Adapted to Florian's recent changes to the AxClass package.
(* $Id$ *)
signature NOMINAL_PACKAGE =
sig
val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
(bstring * string list * mixfix) list) list -> theory -> theory
end
structure NominalPackage : NOMINAL_PACKAGE =
struct
open DatatypeAux;
open NominalAtoms;
(** FIXME: DatatypePackage should export this function **)
local
fun dt_recs (DtTFree _) = []
| dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
| dt_recs (DtRec i) = [i];
fun dt_cases (descr: descr) (_, args, constrs) =
let
fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));
in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
fun induct_cases descr =
DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
in
fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
fun mk_case_names_exhausts descr new =
map (RuleCases.case_names o exhaust_cases descr o #1)
(List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
end;
(*******************************)
val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
fun read_typ sign ((Ts, sorts), str) =
let
val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
(map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
(** taken from HOL/Tools/datatype_aux.ML **)
fun indtac indrule indnames i st =
let
val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
(Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
val getP = if can HOLogic.dest_imp (hd ts) then
(apfst SOME) o HOLogic.dest_imp else pair NONE;
fun abstr (t1, t2) = (case t1 of
NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
(term_frees t2) of
[Free (s, T)] => absfree (s, T, t2)
| _ => sys_error "indtac")
| SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
val cert = cterm_of (Thm.sign_of_thm st);
val Ps = map (cert o head_of o snd o getP) ts;
val indrule' = cterm_instantiate (Ps ~~
(map (cert o abstr o getP) ts')) indrule
in
rtac indrule' i st
end;
fun mk_subgoal i f st =
let
val cg = List.nth (cprems_of st, i - 1);
val g = term_of cg;
val revcut_rl' = Thm.lift_rule cg revcut_rl;
val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
val ps = Logic.strip_params g;
val cert = cterm_of (sign_of_thm st);
in
compose_tac (false,
Thm.instantiate ([], [(cert v, cert (list_abs (ps,
f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
revcut_rl', 2) i st
end;
(** simplification procedure for sorting permutations **)
val dj_cp = thm "dj_cp";
fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
Type ("fun", [_, U])])) = (T, U);
fun permTs_of (Const ("nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
| permTs_of _ = [];
fun perm_simproc' thy ss (Const ("nominal.perm", T) $ t $ (u as Const ("nominal.perm", U) $ r $ s)) =
let
val (aT as Type (a, []), S) = dest_permT T;
val (bT as Type (b, []), _) = dest_permT U
in if aT mem permTs_of u andalso aT <> bT then
let
val a' = Sign.base_name a;
val b' = Sign.base_name b;
val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
val dj_cp' = [cp, dj] MRS dj_cp;
val cert = SOME o cterm_of thy
in
SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
[cert t, cert r, cert s] dj_cp'))
end
else NONE
end
| perm_simproc' thy ss _ = NONE;
val perm_simproc =
Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
val meta_spec = thm "meta_spec";
fun projections rule =
ProjectRule.projections rule
|> map (standard #> RuleCases.save rule);
fun norm_sort thy = Sorts.norm_sort (snd (#classes (Type.rep_tsig (Sign.tsig_of thy))));
fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
let
(* this theory is used just for parsing *)
val tmp_thy = thy |>
Theory.copy |>
Theory.add_types (map (fn (tvs, tname, mx, _) =>
(tname, length tvs, mx)) dts);
val sign = Theory.sign_of tmp_thy;
val atoms = atoms_of thy;
val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
Sign.base_name atom2)) atoms) atoms);
fun augment_sort S = S union classes;
val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
in (constrs @ [(cname, cargs', mx)], sorts') end
fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
in (dts @ [(tvs, tname, mx, constrs')], sorts') end
val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
val sorts' = map (apsnd augment_sort) sorts;
val tyvars = map #1 dts';
val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
val ps = map (fn (_, n, _, _) =>
(Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
val rps = map Library.swap ps;
fun replace_types (Type ("nominal.ABS", [T, U])) =
Type ("fun", [T, Type ("nominal.noption", [replace_types U])])
| replace_types (Type (s, Ts)) =
Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
| replace_types T = T;
fun replace_types' (Type (s, Ts)) =
Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
| replace_types' T = T;
val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
map (fn (cname, cargs, mx) => (cname,
map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
val ({induction, ...},thy1) =
DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
val SOME {descr, ...} = Symtab.lookup
(DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
(**** define permutation functions ****)
val permT = mk_permT (TFree ("'x", HOLogic.typeS));
val pi = Free ("pi", permT);
val perm_types = map (fn (i, _) =>
let val T = nth_dtyp i
in permT --> T --> T end) descr;
val perm_names = replicate (length new_type_names) "nominal.perm" @
DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
("perm_" ^ name_of_typ (nth_dtyp i)))
(length new_type_names upto length descr - 1));
val perm_names_types = perm_names ~~ perm_types;
val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
let val T = nth_dtyp i
in map (fn (cname, dts) =>
let
val Ts = map (typ_of_dtyp descr sorts') dts;
val names = DatatypeProp.make_tnames Ts;
val args = map Free (names ~~ Ts);
val c = Const (cname, Ts ---> T);
fun perm_arg (dt, x) =
let val T = type_of x
in if is_rec_type dt then
let val (Us, _) = strip_type T
in list_abs (map (pair "x") Us,
Const (List.nth (perm_names_types, body_index dt)) $ pi $
list_comb (x, map (fn (i, U) =>
Const ("nominal.perm", permT --> U --> U) $
(Const ("List.rev", permT --> permT) $ pi) $
Bound i) ((length Us - 1 downto 0) ~~ Us)))
end
else Const ("nominal.perm", permT --> T --> T) $ pi $ x
end;
in
(("", HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const (List.nth (perm_names_types, i)) $
Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
list_comb (c, args),
list_comb (c, map perm_arg (dts ~~ args))))), [])
end) constrs
end) descr);
val (thy2, perm_simps) = thy1 |>
Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
(List.drop (perm_names_types, length new_type_names))) |>
PrimrecPackage.add_primrec_i "" perm_eqs;
(**** prove that permutation functions introduced by unfolding are ****)
(**** equivalent to already existing permutation functions ****)
val _ = warning ("length descr: " ^ string_of_int (length descr));
val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
val unfolded_perm_eq_thms =
if length descr = length new_type_names then []
else map standard (List.drop (split_conj_thm
(Goal.prove thy2 [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn (c as (s, T), x) =>
let val [T1, T2] = binder_types T
in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
Const ("nominal.perm", T) $ pi $ Free (x, T2))
end)
(perm_names_types ~~ perm_indnames))))
(fn _ => EVERY [indtac induction perm_indnames 1,
ALLGOALS (asm_full_simp_tac
(simpset_of thy2 addsimps [perm_fun_def]))])),
length new_type_names));
(**** prove [] \<bullet> t = t ****)
val _ = warning "perm_empty_thms";
val perm_empty_thms = List.concat (map (fn a =>
let val permT = mk_permT (Type (a, []))
in map standard (List.take (split_conj_thm
(Goal.prove thy2 [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn ((s, T), x) => HOLogic.mk_eq
(Const (s, permT --> T --> T) $
Const ("List.list.Nil", permT) $ Free (x, T),
Free (x, T)))
(perm_names ~~
map body_type perm_types ~~ perm_indnames))))
(fn _ => EVERY [indtac induction perm_indnames 1,
ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
length new_type_names))
end)
atoms);
(**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
val _ = warning "perm_append_thms";
(*FIXME: these should be looked up statically*)
val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
val pt2 = PureThy.get_thm thy2 (Name "pt2");
val perm_append_thms = List.concat (map (fn a =>
let
val permT = mk_permT (Type (a, []));
val pi1 = Free ("pi1", permT);
val pi2 = Free ("pi2", permT);
val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
val pt2' = pt_inst RS pt2;
val pt2_ax = PureThy.get_thm thy2
(Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
in List.take (map standard (split_conj_thm
(Goal.prove thy2 [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn ((s, T), x) =>
let val perm = Const (s, permT --> T --> T)
in HOLogic.mk_eq
(perm $ (Const ("List.op @", permT --> permT --> permT) $
pi1 $ pi2) $ Free (x, T),
perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
end)
(perm_names ~~
map body_type perm_types ~~ perm_indnames))))
(fn _ => EVERY [indtac induction perm_indnames 1,
ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
length new_type_names)
end) atoms);
(**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
val _ = warning "perm_eq_thms";
val pt3 = PureThy.get_thm thy2 (Name "pt3");
val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
val perm_eq_thms = List.concat (map (fn a =>
let
val permT = mk_permT (Type (a, []));
val pi1 = Free ("pi1", permT);
val pi2 = Free ("pi2", permT);
(*FIXME: not robust - better access these theorems using NominalData?*)
val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
val pt3' = pt_inst RS pt3;
val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
val pt3_ax = PureThy.get_thm thy2
(Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
in List.take (map standard (split_conj_thm
(Goal.prove thy2 [] [] (Logic.mk_implies
(HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn ((s, T), x) =>
let val perm = Const (s, permT --> T --> T)
