(* $Id$ *)
signature NOMINAL_ATOMS =
sig
val create_nom_typedecls : string list -> theory -> theory
val atoms_of : theory -> string list
val mk_permT : typ -> typ
val setup : (theory -> theory) list
end
structure NominalAtoms : NOMINAL_ATOMS =
struct
(* data kind 'HOL/nominal' *)
structure NominalArgs =
struct
val name = "HOL/nominal";
type T = unit Symtab.table;
val empty = Symtab.empty;
val copy = I;
val extend = I;
fun merge _ x = Symtab.merge (K true) x;
fun print sg tab = ();
end;
structure NominalData = TheoryDataFun(NominalArgs);
fun atoms_of thy = map fst (Symtab.dest (NominalData.get thy));
(* FIXME: add to hologic.ML ? *)
fun mk_listT T = Type ("List.list", [T]);
fun mk_permT T = mk_listT (HOLogic.mk_prodT (T, T));
fun mk_Cons x xs =
let val T = fastype_of x
in Const ("List.list.Cons", T --> mk_listT T --> mk_listT T) $ x $ xs end;
(* 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
(* declares a type-decl for every atom-kind: *)
(* that is typedecl <ak> *)
val thy1 = TypedefPackage.add_typedecls (map (fn x => (x,[],NoSyn)) 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 (sign_of thy1)) ak_names;
val ak_names_types = ak_names ~~ map (Type o rpair []) full_ak_names;
(* adds for every atom-kind an axiom *)
(* <ak>_infinite: infinite (UNIV::<ak_type> set) *)
val (inf_axs,thy2) = PureThy.add_axioms_i (map (fn (ak_name, T) =>
let
val name = ak_name ^ "_infinite"
val axiom = HOLogic.mk_Trueprop (HOLogic.mk_not
(HOLogic.mk_mem (HOLogic.mk_UNIV T,
Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT T)))))
in
((name, axiom), [])
end) ak_names_types) thy1;
(* 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 (thy3, swap_eqs) = foldl_map (fn (thy, (ak_name, T)) =>
let
val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
val swap_name = Sign.full_name (sign_of thy) ("swap_" ^ ak_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 ("HOL.If", HOLogic.boolT --> T --> T --> T);
val cswap_akname = Const (swap_name, swapT);
val cswap = Const ("nominal.swap", swapT)
val name = "swap_"^ak_name^"_def";
val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(cswap_akname $ 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 |> Theory.add_consts_i [("swap_" ^ ak_name, swapT, NoSyn)]
|> (#2 o PureThy.add_defs_i true [((name, def2),[])])
|> PrimrecPackage.add_primrec_i "" [(("", def1),[])]
end) (thy2, ak_names_types);
(* 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 (thy4, prm_eqs) = foldl_map (fn (thy, (ak_name, T)) =>
let
val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
val swap_name = Sign.full_name (sign_of thy) ("swap_" ^ ak_name)
val prmT = mk_permT T --> T --> T;
val prm_name = ak_name ^ "_prm_" ^ ak_name;
val qu_prm_name = Sign.full_name (sign_of thy) prm_name;
val x = Free ("x", HOLogic.mk_prodT (T, T));
val xs = Free ("xs", mk_permT T);
val a = Free ("a", T) ;
val cnil = Const ("List.list.Nil", mk_permT T);
val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (qu_prm_name, prmT) $ cnil $ a, a));
val def2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const (qu_prm_name, prmT) $ mk_Cons x xs $ a,
