(* Title: HOL/Nominal/nominal_datatype.ML
Author: Stefan Berghofer and Christian Urban, TU Muenchen
Nominal datatype package for Isabelle/HOL.
*)
signature NOMINAL_DATATYPE =
sig
val nominal_datatype : Datatype_Aux.config -> Datatype.spec list -> theory -> theory
val nominal_datatype_cmd : Datatype_Aux.config -> Datatype.spec_cmd list -> theory -> theory
type descr
type nominal_datatype_info
val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table
val get_nominal_datatype : theory -> string -> nominal_datatype_info option
val mk_perm: typ list -> term -> term -> term
val perm_of_pair: term * term -> term
val mk_not_sym: thm list -> thm list
val perm_simproc: simproc
val fresh_const: typ -> typ -> term
val fresh_star_const: typ -> typ -> term
end
structure NominalDatatype : NOMINAL_DATATYPE =
struct
val finite_emptyI = @{thm finite.emptyI};
val finite_Diff = @{thm finite_Diff};
val finite_Un = @{thm finite_Un};
val Un_iff = @{thm Un_iff};
val In0_eq = @{thm In0_eq};
val In1_eq = @{thm In1_eq};
val In0_not_In1 = @{thm In0_not_In1};
val In1_not_In0 = @{thm In1_not_In0};
val Un_assoc = @{thm Un_assoc};
val Collect_disj_eq = @{thm Collect_disj_eq};
val Collect_False_empty = @{thm empty_def [THEN sym, THEN eq_reflection]};
val empty_iff = @{thm empty_iff};
open NominalAtoms;
(* theory data *)
type descr =
(int * (string * Datatype_Aux.dtyp list *
(string * (Datatype_Aux.dtyp list * Datatype_Aux.dtyp) list) list)) list;
type nominal_datatype_info =
{index : int,
descr : descr,
rec_names : string list,
rec_rewrites : thm list,
induction : thm,
distinct : thm list,
inject : thm list};
structure NominalDatatypesData = Theory_Data
(
type T = nominal_datatype_info Symtab.table;
val empty = Symtab.empty;
val extend = I;
fun merge data = Symtab.merge (K true) data;
);
val get_nominal_datatypes = NominalDatatypesData.get;
val put_nominal_datatypes = NominalDatatypesData.put;
val map_nominal_datatypes = NominalDatatypesData.map;
val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;
(**** make datatype info ****)
fun make_dt_info descr induct reccomb_names rec_thms
(i, (((_, (tname, _, _)), distinct), inject)) =
(tname,
{index = i,
descr = descr,
rec_names = reccomb_names,
rec_rewrites = rec_thms,
induction = induct,
distinct = distinct,
inject = inject});
(*******************************)
val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of Datatype.distinct_lemma);
(** simplification procedure for sorting permutations **)
val dj_cp = @{thm dj_cp};
fun dest_permT (Type (@{type_name fun},
[Type (@{type_name list}, [Type (@{type_name Product_Type.prod}, [T, _])]),
Type (@{type_name fun}, [_, U])])) = (T, U);
fun permTs_of (Const (@{const_name Nominal.perm}, T) $ t $ u) = fst (dest_permT T) :: permTs_of u
| permTs_of _ = [];
fun perm_simproc' ctxt (Const (@{const_name Nominal.perm}, T) $ t $ (u as Const (@{const_name Nominal.perm}, U) $ r $ s)) =
let
val thy = Proof_Context.theory_of ctxt;
val (aT as Type (a, []), S) = dest_permT T;
val (bT as Type (b, []), _) = dest_permT U;
in if member (op =) (permTs_of u) aT andalso aT <> bT then
let
val cp = cp_inst_of thy a b;
val dj = dj_thm_of thy 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' _ _ = NONE;
val perm_simproc =
Simplifier.simproc_global @{theory} "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';
fun projections rule =
Project_Rule.projections (Proof_Context.init_global (Thm.theory_of_thm rule)) rule
|> map (Drule.export_without_context #> Rule_Cases.save rule);
val supp_prod = @{thm supp_prod};
val fresh_prod = @{thm fresh_prod};
val supports_fresh = @{thm supports_fresh};
val supports_def = Simpdata.mk_eq @{thm Nominal.supports_def};
val fresh_def = Simpdata.mk_eq @{thm fresh_def};
val supp_def = Simpdata.mk_eq @{thm supp_def};
val rev_simps = @{thms rev.simps};
val app_simps = @{thms append.simps};
val at_fin_set_supp = @{thm at_fin_set_supp};
val at_fin_set_fresh = @{thm at_fin_set_fresh};
val abs_fun_eq1 = @{thm abs_fun_eq1};
fun collect_simp ctxt = rewrite_rule ctxt [mk_meta_eq mem_Collect_eq];
fun mk_perm Ts t u =
let
val T = fastype_of1 (Ts, t);
val U = fastype_of1 (Ts, u)
in Const (@{const_name Nominal.perm}, T --> U --> U) $ t $ u end;
fun perm_of_pair (x, y) =
let
val T = fastype_of x;
val pT = mk_permT T
in Const (@{const_name Cons}, HOLogic.mk_prodT (T, T) --> pT --> pT) $
HOLogic.mk_prod (x, y) $ Const (@{const_name Nil}, pT)
end;
fun mk_not_sym ths = maps (fn th => case prop_of th of
_ $ (Const (@{const_name Not}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ _)) => [th, th RS not_sym]
| _ => [th]) ths;
fun fresh_const T U = Const (@{const_name Nominal.fresh}, T --> U --> HOLogic.boolT);
fun fresh_star_const T U =
Const (@{const_name Nominal.fresh_star}, HOLogic.mk_setT T --> U --> HOLogic.boolT);
fun gen_nominal_datatype prep_specs config dts thy =
let
val new_type_names = map (fn ((tname, _, _), _) => Binding.name_of tname) dts;
val (dts', _) = prep_specs dts thy;
val atoms = atoms_of thy;
val tyvars = map (fn ((_, tvs, _), _) => tvs) dts';
val sorts = flat tyvars;
fun inter_sort thy S S' = Sign.inter_sort thy (S, S');
fun augment_sort_typ thy S =
let val S = Sign.minimize_sort thy (Sign.certify_sort thy S)
in map_type_tfree (fn (s, S') => TFree (s,
if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S'))
end;
fun augment_sort thy S = map_types (augment_sort_typ thy S);
val types_syntax = map (fn ((tname, tvs, mx), constrs) => (tname, mx)) dts';
val constr_syntax = map (fn (_, constrs) =>
map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
val ps = map (fn ((n, _, _), _) =>
(Sign.full_name thy n, Sign.full_name thy (Binding.suffix_name "_Rep" n))) dts;
val rps = map Library.swap ps;
fun replace_types (Type (@{type_name ABS}, [T, U])) =
Type (@{type_name fun}, [T, Type (@{type_name noption}, [replace_types U])])
| replace_types (Type (s, Ts)) =
Type (the_default s (AList.lookup op = ps s), map replace_types Ts)
| replace_types T = T;
val dts'' = map (fn ((tname, tvs, mx), constrs) =>
((Binding.suffix_name "_Rep" tname, tvs, NoSyn),
map (fn (cname, cargs, mx) => (Binding.suffix_name "_Rep" cname,
map replace_types cargs, NoSyn)) constrs)) dts';
val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
val (full_new_type_names',thy1) = Datatype.add_datatype config dts'' thy;
val {descr, induct, ...} = Datatype_Data.the_info thy1 (hd full_new_type_names');
fun nth_dtyp i = Datatype_Aux.typ_of_dtyp descr (Datatype_Aux.DtRec i);
val big_name = space_implode "_" new_type_names;
(**** define permutation functions ****)
val permT = mk_permT (TFree ("'x", @{sort type}));
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' = Datatype_Prop.indexify_names (map (fn (i, _) =>
"perm_" ^ Datatype_Aux.name_of_typ (nth_dtyp i)) descr);
val perm_names = replicate (length new_type_names) @{const_name Nominal.perm} @
map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names));
val perm_names_types = perm_names ~~ perm_types;
val perm_names_types' = perm_names' ~~ perm_types;
val perm_eqs = maps (fn (i, (_, _, constrs)) =>
let val T = nth_dtyp i
in map (fn (cname, dts) =>
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
val names = Name.variant_list ["pi"] (Datatype_Prop.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 Datatype_Aux.is_rec_type dt then
let val Us = binder_types T
in
fold_rev (Term.abs o pair "x") Us
(Free (nth perm_names_types' (Datatype_Aux.body_index dt)) $ pi $
list_comb (x, map (fn (i, U) =>
Const (@{const_name Nominal.perm}, permT --> U --> U) $
(Const (@{const_name rev}, permT --> permT) $ pi) $
Bound i) ((length Us - 1 downto 0) ~~ Us)))
end
else Const (@{const_name Nominal.perm}, permT --> T --> T) $ pi $ x
end;
in
(Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
(Free (nth perm_names_types' i) $
Free ("pi", mk_permT (TFree ("'x", @{sort type}))) $
list_comb (c, args),
list_comb (c, map perm_arg (dts ~~ args)))))
end) constrs
end) descr;
val (perm_simps, thy2) =
Primrec.add_primrec_overloaded
(map (fn (s, sT) => (s, sT, false))
(List.take (perm_names' ~~ perm_names_types, length new_type_names)))
(map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1;
(**** 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 = Datatype_Prop.make_tnames (map body_type perm_types);
val perm_fun_def = Simpdata.mk_eq @{thm perm_fun_def};
val unfolded_perm_eq_thms =
if length descr = length new_type_names then []
else map Drule.export_without_context (List.drop (Datatype_Aux.split_conj_thm
(Goal.prove_global_future 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 (@{const_name Nominal.perm}, T) $ pi $ Free (x, T2))
end)
