(* Title: HOL/Nominal/nominal_atoms.ML Author: Christian Urban and Stefan Berghofer, TU Muenchen
Declaration of atom types to be used in nominal datatypes.
*)
signature NOMINAL_ATOMS = sig val create_nom_typedecls : stringlist -> theory -> theory type atom_info val get_atom_infos : theory -> atom_info Symtab.table val get_atom_info : theory -> string -> atom_info option val the_atom_info : theory -> string -> atom_info val fs_class_of : theory -> string -> string val pt_class_of : theory -> string -> string val cp_class_of : theory -> string -> string -> string val at_inst_of : theory -> string -> thm val pt_inst_of : theory -> string -> thm val cp_inst_of : theory -> string -> string -> thm val dj_thm_of : theory -> string -> string -> thm val atoms_of : theory -> stringlist val mk_permT : typ -> typ end
structure NominalAtoms : NOMINAL_ATOMS = struct
val finite_emptyI = @{thm "finite.emptyI"}; val Collect_const = @{thm "Collect_const"};
val inductive_forall_def = @{thm HOL.induct_forall_def};
val get_atom_infos = NominalData.get; val get_atom_info = Symtab.lookup o NominalData.get;
fun gen_lookup lookup name = case lookup name of
SOME info => info
| NONE => error ("Unknown atom type " ^ quote name);
fun the_atom_info thy = gen_lookup (get_atom_info thy);
fun gen_lookup' f thy = the_atom_info thy #> f; fun gen_lookup'' f thy =
gen_lookup' (f #> Symtab.lookup #> gen_lookup) thy;
val fs_class_of = gen_lookup' #fs_class; val pt_class_of = gen_lookup' #pt_class; val at_inst_of = gen_lookup' #at_inst; val pt_inst_of = gen_lookup' #pt_inst; val cp_class_of = gen_lookup'' #cp_classes; val cp_inst_of = gen_lookup'' #cp_inst; val dj_thm_of = gen_lookup'' #dj_thms;
fun atoms_of thy = map fst (Symtab.dest (NominalData.get thy));
fun mk_permT T = HOLogic.listT (HOLogic.mk_prodT (T, T));
fun mk_Cons x xs = letval T = fastype_of x inConst (\<^const_name>\<open>Cons\<close>, T --> HOLogic.listT T --> HOLogic.listT T) $ x $ xs end;
fun add_thms_string args = Global_Theory.add_thms ((map o apfst o apfst) Binding.name args); fun add_thmss_string args = Global_Theory.add_thmss ((map o apfst o apfst) Binding.name args);
(* this function sets up all matters related to atom- *) (* kinds; the user specifies a list of atom-kind names *) (* atom_decl <ak1> ... <akn> *) fun create_nom_typedecls ak_names thy = let
val (_,thy1) =
fold_map (fn ak => fn thy => letval dt = ((Binding.name ak, [], NoSyn), [(Binding.name ak, [\<^typ>\<open>nat\<close>], NoSyn)]) val (dt_names, thy1) = BNF_LFP_Compat.add_datatype [BNF_LFP_Compat.Kill_Type_Args] [dt] thy;
val injects = maps (#inject o BNF_LFP_Compat.the_info thy1 []) dt_names; val ak_type = Type (Sign.intern_type thy1 ak,[]) val ak_sign = Sign.intern_const thy1 ak
val inj_type = \<^typ>\<open>nat\<close> --> ak_type val inj_on_type = inj_type --> \<^typ>\<open>nat set\<close> --> \<^typ>\<open>bool\<close>
(* first statement *) val stmnt1 = HOLogic.mk_Trueprop
(Const (\<^const_name>\<open>inj_on\<close>,inj_on_type) $ Const (ak_sign,inj_type) $ HOLogic.mk_UNIV \<^typ>\<open>nat\<close>)
val (inj_thm,thy2) =
add_thms_string [((ak^"_inj",Goal.prove_global thy1 [] [] stmnt1 (proof1 o #context)), [])] thy1
(* second statement *) val y = Free ("y",ak_type) val stmnt2 = HOLogic.mk_Trueprop
(HOLogic.mk_exists ("x",\<^typ>\<open>nat\<close>,HOLogic.mk_eq (y,Const (ak_sign,inj_type) $ Bound 0)))
val proof2 = fn {prems, context = ctxt} =>
Induct_Tacs.case_tac ctxt "y" [] NONE 1 THEN
asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simp1) 1 THEN
resolve_tac ctxt @{thms exI} 1 THEN
resolve_tac ctxt @{thms refl} 1
(* third statement *) val (inject_thm,thy3) =
add_thms_string [((ak^"_injection",Goal.prove_global thy2 [] [] stmnt2 proof2), [])] thy2
val (inf_thm,thy4) =
add_thms_string [((ak^"_infinite",Goal.prove_global thy3 [] [] stmnt3 (proof3 o #context)), [])] thy3 in
((inj_thm,inject_thm,inf_thm),thy4) end) ak_names thy