Const (swap_name, swapT) $ x $ (Const (qu_prm_name, prmT) $ xs $ a)));
in
thy |> Theory.add_consts_i [(prm_name, mk_permT T --> T --> T, NoSyn)]
|> PrimrecPackage.add_primrec_i "" [(("", def1), []),(("", def2), [])]
end) (thy3, ak_names_types);
(* 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 ("nominal.perm", mk_permT T --> T' --> T');
val cperm_def = Const (Sign.full_name (sign_of thy') perm_def_name, mk_permT T --> T' --> T');
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
PureThy.add_defs_i true [((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 |> PureThy.add_thms (map (fn (ak_name, T) =>
let
val ak_name_qu = Sign.full_name (sign_of thy5) (ak_name);
val i_type = Type(ak_name_qu,[]);
val cat = Const ("nominal.at",(Term.itselfT i_type) --> HOLogic.boolT);
val at_type = Logic.mk_type i_type;
val simp_s = HOL_basic_ss addsimps PureThy.get_thmss thy5
[Name "at_def",
Name (ak_name ^ "_prm_" ^ ak_name ^ "_def"),
Name (ak_name ^ "_prm_" ^ ak_name ^ ".simps"),
Name ("swap_" ^ ak_name ^ "_def"),
Name ("swap_" ^ ak_name ^ ".simps"),
Name (ak_name ^ "_infinite")]
val name = "at_"^ak_name^ "_inst";
val statement = HOLogic.mk_Trueprop (cat $ at_type);
val proof = fn _ => auto_tac (claset(),simp_s);
in
((name, standard (Goal.prove thy5 [] [] statement proof)), [])
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",["HOL.type"]);
val x = Free ("x", ty);
val pi1 = Free ("pi1", mk_permT T);
val pi2 = Free ("pi2", mk_permT T);
val cperm = Const ("nominal.perm", mk_permT T --> ty --> ty);
val cnil = Const ("List.list.Nil", mk_permT T);
val cappend = Const ("List.op @",mk_permT T --> mk_permT T --> mk_permT T);
val cprm_eq = Const ("nominal.prm_eq",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.add_axclass_i (cl_name, ["HOL.type"])
[((cl_name^"1", axiom1),[Simplifier.simp_add_global]),
((cl_name^"2", axiom2),[]),
((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 |> PureThy.add_thms (map (fn (ak_name, T) =>
let
val ak_name_qu = Sign.full_name (sign_of thy7) (ak_name);
val pt_name_qu = Sign.full_name (sign_of thy7) ("pt_"^ak_name);
val i_type1 = TFree("'x",[pt_name_qu]);
val i_type2 = Type(ak_name_qu,[]);
val cpt = Const ("nominal.pt",(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;
val simp_s = HOL_basic_ss addsimps PureThy.get_thmss thy7
[Name "pt_def",
Name ("pt_" ^ ak_name ^ "1"),
Name ("pt_" ^ ak_name ^ "2"),
Name ("pt_" ^ ak_name ^ "3")];
val name = "pt_"^ak_name^ "_inst";
val statement = HOLogic.mk_Trueprop (cpt $ pt_type $ at_type);
val proof = fn _ => auto_tac (claset(),simp_s);
in
((name, standard (Goal.prove thy7 [] [] statement proof)), [])
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_name (sign_of thy) ("pt_"^ak_name);
val ty = TFree("'a",["HOL.type"]);
val x = Free ("x", ty);
val csupp = Const ("nominal.supp", ty --> HOLogic.mk_setT T);
val cfinites = Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT T))
val axiom1 = HOLogic.mk_Trueprop (HOLogic.mk_mem (csupp $ x, cfinites));
in
AxClass.add_axclass_i (cl_name, [pt_name]) [((cl_name^"1", axiom1),[])] thy
end) ak_names_types thy8;
