE, i) = + val fv_c = nth fv_frees i; + val alpha = nth alpha_frees i; + fun fv_bind args (NONE, i) = if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else (* TODO we assume that all can be 'atomized' *) + if (is_funtype o fastype_of) (nth args i) then mk_atoms (nth args i) else mk_single_atom (nth args i) - | fv_bind (SOME f, i) = f $ (nth args i); + | fv_bind args (SOME f, i) = f $ (nth args i); fun fv_arg ((dt, x), bindxs) = let val arg = if is_rec_type dt then nth fv_frees (body_index dt) $ x else (* TODO: we just assume everything can be 'atomized' *) - HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x] - val sub = mk_union (map fv_bind bindxs) + if (is_funtype o fastype_of) x then mk_atoms x else + HOLogic.mk_set @{typ atom} [mk_atom (fastype_of x) $ x] + val sub = mk_union (map (fv_bind args) bindxs) in mk_diff arg sub end; - val _ = tracing ("d" ^ string_of_int (length dts)); - val _ = tracing (string_of_int (length args)); - val _ = tracing (string_of_int (length bindcs)); + val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq + (fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ bindcs)))) + val alpha_rhs = + HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2))); + fun alpha_arg ((dt, bindxs), (arg, arg2)) = + if bindxs = [] then ( + if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2) + else (HOLogic.mk_eq (arg, arg2))) + else + if is_rec_type dt then let + (* THE HARD CASE *) + val lhs_binds = mk_union (map (fv_bind args) bindxs); + val lhs = mk_pair (lhs_binds, arg); + val rhs_binds = mk_union (map (fv_bind args2) bindxs); + val rhs = mk_pair (rhs_binds, arg2); + val alpha = nth alpha_frees (body_index dt); + val fv = nth fv_frees (body_index dt); + val alpha_gen_pre = Const (@{const_name alpha_gen}, dummyT) $ lhs $ alpha $ fv $ (Free ("pi", @{typ perm})) $ rhs; + val alpha_gen_t = Syntax.check_term lthy alpha_gen_pre + in + HOLogic.mk_exists ("pi", @{typ perm}, alpha_gen_t) + (* TODO Add some test that is makes sense *) + end else @{term "True"} + val alpha_lhss = map (HOLogic.mk_Trueprop o alpha_arg) (dts ~~ bindcs ~~ (args ~~ args2)) + val alpha_eq = Logic.list_implies (alpha_lhss, alpha_rhs) in - (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq - (fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ bindcs))))) + (fv_eq, alpha_eq) end; - fun fv_eq (i, (_, _, constrs)) binds = map2 (fv_eq_constr i) constrs binds; - val fv_eqs = flat (map2 fv_eq descr bindsall) + fun fv_alpha_eq (i, (_, _, constrs)) binds = map2 (fv_alpha_constr i) constrs binds; + val (fv_eqs, alpha_eqs) = split_list (flat (map2 fv_alpha_eq descr bindsall)) + val add_binds = map (fn x => (Attrib.empty_binding, x)) + val (fvs, lthy') = (Primrec.add_primrec + (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) (add_binds fv_eqs) lthy) + val (alphas, lthy'') = (Inductive.add_inductive_i + {quiet_mode = false, verbose = true, alt_name = Binding.empty, + coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false} + (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) alpha_names alpha_types) [] + (add_binds alpha_eqs) [] lthy') in - snd (Primrec.add_primrec - (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) fv_eqs lthy) + ((fvs, alphas), lthy'') end *} -(* test +(* tests atom_decl name +datatype ty = + Var "name set" + +ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *} + +local_setup {* define_fv_alpha "Fv.ty" [[[[]]]] *} +print_theorems + + datatype rtrm1 = rVr1 "name" | rAp1 "rtrm1" "rtrm1" @@ -113,11 +207,12 @@ | "bv1 (BVr x) = {atom x}" | "bv1 (BPr bp1 bp2) = (bv1 bp1) \ (bv1 bp1)" -local_setup {* define_raw_fv "Fv.rtrm1" - [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(NONE, 0)], [], [(SOME @{term bv1}, 0)]]], +setup {* snd o define_raw_perms ["rtrm1", "bp"] ["Fv.rtrm1", "Fv.bp"] *} + +local_setup {* define_fv_alpha "Fv.rtrm1" + [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(SOME @{term bv1}, 0)], [], [(SOME @{term bv1}, 0)]]], [[], [[]], [[], []]]] *} print_theorems - *) end