(* produces a list consisting of pairs: *) (* fst component is the atom-kind name *) (* snd component is its type *) val full_ak_names = map (Sign.intern_type thy1) ak_names; val ak_names_types = ak_names ~~ map (Type o rpair []) full_ak_names;
(* declares a swapping function for every atom-kind, it is *) (* const swap_<ak> :: <akT> * <akT> => <akT> => <akT> *) (* swap_<ak> (a,b) c = (if a=c then b (else if b=c then a else c)) *) (* overloades then the general swap-function *) val (swap_eqs, thy3) = fold_map (fn (ak_name, T) => fn thy => let val thy' = Sign.add_path "rec" thy; val swapT = HOLogic.mk_prodT (T, T) --> T --> T; val swap_name = "swap_" ^ ak_name; val full_swap_name = Sign.full_bname thy' swap_name; val a = Free ("a", T); val b = Free ("b", T); val c = Free ("c", T); val ab = Free ("ab", HOLogic.mk_prodT (T, T)) val cif = Const (\<^const_name>\<open>If\<close>, HOLogic.boolT --> T --> T --> T); val cswap_akname = Const (full_swap_name, swapT); val cswap = Const (\<^const_name>\<open>Nominal.swap\<close>, swapT)
val name = swap_name ^ "_def"; val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(Free (swap_name, swapT) $ HOLogic.mk_prod (a,b) $ c,
cif $ HOLogic.mk_eq (a,c) $ b $ (cif $ HOLogic.mk_eq (b,c) $ a $ c))) val def2 = Logic.mk_equals (cswap $ ab $ c, cswap_akname $ ab $ c) in
thy' |>
BNF_LFP_Compat.primrec_global
[(Binding.name swap_name, SOME swapT, NoSyn)]
[((Binding.empty_atts, def1), [], [])] ||>
Sign.parent_path ||>>
fold_map Global_Theory.add_def_unchecked_overloaded [(Binding.name name, def2)] |>> (snd o fst) end) ak_names_types thy1;
(* declares a permutation function for every atom-kind acting *) (* on such atoms *) (* const <ak>_prm_<ak> :: (<akT> * <akT>)list => akT => akT *) (* <ak>_prm_<ak> [] a = a *) (* <ak>_prm_<ak> (x#xs) a = swap_<ak> x (perm xs a) *) val (prm_eqs, thy4) = fold_map (fn (ak_name, T) => fn thy => let val thy' = Sign.add_path "rec" thy; val swapT = HOLogic.mk_prodT (T, T) --> T --> T; val swap_name = Sign.full_bname thy' ("swap_" ^ ak_name) val prmT = mk_permT T --> T --> T; val prm_name = ak_name ^ "_prm_" ^ ak_name; val prm = Free (prm_name, prmT); val x = Free ("x", HOLogic.mk_prodT (T, T)); val xs = Free ("xs", mk_permT T); val a = Free ("a", T) ;
val cnil = Const (\<^const_name>\<open>Nil\<close>, mk_permT T);
val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq (prm $ cnil $ a, a));
val def2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(prm $ mk_Cons x xs $ a, Const (swap_name, swapT) $ x $ (prm $ xs $ a))); in
thy' |>
BNF_LFP_Compat.primrec_global
[(Binding.name prm_name, SOME prmT, NoSyn)]
(map (fn def => ((Binding.empty_atts, def), [], [])) [def1, def2]) ||>
Sign.parent_path end) ak_names_types thy3;
(* defines permutation functions for all combinations of atom-kinds; *) (* there are a trivial cases and non-trivial cases *) (* non-trivial case: *) (* <ak>_prm_<ak>_def: perm pi a == <ak>_prm_<ak> pi a *) (* trivial case with <ak> != <ak'> *) (* <ak>_prm<ak'>_def[simp]: perm pi a == a *) (* *) (* the trivial cases are added to the simplifier, while the non- *) (* have their own rules proved below *) val (perm_defs, thy5) = fold_map (fn (ak_name, T) => fn thy =>
fold_map (fn (ak_name', T') => fn thy' => let val perm_def_name = ak_name ^ "_prm_" ^ ak_name'; val pi = Free ("pi", mk_permT T); val a = Free ("a", T'); val cperm = Const (\<^const_name>\<open>Nominal.perm\<close>, mk_permT T --> T' --> T'); val thy'' = Sign.add_path "rec" thy' val cperm_def = Const (Sign.full_bname thy'' perm_def_name, mk_permT T --> T' --> T'); val thy''' = Sign.parent_path thy'';
val name = ak_name ^ "_prm_" ^ ak_name' ^ "_def"; val def = Logic.mk_equals
(cperm $ pi $ a, if ak_name = ak_name' then cperm_def $ pi $ a else a) in
Global_Theory.add_def_unchecked_overloaded (Binding.name name, def) thy''' end) ak_names_types thy) ak_names_types thy4;
(* proves that every atom-kind is an instance of at *) (* lemma at_<ak>_inst: *) (* at TYPE(<ak>) *) val (prm_cons_thms,thy6) =
