--- a/src/HOL/IsaMakefile Fri Jul 03 16:51:07 2009 +0200
+++ b/src/HOL/IsaMakefile Fri Jul 03 16:51:08 2009 +0200
@@ -1003,11 +1003,11 @@
$(OUT)/HOL-Nominal: $(OUT)/HOL Nominal/ROOT.ML \
Nominal/Nominal.thy \
Nominal/nominal_atoms.ML \
+ Nominal/nominal_datatype.ML \
Nominal/nominal_fresh_fun.ML \
Nominal/nominal_induct.ML \
Nominal/nominal_inductive.ML \
Nominal/nominal_inductive2.ML \
- Nominal/nominal.ML \
Nominal/nominal_permeq.ML \
Nominal/nominal_primrec.ML \
Nominal/nominal_thmdecls.ML \
--- a/src/HOL/Nominal/Nominal.thy Fri Jul 03 16:51:07 2009 +0200
+++ b/src/HOL/Nominal/Nominal.thy Fri Jul 03 16:51:08 2009 +0200
@@ -3,7 +3,7 @@
uses
("nominal_thmdecls.ML")
("nominal_atoms.ML")
- ("nominal.ML")
+ ("nominal_datatype.ML")
("nominal_induct.ML")
("nominal_permeq.ML")
("nominal_fresh_fun.ML")
@@ -3670,7 +3670,7 @@
lemma allE_Nil: assumes "\<forall>x. P x" obtains "P []"
using assms ..
-use "nominal.ML"
+use "nominal_datatype.ML"
(******************************************************)
(* primitive recursive functions on nominal datatypes *)
--- a/src/HOL/Nominal/nominal.ML Fri Jul 03 16:51:07 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,2094 +0,0 @@
-(* Title: HOL/Nominal/nominal.ML
- Author: Stefan Berghofer and Christian Urban, TU Muenchen
-
-Nominal datatype package for Isabelle/HOL.
-*)
-
-signature NOMINAL =
-sig
- val add_nominal_datatype : Datatype.config -> string list ->
- (string list * bstring * mixfix *
- (bstring * string list * mixfix) list) 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 Nominal : NOMINAL =
-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 empty_def = thm "empty_def";
-val empty_iff = thm "empty_iff";
-
-open DatatypeAux;
-open NominalAtoms;
-
-(** FIXME: Datatype 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 = Long_Name.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 "_" (Long_Name.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;
-
-(* theory data *)
-
-type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;
-
-type nominal_datatype_info =
- {index : int,
- descr : descr,
- sorts : (string * sort) list,
- rec_names : string list,
- rec_rewrites : thm list,
- induction : thm,
- distinct : thm list,
- inject : thm list};
-
-structure NominalDatatypesData = TheoryDataFun
-(
- type T = nominal_datatype_info Symtab.table;
- val empty = Symtab.empty;
- val copy = I;
- val extend = I;
- fun merge _ tabs : T = Symtab.merge (K true) tabs;
-);
-
-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 sorts induct reccomb_names rec_thms
- (((i, (_, (tname, _, _))), distinct), inject) =
- (tname,
- {index = i,
- descr = descr,
- sorts = sorts,
- rec_names = reccomb_names,
- rec_rewrites = rec_thms,
- induction = induct,
- distinct = distinct,
- inject = inject});
-
-(*******************************)
-
-val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
-
-
-(** 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 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' thy ss _ = NONE;
-
-val perm_simproc =
- Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';
-
-val meta_spec = thm "meta_spec";
-
-fun projections rule =
- ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule
- |> map (standard #> RuleCases.save rule);
-
-val supp_prod = thm "supp_prod";
-val fresh_prod = thm "fresh_prod";
-val supports_fresh = thm "supports_fresh";
-val supports_def = thm "Nominal.supports_def";
-val fresh_def = thm "fresh_def";
-val supp_def = 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";
-
-val collect_simp = rewrite_rule [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 ("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 ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
- HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
- end;
-
-fun mk_not_sym ths = maps (fn th => case prop_of th of
- _ $ (Const ("Not", _) $ (Const ("op =", _) $ _ $ _)) => [th, th RS not_sym]
- | _ => [th]) ths;
-
-fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT);
-fun fresh_star_const T U =
- Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT);
-
-fun gen_add_nominal_datatype prep_typ config new_type_names dts thy =
- let
- (* this theory is used just for parsing *)
-
- val tmp_thy = thy |>
- Theory.copy |>
- Sign.add_types (map (fn (tvs, tname, mx, _) =>
- (Binding.name tname, length tvs, mx)) dts);
-
- val atoms = atoms_of thy;
-
- fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
- let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], 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 tyvars = map (map (fn s =>
- (s, the (AList.lookup (op =) sorts s))) o #1) dts';
-
- fun inter_sort thy S S' = Type.inter_sort (Sign.tsig_of thy) (S, S');
- fun augment_sort_typ thy S =
- let val S = 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 (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_bname tmp_thy n, Sign.full_bname tmp_thy (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;
-
- val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, Binding.name (tname ^ "_Rep"), NoSyn,
- map (fn (cname, cargs, mx) => (Binding.name (cname ^ "_Rep"),
- 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 new_type_names' dts'' thy;
-
- val {descr, induction, ...} =
- Datatype.the_info thy1 (hd full_new_type_names');
- fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-
- val big_name = space_implode "_" new_type_names;