(* 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 |> PureThy.add_thms (map (fn (ak_name, T) =>
let
val ak_name_qu = Sign.full_name (sign_of thy11) (ak_name);
val fs_name_qu = Sign.full_name (sign_of thy11) ("fs_"^ak_name);
val i_type1 = TFree("'x",[fs_name_qu]);
val i_type2 = Type(ak_name_qu,[]);
val cfs = Const ("nominal.fs",
(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;
val simp_s = HOL_basic_ss addsimps PureThy.get_thmss thy11
[Name "fs_def",
Name ("fs_" ^ ak_name ^ "1")];
val name = "fs_"^ak_name^ "_inst";
val statement = HOLogic.mk_Trueprop (cfs $ fs_type $ at_type);
val proof = fn _ => auto_tac (claset(),simp_s);
in
((name, standard (Goal.prove thy11 [] [] statement proof)), [])
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 (_,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",["HOL.type"]);
val x = Free ("x", ty);
val pi1 = Free ("pi1", mk_permT T);
val pi2 = Free ("pi2", mk_permT T');
val cperm1 = Const ("nominal.perm", mk_permT T --> ty --> ty);
val cperm2 = Const ("nominal.perm", mk_permT T' --> ty --> ty);
val cperm3 = Const ("nominal.perm", mk_permT T --> mk_permT T' --> mk_permT T');
val ax1 = HOLogic.mk_Trueprop
(HOLogic.mk_eq (cperm1 $ pi1 $ (cperm2 $ pi2 $ x),
cperm2 $ (cperm3 $ pi1 $ pi2) $ (cperm1 $ pi1 $ x)));
in
AxClass.add_axclass_i (cl_name, ["HOL.type"]) [((cl_name^"1", ax1),[])] thy'
end) ak_names_types thy) ak_names_types thy12;
(* proves for every <ak>-combination a cp_<ak1>_<ak2>_inst theorem; *)
(* lemma cp_<ak1>_<ak2>_inst: *)
(* cp TYPE('a::cp_<ak1>_<ak2>) TYPE(<ak1>) TYPE(<ak2>) *)
val (cp_thms,thy12c) = fold_map (fn (ak_name, T) => fn thy =>
fold_map (fn (ak_name', T') => fn thy' =>
let
val ak_name_qu = Sign.full_name (sign_of thy') (ak_name);
val ak_name_qu' = Sign.full_name (sign_of thy') (ak_name');
val cp_name_qu = Sign.full_name (sign_of thy') ("cp_"^ak_name^"_"^ak_name');
val i_type0 = TFree("'a",[cp_name_qu]);
val i_type1 = Type(ak_name_qu,[]);
val i_type2 = Type(ak_name_qu',[]);
val ccp = Const ("nominal.cp",
(Term.itselfT i_type0)-->(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;
val cp_type = Logic.mk_type i_type0;
val simp_s = HOL_basic_ss addsimps PureThy.get_thmss thy' [(Name "cp_def")];
val cp1 = PureThy.get_thm thy' (Name ("cp_"^ak_name^"_"^ak_name'^"1"));
val name = "cp_"^ak_name^ "_"^ak_name'^"_inst";
val statement = HOLogic.mk_Trueprop (ccp $ cp_type $ at_type $ at_type');
val proof = fn _ => EVERY [auto_tac (claset(),simp_s), rtac cp1 1];
in
PureThy.add_thms [((name, standard (Goal.prove thy' [] [] statement proof)), [])] thy'
end)
ak_names_types thy) ak_names_types thy12b;
(* 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' =>
(if not (ak_name = ak_name')
then
let
val ak_name_qu = Sign.full_name (sign_of thy') (ak_name);
val ak_name_qu' = Sign.full_name (sign_of thy') (ak_name');
val i_type1 = Type(ak_name_qu,[]);
val i_type2 = Type(ak_name_qu',[]);
val cdj = Const ("nominal.disjoint",
(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;
val simp_s = HOL_basic_ss addsimps PureThy.get_thmss thy'
[Name "disjoint_def",
Name (ak_name^"_prm_"^ak_name'^"_def"),