(perm_names_types ~~ perm_indnames))))
(fn {context = ctxt, ...} => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
ALLGOALS (asm_full_simp_tac (ctxt addsimps [perm_fun_def]))])),
length new_type_names));
(**** prove [] \<bullet> t = t ****)
val _ = warning "perm_empty_thms";
val perm_empty_thms = maps (fn a =>
let val permT = mk_permT (Type (a, []))
in map Drule.export_without_context (List.take (Datatype_Aux.split_conj_thm
(Goal.prove_global_future thy2 [] []
(augment_sort thy2 [pt_class_of thy2 a]
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn ((s, T), x) => HOLogic.mk_eq
(Const (s, permT --> T --> T) $
Const (@{const_name Nil}, permT) $ Free (x, T),
Free (x, T)))
(perm_names ~~
map body_type perm_types ~~ perm_indnames)))))
(fn {context = ctxt, ...} => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
ALLGOALS (asm_full_simp_tac ctxt)])),
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 = Global_Theory.get_thm thy2 "at_pt_inst";
val pt2 = Global_Theory.get_thm thy2 "pt2";
val perm_append_thms = maps (fn a =>
let
val permT = mk_permT (Type (a, []));
val pi1 = Free ("pi1", permT);
val pi2 = Free ("pi2", permT);
val pt_inst = pt_inst_of thy2 a;
val pt2' = pt_inst RS pt2;
val pt2_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
(Goal.prove_global_future thy2 [] []
(augment_sort thy2 [pt_class_of thy2 a]
(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 (@{const_name append}, 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 {context = ctxt, ...} => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
ALLGOALS (asm_full_simp_tac (ctxt 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 = Global_Theory.get_thm thy2 "pt3";
val pt3_rev = Global_Theory.get_thm thy2 "pt3_rev";
val perm_eq_thms = maps (fn a =>
let
val permT = mk_permT (Type (a, []));
val pi1 = Free ("pi1", permT);
val pi2 = Free ("pi2", permT);
val at_inst = at_inst_of thy2 a;
val pt_inst = pt_inst_of thy2 a;
val pt3' = pt_inst RS pt3;
val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
val pt3_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
(Goal.prove_global_future thy2 [] []
(augment_sort thy2 [pt_class_of thy2 a] (Logic.mk_implies
(HOLogic.mk_Trueprop (Const (@{const_name 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 {context = ctxt, ...} => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
ALLGOALS (asm_full_simp_tac (ctxt 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 = Global_Theory.get_thm thy2 "cp1";
val dj_cp = Global_Theory.get_thm thy2 "dj_cp";
val pt_perm_compose = Global_Theory.get_thm thy2 "pt_perm_compose";
val pt_perm_compose_rev = Global_Theory.get_thm thy2 "pt_perm_compose_rev";
val dj_perm_perm_forget = Global_Theory.get_thm thy2 "dj_perm_perm_forget";
fun composition_instance name1 name2 thy =
let
val cp_class = cp_class_of thy name1 name2;
val pt_class =
if name1 = name2 then [pt_class_of thy name1]
else [];
val permT1 = mk_permT (Type (name1, []));
val permT2 = mk_permT (Type (name2, []));
val Ts = map body_type perm_types;
val cp_inst = cp_inst_of thy name1 name2;
fun simps ctxt = ctxt addsimps (perm_fun_def ::
(if name1 <> name2 then
let val dj = dj_thm_of thy name2 name1
in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
else
let
val at_inst = at_inst_of thy name1;
val pt_inst = pt_inst_of thy name1;
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 sort = Sign.minimize_sort thy (Sign.certify_sort thy (cp_class :: pt_class));
val thms = Datatype_Aux.split_conj_thm (Goal.prove_global_future thy [] []
(augment_sort thy sort
(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 (@{const_name 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 {context = ctxt, ...} => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
ALLGOALS (asm_full_simp_tac (simps ctxt))]))
in
fold (fn (s, tvs) => fn thy => Axclass.prove_arity
(s, map (inter_sort thy sort o snd) tvs, [cp_class])
(fn _ => Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
(full_new_type_names' ~~ tyvars) thy
end;
val (perm_thmss,thy3) = thy2 |>
fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
fold (fn atom => fn thy =>
let val pt_name = pt_class_of thy atom
in
fold (fn (s, tvs) => fn thy => Axclass.prove_arity
(s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name])
(fn _ => EVERY
[Class.intro_classes_tac [],
resolve_tac perm_empty_thms 1,
resolve_tac perm_append_thms 1,
resolve_tac perm_eq_thms 1, assume_tac 1]) thy)
(full_new_type_names' ~~ tyvars) thy
end) atoms |>
Global_Theory.add_thmss
[((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"),
unfolded_perm_eq_thms), [Simplifier.simp_add]),
((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"),
perm_empty_thms), [Simplifier.simp_add]),
((Binding.name (space_implode "_" new_type_names ^ "_perm_append"),
perm_append_thms), [Simplifier.simp_add]),
((Binding.name (space_implode "_" new_type_names ^ "_perm_eq"),
perm_eq_thms), [Simplifier.simp_add])];
(**** Define representing sets ****)
val _ = warning "representing sets";
val rep_set_names =
Datatype_Prop.indexify_names
(map (fn (i, _) => Datatype_Aux.name_of_typ (nth_dtyp i) ^ "_set") descr);
val big_rep_name =
space_implode "_" (Datatype_Prop.indexify_names (map_filter
(fn (i, (@{type_name noption}, _, _)) => NONE
| (i, _) => SOME (Datatype_Aux.name_of_typ (nth_dtyp i))) descr)) ^ "_set";
val _ = warning ("big_rep_name: " ^ big_rep_name);
fun strip_option (dtf as Datatype_Aux.DtType ("fun", [dt, Datatype_Aux.DtRec i])) =
(case AList.lookup op = descr i of
SOME (@{type_name noption}, _, [(_, [dt']), _]) =>
apfst (cons dt) (strip_option dt')
| _ => ([], dtf))
| strip_option (Datatype_Aux.DtType ("fun",
[dt, Datatype_Aux.DtType (@{type_name noption}, [dt'])])) =
apfst (cons dt) (strip_option dt')
| strip_option dt = ([], dt);
val dt_atomTs = distinct op = (map (Datatype_Aux.typ_of_dtyp descr)
(maps (fn (_, (_, _, cs)) => maps (maps (fst o strip_option) o snd) cs) descr));
val dt_atoms = map (fst o dest_Type) dt_atomTs;
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'') = Datatype_Aux.strip_dtyp dt';
val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
val Us = map (Datatype_Aux.typ_of_dtyp descr) dts';
val T = Datatype_Aux.typ_of_dtyp descr dt'';
val free = Datatype_Aux.mk_Free "x" (Us ---> T) j;
val free' = Datatype_Aux.app_bnds free (length Us);
fun mk_abs_fun T (i, t) =
let val U = fastype_of t
in (i + 1, Const (@{const_name Nominal.abs_fun}, [T, U, T] --->
Type (@{type_name noption}, [U])) $ Datatype_Aux.mk_Free "y" T i $ t)
end
in (j + 1, j' + length Ts,
case dt'' of
Datatype_Aux.DtRec k => Logic.list_all (map (pair "x") Us,
HOLogic.mk_Trueprop (Free (nth rep_set_names k,
T --> HOLogic.boolT) $ free')) :: prems
| _ => prems,
snd (fold_rev mk_abs_fun Ts (j', free)) :: ts)
end;
val (_, _, prems, ts) = fold_rev mk_prem cargs (1, 1, [], []);
val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $
list_comb (Const (cname, map fastype_of ts ---> T), ts))
in Logic.list_implies (prems, concl)
end;
val (intr_ts, (rep_set_names', recTs')) =
apfst flat (apsnd ListPair.unzip (ListPair.unzip (map_filter
(fn ((_, (@{type_name noption}, _, _)), _) => NONE
| ((i, (_, _, constrs)), rep_set_name) =>
let val T = nth_dtyp i
in SOME (map (make_intr rep_set_name T) constrs,
(rep_set_name, T))
end)
(descr ~~ rep_set_names))));
val rep_set_names'' = map (Sign.full_bname thy3) rep_set_names';
val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =
thy3
|> Sign.map_naming Name_Space.conceal
|> Inductive.add_inductive_global
{quiet_mode = false, verbose = false, alt_name = Binding.name big_rep_name,
coind = false, no_elim = true, no_ind = false, skip_mono = true}
(map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))
(rep_set_names' ~~ recTs'))
[] (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
||> Sign.restore_naming thy3;
(**** Prove that representing set is closed under permutation ****)
val _ = warning "proving closure under permutation...";
val abs_perm = Global_Theory.get_thms thy4 "abs_perm";
val perm_indnames' = map_filter
(fn (x, (_, (@{type_name noption}, _, _))) => NONE | (x, _) => SOME x)
(perm_indnames ~~ descr);
fun mk_perm_closed name = map (fn th => Drule.export_without_context (th RS mp))
(List.take (Datatype_Aux.split_conj_thm (Goal.prove_global_future thy4 [] []
(augment_sort thy4
(pt_class_of thy4 name :: map (cp_class_of thy4 name) (remove (op =) name dt_atoms))
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
(fn ((s, T), x) =>
let
val S = Const (s, T --> HOLogic.boolT);
val permT = mk_permT (Type (name, []))
in HOLogic.mk_imp (S $ Free (x, T),
S $ (Const (@{const_name Nominal.perm}, permT --> T --> T) $