in HOLogic.mk_eq
(perm $ pi1 $ Free (x, T),
perm $ pi2 $ Free (x, T))
end)
(perm_names ~~
map body_type perm_types ~~ perm_indnames)))))
(fn _ => EVERY [indtac induction perm_indnames 1,
ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
length new_type_names)
end) atoms);
(**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
val cp1 = PureThy.get_thm thy2 (Name "cp1");
val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
fun composition_instance name1 name2 thy =
let
val name1' = Sign.base_name name1;
val name2' = Sign.base_name name2;
val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
val permT1 = mk_permT (Type (name1, []));
val permT2 = mk_permT (Type (name2, []));
val augment = map_type_tfree
(fn (x, S) => TFree (x, cp_class :: S));
val Ts = map (augment o body_type) perm_types;
val cp_inst = PureThy.get_thm thy
(Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
val simps = simpset_of thy addsimps (perm_fun_def ::
(if name1 <> name2 then
let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
else
let
val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
in
[cp_inst RS cp1 RS sym,
at_inst RS (pt_inst RS pt_perm_compose) RS sym,
at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
end))
val thms = split_conj_thm (standard (Goal.prove thy [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn ((s, T), x) =>
let
val pi1 = Free ("pi1", permT1);
val pi2 = Free ("pi2", permT2);
val perm1 = Const (s, permT1 --> T --> T);
val perm2 = Const (s, permT2 --> T --> T);
val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
in HOLogic.mk_eq
(perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
end)
(perm_names ~~ Ts ~~ perm_indnames))))
(fn _ => EVERY [indtac induction perm_indnames 1,
ALLGOALS (asm_full_simp_tac simps)])))
in
foldl (fn ((s, tvs), thy) => AxClass.add_inst_arity_i I
(s, replicate (length tvs) (cp_class :: classes), [cp_class])
(ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
thy (full_new_type_names' ~~ tyvars)
end;
val (perm_thmss,thy3) = thy2 |>
fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
AxClass.add_inst_arity_i I (tyname, replicate (length args) classes, classes)
(ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
[resolve_tac perm_empty_thms 1,
resolve_tac perm_append_thms 1,
resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
(List.take (descr, length new_type_names)) |>
PureThy.add_thmss
[((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
unfolded_perm_eq_thms), [Simplifier.simp_add]),
((space_implode "_" new_type_names ^ "_perm_empty",
perm_empty_thms), [Simplifier.simp_add]),
((space_implode "_" new_type_names ^ "_perm_append",
perm_append_thms), [Simplifier.simp_add]),
((space_implode "_" new_type_names ^ "_perm_eq",
perm_eq_thms), [Simplifier.simp_add])];
(**** Define representing sets ****)
val _ = warning "representing sets";
val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
(map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
val big_rep_name =
space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
(fn (i, ("nominal.noption", _, _)) => NONE
| (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
val _ = warning ("big_rep_name: " ^ big_rep_name);
fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
(case AList.lookup op = descr i of
SOME ("nominal.noption", _, [(_, [dt']), _]) =>
apfst (cons dt) (strip_option dt')
| _ => ([], dtf))
| strip_option (DtType ("fun", [dt, DtType ("nominal.noption", [dt'])])) =
apfst (cons dt) (strip_option dt')
| strip_option dt = ([], dt);
val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
(List.concat (map (fn (_, (_, _, cs)) => List.concat
(map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
fun make_intr s T (cname, cargs) =
let
fun mk_prem (dt, (j, j', prems, ts)) =
let
val (dts, dt') = strip_option dt;
val (dts', dt'') = strip_dtyp dt';
val Ts = map (typ_of_dtyp descr sorts') dts;
val Us = map (typ_of_dtyp descr sorts') dts';
val T = typ_of_dtyp descr sorts' dt'';
val free = mk_Free "x" (Us ---> T) j;
val free' = app_bnds free (length Us);
fun mk_abs_fun (T, (i, t)) =
let val U = fastype_of t
in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
Type ("nominal.noption", [U])) $ mk_Free "y" T i $ t)
end
in (j + 1, j' + length Ts,
case dt'' of
DtRec k => list_all (map (pair "x") Us,
HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
Const (List.nth (rep_set_names, k),
HOLogic.mk_setT T)))) :: prems
| _ => prems,
snd (foldr mk_abs_fun (j', free) Ts) :: ts)
end;
val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
(list_comb (Const (cname, map fastype_of ts ---> T), ts),
Const (s, HOLogic.mk_setT T)))
in Logic.list_implies (prems, concl)
end;
val (intr_ts, ind_consts) =
apfst List.concat (ListPair.unzip (List.mapPartial
(fn ((_, ("nominal.noption", _, _)), _) => NONE
| ((i, (_, _, constrs)), rep_set_name) =>
let val T = nth_dtyp i
in SOME (map (make_intr rep_set_name T) constrs,
Const (rep_set_name, HOLogic.mk_setT T))
end)
(descr ~~ rep_set_names)));