thy5 |> add_thms_string (map (fn (ak_name, T) => let val ak_name_qu = Sign.full_bname thy5 (ak_name); val i_type = Type(ak_name_qu,[]); val cat = Const (\<^const_name>\<open>Nominal.at\<close>, Term.itselfT i_type --> HOLogic.boolT); val at_type = Logic.mk_type i_type; fun proof ctxt =
simp_tac (put_simpset HOL_ss ctxt
addsimps maps (Global_Theory.get_thms thy5)
["at_def",
ak_name ^ "_prm_" ^ ak_name ^ "_def",
ak_name ^ "_prm_" ^ ak_name ^ ".simps", "swap_" ^ ak_name ^ "_def", "swap_" ^ ak_name ^ ".simps",
ak_name ^ "_infinite"]) 1; val name = "at_"^ak_name^ "_inst"; val statement = HOLogic.mk_Trueprop (cat $ at_type); in
((name, Goal.prove_global thy5 [] [] statement (proof o #context)), []) end) ak_names_types);
(* declares a perm-axclass for every atom-kind *) (* axclass pt_<ak> *) (* pt_<ak>1[simp]: perm [] x = x *) (* pt_<ak>2: perm (pi1@pi2) x = perm pi1 (perm pi2 x) *) (* pt_<ak>3: pi1 ~ pi2 ==> perm pi1 x = perm pi2 x *) val (pt_ax_classes,thy7) = fold_map (fn (ak_name, T) => fn thy => let val cl_name = "pt_"^ak_name; val ty = TFree("'a", \<^sort>\<open>type\<close>); val x = Free ("x", ty); val pi1 = Free ("pi1", mk_permT T); val pi2 = Free ("pi2", mk_permT T); val cperm = Const (\<^const_name>\<open>Nominal.perm\<close>, mk_permT T --> ty --> ty); val cnil = Const (\<^const_name>\<open>Nil\<close>, mk_permT T); val cappend = Const (\<^const_name>\<open>append\<close>, mk_permT T --> mk_permT T --> mk_permT T); val cprm_eq = Const (\<^const_name>\<open>Nominal.prm_eq\<close>, mk_permT T --> mk_permT T --> HOLogic.boolT); (* nil axiom *) val axiom1 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(cperm $ cnil $ x, x)); (* append axiom *) val axiom2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(cperm $ (cappend $ pi1 $ pi2) $ x, cperm $ pi1 $ (cperm $ pi2 $ x))); (* perm-eq axiom *) val axiom3 = Logic.mk_implies
(HOLogic.mk_Trueprop (cprm_eq $ pi1 $ pi2),
HOLogic.mk_Trueprop (HOLogic.mk_eq (cperm $ pi1 $ x, cperm $ pi2 $ x))); in
Axclass.define_class (Binding.name cl_name, \<^sort>\<open>type\<close>) []
[((Binding.name (cl_name ^ "1"), [Simplifier.simp_add]), [axiom1]),
((Binding.name (cl_name ^ "2"), []), [axiom2]),
((Binding.name (cl_name ^ "3"), []), [axiom3])] thy end) ak_names_types thy6;
(* proves that every pt_<ak>-type together with <ak>-type *) (* instance of pt *) (* lemma pt_<ak>_inst: *) (* pt TYPE('x::pt_<ak>) TYPE(<ak>) *) val (prm_inst_thms,thy8) =
thy7 |> add_thms_string (map (fn (ak_name, T) => let val ak_name_qu = Sign.full_bname thy7 ak_name; val pt_name_qu = Sign.full_bname thy7 ("pt_"^ak_name); val i_type1 = TFree("'x",[pt_name_qu]); val i_type2 = Type(ak_name_qu,[]); val cpt = Const (\<^const_name>\<open>Nominal.pt\<close>, (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT); val pt_type = Logic.mk_type i_type1; val at_type = Logic.mk_type i_type2; fun proof ctxt =
simp_tac (put_simpset HOL_ss ctxt addsimps maps (Global_Theory.get_thms thy7)
["pt_def", "pt_" ^ ak_name ^ "1", "pt_" ^ ak_name ^ "2", "pt_" ^ ak_name ^ "3"]) 1;
val name = "pt_"^ak_name^ "_inst"; val statement = HOLogic.mk_Trueprop (cpt $ pt_type $ at_type); in
((name, Goal.prove_global thy7 [] [] statement (proof o #context)), []) end) ak_names_types);
(* declares an fs-axclass for every atom-kind *) (* axclass fs_<ak> *) (* fs_<ak>1: finite ((supp x)::<ak> set) *) val (fs_ax_classes,thy11) = fold_map (fn (ak_name, T) => fn thy => let val cl_name = "fs_"^ak_name; val pt_name = Sign.full_bname thy ("pt_"^ak_name); val ty = TFree("'a",\<^sort>\<open>type\<close>); val x = Free ("x", ty); val csupp = Const (\<^const_name>\<open>Nominal.supp\<close>, ty --> HOLogic.mk_setT T); val cfinite = Const (\<^const_name>\<open>finite\<close>, HOLogic.mk_setT T --> HOLogic.boolT)
val axiom1 = HOLogic.mk_Trueprop (cfinite $ (csupp $ x));
(* proves that every fs_<ak>-type together with <ak>-type *) (* instance of fs-type *) (* lemma abst_<ak>_inst: *) (* fs TYPE('x::pt_<ak>) TYPE (<ak>) *) val (fs_inst_thms,thy12) =