Free ("pi", permT) $ Free (x, T)))
end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))
(fn {context = ctxt, ...} => EVERY
[Datatype_Aux.ind_tac rep_induct [] 1,
ALLGOALS (simp_tac (ctxt addsimps
(Thm.symmetric perm_fun_def :: abs_perm))),
ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])),
length new_type_names));
val perm_closed_thmss = map mk_perm_closed atoms;
(**** typedef ****)
val _ = warning "defining type...";
val (typedefs, thy6) =
thy4
|> fold_map (fn (((name, mx), tvs), (cname, U)) => fn thy =>
Typedef.add_typedef_global
(name, map (fn (v, _) => (v, dummyS)) tvs, mx) (* FIXME keep constraints!? *)
(Const (@{const_name Collect}, (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $
Const (cname, U --> HOLogic.boolT)) NONE
(rtac exI 1 THEN rtac CollectI 1 THEN
QUIET_BREADTH_FIRST (has_fewer_prems 1)
(resolve_tac rep_intrs 1)) thy |> (fn ((_, r), thy) =>
let
val permT = mk_permT
(TFree (singleton (Name.variant_list (map fst tvs)) "'a", @{sort type}));
val pi = Free ("pi", permT);
val T = Type (Sign.full_name thy name, map TFree tvs);
in apfst (pair r o hd)
(Global_Theory.add_defs_unchecked true
[((Binding.map_name (fn n => "prm_" ^ n ^ "_def") name, Logic.mk_equals
(Const (@{const_name Nominal.perm}, permT --> T --> T) $ pi $ Free ("x", T),
Const (Sign.intern_const thy ("Abs_" ^ Binding.name_of name), U --> T) $
(Const (@{const_name Nominal.perm}, permT --> U --> U) $ pi $
(Const (Sign.intern_const thy ("Rep_" ^ Binding.name_of name), T --> U) $
Free ("x", T))))), [])] thy)
end))
(types_syntax ~~ tyvars ~~ List.take (rep_set_names'' ~~ recTs', length new_type_names));
val ctxt6 = Proof_Context.init_global thy6;
val perm_defs = map snd typedefs;
val Abs_inverse_thms = map (collect_simp ctxt6 o #Abs_inverse o snd o fst) typedefs;
val Rep_inverse_thms = map (#Rep_inverse o snd o fst) typedefs;
val Rep_thms = map (collect_simp ctxt6 o #Rep o snd o fst) typedefs;
(** prove that new types are in class pt_<name> **)
val _ = warning "prove that new types are in class pt_<name> ...";
fun pt_instance (atom, perm_closed_thms) =
fold (fn ((((((Abs_inverse, Rep_inverse), Rep),
perm_def), name), tvs), perm_closed) => fn thy =>
let
val pt_class = pt_class_of thy atom;
val sort = Sign.minimize_sort thy (Sign.certify_sort thy
(pt_class :: map (cp_class_of thy atom) (remove (op =) atom dt_atoms)))
in Axclass.prove_arity
(Sign.intern_type thy name,
map (inter_sort thy sort o snd) tvs, [pt_class])
(fn ctxt => EVERY [Class.intro_classes_tac [],
rewrite_goals_tac ctxt [perm_def],
asm_full_simp_tac (ctxt addsimps [Rep_inverse]) 1,
asm_full_simp_tac (ctxt addsimps
[Rep RS perm_closed RS Abs_inverse]) 1,
asm_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps [Global_Theory.get_thm thy
("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy
end)
(Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~
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 cp_class = cp_class_of thy atom1 atom2;
val sort = Sign.minimize_sort thy (Sign.certify_sort thy
(pt_class_of thy atom1 :: map (cp_class_of thy atom1) (remove (op =) atom1 dt_atoms) @
(if atom1 = atom2 then [cp_class_of thy atom1 atom1] else
pt_class_of thy atom2 :: map (cp_class_of thy atom2) (remove (op =) atom2 dt_atoms))));
val cp1' = cp_inst_of thy atom1 atom2 RS cp1
in fold (fn ((((((Abs_inverse, Rep),
perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
Axclass.prove_arity
(Sign.intern_type thy name,
map (inter_sort thy sort o snd) tvs, [cp_class])
(fn ctxt => EVERY [Class.intro_classes_tac [],
rewrite_goals_tac ctxt [perm_def],
asm_full_simp_tac (ctxt 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)
(Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ 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 x) (atoms ~~ perm_closed_thmss))
(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 (@{const_name Nominal.abs_fun}, T --> U --> T -->
Type (@{type_name noption}, [U])) $ x $ t
end;
val (ty_idxs, _) = List.foldl
(fn ((i, (@{type_name noption}, _, _)), p) => p
| ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
fun reindex (Datatype_Aux.DtType (s, dts)) = Datatype_Aux.DtType (s, map reindex dts)
| reindex (Datatype_Aux.DtRec i) = Datatype_Aux.DtRec (the (AList.lookup op = ty_idxs i))
| reindex dt = dt;
fun strip_suffix i s = implode (List.take (raw_explode s, size s - i)); (* FIXME Symbol.explode (?) *)
(** strips the "_Rep" in type names *)
fun strip_nth_name i s =
let val xs = Long_Name.explode s;
in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
val (descr'', ndescr) = ListPair.unzip (map_filter
(fn (i, (@{type_name noption}, _, _)) => NONE
| (i, (s, dts, constrs)) =>
let
val SOME index = AList.lookup op = ty_idxs i;
val (constrs2, constrs1) =
map_split (fn (cname, cargs) =>
apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
(fold_map (fn dt => fn dts =>
let val (dts', dt') = strip_option dt
in ((length dts, length dts'), dts @ dts' @ [reindex dt']) end)
cargs [])) constrs
in SOME ((index, (strip_nth_name 1 s, map reindex dts, constrs1)),
(index, constrs2))
end) descr);
val (descr1, descr2) = chop (length new_type_names) descr'';
val descr' = [descr1, descr2];
fun partition_cargs idxs xs = map (fn (i, j) =>
(List.take (List.drop (xs, i), j), nth xs (i + j))) idxs;
val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
(constrs ~~ idxss)))) (descr'' ~~ ndescr);
fun nth_dtyp' i = Datatype_Aux.typ_of_dtyp descr'' (Datatype_Aux.DtRec i);
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 = Datatype_Aux.get_rec_types descr'';
val newTs' = take (length new_type_names) recTs';
val newTs = take (length new_type_names) recTs;
val full_new_type_names = map (Sign.full_bname thy) new_type_names;
fun make_constr_def tname T T' (((cname_rep, _), (cname, cargs)), (cname', mx))
(thy, defs, eqns) =
let
fun constr_arg (dts, dt) (j, l_args, r_args) =
let
val xs =
map (fn (dt, i) => Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) i)
(dts ~~ (j upto j + length dts - 1))
val x = Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts)
in
(j + length dts + 1,
xs @ x :: l_args,
fold_rev mk_abs_fun xs
(case dt of
Datatype_Aux.DtRec k => if k < length new_type_names then
Const (nth rep_names k, Datatype_Aux.typ_of_dtyp descr'' dt -->
Datatype_Aux.typ_of_dtyp descr dt) $ x
else error "nested recursion not (yet) supported"
| _ => x) :: r_args)
end
val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);
val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
val constrT = map fastype_of l_args ---> T;
val lhs = list_comb (Const (cname, constrT), l_args);
val rhs = list_comb (Const (cname_rep, 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 = (Long_Name.base_name cname) ^ "_def";
val ([def_thm], thy') = thy |>
Sign.add_consts [(cname', constrT, mx)] |>
(Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
in (thy', defs @ [def_thm], eqns @ [eqn]) end;
fun dt_constr_defs ((((((_, (_, _, constrs)),
(_, (_, _, constrs'))), tname), T), T'), constr_syntax) (thy, defs, eqns, dist_lemmas) =
let
val rep_const = cterm_of thy
(Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
val dist =
Drule.export_without_context
(cterm_instantiate [(cterm_of thy distinct_f, rep_const)] Datatype.distinct_lemma);
val (thy', defs', eqns') = fold (make_constr_def tname T T')
(constrs ~~ constrs' ~~ constr_syntax) (Sign.add_path tname thy, defs, [])
in
(Sign.parent_path thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
end;
val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = fold dt_constr_defs
(List.take (descr, length new_type_names) ~~
List.take (pdescr, length new_type_names) ~~
new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax)
(thy7, [], [], []);
val abs_inject_thms = map (collect_simp ctxt6 o #Abs_inject o snd o fst) typedefs
val rep_inject_thms = map (#Rep_inject o snd o fst) typedefs
(* 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 Goal.prove_global_future thy8 [] [] eqn (fn {context = ctxt, ...} => EVERY
[resolve_tac inj_thms 1,
rewrite_goals_tac ctxt 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_global (prop_of th);
val Type ("fun", [T, U]) = fastype_of Rep;
val permT = mk_permT (Type (atom, []));
val pi = Free ("pi", permT);
in
Goal.prove_global_future thy8 [] []
(augment_sort thy8
(pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(Const (@{const_name Nominal.perm}, permT --> U --> U) $ pi $ (Rep $ x),
Rep $ (Const (@{const_name Nominal.perm}, permT --> T --> T) $ pi $ x)))))
(fn {context = ctxt, ...} =>
simp_tac (put_simpset HOL_basic_ss ctxt addsimps (perm_defs @ Abs_inverse_thms @
perm_closed_thms @ Rep_thms)) 1)
end) Rep_thms;