val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
setmp InductivePackage.quiet_mode false
(InductivePackage.add_inductive_i false true big_rep_name false true false
ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
(**** Prove that representing set is closed under permutation ****)
val _ = warning "proving closure under permutation...";
val perm_indnames' = List.mapPartial
(fn (x, (_, ("nominal.noption", _, _))) => NONE | (x, _) => SOME x)
(perm_indnames ~~ descr);
fun mk_perm_closed name = map (fn th => standard (th RS mp))
(List.take (split_conj_thm (Goal.prove thy4 [] []
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
(fn (S, x) =>
let
val S = map_term_types (map_type_tfree
(fn (s, cs) => TFree (s, cs union cp_classes))) S;
val T = HOLogic.dest_setT (fastype_of S);
val permT = mk_permT (Type (name, []))
in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
Free ("pi", permT) $ Free (x, T), S))
end) (ind_consts ~~ perm_indnames'))))
(fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
[indtac rep_induct [] 1,
ALLGOALS (simp_tac (simpset_of thy4 addsimps
(symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
ALLGOALS (resolve_tac rep_intrs
THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
length new_type_names));
(* FIXME: theorems are stored in database for testing only *)
val perm_closed_thmss = map mk_perm_closed atoms;
val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
(**** typedef ****)
val _ = warning "defining type...";
val (typedefs, thy6) =
fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
setmp TypedefPackage.quiet_mode true
(TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
(rtac exI 1 THEN
QUIET_BREADTH_FIRST (has_fewer_prems 1)
(resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
let
val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
val pi = Free ("pi", permT);
val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
val T = Type (Sign.intern_type thy name, tvs');
val Const (_, Type (_, [U])) = c
in apfst (pair r o hd)
(PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
(Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
(Const ("nominal.perm", permT --> U --> U) $ pi $
(Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
Free ("x", T))))), [])] thy)
end))
(types_syntax ~~ tyvars ~~
(List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
val perm_defs = map snd typedefs;
val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
val Rep_thms = map (#Rep o fst) typedefs;
val big_name = space_implode "_" new_type_names;
(** prove that new types are in class pt_<name> **)
val _ = warning "prove that new types are in class pt_<name> ...";
fun pt_instance ((class, atom), perm_closed_thms) =
fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
perm_def), name), tvs), perm_closed) => fn thy =>
AxClass.add_inst_arity_i I
(Sign.intern_type thy name,
replicate (length tvs) (classes @ cp_classes), [class])
(EVERY [ClassPackage.intro_classes_tac [],
rewrite_goals_tac [perm_def],
asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
asm_full_simp_tac (simpset_of thy addsimps
[Rep RS perm_closed RS Abs_inverse]) 1,
asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
(Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
(typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
(** prove that new types are in class cp_<name1>_<name2> **)
val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
let
val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
val class = Sign.intern_class thy name;
val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
AxClass.add_inst_arity_i I
(Sign.intern_type thy name,
replicate (length tvs) (classes @ cp_classes), [class])
(EVERY [ClassPackage.intro_classes_tac [],
rewrite_goals_tac [perm_def],
asm_full_simp_tac (simpset_of thy addsimps
((Rep RS perm_closed1 RS Abs_inverse) ::
(if atom1 = atom2 then []
else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
cong_tac 1,
rtac refl 1,
rtac cp1' 1]) thy)
(typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
perm_closed_thms2) thy
end;
val thy7 = fold (fn x => fn thy => thy |>
pt_instance x |>
fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
(classes ~~ atoms ~~ perm_closed_thmss) thy6;
(**** constructors ****)
fun mk_abs_fun (x, t) =
let
val T = fastype_of x;
val U = fastype_of t
in
Const ("nominal.abs_fun", T --> U --> T -->
Type ("nominal.noption", [U])) $ x $ t
end;
val (ty_idxs, _) = foldl
(fn ((i, ("nominal.noption", _, _)), p) => p
| ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
| reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
| reindex dt = dt;
fun strip_suffix i s = implode (List.take (explode s, size s - i));
(** strips the "_Rep" in type names *)
fun strip_nth_name i s =
let val xs = NameSpace.unpack s;
in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
val (descr'', ndescr) = ListPair.unzip (List.mapPartial
(fn (i, ("nominal.noption", _, _)) => NONE
| (i, (s, dts, constrs)) =>
let
val SOME index = AList.lookup op = ty_idxs i;