thy11 |> add_thms_string (map (fn (ak_name, T) => let val ak_name_qu = Sign.full_bname thy11 ak_name; val fs_name_qu = Sign.full_bname thy11 ("fs_"^ak_name); val i_type1 = TFree("'x",[fs_name_qu]); val i_type2 = Type(ak_name_qu,[]); val cfs = Const (\<^const_name>\<open>Nominal.fs\<close>,
(Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT); val fs_type = Logic.mk_type i_type1; val at_type = Logic.mk_type i_type2; fun proof ctxt =
simp_tac (put_simpset HOL_ss ctxt addsimps maps (Global_Theory.get_thms thy11)
["fs_def", "fs_" ^ ak_name ^ "1"]) 1;
val name = "fs_"^ak_name^ "_inst"; val statement = HOLogic.mk_Trueprop (cfs $ fs_type $ at_type); in
((name, Goal.prove_global thy11 [] [] statement (proof o #context)), []) end) ak_names_types);
(* declares for every atom-kind combination an axclass *) (* cp_<ak1>_<ak2> giving a composition property *) (* cp_<ak1>_<ak2>1: pi1 o pi2 o x = (pi1 o pi2) o (pi1 o x) *) val (cp_ax_classes,thy12b) = fold_map (fn (ak_name, T) => fn thy =>
fold_map (fn (ak_name', T') => fn thy' => let val cl_name = "cp_"^ak_name^"_"^ak_name'; val ty = TFree("'a",\<^sort>\<open>type\<close>); val x = Free ("x", ty); val pi1 = Free ("pi1", mk_permT T); val pi2 = Free ("pi2", mk_permT T'); val cperm1 = Const (\<^const_name>\<open>Nominal.perm\<close>, mk_permT T --> ty --> ty); val cperm2 = Const (\<^const_name>\<open>Nominal.perm\<close>, mk_permT T' --> ty --> ty); val cperm3 = Const (\<^const_name>\<open>Nominal.perm\<close>, mk_permT T --> mk_permT T' --> mk_permT T');
(* proves for every non-trivial <ak>-combination a disjointness *) (* theorem; i.e. <ak1> != <ak2> *) (* lemma ds_<ak1>_<ak2>: *) (* dj TYPE(<ak1>) TYPE(<ak2>) *) val (dj_thms, thy12d) = fold_map (fn (ak_name,T) => fn thy =>
fold_map (fn (ak_name',T') => fn thy' =>
(ifnot (ak_name = ak_name') then let val ak_name_qu = Sign.full_bname thy' ak_name; val ak_name_qu' = Sign.full_bname thy' ak_name'; val i_type1 = Type(ak_name_qu,[]); val i_type2 = Type(ak_name_qu',[]); val cdj = Const (\<^const_name>\<open>Nominal.disjoint\<close>,
(Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT); val at_type = Logic.mk_type i_type1; val at_type' = Logic.mk_type i_type2; fun proof ctxt =
simp_tac (put_simpset HOL_ss ctxt
addsimps maps (Global_Theory.get_thms thy')
["disjoint_def",
ak_name ^ "_prm_" ^ ak_name' ^ "_def",
ak_name' ^ "_prm_" ^ ak_name ^ "_def"]) 1;
val name = "dj_"^ak_name^"_"^ak_name'; val statement = HOLogic.mk_Trueprop (cdj $ at_type $ at_type'); in
add_thms_string [((name, Goal.prove_global thy' [] [] statement (proof o #context)), [])] thy' end else
([],thy'))) (* do nothing branch, if ak_name = ak_name' *)
ak_names_types thy) ak_names_types thy12c;
(******** pt_<ak> class instances ********) (*=========================================*) (* some abbreviations for theorems *) val pt1 = @{thm "pt1"}; val pt2 = @{thm "pt2"}; val pt3 = @{thm "pt3"}; val at_pt_inst = @{thm "at_pt_inst"}; val pt_unit_inst = @{thm "pt_unit_inst"}; val pt_prod_inst = @{thm "pt_prod_inst"}; val pt_nprod_inst = @{thm "pt_nprod_inst"}; val pt_list_inst = @{thm "pt_list_inst"}; val pt_optn_inst = @{thm "pt_option_inst"}; val pt_noptn_inst = @{thm "pt_noption_inst"}; val pt_fun_inst = @{thm "pt_fun_inst"}; val pt_set_inst = @{thm "pt_set_inst"};
(* for all atom-kind combinations <ak>/<ak'> show that *) (* every <ak> is an instance of pt_<ak'>; the proof for *) (* ak!=ak' is by definition; the case ak=ak' uses at_pt_inst. *) val thy13 = fold (fn ak_name => fn thy =>
fold (fn ak_name' => fn thy' => let val qu_name = Sign.full_bname thy' ak_name'; val cls_name = Sign.full_bname thy' ("pt_"^ak_name); val at_inst = Global_Theory.get_thm thy' ("at_" ^ ak_name' ^ "_inst");
(******** fs_<ak> class instances ********) (*=========================================*) (* abbreviations for some lemmas *) val fs1 = @{thm "fs1"}; val fs_at_inst = @{thm "fs_at_inst"}; val fs_unit_inst = @{thm "fs_unit_inst"}; val fs_prod_inst = @{thm "fs_prod_inst"}; val fs_nprod_inst = @{thm "fs_nprod_inst"}; val fs_list_inst = @{thm "fs_list_inst"}; val fs_option_inst = @{thm "fs_option_inst"}; val dj_supp = @{thm "dj_supp"};