Name (ak_name'^"_prm_"^ak_name^"_def")];
val name = "dj_"^ak_name^"_"^ak_name';
val statement = HOLogic.mk_Trueprop (cdj $ at_type $ at_type');
val proof = fn _ => auto_tac (claset(),simp_s);
in
PureThy.add_thms [((name, standard (Goal.prove thy' [] [] statement proof)), [])] 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_set_inst = thm "pt_set_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";
(* 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_name (sign_of thy') ak_name';
val cls_name = Sign.full_name (sign_of thy') ("pt_"^ak_name);
val at_inst = PureThy.get_thm thy' (Name ("at_"^ak_name'^"_inst"));
val proof1 = EVERY [AxClass.intro_classes_tac [],
rtac ((at_inst RS at_pt_inst) RS pt1) 1,
rtac ((at_inst RS at_pt_inst) RS pt2) 1,
rtac ((at_inst RS at_pt_inst) RS pt3) 1,
atac 1];
val simp_s = HOL_basic_ss addsimps
PureThy.get_thmss thy' [Name (ak_name^"_prm_"^ak_name'^"_def")];
val proof2 = EVERY [AxClass.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
in
thy'
|> AxClass.add_inst_arity_i (qu_name,[],[cls_name])
(if ak_name = ak_name' then proof1 else proof2)
end) ak_names thy) ak_names thy12c;
(* show that *)
(* fun(pt_<ak>,pt_<ak>) *)
(* noption(pt_<ak>) *)
(* option(pt_<ak>) *)
(* list(pt_<ak>) *)
(* *(pt_<ak>,pt_<ak>) *)
(* nprod(pt_<ak>,pt_<ak>) *)
(* unit *)
(* set(pt_<ak>) *)
(* are instances of pt_<ak> *)
val thy18 = fold (fn ak_name => fn thy =>
let
val cls_name = Sign.full_name (sign_of thy) ("pt_"^ak_name);
val at_thm = PureThy.get_thm thy (Name ("at_"^ak_name^"_inst"));
val pt_inst = PureThy.get_thm thy (Name ("pt_"^ak_name^"_inst"));
fun pt_proof thm =
EVERY [AxClass.intro_classes_tac [],
rtac (thm RS pt1) 1, rtac (thm RS pt2) 1, rtac (thm RS pt3) 1, atac 1];
val pt_thm_fun = at_thm RS (pt_inst RS (pt_inst RS pt_fun_inst));
val pt_thm_noptn = pt_inst RS pt_noptn_inst;
val pt_thm_optn = pt_inst RS pt_optn_inst;
val pt_thm_list = pt_inst RS pt_list_inst;
val pt_thm_prod = pt_inst RS (pt_inst RS pt_prod_inst);
val pt_thm_nprod = pt_inst RS (pt_inst RS pt_nprod_inst);
val pt_thm_unit = pt_unit_inst;
val pt_thm_set = pt_inst RS pt_set_inst
in
thy
|> AxClass.add_inst_arity_i ("fun",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_fun)
|> AxClass.add_inst_arity_i ("nominal.noption",[[cls_name]],[cls_name]) (pt_proof pt_thm_noptn)
|> AxClass.add_inst_arity_i ("Datatype.option",[[cls_name]],[cls_name]) (pt_proof pt_thm_optn)
|> AxClass.add_inst_arity_i ("List.list",[[cls_name]],[cls_name]) (pt_proof pt_thm_list)
|> AxClass.add_inst_arity_i ("*",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_prod)
|> AxClass.add_inst_arity_i ("nominal.nprod",[[cls_name],[cls_name]],[cls_name])
(pt_proof pt_thm_nprod)
|> AxClass.add_inst_arity_i ("Product_Type.unit",[],[cls_name]) (pt_proof pt_thm_unit)
|> AxClass.add_inst_arity_i ("set",[[cls_name]],[cls_name]) (pt_proof pt_thm_set)
end) ak_names thy13;
(*<<<<<<< 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_name (sign_of thy') ak_name';
val qu_class = Sign.full_name (sign_of thy') ("fs_"^ak_name);
val proof =
(if ak_name = ak_name'
then
let val at_thm = PureThy.get_thm thy' (Name ("at_"^ak_name^"_inst"));