-
-
- (**** 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' = DatatypeProp.indexify_names (map (fn (i, _) =>
- "perm_" ^ name_of_typ (nth_dtyp i)) descr);
- val perm_names = replicate (length new_type_names) "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 (typ_of_dtyp descr sorts) dts;
- val names = Name.variant_list ["pi"] (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,
- Free (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
- (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
- (Free (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 (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 = DatatypeProp.make_tnames (map body_type perm_types);
- val perm_fun_def = PureThy.get_thm thy2 "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_global 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_global 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 ("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 "at_pt_inst";
- val pt2 = PureThy.get_thm thy2 "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 = pt_inst_of thy2 a;
- val pt2' = pt_inst RS pt2;
- val pt2_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
- in List.take (map standard (split_conj_thm
- (Goal.prove_global 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 ("List.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 _ => 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 "pt3";
- val pt3_rev = PureThy.get_thm thy2 "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);
- 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 = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
- in List.take (map standard (split_conj_thm
- (Goal.prove_global thy2 [] []
- (augment_sort thy2 [pt_class_of thy2 a] (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 "cp1";
- val dj_cp = PureThy.get_thm thy2 "dj_cp";
- val pt_perm_compose = PureThy.get_thm thy2 "pt_perm_compose";
- val pt_perm_compose_rev = PureThy.get_thm thy2 "pt_perm_compose_rev";
- val dj_perm_perm_forget = PureThy.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;
- val simps = simpset_of thy 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.certify_sort thy (cp_class :: pt_class);
- val thms = split_conj_thm (Goal.prove_global 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 ("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
- fold (fn (s, tvs) => fn thy => AxClass.prove_arity
- (s, map (inter_sort thy sort o snd) tvs, [cp_class])
- (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])
- (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 |>
- PureThy.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 = 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)));
- 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'') = 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 (Free (List.nth (rep_set_names, k),
- T --> HOLogic.boolT) $ free')) :: prems
- | _ => prems,
- snd (List.foldr mk_abs_fun (j', free) Ts) :: ts)
- end;
-
- val (_, _, prems, ts) = List.foldr mk_prem (1, 1, [], []) cargs;
- 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 List.concat (apsnd ListPair.unzip (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,
- (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) =
- Inductive.add_inductive_global (serial_string ())
- {quiet_mode = false, verbose = false, kind = Thm.internalK,
- alt_name = Binding.name big_rep_name, coind = false, no_elim = true, no_ind = false,
- skip_mono = true, fork_mono = false}
- (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))
- (rep_set_names' ~~ recTs'))
- [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy3;
-
- (**** Prove that representing set is closed under permutation ****)
-
- val _ = warning "proving closure under permutation...";
-
- val abs_perm = PureThy.get_thms thy4 "abs_perm";
-
- 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_global thy4 [] []
- (augment_sort thy4
- (pt_class_of thy4 name :: map (cp_class_of thy4 name) (dt_atoms \ name))
- (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 ("Nominal.perm", permT --> T --> T) $
- Free ("pi", permT) $ Free (x, T)))
- end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))
- (fn _ => EVERY
- [indtac rep_induct [] 1,
- ALLGOALS (simp_tac (simpset_of thy4 addsimps
- (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)), name') => fn thy =>
- Typedef.add_typedef false (SOME (Binding.name name'))
- (Binding.name name, map fst tvs, mx)
- (Const ("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 (Name.variant (map fst tvs) "'a", HOLogic.typeS));
- val pi = Free ("pi", permT);
- val T = Type (Sign.intern_type thy name, map TFree tvs);
- in apfst (pair r o hd)
- (PureThy.add_defs_unchecked true [((Binding.name ("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 (rep_set_names'' ~~ recTs', length new_type_names) ~~
- new_type_names);
-
- val perm_defs = map snd typedefs;
- val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs;
- val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
- val Rep_thms = map (collect_simp o #Rep 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.certify_sort thy
- (pt_class :: map (cp_class_of thy atom) (dt_atoms \ atom))
- in AxClass.prove_arity
- (Sign.intern_type thy name,
- map (inter_sort thy sort o snd) tvs, [pt_class])
- (EVERY [Class.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