(* shows that <ak> is an instance of fs_<ak> *) (* uses the theorem at_<ak>_inst *) val thy20 = fold (fn ak_name => fn thy =>
fold (fn ak_name' => fn thy' => let val qu_name = Sign.full_bname thy' ak_name'; val qu_class = Sign.full_bname thy' ("fs_"^ak_name); fun proof ctxt =
(if ak_name = ak_name' then letval at_thm = Global_Theory.get_thm thy' ("at_"^ak_name^"_inst") in
EVERY [Class.intro_classes_tac ctxt [],
resolve_tac ctxt [(at_thm RS fs_at_inst) RS fs1] 1] end else letval dj_inst = Global_Theory.get_thm thy' ("dj_"^ak_name'^"_"^ak_name); val simp_s =
put_simpset HOL_basic_ss ctxt addsimps [dj_inst RS dj_supp, finite_emptyI]; in EVERY [Class.intro_classes_tac ctxt [], asm_simp_tac simp_s 1] end) in
Axclass.prove_arity (qu_name,[],[qu_class]) proof thy' end) ak_names thy) ak_names thy18;
(* shows that *) (* unit *) (* *(fs_<ak>,fs_<ak>) *) (* nprod(fs_<ak>,fs_<ak>) *) (* list(fs_<ak>) *) (* option(fs_<ak>) *) (* are instances of fs_<ak> *)
val thy24 = fold (fn ak_name => fn thy => let val cls_name = Sign.full_bname thy ("fs_"^ak_name); val fs_inst = Global_Theory.get_thm thy ("fs_"^ak_name^"_inst"); fun fs_proof thm ctxt =
EVERY [Class.intro_classes_tac ctxt [], resolve_tac ctxt [thm RS fs1] 1];
(******** cp_<ak>_<ai> class instances ********) (*==============================================*) (* abbreviations for some lemmas *) val cp1 = @{thm "cp1"}; val cp_unit_inst = @{thm "cp_unit_inst"}; val cp_bool_inst = @{thm "cp_bool_inst"}; val cp_prod_inst = @{thm "cp_prod_inst"}; val cp_list_inst = @{thm "cp_list_inst"}; val cp_fun_inst = @{thm "cp_fun_inst"}; val cp_option_inst = @{thm "cp_option_inst"}; val cp_noption_inst = @{thm "cp_noption_inst"}; val cp_set_inst = @{thm "cp_set_inst"}; val pt_perm_compose = @{thm "pt_perm_compose"};
val dj_pp_forget = @{thm "dj_perm_perm_forget"};
(* shows that <aj> is an instance of cp_<ak>_<ai> *) (* for every <ak>/<ai>-combination *) val thy25 = fold (fn ak_name => fn thy =>
fold (fn ak_name' => fn thy' =>
fold (fn ak_name'' => fn thy'' => let val name = Sign.full_bname thy'' ak_name; val cls_name = Sign.full_bname thy'' ("cp_"^ak_name'^"_"^ak_name''); fun proof ctxt =
(if (ak_name'=ak_name'') then
(let val pt_inst = Global_Theory.get_thm thy'' ("pt_"^ak_name''^"_inst"); val at_inst = Global_Theory.get_thm thy'' ("at_"^ak_name''^"_inst"); in
EVERY [Class.intro_classes_tac ctxt [],
resolve_tac ctxt [at_inst RS (pt_inst RS pt_perm_compose)] 1] end) else
(let val dj_inst = Global_Theory.get_thm thy'' ("dj_"^ak_name''^"_"^ak_name'); val simp_s = put_simpset HOL_basic_ss ctxt addsimps
((dj_inst RS dj_pp_forget)::
(maps (Global_Theory.get_thms thy'')
[ak_name' ^"_prm_"^ak_name^"_def",
ak_name''^"_prm_"^ak_name^"_def"])); in
EVERY [Class.intro_classes_tac ctxt [], simp_tac simp_s 1] end)) in
Axclass.prove_arity (name,[],[cls_name]) proof thy'' end) ak_names thy') ak_names thy) ak_names thy24;
(* shows that *) (* units *) (* products *) (* lists *) (* functions *) (* options *) (* noptions *) (* sets *) (* are instances of cp_<ak>_<ai> for every <ak>/<ai>-combination *) val thy26 = fold (fn ak_name => fn thy =>
fold (fn ak_name' => fn thy' => let val cls_name = Sign.full_bname thy' ("cp_"^ak_name^"_"^ak_name'); val cp_inst = Global_Theory.get_thm thy' ("cp_"^ak_name^"_"^ak_name'^"_inst"); val pt_inst = Global_Theory.get_thm thy' ("pt_"^ak_name^"_inst"); val at_inst = Global_Theory.get_thm thy' ("at_"^ak_name^"_inst");
fun cp_proof thm ctxt =
EVERY [Class.intro_classes_tac ctxt [], resolve_tac ctxt [thm RS cp1] 1];
(* show that discrete nominal types are permutation types, finitely *) (* supported and have the commutation property *) (* discrete types have a permutation operation defined as pi o x = x; *) (* which renders the proofs to be simple "simp_all"-proofs. *) val thy32 = let fun discrete_pt_inst discrete_ty defn =
fold (fn ak_name => fn thy => let val qu_class = Sign.full_bname thy ("pt_"^ak_name); fun proof ctxt =