in EVERY [AxClass.intro_classes_tac [],
rtac ((at_thm RS fs_at_inst) RS fs1) 1] end
else
let val dj_inst = PureThy.get_thm thy' (Name ("dj_"^ak_name'^"_"^ak_name));
val simp_s = HOL_basic_ss addsimps [dj_inst RS dj_supp, Finites.emptyI];
in EVERY [AxClass.intro_classes_tac [], asm_simp_tac simp_s 1] end)
in
AxClass.add_inst_arity_i (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_name (sign_of thy) ("fs_"^ak_name);
val fs_inst = PureThy.get_thm thy (Name ("fs_"^ak_name^"_inst"));
fun fs_proof thm = EVERY [AxClass.intro_classes_tac [], rtac (thm RS fs1) 1];
val fs_thm_unit = fs_unit_inst;
val fs_thm_prod = fs_inst RS (fs_inst RS fs_prod_inst);
val fs_thm_nprod = fs_inst RS (fs_inst RS fs_nprod_inst);
val fs_thm_list = fs_inst RS fs_list_inst;
val fs_thm_optn = fs_inst RS fs_option_inst;
in
thy
|> AxClass.add_inst_arity_i ("Product_Type.unit",[],[cls_name]) (fs_proof fs_thm_unit)
|> AxClass.add_inst_arity_i ("*",[[cls_name],[cls_name]],[cls_name]) (fs_proof fs_thm_prod)
|> AxClass.add_inst_arity_i ("nominal.nprod",[[cls_name],[cls_name]],[cls_name])
(fs_proof fs_thm_nprod)
|> AxClass.add_inst_arity_i ("List.list",[[cls_name]],[cls_name]) (fs_proof fs_thm_list)
|> AxClass.add_inst_arity_i ("Datatype.option",[[cls_name]],[cls_name]) (fs_proof fs_thm_optn)
end) ak_names thy20;
(*<<<<<<< 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 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_name (sign_of thy'') ak_name;
val cls_name = Sign.full_name (sign_of thy'') ("cp_"^ak_name'^"_"^ak_name'');
val proof =
(if (ak_name'=ak_name'') then
(let
val pt_inst = PureThy.get_thm thy'' (Name ("pt_"^ak_name''^"_inst"));
val at_inst = PureThy.get_thm thy'' (Name ("at_"^ak_name''^"_inst"));
in
EVERY [AxClass.intro_classes_tac [],
rtac (at_inst RS (pt_inst RS pt_perm_compose)) 1]
end)
else
(let
val dj_inst = PureThy.get_thm thy'' (Name ("dj_"^ak_name''^"_"^ak_name'));
val simp_s = HOL_basic_ss addsimps
((dj_inst RS dj_pp_forget)::
(PureThy.get_thmss thy''
[Name (ak_name' ^"_prm_"^ak_name^"_def"),
Name (ak_name''^"_prm_"^ak_name^"_def")]));
in
EVERY [AxClass.intro_classes_tac [], simp_tac simp_s 1]
end))
in
AxClass.add_inst_arity_i (name,[],[cls_name]) proof thy''
end) ak_names thy') ak_names thy) ak_names thy24;
(* shows that *)
(* units *)
(* products *)
(* lists *)
(* functions *)
(* options *)
(* noptions *)
(* 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_name (sign_of thy') ("cp_"^ak_name^"_"^ak_name');
val cp_inst = PureThy.get_thm thy' (Name ("cp_"^ak_name^"_"^ak_name'^"_inst"));
val pt_inst = PureThy.get_thm thy' (Name ("pt_"^ak_name^"_inst"));
val at_inst = PureThy.get_thm thy' (Name ("at_"^ak_name^"_inst"));
fun cp_proof thm = EVERY [AxClass.intro_classes_tac [],rtac (thm RS cp1) 1];
val cp_thm_unit = cp_unit_inst;
val cp_thm_prod = cp_inst RS (cp_inst RS cp_prod_inst);
val cp_thm_list = cp_inst RS cp_list_inst;
val cp_thm_fun = at_inst RS (pt_inst RS (cp_inst RS (cp_inst RS cp_fun_inst)));
val cp_thm_optn = cp_inst RS cp_option_inst;
val cp_thm_noptn = cp_inst RS cp_noption_inst;
in
thy'