val perm_rep_perm_thms = maps prove_perm_rep_perm (atoms ~~ perm_closed_thmss);
(* prove distinctness theorems *)
val distinct_props = Datatype_Prop.make_distincts descr';
val dist_rewrites = map2 (fn rep_thms => fn 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 = Goal.prove_global_future thy8 [] [] t (fn {context = ctxt, ...} =>
simp_tac (ctxt addsimps (dist_lemma :: rep_thms)) 1)
in
dist_thm :: Drule.export_without_context (dist_thm RS not_sym) ::
prove_distinct_thms p ts
end;
val distinct_thms = map2 prove_distinct_thms
(constr_rep_thmss ~~ dist_lemmas) distinct_props;
(** prove equations for permutation functions **)
val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
let val T = nth_dtyp' i
in maps (fn (atom, perm_closed_thms) =>
map (fn ((cname, dts), constr_rep_thm) =>
let
val cname = Sign.intern_const thy8
(Long_Name.append tname (Long_Name.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 (@{const_name Nominal.perm}, permT --> T --> T) $ pi $ t end;
fun constr_arg (dts, dt) (j, l_args, r_args) =
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr'') dts;
val xs =
map (fn (T, i) => Datatype_Aux.mk_Free "x" T i)
(Ts ~~ (j upto j + length dts - 1));
val x =
Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
in
(j + length dts + 1,
xs @ x :: l_args,
map perm (xs @ [x]) @ r_args)
end
val (_, l_args, r_args) = fold_rev constr_arg dts (1, [], []);
val c = Const (cname, map fastype_of l_args ---> T)
in
Goal.prove_global_future thy8 [] []
(augment_sort thy8
(pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(perm (list_comb (c, l_args)), list_comb (c, r_args)))))
(fn {context = ctxt, ...} => EVERY
[simp_tac (ctxt addsimps (constr_rep_thm :: perm_defs)) 1,
simp_tac (put_simpset HOL_basic_ss ctxt addsimps (Rep_thms @ Abs_inverse_thms @
constr_defs @ perm_closed_thms)) 1,
TRY (simp_tac (put_simpset HOL_basic_ss ctxt addsimps
(Thm.symmetric perm_fun_def :: abs_perm)) 1),
TRY (simp_tac (put_simpset HOL_basic_ss ctxt 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 (pdescr, 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 = Global_Theory.get_thms thy8 "alpha";
val abs_fresh = Global_Theory.get_thms thy8 "abs_fresh";
val pt_cp_sort =
map (pt_class_of thy8) dt_atoms @
maps (fn s => map (cp_class_of thy8 s) (remove (op =) s dt_atoms)) dt_atoms;
val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
let val T = nth_dtyp' i
in map_filter (fn ((cname, dts), constr_rep_thm) =>
if null dts then NONE else SOME
let
val cname = Sign.intern_const thy8
(Long_Name.append tname (Long_Name.base_name cname));
fun make_inj (dts, dt) (j, args1, args2, eqs) =
let
val Ts_idx =
map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
val ys = map (fn (T, i) => Datatype_Aux.mk_Free "y" T i) Ts_idx;
val x =
Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
val y =
Datatype_Aux.mk_Free "y" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
in
(j + length dts + 1,
xs @ (x :: args1), ys @ (y :: args2),
HOLogic.mk_eq
(fold_rev mk_abs_fun xs x, fold_rev mk_abs_fun ys y) :: eqs)
end;
val (_, args1, args2, eqs) = fold_rev make_inj dts (1, [], [], []);
val Ts = map fastype_of args1;
val c = Const (cname, Ts ---> T)
in
Goal.prove_global_future thy8 [] []
(augment_sort thy8 pt_cp_sort
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
foldr1 HOLogic.mk_conj eqs))))
(fn {context = ctxt, ...} => EVERY
[asm_full_simp_tac (ctxt addsimps (constr_rep_thm ::
rep_inject_thms')) 1,
TRY (asm_full_simp_tac (put_simpset HOL_basic_ss ctxt
addsimps (fresh_def :: supp_def ::
alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
perm_rep_perm_thms)) 1)])
end) (constrs ~~ constr_rep_thms)
end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
(** equations for support and freshness **)
val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
(map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
let val T = nth_dtyp' i
in maps (fn (cname, dts) => map (fn atom =>
let
val cname = Sign.intern_const thy8
(Long_Name.append tname (Long_Name.base_name cname));
val atomT = Type (atom, []);
fun process_constr (dts, dt) (j, args1, args2) =
let
val Ts_idx =
map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
val x =
Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
in
(j + length dts + 1,
xs @ (x :: args1), fold_rev mk_abs_fun xs x :: args2)
end;
val (_, args1, args2) = fold_rev process_constr dts (1, [], []);
val Ts = map fastype_of args1;
val c = list_comb (Const (cname, Ts ---> T), args1);
fun supp t =
Const (@{const_name Nominal.supp}, fastype_of t --> HOLogic.mk_setT atomT) $ t;
fun fresh t = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;
val supp_thm = Goal.prove_global_future thy8 [] []
(augment_sort thy8 pt_cp_sort
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(supp c,
if null dts then HOLogic.mk_set atomT []
else foldr1 (HOLogic.mk_binop @{const_abbrev union}) (map supp args2)))))
(fn {context = ctxt, ...} =>
simp_tac (put_simpset HOL_basic_ss ctxt addsimps (supp_def ::
Un_assoc :: @{thm de_Morgan_conj} :: Collect_disj_eq :: finite_Un ::
Collect_False_empty :: finite_emptyI :: @{thms simp_thms} @
abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
in
(supp_thm,
Goal.prove_global_future thy8 [] [] (augment_sort thy8 pt_cp_sort
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fresh c,
if null dts then @{term True}
else foldr1 HOLogic.mk_conj (map fresh args2)))))
(fn {context = ctxt, ...} =>
simp_tac (put_simpset HOL_ss ctxt addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1))
end) atoms) constrs
end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
(**** weak induction theorem ****)
fun mk_indrule_lemma (((i, _), T), U) (prems, concls) =
let
val Rep_t = Const (nth rep_names i, T --> U) $ Datatype_Aux.mk_Free "x" T i;
val Abs_t = Const (nth abs_names i, U --> T);
in
(prems @ [HOLogic.imp $
(Const (nth rep_set_names'' i, U --> HOLogic.boolT) $ Rep_t) $
(Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
concls @
[Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i])
end;
val (indrule_lemma_prems, indrule_lemma_concls) =
fold mk_indrule_lemma (descr'' ~~ recTs ~~ recTs') ([], []);
val indrule_lemma = Goal.prove_global_future thy8 [] []
(Logic.mk_implies
(HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls)))
(fn {context = ctxt, ...} => EVERY
[REPEAT (etac conjE 1),
REPEAT (EVERY
[TRY (rtac conjI 1),
full_simp_tac (put_simpset HOL_basic_ss ctxt 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' = map (fn r => r RS subst) Abs_inverse_thms;
val dt_induct_prop = Datatype_Prop.make_ind descr';
val dt_induct = Goal.prove_global_future thy8 []
(Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
(fn {prems, context = ctxt} => EVERY
[rtac indrule_lemma' 1,
(Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac ctxt) 1,
EVERY (map (fn (prem, r) => (EVERY
[REPEAT (eresolve_tac Abs_inverse_thms' 1),
simp_tac (put_simpset HOL_basic_ss ctxt addsimps [Thm.symmetric r]) 1,
DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
(prems ~~ constr_defs))]);
val case_names_induct = Datatype_Data.mk_case_names_induct descr'';
(**** prove that new datatypes have finite support ****)
val _ = warning "proving finite support for the new datatype";
val indnames = Datatype_Prop.make_tnames recTs;
val abs_supp = Global_Theory.get_thms thy8 "abs_supp";
val supp_atm = Global_Theory.get_thms thy8 "supp_atm";
val finite_supp_thms = map (fn atom =>
let val atomT = Type (atom, [])
in map Drule.export_without_context (List.take
(Datatype_Aux.split_conj_thm (Goal.prove_global_future thy8 [] []
(augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
(HOLogic.mk_Trueprop
(foldr1 HOLogic.mk_conj (map (fn (s, T) =>
Const (@{const_name finite}, HOLogic.mk_setT atomT --> HOLogic.boolT) $
(Const (@{const_name Nominal.supp}, T --> HOLogic.mk_setT atomT) $ Free (s, T)))
(indnames ~~ recTs)))))
(fn {context = ctxt, ...} => Datatype_Aux.ind_tac dt_induct indnames 1 THEN
ALLGOALS (asm_full_simp_tac (ctxt addsimps
(abs_supp @ supp_atm @
Global_Theory.get_thms thy8 ("fs_" ^ Long_Name.base_name atom ^ "1") @
flat supp_thms))))),
length new_type_names))
end) atoms;
val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
(* Function to add both the simp and eqvt attributes *)
(* These two attributes are duplicated on all the types in the mutual nominal datatypes *)
val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add];
val (_, thy9) = thy8 |>
Sign.add_path big_name |>
Global_Theory.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>
Global_Theory.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>
Sign.parent_path ||>>