- ("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.certify_sort thy
- (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (dt_atoms \ atom1) @
- (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else
- pt_class_of thy atom2 :: map (cp_class_of thy atom2) (dt_atoms \ atom2)));
- 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])
- (EVERY [Class.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)
- (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 ("Nominal.abs_fun", T --> U --> T -->
- Type ("Nominal.noption", [U])) $ x $ t
- end;
-
- val (ty_idxs, _) = List.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 = 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, ("Nominal.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), List.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 = typ_of_dtyp descr'' sorts (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 = 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_bname thy) new_type_names;
-
- fun make_constr_def tname T T' ((thy, defs, eqns),
- (((cname_rep, _), (cname, cargs)), (cname', mx))) =
- let
- fun constr_arg ((dts, dt), (j, l_args, r_args)) =
- let
- val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts dt) i)
- (dts ~~ (j upto j + length dts - 1))
- val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)
- in
- (j + length dts + 1,
- xs @ x :: l_args,
- List.foldr mk_abs_fun
- (case dt of
- DtRec k => if k < length new_type_names then
- Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts dt -->
- typ_of_dtyp descr sorts dt) $ x
- else error "nested recursion not (yet) supported"
- | _ => x) xs :: r_args)
- end
-
- val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs;
- 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_i [(Binding.name cname', constrT, mx)] |>
- (PureThy.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 ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
- (_, (_, _, 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')
- ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
- in
- (parent_path (#flat_names config) 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) ~~
- List.take (pdescr, length new_type_names) ~~
- new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
-
- val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs
- val rep_inject_thms = map (#Rep_inject 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 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
- Goal.prove_global thy8 [] []
- (augment_sort thy8
- (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))
- (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 = DatatypeProp.make_distincts descr' sorts;
- 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)) (k, t :: ts) =
- let
- val dist_thm = Goal.prove_global 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 (k, 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 List.concat (map (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 ("Nominal.perm", permT --> T --> T) $ pi $ t end;
-
- fun constr_arg ((dts, dt), (j, l_args, r_args)) =
- let
- val Ts = map (typ_of_dtyp descr'' sorts) 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 descr'' sorts 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) = List.foldr constr_arg (1, [], []) dts;
- val c = Const (cname, map fastype_of l_args ---> T)
- in
- Goal.prove_global thy8 [] []
- (augment_sort thy8
- (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))
- (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 (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 = PureThy.get_thms thy8 "alpha";
- val abs_fresh = PureThy.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) (dt_atoms \ s)) dt_atoms;
-
- val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
- let val T = 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
- (Long_Name.append tname (Long_Name.base_name cname));
-
- fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
- let
- val Ts_idx = map (typ_of_dtyp descr'' sorts) 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 descr'' sorts dt) (j + length dts);
- val y = mk_Free "y" (typ_of_dtyp descr'' sorts dt) (j + length dts)
- in
- (j + length dts + 1,
- xs @ (x :: args1), ys @ (y :: args2),
- HOLogic.mk_eq
- (List.foldr mk_abs_fun x xs, List.foldr mk_abs_fun y ys) :: eqs)
- end;
-
- val (_, args1, args2, eqs) = List.foldr make_inj (1, [], [], []) dts;
- val Ts = map fastype_of args1;
- val c = Const (cname, Ts ---> T)
- in
- Goal.prove_global 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 _ => 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)])
- 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 List.concat (map (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 (typ_of_dtyp descr'' sorts) 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 descr'' sorts dt) (j + length dts)
- in
- (j + length dts + 1,
- xs @ (x :: args1), List.foldr mk_abs_fun x xs :: args2)
- end;
-
- val (_, args1, args2) = List.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 = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;
- val supp_thm = Goal.prove_global 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_name 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 :: finite_emptyI :: simp_thms @
- abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
- in
- (supp_thm,
- Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort
- (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 (HOL_ss 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 ((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 $
- (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $
- (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 = Goal.prove_global 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' = map (fn r => r RS subst) Abs_inverse_thms;
-
- val dt_induct_prop = DatatypeProp.make_ind descr' sorts;
- val dt_induct = Goal.prove_global thy8 []
- (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
- (fn {prems, ...} => EVERY
- [rtac indrule_lemma' 1,
- (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_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 "abs_supp";
- val supp_atm = PureThy.get_thms thy8 "supp_atm";
-
- val finite_supp_thms = map (fn atom =>
- let val atomT = Type (atom, [])
- in map standard (List.take
- (split_conj_thm (Goal.prove_global thy8 [] []
- (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
- (HOLogic.mk_Trueprop
- (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
- Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
- (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))
- (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 ("fs_" ^ Long_Name.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];
-
- (* 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 |>
- PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>
- PureThy.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>
- Sign.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_eqvt_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 = fs_class_of thy atom;
- val sort = 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])
- (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 HOLogic.typeS
- else Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms);
- 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 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) = List.concat (map (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 = 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 = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
- val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
- val frees = tnames ~~ Ts;
- val frees' = partition_cargs idxs frees;
- val z = (Name.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))) (List.concat (map fst frees')) @
- mk_fresh1 [] (List.concat (map fst frees')) @
- mk_fresh2 [] frees'
-
- 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 =>
- fresh_const T fsT $ t $ u) i T)
- (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
- val tnames = DatatypeProp.make_tnames recTs;
- val zs = Name.variant_list tnames (replicate (length descr'') "z");
- 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 (Const ("Finite_Set.finite",
- (snd (split_last (binder_types T)) --> HOLogic.boolT) -->
- HOLogic.boolT) $ (f $ Free ("x", fsT'))))) 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');
-
- 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 $
- 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, _) =>
- PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;
- val pt2_atoms = map (fn Type (s, _) =>
- PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;
- val exists_fresh' = PureThy.get_thms thy9 "exists_fresh'";
- val fs_atoms = PureThy.get_thms thy9 "fin_supp";
- val abs_supp = PureThy.get_thms thy9 "abs_supp";
- val perm_fresh_fresh = PureThy.get_thms thy9 "perm_fresh_fresh";
- val calc_atm = PureThy.get_thms thy9 "calc_atm";
- val fresh_atm = PureThy.get_thms thy9 "fresh_atm";
- val fresh_left = PureThy.get_thms thy9 "fresh_left";
- val perm_swap = PureThy.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 (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @
- fin_set_supp @ ths)) 1]);
- val (([cx], ths), ctxt') = Obtain.result
- (fn _ => EVERY
- [etac exE 1,
- full_simp_tac (HOL_ss 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) (dt_atoms \ s)) 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 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 = HOL_ss 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 = HOL_ss addsimps abs_fun_eq1 ::
- abs_perm @ calc_atm @ perm_swap;
- val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @
- fin_set_fresh @ calc_atm;
- val ind_ss5 = HOL_basic_ss addsimps pt1_atoms;
- val ind_ss6 = HOL_basic_ss addsimps flat perm_simps';
- val th = Goal.prove context [] []
- (augment_sort thy9 fs_cp_sort aux_ind_concl)
- (fn {context = context1, ...} =>
- EVERY (indtac 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 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 (HOL_ss 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 ("Nominal.fresh", _) $ _ $ _) =>
- simp_tac ind_ss1' i
- | _ $ (Const ("Not", _) $ _) =>
- resolve_tac freshs2' i
- | _ => asm_simp_tac (HOL_basic_ss addsimps
- pt2_atoms addsimprocs [perm_simproc]) i)) 1])
- val final = ProofContext.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 f
- in (cterm_of thy9 f',
- cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
- end) fresh_fs) induct_aux;
-
- val induct = Goal.prove_global thy9 []
- (map (augment_sort thy9 fs_cp_sort) ind_prems)
- (augment_sort thy9 fs_cp_sort ind_concl)
- (fn {prems, ...} => EVERY
- [rtac induct_aux' 1,
- REPEAT (resolve_tac fs_atoms 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 |>