|> AxClass.add_inst_arity_i ("Product_Type.unit",[],[cls_name]) (cp_proof cp_thm_unit)
|> AxClass.add_inst_arity_i ("*",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_prod)
|> AxClass.add_inst_arity_i ("List.list",[[cls_name]],[cls_name]) (cp_proof cp_thm_list)
|> AxClass.add_inst_arity_i ("fun",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_fun)
|> AxClass.add_inst_arity_i ("Datatype.option",[[cls_name]],[cls_name]) (cp_proof cp_thm_optn)
|> AxClass.add_inst_arity_i ("nominal.noption",[[cls_name]],[cls_name]) (cp_proof cp_thm_noptn)
end) ak_names thy) ak_names thy25;
(* 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_name (sign_of thy) ("pt_"^ak_name);
val simp_s = HOL_basic_ss addsimps [defn];
val proof = EVERY [AxClass.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
in
AxClass.add_inst_arity_i (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_name (sign_of thy) ("fs_"^ak_name);
val supp_def = thm "nominal.supp_def";
val simp_s = HOL_ss addsimps [supp_def,Collect_const,Finites.emptyI,defn];
val proof = EVERY [AxClass.intro_classes_tac [], asm_simp_tac simp_s 1];
in
AxClass.add_inst_arity_i (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_name (sign_of thy) ("cp_"^ak_name^"_"^ak_name');
val supp_def = thm "nominal.supp_def";
val simp_s = HOL_ss addsimps [defn];
val proof = EVERY [AxClass.intro_classes_tac [], asm_simp_tac simp_s 1];
in
AxClass.add_inst_arity_i (discrete_ty,[],[qu_class]) proof thy
end) ak_names)) ak_names;
in
thy26
|> discrete_pt_inst "nat" (thm "perm_nat_def")
|> discrete_fs_inst "nat" (thm "perm_nat_def")
|> discrete_cp_inst "nat" (thm "perm_nat_def")
|> discrete_pt_inst "bool" (thm "perm_bool")
|> discrete_fs_inst "bool" (thm "perm_bool")
|> discrete_cp_inst "bool" (thm "perm_bool")
|> discrete_pt_inst "IntDef.int" (thm "perm_int_def")
|> discrete_fs_inst "IntDef.int" (thm "perm_int_def")
|> discrete_cp_inst "IntDef.int" (thm "perm_int_def")
|> discrete_pt_inst "List.char" (thm "perm_char_def")
|> discrete_fs_inst "List.char" (thm "perm_char_def")
|> discrete_cp_inst "List.char" (thm "perm_char_def")
end;
(* 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 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_fresh_fresh = thm "nominal.pt_fresh_fresh";
val pt_bij = thm "nominal.pt_bij";
val pt_perm_compose = thm "nominal.pt_perm_compose";
val perm_eq_app = thm "nominal.perm_eq_app";
val at_fresh = thm "nominal.at_fresh";
val at_calc = thms "nominal.at_calc";
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_bij_ineq = thm "nominal.pt_fresh_bij_ineq";
val fresh_bij = thm "nominal.pt_fresh_bij";
(* 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
(* 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 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 => PureThy.get_thm thy32 (Name ("at_"^ak^"_inst"))) ak_names
(* list of all pt_inst-theorems *)
val pts = map (fn ak => PureThy.get_thm thy32 (Name ("pt_"^ak^"_inst"))) ak_names
(* list of all cp_inst-theorems as a collection of lists*)
val cps =
let fun cps_fun ak1 ak2 = PureThy.get_thm thy32 (Name ("cp_"^ak1^"_"^ak2^"_inst"))
in map (fn aki => (map (cps_fun aki) ak_names)) ak_names end;