val (constrs1, constrs2) = ListPair.unzip
(map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
(foldl_map (fn (dts, dt) =>
let val (dts', dt') = strip_option dt
in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
([], cargs))) constrs)
in SOME ((index, (strip_nth_name 1 s, map reindex dts, constrs1)),
(index, constrs2))
end) descr);
val (descr1, descr2) = splitAt (length new_type_names, descr'');
val descr' = [descr1, descr2];
val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
val rep_names = map (fn s =>
Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
val abs_names = map (fn s =>
Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
val recTs' = List.mapPartial
(fn ((_, ("nominal.noption", _, _)), T) => NONE
| (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
val recTs = get_rec_types descr'' sorts';
val newTs' = Library.take (length new_type_names, recTs');
val newTs = Library.take (length new_type_names, recTs);
val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
let
fun constr_arg (dt, (j, l_args, r_args)) =
let
val x' = mk_Free "x" (typ_of_dtyp' dt) j;
val (dts, dt') = strip_option dt;
val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
(dts ~~ (j upto j + length dts - 1))
val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
val (dts', dt'') = strip_dtyp dt'
in
(j + length dts + 1,
xs @ x :: l_args,
foldr mk_abs_fun
(case dt'' of
DtRec k => if k < length new_type_names then
list_abs (map (pair "z" o typ_of_dtyp') dts',
Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts'))
else error "nested recursion not (yet) supported"
| _ => x) xs :: r_args)
end
val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
val constrT = map fastype_of l_args ---> T;
val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
constrT), l_args);
val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const (rep_name, T --> T') $ lhs, rhs));
val def_name = (Sign.base_name cname) ^ "_def";
val ([def_thm], thy') = thy |>
Theory.add_consts_i [(cname', constrT, mx)] |>
(PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
in (thy', defs @ [def_thm], eqns @ [eqn]) end;
fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
(((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
let
val rep_const = cterm_of thy
(Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
in
(parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
end;
val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
((thy7, [], [], []), List.take (descr, length new_type_names) ~~
new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
val abs_inject_thms = map (fn tname =>
PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
val rep_inject_thms = map (fn tname =>
PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
val rep_thms = map (fn tname =>
PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
val rep_inverse_thms = map (fn tname =>
PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
(* prove theorem Rep_i (Constr_j ...) = Constr'_j ... *)
fun prove_constr_rep_thm eqn =
let
val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
[resolve_tac inj_thms 1,
rewrite_goals_tac rewrites,
rtac refl 3,
resolve_tac rep_intrs 2,
REPEAT (resolve_tac rep_thms 1)]))
end;
val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
(* prove theorem pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
let
val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
val Type ("fun", [T, U]) = fastype_of Rep;
val permT = mk_permT (Type (atom, []));
val pi = Free ("pi", permT);
in
standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
(fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
perm_closed_thms @ Rep_thms)) 1))
end) Rep_thms;
val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
(atoms ~~ perm_closed_thmss));
(* prove distinctness theorems *)
val distinct_props = setmp DatatypeProp.dtK 1000
(DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
(constr_rep_thmss ~~ dist_lemmas);
fun prove_distinct_thms (_, []) = []
| prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
let
val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
in dist_thm::(standard (dist_thm RS not_sym))::
(prove_distinct_thms (p, ts))
end;
val distinct_thms = map prove_distinct_thms
(constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
(** prove equations for permutation functions **)
val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
let val T = replace_types' (nth_dtyp i)
in List.concat (map (fn (atom, perm_closed_thms) =>
map (fn ((cname, dts), constr_rep_thm) =>
let
val cname = Sign.intern_const thy8
(NameSpace.append tname (Sign.base_name cname));
val permT = mk_permT (Type (atom, []));
val pi = Free ("pi", permT);
fun perm t =
let val T = fastype_of t
in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
fun constr_arg (dt, (j, l_args, r_args)) =
let
val x' = mk_Free "x" (typ_of_dtyp' dt) j;
val (dts, dt') = strip_option dt;
val Ts = map typ_of_dtyp' dts;
val xs = map (fn (T, i) => mk_Free "x" T i)
(Ts ~~ (j upto j + length dts - 1))