- Sign.add_path big_name |>
- PureThy.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>
- PureThy.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>
- PureThy.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];
-
- (**** recursion combinator ****)
-
- val _ = warning "defining recursion combinator ...";
-
- val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
-
- val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts used;
-
- val rec_sort = if null dt_atomTs then HOLogic.typeS else
- 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 (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) = List.concat (map (fn y as (_, T) =>
- List.mapPartial (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 ((rec_intr_ts, rec_prems, rec_prems',
- rec_eq_prems, l), ((cname, cargs), idxs)) =
- let
- val Ts = map (typ_of_dtyp descr'' sorts) cargs;
- val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
- val frees' = partition_cargs idxs frees;
- val binders = List.concat (map fst frees');
- val atomTs = distinct op = (maps (map snd o fst) frees');
- val recs = List.mapPartial
- (fn ((_, 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, _) => List.nth (rec_result_Ts, i)) recs;
- val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
- (List.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 (List.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 ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
- (Const ("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 (List.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 (List.concat prems2 @ prems3 @ prems1,
- HOLogic.mk_Trueprop (rec_set $
- list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
- rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
- rec_prems' @ map (fn fr => list_all_free (frees @ frees'',
- Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
- HOLogic.mk_Trueprop fr))) result_freshs,
- rec_eq_prems @ [List.concat prems2 @ prems3],
- l + 1)
- end;
-
- val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
- Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
- Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
- (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets');
-
- val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
- thy10 |>
- Inductive.add_inductive_global (serial_string ())
- {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK,
- alt_name = Binding.name big_rec_name, coind = false, no_elim = false, no_ind = false,
- skip_mono = true, fork_mono = false}
- (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) [] ||>
- PureThy.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct");
-
- (** equivariance **)
-
- val fresh_bij = PureThy.get_thms thy11 "fresh_bij";
- val perm_bij = PureThy.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 (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 => standard (th RS mp)) (split_conj_thm
- (Goal.prove_global thy11 [] []
- (augment_sort thy1 pt_cp_sort
- (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
- (fn _ => rtac rec_induct 1 THEN REPEAT
- (simp_tac (Simplifier.theory_context thy11 HOL_basic_ss
- addsimps flat perm_simps'
- addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
- (resolve_tac rec_intrs THEN_ALL_NEW
- asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
- val ths' = map (fn ((P, Q), th) =>
- Goal.prove_global thy11 [] []
- (augment_sort thy1 pt_cp_sort
- (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))
- (fn _ => dtac (Thm.instantiate ([],
- [(cterm_of thy11 (Var (("pi", 0), permT)),
- cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
- NominalPermeq.perm_simp_tac HOL_ss 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 = PureThy.get_thm thy11 ("fs_" ^ name ^ "1");
- val aset = HOLogic.mk_setT aT;
- val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
- val fins = map (fn (f, T) => HOLogic.mk_Trueprop
- (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
- (rec_fns ~~ rec_fn_Ts)
- in
- map (fn th => standard (th RS mp)) (split_conj_thm
- (Goal.prove_global 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 ("Nominal.supp", U --> aset) $ y))
- end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~
- (1 upto length recTs))))))
- (fn {prems = fins, ...} =>
- (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
- (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
- end) dt_atomTs;
-
- (** freshness **)
-
- val finite_premss = map (fn aT =>
- map (fn (f, T) => HOLogic.mk_Trueprop
- (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
- (Const ("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 = PureThy.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
- standard (Goal.prove (ProofContext.init 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 ("Nominal.supp",
- fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
- supports_fresh) 1,
- simp_tac (HOL_basic_ss addsimps
- [supports_def, 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 (HOL_ss addsimps
- (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
- rtac rec_prem 1,
- simp_tac (HOL_ss addsimps (fs_name ::
- supp_prod :: finite_Un :: finite_prems)) 1,
- simp_tac (HOL_ss addsimps (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 =
- DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
- val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
- val rec_unique_frees' =
- DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
- val rec_unique_concls = map (fn ((x, U), R) =>
- Const ("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 f, Abs ("z", HOLogic.unitT,
- Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
- fresh_fs @
- map (fn (((P, T), (x, U)), Q) =>
- (Var ((P, 0), Logic.varifyT (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 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 (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
- val (([cx], ths), ctxt') = Obtain.result
- (fn _ => EVERY
- [etac exE 1,
- full_simp_tac (HOL_ss 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 ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
- (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
-
- val rec_unique_thms = split_conj_thm (Goal.prove
- (ProofContext.init thy11) (map fst rec_unique_frees)
- (map (augment_sort thy11 fs_cp_sort)
- (List.concat 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) (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 (HOL_ss addsimps [supp_prod, finite_Un]) 1])
- (finite_thss ~~ finite_ctxt_ths) @
- maps (fn ((_, idxss), elim) => maps (fn idxs =>
- [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
- REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
- rtac 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
- (HOL_ss addsimps List.concat distinct_thms)) 1] @
- (if null idxs then [] else [hyp_subst_tac 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 = List.concat (map fst cargsl');
- val (_, cargsr) = strip_comb rhs;
- val cargsr' = partition_cargs idxs cargsr;
- val boundsr = List.concat (map fst cargsr');
- val (params1, _ :: params2) =
- chop (length params div 2) (map term_of params);
- val params' = params1 @ params2;
- val rec_prems = filter (fn th => case prop_of th of
- _ $ p => (case head_of p of
- Const (s, _) => s mem rec_set_names
- | _ => false)
- | _ => false) prems';
- val fresh_prems = filter (fn th => case prop_of th of
- _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
- | _ $ (Const ("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')
- (List.concat (map 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_ss = HOL_ss addsimps
- (finite_Diff :: List.concat 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_ss 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 (HOL_basic_ss 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 (HOL_ss addsimps
- (calc_atm @ List.concat perm_simps' @
- fresh_prems' @ freshs2' @ abs_perm @
- alpha @ List.concat 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 (HOL_ss 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 = List.nth (List.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 (HOL_ss addsimps (fresh_prems' @ freshs2' @
- perm_swap @ perm_fresh_fresh)) 1,
- rtac th 1]))
- in
- Simplifier.simplify
- (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
- end) rec_prems2;
-
- val ihs = filter (fn th => case prop_of th of
- _ $ (Const ("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 (HOL_basic_ss addsimps
- (th' :: perm_swap)) 1)
- end) (rec_prems' ~~ ihs);
-
- (** as # rs **)
- val _ = warning "step 8: as # rs";
- val rec_freshs = List.concat
- (map (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 = List.concat (List.mapPartial (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 (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
- map (fn th => rtac th 1)
- (snd (List.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 (HOL_ss addsimps fresh_bij) 1 THEN
- rtac th 1)
- in
- Goal.prove context'' [] []
- (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))
- (fn _ => EVERY
- [cut_facts_tac [th'] 1,
- full_simp_tac (Simplifier.theory_context thy11 HOL_ss
- addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap
- addsimprocs [NominalPermeq.perm_simproc_app]) 1,
- full_simp_tac (HOL_ss 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
- (HOL_ss addsimps (freshs2 @
- fs_atoms @ fresh_atm @
- List.concat (map 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 (Simplifier.context context'' HOL_ss
- addsimps pi1_pi2_eqs @ rec_eqns
- addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
- TRY (simp_tac (HOL_ss 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 (HOL_basic_ss addsimps perm_fresh_fresh @
- fresh_results @ fresh_results') 1);
- val final' = ProofContext.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 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_i (map (fn ((name, T), T') =>
- (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))
- (reccomb_names ~~ recTs ~~ rec_result_Ts))
- |> (PureThy.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 ("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' = List.concat 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 thy12 []
- (map (augment_sort thy12 fs_cp_sort) prems')
- (augment_sort thy12 fs_cp_sort concl')
- (fn {prems, ...} => EVERY
- [rewrite_goals_tac reccomb_defs,
- rtac 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 ~~
- DatatypeProp.make_primrecs new_type_names descr' sorts thy12);
-
- val dt_infos = map (make_dt_info pdescr sorts induct reccomb_names rec_thms)
- ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms);
-
- (* FIXME: theorems are stored in database for testing only *)
- val (_, thy13) = thy12 |>
- PureThy.add_thmss
- [((Binding.name "rec_equiv", List.concat rec_equiv_thms), []),