(* list of all cp_inst-theorems that have different atom types *)
val cps' =
let fun cps'_fun ak1 ak2 =
if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 (Name ("cp_"^ak1^"_"^ak2^"_inst")))
in map (fn aki => (List.mapPartial I (map (cps'_fun aki) ak_names))) ak_names end;
(* list of all dj_inst-theorems *)
val djs =
let fun djs_fun (ak1,ak2) =
if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 (Name ("dj_"^ak2^"_"^ak1)))
in List.mapPartial I (map djs_fun (Library.product ak_names ak_names)) end;
(* list of all fs_inst-theorems *)
val fss = map (fn ak => PureThy.get_thm thy32 (Name ("fs_"^ak^"_inst"))) ak_names
fun inst_pt thms = Library.flat (map (fn ti => instR ti pts) thms);
fun inst_at thms = Library.flat (map (fn ti => instR ti ats) thms);
fun inst_fs thms = Library.flat (map (fn ti => instR ti fss) thms);
fun inst_cp thms cps = Library.flat (inst_mult thms cps);
fun inst_pt_at thms = inst_zip ats (inst_pt thms);
fun inst_dj thms = Library.flat (map (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 =
let val 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
|> PureThy.add_thmss [(("alpha", inst_pt_at [abs_fun_eq]),[])]
||>> PureThy.add_thmss [(("perm_swap", inst_pt_at [pt_swap_bij]),[])]
||>> PureThy.add_thmss [(("perm_fresh_fresh", inst_pt_at [pt_fresh_fresh]),[])]
||>> PureThy.add_thmss [(("perm_bij", inst_pt_at [pt_bij]),[])]
||>> PureThy.add_thmss
let val thms1 = inst_pt_at [pt_perm_compose];
val thms2 = instR cp1 (Library.flat cps');
in [(("perm_compose",thms1 @ thms2),[])] end
||>> PureThy.add_thmss [(("perm_app_eq", inst_pt_at [perm_eq_app]),[])]
||>> PureThy.add_thmss [(("supp_atm", (inst_at [at_supp]) @ (inst_dj [dj_supp])),[])]
||>> PureThy.add_thmss [(("fresh_atm", inst_at [at_fresh]),[])]
||>> PureThy.add_thmss [(("calc_atm", inst_at at_calc),[])]
||>> PureThy.add_thmss
let val thms1 = inst_pt_at [abs_fun_pi]
and thms2 = inst_pt_pt_at_cp [abs_fun_pi_ineq]
in [(("abs_perm", thms1 @ thms2),[])] end
||>> PureThy.add_thmss
let val thms1 = inst_dj [dj_perm_forget]
and thms2 = inst_dj [dj_pp_forget]
in [(("perm_dj", thms1 @ thms2),[])] end
||>> PureThy.add_thmss
let val thms1 = inst_pt_at_fs [fresh_iff]
and thms2 = inst_pt_pt_at_cp_dj [fresh_iff_ineq]
in [(("abs_fresh", thms1 @ thms2),[])] end
||>> PureThy.add_thmss
let val 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
||>> PureThy.add_thmss
let val thms1 = inst_pt_at [fresh_left]
and thms2 = inst_pt_pt_at_cp [fresh_left_ineq]
in [(("fresh_left", thms1 @ thms2),[])] end
||>> PureThy.add_thmss
let val thms1 = inst_pt_at [fresh_bij]
and thms2 = inst_pt_pt_at_cp [fresh_bij_ineq]
in [(("fresh_eqvt", thms1 @ thms2),[])] end
end;
in NominalData.put (fold Symtab.update (map (rpair ()) full_ak_names)
(NominalData.get thy11)) thy33
end;
(* syntax und parsing *)
structure P = OuterParse and K = OuterKeyword;
val atom_declP =
OuterSyntax.command "atom_decl" "Declare new kinds of atoms" K.thy_decl
(Scan.repeat1 P.name >> (Toplevel.theory o create_nom_typedecls));
val _ = OuterSyntax.add_parsers [atom_declP];
val setup = [NominalData.init];
end;