val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
val (dts', dt'') = strip_dtyp dt';
in
(j + length dts + 1,
xs @ x :: l_args,
map perm (xs @ [x]) @ r_args)
end
val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
val c = Const (cname, map fastype_of l_args ---> T)
in
standard (Goal.prove thy8 [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(perm (list_comb (c, l_args)), list_comb (c, r_args))))
(fn _ => EVERY
[simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
constr_defs @ perm_closed_thms)) 1,
TRY (simp_tac (HOL_basic_ss addsimps
(symmetric perm_fun_def :: abs_perm)) 1),
TRY (simp_tac (HOL_basic_ss addsimps
(perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
perm_closed_thms)) 1)]))
end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
(** prove injectivity of constructors **)
val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
val alpha = PureThy.get_thms thy8 (Name "alpha");
val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
val supp_def = PureThy.get_thm thy8 (Name "supp_def");
val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
let val T = replace_types' (nth_dtyp i)
in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
if null dts then NONE else SOME
let
val cname = Sign.intern_const thy8
(NameSpace.append tname (Sign.base_name cname));
fun make_inj (dt, (j, args1, args2, eqs)) =
let
val x' = mk_Free "x" (typ_of_dtyp' dt) j;
val y' = mk_Free "y" (typ_of_dtyp' dt) j;
val (dts, dt') = strip_option dt;
val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
val (dts', dt'') = strip_dtyp dt';
in
(j + length dts + 1,
xs @ (x :: args1), ys @ (y :: args2),
HOLogic.mk_eq
(foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
end;
val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
val Ts = map fastype_of args1;
val c = Const (cname, Ts ---> T)
in
standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
(HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
foldr1 HOLogic.mk_conj eqs)))
(fn _ => EVERY
[asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
rep_inject_thms')) 1,
TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
perm_rep_perm_thms)) 1),
TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
end) (constrs ~~ constr_rep_thms)
end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
(** equations for support and freshness **)
val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
(map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
let val T = replace_types' (nth_dtyp i)
in List.concat (map (fn (cname, dts) => map (fn atom =>
let
val cname = Sign.intern_const thy8
(NameSpace.append tname (Sign.base_name cname));
val atomT = Type (atom, []);
fun process_constr (dt, (j, args1, args2)) =
let
val x' = mk_Free "x" (typ_of_dtyp' dt) j;
val (dts, dt') = strip_option dt;
val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
val (dts', dt'') = strip_dtyp dt';
in
(j + length dts + 1,
xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
end;
val (_, args1, args2) = foldr process_constr (1, [], []) dts;
val Ts = map fastype_of args1;
val c = list_comb (Const (cname, Ts ---> T), args1);
fun supp t =
Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
fun fresh t =
Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
Free ("a", atomT) $ t;
val supp_thm = standard (Goal.prove thy8 [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(supp c,
if null dts then Const ("{}", HOLogic.mk_setT atomT)
else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
(fn _ =>
simp_tac (HOL_basic_ss addsimps (supp_def ::
Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
symmetric empty_def :: Finites.emptyI :: simp_thms @
abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
in
(supp_thm,
standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
(fresh c,
if null dts then HOLogic.true_const
else foldr1 HOLogic.mk_conj (map fresh args2))))
(fn _ =>
simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
end) atoms) constrs)
end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
(**** weak induction theorem ****)
val arities = get_arities descr'';
fun mk_funs_inv thm =
let
val {sign, prop, ...} = rep_thm thm;
val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
(_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
val used = add_term_tfree_names (a, []);
fun mk_thm i =
let
val Ts = map (TFree o rpair HOLogic.typeS)
(variantlist (replicate i "'t", used));
val f = Free ("f", Ts ---> U)
in standard (Goal.prove sign [] [] (Logic.mk_implies
(HOLogic.mk_Trueprop (HOLogic.list_all
(map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
r $ (a $ app_bnds f i)), f))))
(fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
end
in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
let
val Rep_t = Const (List.nth (rep_names, i), T --> U) $
mk_Free "x" T i;
val Abs_t = Const (List.nth (abs_names, i), U --> T)
in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
(mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
end;