Class.intro_classes_tac ctxt [] THEN
REPEAT (asm_simp_tac (put_simpset HOL_basic_ss ctxt addsimps [Simpdata.mk_eq defn]) 1); in
Axclass.prove_arity (discrete_ty, [], [qu_class]) proof thy end) ak_names;
fun discrete_fs_inst discrete_ty defn =
fold (fn ak_name => fn thy => let val qu_class = Sign.full_bname thy ("fs_"^ak_name); val supp_def = Simpdata.mk_eq @{thm "Nominal.supp_def"}; fun proof ctxt =
Class.intro_classes_tac ctxt [] THEN
asm_simp_tac (put_simpset HOL_ss ctxt
addsimps [supp_def, Collect_const, finite_emptyI, Simpdata.mk_eq defn]) 1; in
Axclass.prove_arity (discrete_ty, [], [qu_class]) proof thy end) ak_names;
fun discrete_cp_inst discrete_ty defn =
fold (fn ak_name' => (fold (fn ak_name => fn thy => let val qu_class = Sign.full_bname thy ("cp_"^ak_name^"_"^ak_name'); val supp_def = Simpdata.mk_eq @{thm "Nominal.supp_def"}; fun proof ctxt =
Class.intro_classes_tac ctxt [] THEN
asm_simp_tac (put_simpset HOL_ss ctxt addsimps [Simpdata.mk_eq defn]) 1; in
Axclass.prove_arity (discrete_ty, [], [qu_class]) proof thy end) ak_names)) ak_names;
(* abbreviations for some lemmas *) (*===============================*) val abs_fun_pi = @{thm "Nominal.abs_fun_pi"}; val abs_fun_pi_ineq = @{thm "Nominal.abs_fun_pi_ineq"}; val abs_fun_eq = @{thm "Nominal.abs_fun_eq"}; val abs_fun_eq' = @{thm "Nominal.abs_fun_eq'"}; val abs_fun_fresh = @{thm "Nominal.abs_fun_fresh"}; val abs_fun_fresh' = @{thm "Nominal.abs_fun_fresh'"}; val dj_perm_forget = @{thm "Nominal.dj_perm_forget"}; val dj_pp_forget = @{thm "Nominal.dj_perm_perm_forget"}; val fresh_iff = @{thm "Nominal.fresh_abs_fun_iff"}; val fresh_iff_ineq = @{thm "Nominal.fresh_abs_fun_iff_ineq"}; val abs_fun_supp = @{thm "Nominal.abs_fun_supp"}; val abs_fun_supp_ineq = @{thm "Nominal.abs_fun_supp_ineq"}; val pt_swap_bij = @{thm "Nominal.pt_swap_bij"}; val pt_swap_bij' = @{thm "Nominal.pt_swap_bij'"}; val pt_fresh_fresh = @{thm "Nominal.pt_fresh_fresh"}; val pt_bij = @{thm "Nominal.pt_bij"}; val pt_perm_compose = @{thm "Nominal.pt_perm_compose"}; val pt_perm_compose' = @{thm "Nominal.pt_perm_compose'"}; val perm_app = @{thm "Nominal.pt_fun_app_eq"}; val at_fresh = @{thm "Nominal.at_fresh"}; val at_fresh_ineq = @{thm "Nominal.at_fresh_ineq"}; val at_calc = @{thms "Nominal.at_calc"}; val at_swap_simps = @{thms "Nominal.at_swap_simps"}; val at_supp = @{thm "Nominal.at_supp"}; val dj_supp = @{thm "Nominal.dj_supp"}; val fresh_left_ineq = @{thm "Nominal.pt_fresh_left_ineq"}; val fresh_left = @{thm "Nominal.pt_fresh_left"}; val fresh_right_ineq = @{thm "Nominal.pt_fresh_right_ineq"}; val fresh_right = @{thm "Nominal.pt_fresh_right"}; val fresh_bij_ineq = @{thm "Nominal.pt_fresh_bij_ineq"}; val fresh_bij = @{thm "Nominal.pt_fresh_bij"}; val fresh_star_bij_ineq = @{thms "Nominal.pt_fresh_star_bij_ineq"}; val fresh_star_bij = @{thms "Nominal.pt_fresh_star_bij"}; val fresh_eqvt = @{thm "Nominal.pt_fresh_eqvt"}; val fresh_eqvt_ineq = @{thm "Nominal.pt_fresh_eqvt_ineq"}; val fresh_star_eqvt = @{thms "Nominal.pt_fresh_star_eqvt"}; val fresh_star_eqvt_ineq= @{thms "Nominal.pt_fresh_star_eqvt_ineq"}; val set_diff_eqvt = @{thm "Nominal.pt_set_diff_eqvt"}; val in_eqvt = @{thm "Nominal.pt_in_eqvt"}; val eq_eqvt = @{thm "Nominal.pt_eq_eqvt"}; val all_eqvt = @{thm "Nominal.pt_all_eqvt"}; val ex_eqvt = @{thm "Nominal.pt_ex_eqvt"}; val ex1_eqvt = @{thm "Nominal.pt_ex1_eqvt"}; val the_eqvt = @{thm "Nominal.pt_the_eqvt"}; val pt_pi_rev = @{thm "Nominal.pt_pi_rev"}; val pt_rev_pi = @{thm "Nominal.pt_rev_pi"}; val at_exists_fresh = @{thm "Nominal.at_exists_fresh"}; val at_exists_fresh' = @{thm "Nominal.at_exists_fresh'"}; val fresh_perm_app_ineq = @{thm "Nominal.pt_fresh_perm_app_ineq"}; val fresh_perm_app = @{thm "Nominal.pt_fresh_perm_app"}; val fresh_aux = @{thm "Nominal.pt_fresh_aux"}; val pt_perm_supp_ineq = @{thm "Nominal.pt_perm_supp_ineq"}; val pt_perm_supp = @{thm "Nominal.pt_perm_supp"}; val subseteq_eqvt = @{thm "Nominal.pt_subseteq_eqvt"};