Datatype_Aux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
Datatype_Aux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
Datatype_Aux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>>
Datatype_Aux.store_thmss "inject" new_type_names inject_thms ||>>
Datatype_Aux.store_thmss "supp" new_type_names supp_thms ||>>
Datatype_Aux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
fold (fn (atom, ths) => fn thy =>
let
val class = fs_class_of thy atom;
val sort = Sign.minimize_sort thy (Sign.certify_sort thy (class :: pt_cp_sort));
in fold (fn Type (s, Ts) => Axclass.prove_arity
(s, map (inter_sort thy sort o snd o dest_TFree) Ts, [class])
(fn _ => Class.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 @{sort type}
else Sign.minimize_sort thy9 (Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms));
val fsT = TFree ("'n", ind_sort);
val fsT' = TFree ("'n", @{sort type});
val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
(Datatype_Prop.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
fun make_pred fsT i T = Free (nth pnames i, fsT --> T --> HOLogic.boolT);
fun mk_fresh1 xs [] = []
| mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
(HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
(filter (fn (_, U) => T = U) (rev xs)) @
mk_fresh1 (y :: xs) ys;
fun mk_fresh2 xss [] = []
| mk_fresh2 xss ((p as (ys, _)) :: yss) = maps (fn y as (_, T) =>
map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
(fresh_const T U $ Free y $ Free x)) (rev xss @ yss)) ys @
mk_fresh2 (p :: xss) yss;
fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
let
val recs = filter Datatype_Aux.is_rec_type cargs;
val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
val recTs' = map (Datatype_Aux.typ_of_dtyp descr'') recs;
val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);
val rec_tnames = map fst (filter (Datatype_Aux.is_rec_type o snd) (tnames ~~ cargs));
val frees = tnames ~~ Ts;
val frees' = partition_cargs idxs frees;
val z = (singleton (Name.variant_list tnames) "z", fsT);
fun mk_prem ((dt, s), T) =
let
val (Us, U) = strip_type T;
val l = length Us;
in
Logic.list_all (z :: map (pair "x") Us,
HOLogic.mk_Trueprop
(make_pred fsT (Datatype_Aux.body_index dt) U $ Bound l $
Datatype_Aux.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))) (maps fst frees') @
mk_fresh1 [] (maps fst frees') @
mk_fresh2 [] frees'
in
fold_rev (Logic.all o 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 = maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
map (make_ind_prem fsT (fn T => fn t => fn u =>
fresh_const T fsT $ t $ u) i T)
(constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);
val tnames = Datatype_Prop.make_tnames recTs;
val zs = Name.variant_list tnames (replicate (length descr'') "z");
val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
(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)) => Logic.all (Free ("x", fsT'))
(HOLogic.mk_Trueprop (Const (@{const_name finite},
Term.range_type T -->
HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @
maps (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 @{const_name HOL.conj})
(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 aux_ind_vars =
(Datatype_Prop.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 @{const_name HOL.conj})
(map (fn (((i, _), T), tname) =>
HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1))
(Free (tname, T))))
(descr'' ~~ recTs ~~ tnames)));
val fin_set_supp = map (fn s =>
at_inst_of thy9 s RS at_fin_set_supp) dt_atoms;
val fin_set_fresh = map (fn s =>
at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms;
val pt1_atoms = map (fn Type (s, _) =>
Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;
val pt2_atoms = map (fn Type (s, _) =>
Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;
val exists_fresh' = Global_Theory.get_thms thy9 "exists_fresh'";
val fs_atoms = Global_Theory.get_thms thy9 "fin_supp";
val abs_supp = Global_Theory.get_thms thy9 "abs_supp";
val perm_fresh_fresh = Global_Theory.get_thms thy9 "perm_fresh_fresh";
val calc_atm = Global_Theory.get_thms thy9 "calc_atm";
val fresh_atm = Global_Theory.get_thms thy9 "fresh_atm";
val fresh_left = Global_Theory.get_thms thy9 "fresh_left";
val perm_swap = Global_Theory.get_thms thy9 "perm_swap";
fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) =
let
val p = foldr1 HOLogic.mk_prod (ts @ freshs1);
val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
(HOLogic.exists_const T $ Abs ("x", T,
fresh_const T (fastype_of p) $
Bound 0 $ p)))
(fn _ => EVERY
[resolve_tac exists_fresh' 1,
simp_tac (put_simpset HOL_ss ctxt addsimps (supp_prod :: finite_Un :: fs_atoms @
fin_set_supp @ ths)) 1]);
val (([(_, cx)], ths), ctxt') = Obtain.result
(fn ctxt' => EVERY
[etac exE 1,
full_simp_tac (put_simpset HOL_ss ctxt' addsimps (fresh_prod :: fresh_atm)) 1,
REPEAT (etac conjE 1)])
[ex] ctxt
in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
fun fresh_fresh_inst thy a b =
let
val T = fastype_of a;
val SOME th = find_first (fn th => case prop_of th of
_ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T
| _ => false) perm_fresh_fresh
in
Drule.instantiate' []
[SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th
end;
val fs_cp_sort =
map (fs_class_of thy9) dt_atoms @
maps (fn s => map (cp_class_of thy9 s) (remove (op =) s dt_atoms)) dt_atoms;
(**********************************************************************
The subgoals occurring in the proof of induct_aux have the
following parameters:
x_1 ... x_k p_1 ... p_m z
where
x_i : constructor arguments (introduced by weak induction rule)
p_i : permutations (one for each atom type in the data type)
z : freshness context
***********************************************************************)
val _ = warning "proving strong induction theorem ...";
val induct_aux = Goal.prove_global_future thy9 []
(map (augment_sort thy9 fs_cp_sort) ind_prems')
(augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} =>
let
val (prems1, prems2) = chop (length dt_atomTs) prems;
val ind_ss2 = put_simpset HOL_ss context addsimps
finite_Diff :: abs_fresh @ abs_supp @ fs_atoms;
val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @
fresh_atm @ rev_simps @ app_simps;
val ind_ss3 = put_simpset HOL_ss context addsimps abs_fun_eq1 ::
abs_perm @ calc_atm @ perm_swap;
val ind_ss4 = put_simpset HOL_basic_ss context addsimps fresh_left @ prems1 @
fin_set_fresh @ calc_atm;
val ind_ss5 = put_simpset HOL_basic_ss context addsimps pt1_atoms;
val ind_ss6 = put_simpset HOL_basic_ss context addsimps flat perm_simps';
val th = Goal.prove context [] []
(augment_sort thy9 fs_cp_sort aux_ind_concl)
(fn {context = context1, ...} =>
EVERY (Datatype_Aux.ind_tac dt_induct tnames 1 ::
maps (fn ((_, (_, _, constrs)), (_, constrs')) =>
map (fn ((cname, cargs), is) =>
REPEAT (rtac allI 1) THEN
SUBPROOF (fn {prems = iprems, params, concl,
context = context2, ...} =>
let
val concl' = term_of concl;
val _ $ (_ $ _ $ u) = concl';
val U = fastype_of u;
val (xs, params') =
chop (length cargs) (map (term_of o #2) params);
val Ts = map fastype_of xs;
val cnstr = Const (cname, Ts ---> U);
val (pis, z) = split_last params';
val mk_pi = fold_rev (mk_perm []) pis;
val xs' = partition_cargs is xs;
val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs';
val ts = maps (fn (ts, u) => ts @ [u]) xs'';
val (freshs1, freshs2, context3) = fold (fn t =>
let val T = fastype_of t
in obtain_fresh_name' prems1
(the (AList.lookup op = fresh_fs T) $ z :: ts) T
end) (maps fst xs') ([], [], context2);
val freshs1' = unflat (map fst xs') freshs1;
val freshs2' = map (Simplifier.simplify ind_ss4)
(mk_not_sym freshs2);
val ind_ss1' = ind_ss1 addsimps freshs2';
val ind_ss3' = ind_ss3 addsimps freshs2';
val rename_eq =
if forall (null o fst) xs' then []
else [Goal.prove context3 [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(list_comb (cnstr, ts),
list_comb (cnstr, maps (fn ((bs, t), cs) =>
cs @ [fold_rev (mk_perm []) (map perm_of_pair
(bs ~~ cs)) t]) (xs'' ~~ freshs1')))))
(fn _ => EVERY
(simp_tac (put_simpset HOL_ss context3 addsimps flat inject_thms) 1 ::
REPEAT (FIRSTGOAL (rtac conjI)) ::
maps (fn ((bs, t), cs) =>
if null bs then []
else rtac sym 1 :: maps (fn (b, c) =>
[rtac trans 1, rtac sym 1,
rtac (fresh_fresh_inst thy9 b c) 1,
simp_tac ind_ss1' 1,
simp_tac ind_ss2 1,
simp_tac ind_ss3' 1]) (bs ~~ cs))
(xs'' ~~ freshs1')))];
val th = Goal.prove context3 [] [] concl' (fn _ => EVERY
[simp_tac (ind_ss6 addsimps rename_eq) 1,
cut_facts_tac iprems 1,
(resolve_tac prems THEN_ALL_NEW
SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
_ $ (Const (@{const_name Nominal.fresh}, _) $ _ $ _) =>