val (indrule_lemma_prems, indrule_lemma_concls) =
Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
val indrule_lemma = standard (Goal.prove thy8 [] []
(Logic.mk_implies
(HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
[REPEAT (etac conjE 1),
REPEAT (EVERY
[TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
etac mp 1, resolve_tac Rep_thms 1])]));
val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
map (Free o apfst fst o dest_Var) Ps;
val indrule_lemma' = cterm_instantiate
(map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
val dt_induct = standard (Goal.prove thy8 []
(Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
(fn prems => EVERY
[rtac indrule_lemma' 1,
(DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
EVERY (map (fn (prem, r) => (EVERY
[REPEAT (eresolve_tac Abs_inverse_thms' 1),
simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
(prems ~~ constr_defs))]));
val case_names_induct = mk_case_names_induct descr'';
(**** prove that new datatypes have finite support ****)
val _ = warning "proving finite support for the new datatype";
val indnames = DatatypeProp.make_tnames recTs;
val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
val finite_supp_thms = map (fn atom =>
let val atomT = Type (atom, [])
in map standard (List.take
(split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
(foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
(Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
(indnames ~~ recTs))))
(fn _ => indtac dt_induct indnames 1 THEN
ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
(abs_supp @ supp_atm @
PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
List.concat supp_thms))))),
length new_type_names))
end) atoms;
val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
val (_, thy9) = thy8 |>
Theory.add_path big_name |>
PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
Theory.parent_path ||>>
DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
fold (fn (atom, ths) => fn thy =>
let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
in fold (fn T => AxClass.add_inst_arity_i I
(fst (dest_Type T),
replicate (length sorts) [class], [class])
(ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
end) (atoms ~~ finite_supp_thms);
(**** strong induction theorem ****)
val pnames = if length descr'' = 1 then ["P"]
else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
val ind_sort = if null dt_atomTs then HOLogic.typeS
else norm_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
Sign.base_name (fst (dest_Type T)))) dt_atomTs);
val fsT = TFree ("'n", ind_sort);
val fsT' = TFree ("'n", HOLogic.typeS);
val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
(DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
fun make_pred fsT i T =
Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
let
val recs = List.filter is_rec_type cargs;
val Ts = map (typ_of_dtyp descr'' sorts') cargs;
val recTs' = map (typ_of_dtyp descr'' sorts') recs;
val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
val frees = tnames ~~ Ts;
val z = (variant tnames "z", fsT);
fun mk_prem ((dt, s), T) =
let
val (Us, U) = strip_type T;
val l = length Us
in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
(make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
end;
val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
(f T (Free p) (Free z)))
(map (curry List.nth frees) (List.concat (map (fn (m, n) =>
m upto m + n - 1) idxs)))
in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
list_comb (Const (cname, Ts ---> T), map Free frees))))
end;
val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
map (make_ind_prem fsT (fn T => fn t => fn u =>
Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
(constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
val tnames = DatatypeProp.make_tnames recTs;
val zs = variantlist (replicate (length descr'') "z", tnames);
val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
(map (fn ((((i, _), T), tname), z) =>
make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
(descr'' ~~ recTs ~~ tnames ~~ zs)));
val induct = Logic.list_implies (ind_prems, ind_concl);
val ind_prems' =
map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
(constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
(map (fn ((((i, _), T), tname), z) =>
make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
(descr'' ~~ recTs ~~ tnames ~~ zs)));
val induct' = Logic.list_implies (ind_prems', ind_concl');
fun mk_perm Ts (t, u) =
let
val T = fastype_of1 (Ts, t);
val U = fastype_of1 (Ts, u)
in Const ("nominal.perm", T --> U --> U) $ t $ u end;
val aux_ind_vars =
(DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
map mk_permT dt_atomTs) @ [("z", fsT')];
val aux_ind_Ts = rev (map snd aux_ind_vars);
val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
(map (fn (((i, _), T), tname) =>
HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
foldr (mk_perm aux_ind_Ts) (Free (tname, T))
(map Bound (length dt_atomTs downto 1))))