(* Now we collect and instantiate some lemmas w.r.t. all atom *) (* types; this allows for example to use abs_perm (which is a *) (* collection of theorems) instead of thm abs_fun_pi with explicit *) (* instantiations. *) val (_, thy33) = let val ctxt32 = Proof_Context.init_global thy32;
(* takes a theorem thm and a list of theorems [t1,..,tn] *) (* produces a list of theorems of the form [t1 RS thm,..,tn RS thm] *) fun instR thm thms = map (fn ti => ti RS thm) thms;
(* takes a theorem thm and a list of theorems [(t1,m1),..,(tn,mn)] *) (* produces a list of theorems of the form [[t1,m1] MRS thm,..,[tn,mn] MRS thm] *) fun instRR thm thms = map (fn (ti,mi) => [ti,mi] MRS thm) thms;
(* takes two theorem lists (hopefully of the same length ;o) *) (* produces a list of theorems of the form *) (* [t1 RS m1,..,tn RS mn] where [t1,..,tn] is thms1 and [m1,..,mn] is thms2 *) fun inst_zip thms1 thms2 = map (fn (t1,t2) => t1 RS t2) (thms1 ~~ thms2);
(* takes a theorem list of the form [l1,...,ln] *) (* and a list of theorem lists of the form *) (* [[h11,...,h1m],....,[hk1,....,hkm] *) (* produces the list of theorem lists *) (* [[l1 RS h11,...,l1 RS h1m],...,[ln RS hk1,...,ln RS hkm]] *) fun inst_mult thms thmss = map (fn (t,ts) => instR t ts) (thms ~~ thmss);
(* FIXME: these lists do not need to be created dynamically again *)
(* list of all at_inst-theorems *) val ats = map (fn ak => Global_Theory.get_thm thy32 ("at_"^ak^"_inst")) ak_names (* list of all pt_inst-theorems *) val pts = map (fn ak => Global_Theory.get_thm thy32 ("pt_"^ak^"_inst")) ak_names (* list of all cp_inst-theorems as a collection of lists*) val cps = letfun cps_fun ak1 ak2 = Global_Theory.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst") inmap (fn aki => (map (cps_fun aki) ak_names)) ak_names end; (* list of all cp_inst-theorems that have different atom types *) val cps' = letfun cps'_fun ak1 ak2 = if ak1=ak2 then NONE else SOME (Global_Theory.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst")) inmap (fn aki => (map_filter I (map (cps'_fun aki) ak_names))) ak_names end; (* list of all dj_inst-theorems *) val djs = letfun djs_fun ak1 ak2 = if ak1=ak2 then NONE else SOME(Global_Theory.get_thm thy32 ("dj_"^ak2^"_"^ak1)) in map_filter I (map_product djs_fun ak_names ak_names) end; (* list of all fs_inst-theorems *) val fss = map (fn ak => Global_Theory.get_thm thy32 ("fs_"^ak^"_inst")) ak_names (* list of all at_inst-theorems *) val fs_axs = map (fn ak => Global_Theory.get_thm thy32 ("fs_"^ak^"1")) ak_names
fun inst_pt thms = maps (fn ti => instR ti pts) thms; fun inst_at thms = maps (fn ti => instR ti ats) thms; fun inst_fs thms = maps (fn ti => instR ti fss) thms; fun inst_cp thms cps = flat (inst_mult thms cps); fun inst_pt_at thms = maps (fn ti => instRR ti (pts ~~ ats)) thms; fun inst_dj thms = maps (fn ti => instR ti djs) thms; fun inst_pt_pt_at_cp thms = inst_cp (inst_zip ats (inst_zip pts (inst_pt thms))) cps; fun inst_pt_at_fs thms = inst_zip (inst_fs [fs1]) (inst_zip ats (inst_pt thms)); fun inst_pt_pt_at_cp thms = letval i_pt_pt_at = inst_zip ats (inst_zip pts (inst_pt thms)); val i_pt_pt_at_cp = inst_cp i_pt_pt_at cps'; in i_pt_pt_at_cp end; fun inst_pt_pt_at_cp_dj thms = inst_zip djs (inst_pt_pt_at_cp thms); in
thy32
|> add_thmss_string [(("alpha", inst_pt_at [abs_fun_eq]),[])]
||>> add_thmss_string [(("alpha'", inst_pt_at [abs_fun_eq']),[])]
||>> add_thmss_string [(("alpha_fresh", inst_pt_at [abs_fun_fresh]),[])]
||>> add_thmss_string [(("alpha_fresh'", inst_pt_at [abs_fun_fresh']),[])]
||>> add_thmss_string [(("perm_swap", inst_pt_at [pt_swap_bij] @ inst_pt_at [pt_swap_bij']),[])]
||>> add_thmss_string letval thms1 = inst_at at_swap_simps and thms2 = inst_dj [dj_perm_forget] in [(("swap_simps", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [pt_pi_rev]; val thms2 = inst_pt_at [pt_rev_pi]; in [(("perm_pi_simp",thms1 @ thms2),[])] end
||>> add_thmss_string [(("perm_fresh_fresh", inst_pt_at [pt_fresh_fresh]),[])]
||>> add_thmss_string [(("perm_bij", inst_pt_at [pt_bij]),[])]