simp_tac ind_ss1' i
| _ $ (Const (@{const_name Not}, _) $ _) =>
resolve_tac freshs2' i
| _ => asm_simp_tac (put_simpset HOL_basic_ss context3 addsimps
pt2_atoms addsimprocs [perm_simproc]) i)) 1])
val final = Proof_Context.export context3 context2 [th]
in
resolve_tac final 1
end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr)))
in
EVERY
[cut_facts_tac [th] 1,
REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1),
REPEAT (etac allE 1),
REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)]
end);
val induct_aux' = Thm.instantiate ([],
map (fn (s, v as Var (_, T)) =>
(cterm_of thy9 v, cterm_of thy9 (Free (s, T))))
(pnames ~~ map head_of (HOLogic.dest_conj
(HOLogic.dest_Trueprop (concl_of induct_aux)))) @
map (fn (_, f) =>
let val f' = Logic.varify_global f
in (cterm_of thy9 f',
cterm_of thy9 (Const (@{const_name Nominal.supp}, fastype_of f')))
end) fresh_fs) induct_aux;
val induct = Goal.prove_global_future thy9 []
(map (augment_sort thy9 fs_cp_sort) ind_prems)
(augment_sort thy9 fs_cp_sort ind_concl)
(fn {prems, context = ctxt} => EVERY
[rtac induct_aux' 1,
REPEAT (resolve_tac fs_atoms 1),
REPEAT ((resolve_tac prems THEN_ALL_NEW
(etac @{thm meta_spec} ORELSE'
full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps [fresh_def]))) 1)])
val (_, thy10) = thy9 |>
Sign.add_path big_name |>
Global_Theory.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>
Global_Theory.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>
Global_Theory.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];
(**** recursion combinator ****)
val _ = warning "defining recursion combinator ...";
val used = fold Term.add_tfree_namesT recTs [];
val (rec_result_Ts', rec_fn_Ts') = Datatype_Prop.make_primrec_Ts descr' used;
val rec_sort = if null dt_atomTs then @{sort type} else
Sign.minimize_sort thy10 (Sign.certify_sort thy10 pt_cp_sort);
val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
val rec_set_Ts = map (fn (T1, T2) =>
rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
val big_rec_name = big_name ^ "_rec_set";
val rec_set_names' =
if length descr'' = 1 then [big_rec_name] else
map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
(1 upto (length descr''));
val rec_set_names = map (Sign.full_bname thy10) rec_set_names';
val rec_fns = map (uncurry (Datatype_Aux.mk_Free "f"))
(rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
(rec_set_names' ~~ rec_set_Ts);
val rec_sets = map (fn c => list_comb (Const c, rec_fns))
(rec_set_names ~~ rec_set_Ts);
(* introduction rules for graph of recursion function *)
val rec_preds = map (fn (a, T) =>
Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
fun mk_fresh3 rs [] = []
| mk_fresh3 rs ((p as (ys, z)) :: yss) = maps (fn y as (_, T) =>
map_filter (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
else SOME (HOLogic.mk_Trueprop
(fresh_const T U $ Free y $ Free r))) rs) ys @
mk_fresh3 rs yss;
(* FIXME: avoid collisions with other variable names? *)
val rec_ctxt = Free ("z", fsT');
fun make_rec_intr T p rec_set ((cname, cargs), idxs)
(rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, l) =
let
val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
val frees' = partition_cargs idxs frees;
val binders = maps fst frees';
val atomTs = distinct op = (maps (map snd o fst) frees');
val recs = map_filter
(fn ((_, Datatype_Aux.DtRec i), p) => SOME (i, p) | _ => NONE)
(partition_cargs idxs cargs ~~ frees');
val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
map (fn (i, _) => nth rec_result_Ts i) recs;
val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
(nth rec_sets' i $ Free x $ Free y)) (recs ~~ frees'');
val prems2 =
map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
(fresh_const T (fastype_of f) $ Free p $ f)) binders) rec_fns;
val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';
val prems4 = map (fn ((i, _), y) =>
HOLogic.mk_Trueprop (nth rec_preds i $ Free y)) (recs ~~ frees'');
val prems5 = mk_fresh3 (recs ~~ frees'') frees';
val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
(Const (@{const_name finite}, HOLogic.mk_setT aT --> HOLogic.boolT) $
(Const (@{const_name Nominal.supp}, T --> HOLogic.mk_setT aT) $ Free y)))
frees'') atomTs;
val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop
(fresh_const T fsT' $ Free x $ rec_ctxt)) binders;
val result = list_comb (nth rec_fns l, map Free (frees @ frees''));
val result_freshs = map (fn p as (_, T) =>
fresh_const T (fastype_of result) $ Free p $ result) binders;
val P = HOLogic.mk_Trueprop (p $ result)
in
(rec_intr_ts @ [Logic.list_implies (flat prems2 @ prems3 @ prems1,
HOLogic.mk_Trueprop (rec_set $
list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
rec_prems @ [fold_rev (Logic.all o Free) (frees @ frees'') (Logic.list_implies (prems4, P))],
rec_prems' @ map (fn fr => fold_rev (Logic.all o Free) (frees @ frees'')
(Logic.list_implies (nth prems2 l @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
HOLogic.mk_Trueprop fr))) result_freshs,
rec_eq_prems @ [flat prems2 @ prems3],
l + 1)
end;
val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
fold (fn ((((d, d'), T), p), rec_set) =>
fold (make_rec_intr T p rec_set) (#3 (snd d) ~~ snd d'))
(descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets')
([], [], [], [], 0);
val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
thy10
|> Sign.map_naming Name_Space.conceal
|> Inductive.add_inductive_global
{quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
coind = false, no_elim = false, no_ind = false, skip_mono = true}
(map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
(map dest_Free rec_fns)
(map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []
||> Global_Theory.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct")
||> Sign.restore_naming thy10;
(** equivariance **)
val fresh_bij = Global_Theory.get_thms thy11 "fresh_bij";
val perm_bij = Global_Theory.get_thms thy11 "perm_bij";
val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
let
val permT = mk_permT aT;
val pi = Free ("pi", permT);
val rec_fns_pi = map (mk_perm [] pi o uncurry (Datatype_Aux.mk_Free "f"))
(rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
(rec_set_names ~~ rec_set_Ts);
val ps = map (fn ((((T, U), R), R'), i) =>
let
val x = Free ("x" ^ string_of_int i, T);
val y = Free ("y" ^ string_of_int i, U)
in
(R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y)
end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
val ths = map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
(Goal.prove_global_future thy11 [] []
(augment_sort thy1 pt_cp_sort
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
(fn {context = ctxt, ...} => rtac rec_induct 1 THEN REPEAT
(simp_tac (put_simpset HOL_basic_ss ctxt
addsimps flat perm_simps'
addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
(resolve_tac rec_intrs THEN_ALL_NEW
asm_simp_tac (put_simpset HOL_ss ctxt addsimps (fresh_bij @ perm_bij))) 1))))
val ths' = map (fn ((P, Q), th) =>
Goal.prove_global_future thy11 [] []
(augment_sort thy1 pt_cp_sort
(Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))
(fn {context = ctxt, ...} => dtac (Thm.instantiate ([],
[(cterm_of thy11 (Var (("pi", 0), permT)),
cterm_of thy11 (Const (@{const_name rev}, permT --> permT) $ pi))]) th) 1 THEN
NominalPermeq.perm_simp_tac (put_simpset HOL_ss ctxt) 1)) (ps ~~ ths)
in (ths, ths') end) dt_atomTs);
(** finite support **)
val rec_fin_supp_thms = map (fn aT =>
let
val name = Long_Name.base_name (fst (dest_Type aT));
val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
val aset = HOLogic.mk_setT aT;
val finite = Const (@{const_name finite}, aset --> HOLogic.boolT);
val fins = map (fn (f, T) => HOLogic.mk_Trueprop
(finite $ (Const (@{const_name Nominal.supp}, T --> aset) $ f)))
(rec_fns ~~ rec_fn_Ts)
in
map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
(Goal.prove_global_future thy11 []
(map (augment_sort thy11 fs_cp_sort) fins)
(augment_sort thy11 fs_cp_sort
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn (((T, U), R), i) =>
let
val x = Free ("x" ^ string_of_int i, T);
val y = Free ("y" ^ string_of_int i, U)
in
HOLogic.mk_imp (R $ x $ y,
finite $ (Const (@{const_name Nominal.supp}, U --> aset) $ y))
end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~
(1 upto length recTs))))))
(fn {prems = fins, context = ctxt} =>
(rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
(NominalPermeq.finite_guess_tac (put_simpset HOL_ss ctxt addsimps [fs_name]) 1))))
end) dt_atomTs;
(** freshness **)
val finite_premss = map (fn aT =>
map (fn (f, T) => HOLogic.mk_Trueprop
(Const (@{const_name finite}, HOLogic.mk_setT aT --> HOLogic.boolT) $