(descr'' ~~ recTs ~~ tnames)));
fun mk_ind_perm i k p l vs j =
let
val n = length vs;
val Ts = map snd vs;
val T = List.nth (Ts, i - j);
val pT = NominalAtoms.mk_permT T
in
Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
(HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
(Bound (i - j))
(map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
map Bound (n - k - 1 downto n - k - p))) $
Const ("List.list.Nil", pT)
end;
fun mk_fresh i i' j k p l vs _ _ =
let
val n = length vs;
val Ts = map snd vs;
val T = List.nth (Ts, n - i - 1 - j);
val f = the (AList.lookup op = fresh_fs T);
val U = List.nth (Ts, n - i' - 1);
val S = HOLogic.mk_setT T;
val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
(j - 1 downto 0) @
map Bound (n - k downto n - k - p + 1)
in
HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
Abs ("a", T, HOLogic.Not $
(Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
(Const ("insert", T --> S --> S) $
(foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms) $
(Const ("op Un", S --> S --> S) $ (f $ Bound (n - k - p)) $
(Const ("nominal.supp", U --> S) $
foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms))))))
end;
fun mk_fresh_constr is p vs _ concl =
let
val n = length vs;
val Ts = map snd vs;
val _ $ (_ $ _ $ t) = concl;
val c = head_of t;
val T = body_type (fastype_of c);
val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
val ps = map Bound (n - k - 1 downto n - k - p);
val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
(m - i - 1, n - i,
ts @ map Bound (n downto n - i + 1) @
[foldr (mk_perm Ts) (Bound (m - i))
(map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
(n - 1, n - k - p - 2, [], []) is
in
HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
end;
val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
[PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
val induct_aux_lemmas' = map (fn Type (s, _) =>
PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
val induct_aux = standard (Goal.prove thy9 [] ind_prems' ind_concl'
(fn prems => EVERY
([mk_subgoal 1 (K (K (K aux_ind_concl))),
indtac dt_induct tnames 1] @
List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
List.concat (map (fn ((cname, cargs), is) =>
[simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
REPEAT (rtac allI 1)] @
List.concat (map
(fn ((_, 0), _) => []
| ((i, j), k) => List.concat (map (fn j' =>
let
val DtType (tname, _) = List.nth (cargs, i + j');
val atom = Sign.base_name tname
in
[mk_subgoal 1 (mk_fresh i (i + j) j'
(length cargs) (length dt_atomTs)
(length cargs + length dt_atomTs + 1 + i - k)),
rtac at_finite_select 1,
rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
asm_full_simp_tac (simpset_of thy9 addsimps
[PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
resolve_tac prems 1,
etac exE 1,
asm_full_simp_tac (simpset_of thy9 addsimps
[symmetric fresh_def]) 1]
end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
(if exists (not o equal 0 o snd) is then
[mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
asm_full_simp_tac (simpset_of thy9 addsimps
(List.concat inject_thms @
alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
induct_aux_lemmas)) 1,
dtac sym 1,
asm_full_simp_tac (simpset_of thy9
addsimps induct_aux_lemmas'
addsimprocs [perm_simproc]) 1,
REPEAT (etac conjE 1)]
else
[]) @
[(resolve_tac prems THEN_ALL_NEW
(atac ORELSE' ((REPEAT o etac allE) THEN' atac))) 1])
(constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
[REPEAT (eresolve_tac [conjE, allE_Nil] 1),
REPEAT (etac allE 1),
REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)])));
val induct_aux' = Thm.instantiate ([],
map (fn (s, T) =>
let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
(pnames ~~ recTs) @
map (fn (_, f) =>
let val f' = Logic.varify f
in (cterm_of thy9 f',
cterm_of thy9 (Const ("nominal.supp", fastype_of f')))
end) fresh_fs) induct_aux;
val induct = standard (Goal.prove thy9 [] ind_prems ind_concl
(fn prems => EVERY
[rtac induct_aux' 1,
REPEAT (resolve_tac induct_aux_lemmas 1),
REPEAT ((resolve_tac prems THEN_ALL_NEW
(etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]))
val (_, thy10) = thy9 |>
Theory.add_path big_name |>
PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])] ||>
Theory.parent_path;
in
thy10
end;
val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
(* FIXME: The following stuff should be exported by DatatypePackage *)
local structure P = OuterParse and K = OuterKeyword in
val datatype_decl =
Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
(P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
fun mk_datatype args =
let
val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
val specs = map (fn ((((_, vs), t), mx), cons) =>
(vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
in add_nominal_datatype false names specs end;
val nominal_datatypeP =
OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
(P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
val _ = OuterSyntax.add_parsers [nominal_datatypeP];
end;
end