||>> add_thmss_string letval thms1 = inst_pt_at [pt_perm_compose]; val thms2 = instR cp1 (Library.flat cps'); in [(("perm_compose",thms1 @ thms2),[])] end
||>> add_thmss_string [(("perm_compose'",inst_pt_at [pt_perm_compose']),[])]
||>> add_thmss_string [(("perm_app", inst_pt_at [perm_app]),[])]
||>> add_thmss_string [(("supp_atm", (inst_at [at_supp]) @ (inst_dj [dj_supp])),[])]
||>> add_thmss_string [(("exists_fresh", inst_at [at_exists_fresh]),[])]
||>> add_thmss_string [(("exists_fresh'", inst_at [at_exists_fresh']),[])]
||>> add_thmss_string let val thms1 = inst_pt_at [all_eqvt]; val thms2 = map (fold_rule ctxt32 [inductive_forall_def]) thms1 in
[(("all_eqvt", thms1 @ thms2), [NominalThmDecls.eqvt_force_add])] end
||>> add_thmss_string [(("ex_eqvt", inst_pt_at [ex_eqvt]),[NominalThmDecls.eqvt_force_add])]
||>> add_thmss_string [(("ex1_eqvt", inst_pt_at [ex1_eqvt]),[NominalThmDecls.eqvt_force_add])]
||>> add_thmss_string [(("the_eqvt", inst_pt_at [the_eqvt]),[NominalThmDecls.eqvt_force_add])]
||>> add_thmss_string letval thms1 = inst_at [at_fresh] val thms2 = inst_dj [at_fresh_ineq] in [(("fresh_atm", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_at at_calc and thms2 = inst_dj [dj_perm_forget] in [(("calc_atm", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [abs_fun_pi] and thms2 = inst_pt_pt_at_cp [abs_fun_pi_ineq] in [(("abs_perm", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_dj [dj_perm_forget] and thms2 = inst_dj [dj_pp_forget] in [(("perm_dj", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at_fs [fresh_iff] and thms2 = inst_pt_at [fresh_iff] and thms3 = inst_pt_pt_at_cp_dj [fresh_iff_ineq] in [(("abs_fresh", thms1 @ thms2 @ thms3),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [abs_fun_supp] and thms2 = inst_pt_at_fs [abs_fun_supp] and thms3 = inst_pt_pt_at_cp_dj [abs_fun_supp_ineq] in [(("abs_supp", thms1 @ thms2 @ thms3),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_left] and thms2 = inst_pt_pt_at_cp [fresh_left_ineq] in [(("fresh_left", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_right] and thms2 = inst_pt_pt_at_cp [fresh_right_ineq] in [(("fresh_right", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_bij] and thms2 = inst_pt_pt_at_cp [fresh_bij_ineq] in [(("fresh_bij", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at fresh_star_bij and thms2 = maps (fn ti => inst_pt_pt_at_cp [ti]) fresh_star_bij_ineq in [(("fresh_star_bij", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_eqvt] and thms2 = inst_pt_pt_at_cp_dj [fresh_eqvt_ineq] in [(("fresh_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at fresh_star_eqvt and thms2 = maps (fn ti => inst_pt_pt_at_cp_dj [ti]) fresh_star_eqvt_ineq in [(("fresh_star_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at [in_eqvt] in [(("in_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at [eq_eqvt] in [(("eq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at [set_diff_eqvt] in [(("set_diff_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at [subseteq_eqvt] in [(("subseteq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_aux] and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] in [(("fresh_aux", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [fresh_perm_app] and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] in [(("fresh_perm_app", thms1 @ thms2),[])] end
||>> add_thmss_string letval thms1 = inst_pt_at [pt_perm_supp] and thms2 = inst_pt_pt_at_cp [pt_perm_supp_ineq] in [(("supp_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
||>> add_thmss_string [(("fin_supp",fs_axs),[])] end;
val _ =
Outer_Syntax.command \<^command_keyword>\<open>atom_decl\<close> "declare new kinds of atoms"
(Scan.repeat1 Parse.name >> (Toplevel.theory o create_nom_typedecls));
end;
¤ Diese beiden folgenden Angebotsgruppen bietet das Unternehmen0.36Angebot
Wie Sie bei der Firma Beratungs- und Dienstleistungen beauftragen können
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