(Const (@{const_name Nominal.supp}, T --> HOLogic.mk_setT aT) $ f)))
(rec_fns ~~ rec_fn_Ts)) dt_atomTs;
val rec_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns;
val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
let
val name = Long_Name.base_name (fst (dest_Type aT));
val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
val a = Free ("a", aT);
val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
(fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts)
in
map (fn (((T, U), R), eqvt_th) =>
let
val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T);
val y = Free ("y", U);
val y' = Free ("y'", U)
in
Drule.export_without_context (Goal.prove (Proof_Context.init_global thy11) []
(map (augment_sort thy11 fs_cp_sort)
(finite_prems @
[HOLogic.mk_Trueprop (R $ x $ y),
HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),
HOLogic.mk_Trueprop (fresh_const aT T $ a $ x)] @
freshs))
(HOLogic.mk_Trueprop (fresh_const aT U $ a $ y))
(fn {prems, context} =>
let
val (finite_prems, rec_prem :: unique_prem ::
fresh_prems) = chop (length finite_prems) prems;
val unique_prem' = unique_prem RS spec RS mp;
val unique = [unique_prem', unique_prem' RS sym] MRS trans;
val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns')
in EVERY
[rtac (Drule.cterm_instantiate
[(cterm_of thy11 S,
cterm_of thy11 (Const (@{const_name Nominal.supp},
fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
supports_fresh) 1,
simp_tac (put_simpset HOL_basic_ss context addsimps
[supports_def, Thm.symmetric fresh_def, fresh_prod]) 1,
REPEAT_DETERM (resolve_tac [allI, impI] 1),
REPEAT_DETERM (etac conjE 1),
rtac unique 1,
SUBPROOF (fn {prems = prems', params = [(_, a), (_, b)], ...} => EVERY
[cut_facts_tac [rec_prem] 1,
rtac (Thm.instantiate ([],
[(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
asm_simp_tac (put_simpset HOL_ss context addsimps
(prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
rtac rec_prem 1,
simp_tac (put_simpset HOL_ss context addsimps (fs_name ::
supp_prod :: finite_Un :: finite_prems)) 1,
simp_tac (put_simpset HOL_ss context addsimps (Thm.symmetric fresh_def ::
fresh_prod :: fresh_prems)) 1]
end))
end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
(** uniqueness **)
val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);
val fun_tupleT = fastype_of fun_tuple;
val rec_unique_frees =
Datatype_Prop.indexify_names (replicate (length recTs) "x") ~~ recTs;
val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
val rec_unique_frees' =
Datatype_Prop.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
val rec_unique_concls = map (fn ((x, U), R) =>
Const (@{const_name Ex1}, (U --> HOLogic.boolT) --> HOLogic.boolT) $
Abs ("y", U, R $ Free x $ Bound 0))
(rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
val induct_aux_rec = Drule.cterm_instantiate
(map (pairself (cterm_of thy11) o apsnd (augment_sort thy11 fs_cp_sort))
(map (fn (aT, f) => (Logic.varify_global f, Abs ("z", HOLogic.unitT,
Const (@{const_name Nominal.supp}, fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
fresh_fs @
map (fn (((P, T), (x, U)), Q) =>
(Var ((P, 0), Logic.varifyT_global (fsT' --> T --> HOLogic.boolT)),
Abs ("z", HOLogic.unitT, absfree (x, U) Q)))
(pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
map (fn (s, T) => (Var ((s, 0), Logic.varifyT_global T), Free (s, T)))
rec_unique_frees)) induct_aux;
fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
let
val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
(HOLogic.exists_const T $ Abs ("x", T,
fresh_const T (fastype_of p) $ Bound 0 $ p)))
(fn _ => EVERY
[cut_facts_tac ths 1,
REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
resolve_tac exists_fresh' 1,
asm_simp_tac (put_simpset HOL_ss ctxt addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
val (([(_, cx)], ths), ctxt') = Obtain.result
(fn _ => EVERY
[etac exE 1,
full_simp_tac (put_simpset HOL_ss ctxt addsimps (fresh_prod :: fresh_atm)) 1,
REPEAT (etac conjE 1)])
[ex] ctxt
in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
val finite_ctxt_prems = map (fn aT =>
HOLogic.mk_Trueprop
(Const (@{const_name finite}, HOLogic.mk_setT aT --> HOLogic.boolT) $
(Const (@{const_name Nominal.supp}, fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
val rec_unique_thms = Datatype_Aux.split_conj_thm (Goal.prove
(Proof_Context.init_global thy11) (map fst rec_unique_frees)
(map (augment_sort thy11 fs_cp_sort)
(flat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems'))
(augment_sort thy11 fs_cp_sort
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls)))
(fn {prems, context} =>
let
val k = length rec_fns;
val (finite_thss, ths1) = fold_map (fn T => fn xs =>
apfst (pair T) (chop k xs)) dt_atomTs prems;
val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
val (P_ind_ths, fcbs) = chop k ths2;
val P_ths = map (fn th => th RS mp) (Datatype_Aux.split_conj_thm
(Goal.prove context
(map fst (rec_unique_frees'' @ rec_unique_frees')) []
(augment_sort thy11 fs_cp_sort
(HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
(map (fn (((x, y), S), P) => HOLogic.mk_imp
(S $ Free x $ Free y, P $ (Free y)))
(rec_unique_frees'' ~~ rec_unique_frees' ~~
rec_sets ~~ rec_preds)))))
(fn _ =>
rtac rec_induct 1 THEN
REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
val rec_fin_supp_thms' = map
(fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
(rec_fin_supp_thms ~~ finite_thss);
in EVERY
([rtac induct_aux_rec 1] @
maps (fn ((_, finite_ths), finite_th) =>
[cut_facts_tac (finite_th :: finite_ths) 1,
asm_simp_tac (put_simpset HOL_ss context addsimps [supp_prod, finite_Un]) 1])
(finite_thss ~~ finite_ctxt_ths) @
maps (fn ((_, idxss), elim) => maps (fn idxs =>
[full_simp_tac (put_simpset HOL_ss context addsimps [Thm.symmetric fresh_def, supp_prod, Un_iff]) 1,
REPEAT_DETERM (eresolve_tac @{thms conjE ex1E} 1),
rtac @{thm ex1I} 1,
(resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
rotate_tac ~1 1,
((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
(put_simpset HOL_ss context addsimps flat distinct_thms)) 1] @
(if null idxs then [] else [hyp_subst_tac context 1,
SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
let
val SOME prem = find_first (can (HOLogic.dest_eq o
HOLogic.dest_Trueprop o prop_of)) prems';
val _ $ (_ $ lhs $ rhs) = prop_of prem;
val _ $ (_ $ lhs' $ rhs') = term_of concl;
val rT = fastype_of lhs';
val (c, cargsl) = strip_comb lhs;
val cargsl' = partition_cargs idxs cargsl;
val boundsl = maps fst cargsl';
val (_, cargsr) = strip_comb rhs;
val cargsr' = partition_cargs idxs cargsr;
val boundsr = maps fst cargsr';
val (params1, _ :: params2) =
chop (length params div 2) (map (term_of o #2) params);
val params' = params1 @ params2;
val rec_prems = filter (fn th => case prop_of th of
_ $ p => (case head_of p of
Const (s, _) => member (op =) rec_set_names s
| _ => false)
| _ => false) prems';
val fresh_prems = filter (fn th => case prop_of th of
_ $ (Const (@{const_name Nominal.fresh}, _) $ _ $ _) => true
| _ $ (Const (@{const_name Not}, _) $ _) => true
| _ => false) prems';
val Ts = map fastype_of boundsl;
val _ = warning "step 1: obtaining fresh names";
val (freshs1, freshs2, context'') = fold
(obtain_fresh_name (rec_ctxt :: rec_fns' @ params')
(maps snd finite_thss @ finite_ctxt_ths @ rec_prems)
rec_fin_supp_thms')
Ts ([], [], context');
val pi1 = map perm_of_pair (boundsl ~~ freshs1);
val rpi1 = rev pi1;
val pi2 = map perm_of_pair (boundsr ~~ freshs1);
val rpi2 = rev pi2;
val fresh_prems' = mk_not_sym fresh_prems;
val freshs2' = mk_not_sym freshs2;
(** as, bs, cs # K as ts, K bs us **)
val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
val prove_fresh_simpset = put_simpset HOL_ss context'' addsimps
(finite_Diff :: flat fresh_thms @
fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
(* FIXME: avoid asm_full_simp_tac ? *)
fun prove_fresh ths y x = Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const
(fastype_of x) (fastype_of y) $ x $ y))
(fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_simpset 1);
val constr_fresh_thms =
map (prove_fresh fresh_prems lhs) boundsl @
map (prove_fresh fresh_prems rhs) boundsr @
map (prove_fresh freshs2 lhs) freshs1 @
map (prove_fresh freshs2 rhs) freshs1;
(** pi1 o (K as ts) = pi2 o (K bs us) **)
val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
val pi1_pi2_eq = Goal.prove context'' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs)))
(fn _ => EVERY
[cut_facts_tac constr_fresh_thms 1,
asm_simp_tac (put_simpset HOL_basic_ss context'' addsimps perm_fresh_fresh) 1,
rtac prem 1]);
(** pi1 o ts = pi2 o us **)
val _ = warning "step 4: pi1 o ts = pi2 o us";
val pi1_pi2_eqs = map (fn (t, u) =>
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u)))
(fn _ => EVERY
[cut_facts_tac [pi1_pi2_eq] 1,
asm_full_simp_tac (put_simpset HOL_ss context'' addsimps
(calc_atm @ flat perm_simps' @
fresh_prems' @ freshs2' @ abs_perm @
alpha @ flat inject_thms)) 1]))
(map snd cargsl' ~~ map snd cargsr');
(** pi1^-1 o pi2 o us = ts **)
val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fold_rev (mk_perm []) (rpi1 @ pi2) u, t)))
(fn _ => simp_tac (put_simpset HOL_ss context'' addsimps
((eq RS sym) :: perm_swap)) 1))
(map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
val (rec_prems1, rec_prems2) =
chop (length rec_prems div 2) rec_prems;
(** (ts, pi1^-1 o pi2 o vs) in rec_set **)
val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
val rec_prems' = map (fn th =>
let
val _ $ (S $ x $ y) = prop_of th;
val Const (s, _) = head_of S;
val k = find_index (equal s) rec_set_names;
val pi = rpi1 @ pi2;
fun mk_pi z = fold_rev (mk_perm []) pi z;
fun eqvt_tac p =
let
val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
val l = find_index (equal T) dt_atomTs;
val th = nth (nth rec_equiv_thms' l) k;
val th' = Thm.instantiate ([],
[(cterm_of thy11 (Var (("pi", 0), U)),
cterm_of thy11 p)]) th;
in rtac th' 1 end;
val th' = Goal.prove context'' [] []
(HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))
(fn _ => EVERY
(map eqvt_tac pi @
[simp_tac (put_simpset HOL_ss context'' addsimps (fresh_prems' @ freshs2' @
perm_swap @ perm_fresh_fresh)) 1,
rtac th 1]))
in
Simplifier.simplify
(put_simpset HOL_basic_ss context'' addsimps rpi1_pi2_eqs) th'
end) rec_prems2;
val ihs = filter (fn th => case prop_of th of
_ $ (Const (@{const_name All}, _) $ _) => true | _ => false) prems';
(** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)
val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";
val rec_eqns = map (fn (th, ih) =>
let
val th' = th RS (ih RS spec RS mp) RS sym;
val _ $ (_ $ lhs $ rhs) = prop_of th';
fun strip_perm (_ $ _ $ t) = strip_perm t
| strip_perm t = t;
in
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fold_rev (mk_perm []) pi1 lhs,
fold_rev (mk_perm []) pi2 (strip_perm rhs))))
(fn _ => simp_tac (put_simpset HOL_basic_ss context'' addsimps
(th' :: perm_swap)) 1)
end) (rec_prems' ~~ ihs);
(** as # rs **)
val _ = warning "step 8: as # rs";
val rec_freshs =
maps (fn (rec_prem, ih) =>
let
val _ $ (S $ x $ (y as Free (_, T))) =
prop_of rec_prem;
val k = find_index (equal S) rec_sets;
val atoms = flat (map_filter (fn (bs, z) =>
if z = x then NONE else SOME bs) cargsl')
in
map (fn a as Free (_, aT) =>
let val l = find_index (equal aT) dt_atomTs;
in
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const aT T $ a $ y))
(fn _ => EVERY
(rtac (nth (nth rec_fresh_thms l) k) 1 ::
map (fn th => rtac th 1)
(snd (nth finite_thss l)) @
[rtac rec_prem 1, rtac ih 1,
REPEAT_DETERM (resolve_tac fresh_prems 1)]))
end) atoms
end) (rec_prems1 ~~ ihs);
(** as # fK as ts rs , bs # fK bs us vs **)
val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
fun prove_fresh_result (a as Free (_, aT)) =
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const aT rT $ a $ rhs'))
(fn _ => EVERY
[resolve_tac fcbs 1,
REPEAT_DETERM (resolve_tac
(fresh_prems @ rec_freshs) 1),
REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
THEN resolve_tac rec_prems 1),
resolve_tac P_ind_ths 1,
REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
val fresh_results'' = map prove_fresh_result boundsl;
fun prove_fresh_result'' ((a as Free (_, aT), b), th) =
let val th' = Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const aT rT $
fold_rev (mk_perm []) (rpi2 @ pi1) a $
fold_rev (mk_perm []) (rpi2 @ pi1) rhs'))
(fn _ => simp_tac (put_simpset HOL_ss context'' addsimps fresh_bij) 1 THEN
rtac th 1)
in
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))
(fn {context = ctxt, ...} => EVERY
[cut_facts_tac [th'] 1,
full_simp_tac (put_simpset HOL_ss ctxt
addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap
addsimprocs [NominalPermeq.perm_simproc_app]) 1,
full_simp_tac (put_simpset HOL_ss context'' addsimps (calc_atm @
fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])
end;
val fresh_results = fresh_results'' @ map prove_fresh_result''
(boundsl ~~ boundsr ~~ fresh_results'');
(** cs # fK as ts rs , cs # fK bs us vs **)
val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
fun prove_fresh_result' recs t (a as Free (_, aT)) =
Goal.prove context'' [] []
(HOLogic.mk_Trueprop (fresh_const aT rT $ a $ t))
(fn _ => EVERY
[cut_facts_tac recs 1,
REPEAT_DETERM (dresolve_tac
(the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
NominalPermeq.fresh_guess_tac
(put_simpset HOL_ss context'' addsimps (freshs2 @
fs_atoms @ fresh_atm @
maps snd finite_thss)) 1]);
val fresh_results' =
map (prove_fresh_result' rec_prems1 rhs') freshs1 @
map (prove_fresh_result' rec_prems2 lhs') freshs1;
(** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
val pi1_pi2_result = Goal.prove context'' [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs')))
(fn _ => simp_tac (put_simpset HOL_ss context''
addsimps pi1_pi2_eqs @ rec_eqns
addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
TRY (simp_tac (put_simpset HOL_ss context'' addsimps
(fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
val _ = warning "final result";
val final = Goal.prove context'' [] [] (term_of concl)
(fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
full_simp_tac (put_simpset HOL_basic_ss context'' addsimps perm_fresh_fresh @
fresh_results @ fresh_results') 1);
val final' = Proof_Context.export context'' context' [final];
val _ = warning "finished!"
in
resolve_tac final' 1
end) context 1])) idxss) (ndescr ~~ rec_elims))
end));
val rec_total_thms = map (fn r => r RS @{thm theI'}) rec_unique_thms;
(* define primrec combinators *)
val big_reccomb_name = space_implode "_" new_type_names ^ "_rec";
val reccomb_names = map (Sign.full_bname thy11)
(if length descr'' = 1 then [big_reccomb_name] else
(map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
(1 upto (length descr''))));
val reccombs = map (fn ((name, T), T') => list_comb
(Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
(reccomb_names ~~ recTs ~~ rec_result_Ts);
val (reccomb_defs, thy12) =
thy11
|> Sign.add_consts (map (fn ((name, T), T') =>
(Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))
(reccomb_names ~~ recTs ~~ rec_result_Ts))
|> (Global_Theory.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
(Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T)
(Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T')
(set $ Free ("x", T) $ Free ("y", T'))))))
(reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
(* prove characteristic equations for primrec combinators *)
val rec_thms = map (fn (prems, concl) =>
let
val _ $ (_ $ (_ $ x) $ _) = concl;
val (_, cargs) = strip_comb x;
val ps = map (fn (x as Free (_, T), i) =>
(Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
val prems' = flat finite_premss @ finite_ctxt_prems @
rec_prems @ rec_prems' @ map (subst_atomic ps) prems;
fun solve rules prems = resolve_tac rules THEN_ALL_NEW
(resolve_tac prems THEN_ALL_NEW atac)
in
Goal.prove_global_future thy12 []
(map (augment_sort thy12 fs_cp_sort) prems')
(augment_sort thy12 fs_cp_sort concl')
(fn {context = ctxt, prems} => EVERY
[rewrite_goals_tac ctxt reccomb_defs,
rtac @{thm the1_equality} 1,
solve rec_unique_thms prems 1,
resolve_tac rec_intrs 1,
REPEAT (solve (prems @ rec_total_thms) prems 1)])
end) (rec_eq_prems ~~
Datatype_Prop.make_primrecs reccomb_names descr' thy12);
val dt_infos = map_index (make_dt_info pdescr induct reccomb_names rec_thms)
(descr1 ~~ distinct_thms ~~ inject_thms);
(* FIXME: theorems are stored in database for testing only *)
val (_, thy13) = thy12 |>
Global_Theory.add_thmss
[((Binding.name "rec_equiv", flat rec_equiv_thms), []),
((Binding.name "rec_equiv'", flat rec_equiv_thms'), []),
((Binding.name "rec_fin_supp", flat rec_fin_supp_thms), []),
((Binding.name "rec_fresh", flat rec_fresh_thms), []),
((Binding.name "rec_unique", map Drule.export_without_context rec_unique_thms), []),
((Binding.name "recs", rec_thms), [])] ||>
Sign.parent_path ||>
map_nominal_datatypes (fold Symtab.update dt_infos);
in
thy13
end;
val nominal_datatype = gen_nominal_datatype Datatype.check_specs;
val nominal_datatype_cmd = gen_nominal_datatype Datatype.read_specs;
val _ =
Outer_Syntax.command @{command_spec "nominal_datatype"} "define nominal datatypes"
(Parse.and_list1 Datatype.spec_cmd >>
(Toplevel.theory o nominal_datatype_cmd Datatype_Aux.default_config));
end