- ((Binding.name "rec_equiv'", List.concat rec_equiv_thms'), []),
- ((Binding.name "rec_fin_supp", List.concat rec_fin_supp_thms), []),
- ((Binding.name "rec_fresh", List.concat rec_fresh_thms), []),
- ((Binding.name "rec_unique", map standard rec_unique_thms), []),
- ((Binding.name "recs", rec_thms), [])] ||>
- Sign.parent_path ||>
- map_nominal_datatypes (fold Symtab.update dt_infos);
-
- in
- thy13
- end;
-
-val add_nominal_datatype = gen_add_nominal_datatype Datatype.read_typ;
-
-
-(* FIXME: The following stuff should be exported by Datatype *)
-
-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 Datatype.default_config names specs end;
-
-val _ =
- OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
- (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
-
-end;
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Nominal/nominal_datatype.ML Fri Jul 03 16:51:08 2009 +0200
@@ -0,0 +1,2094 @@
+(* 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 add_nominal_datatype : Datatype.config -> string list ->
+ (string list * bstring * mixfix *
+ (bstring * string list * mixfix) list) 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 empty_def = thm "empty_def";
+val empty_iff = thm "empty_iff";
+
+open DatatypeAux;
+open NominalAtoms;
+
+(** FIXME: Datatype 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 = Long_Name.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 "_" (Long_Name.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;
+
+(* theory data *)
+
+type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;
+
+type nominal_datatype_info =
+ {index : int,
+ descr : descr,
+ sorts : (string * sort) list,
+ rec_names : string list,
+ rec_rewrites : thm list,
+ induction : thm,
+ distinct : thm list,
+ inject : thm list};
+
+structure NominalDatatypesData = TheoryDataFun
+(
+ type T = nominal_datatype_info Symtab.table;
+ val empty = Symtab.empty;
+ val copy = I;
+ val extend = I;
+ fun merge _ tabs : T = Symtab.merge (K true) tabs;
+);
+
+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 sorts induct reccomb_names rec_thms
+ (((i, (_, (tname, _, _))), distinct), inject) =
+ (tname,
+ {index = i,
+ descr = descr,
+ sorts = sorts,
+ rec_names = reccomb_names,
+ rec_rewrites = rec_thms,
+ induction = induct,
+ distinct = distinct,
+ inject = inject});
+
+(*******************************)
+
+val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
+
+
+(** 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 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' thy ss _ = NONE;
+
+val perm_simproc =
+ Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';
+
+val meta_spec = thm "meta_spec";
+
+fun projections rule =
+ ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule
+ |> map (standard #> RuleCases.save rule);
+
+val supp_prod = thm "supp_prod";
+val fresh_prod = thm "fresh_prod";
+val supports_fresh = thm "supports_fresh";
+val supports_def = thm "Nominal.supports_def";
+val fresh_def = thm "fresh_def";
+val supp_def = 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";
+
+val collect_simp = rewrite_rule [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 ("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 ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
+ HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
+ end;
+
+fun mk_not_sym ths = maps (fn th => case prop_of th of
+ _ $ (Const ("Not", _) $ (Const ("op =", _) $ _ $ _)) => [th, th RS not_sym]
+ | _ => [th]) ths;
+
+fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT);
+fun fresh_star_const T U =
+ Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT);
+
+fun gen_add_nominal_datatype prep_typ config new_type_names dts thy =
+ let
+ (* this theory is used just for parsing *)
+
+ val tmp_thy = thy |>
+ Theory.copy |>
+ Sign.add_types (map (fn (tvs, tname, mx, _) =>
+ (Binding.name tname, length tvs, mx)) dts);
+
+ val atoms = atoms_of thy;
+
+ fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
+ let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], 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 tyvars = map (map (fn s =>
+ (s, the (AList.lookup (op =) sorts s))) o #1) dts';
+
+ fun inter_sort thy S S' = Type.inter_sort (Sign.tsig_of thy) (S, S');
+ fun augment_sort_typ thy S =
+ let val S = 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 (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_bname tmp_thy n, Sign.full_bname tmp_thy (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;
+
+ val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, Binding.name (tname ^ "_Rep"), NoSyn,
+ map (fn (cname, cargs, mx) => (Binding.name (cname ^ "_Rep"),
+ 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 new_type_names' dts'' thy;
+
+ val {descr, induction, ...} =
+ Datatype.the_info thy1 (hd full_new_type_names');
+ fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
+
+ val big_name = space_implode "_" new_type_names;
+
+
+ (**** 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' = DatatypeProp.indexify_names (map (fn (i, _) =>
+ "perm_" ^ name_of_typ (nth_dtyp i)) descr);
+ val perm_names = replicate (length new_type_names) "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 (typ_of_dtyp descr sorts) dts;
+ val names = Name.variant_list ["pi"] (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,
+ Free (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
+ (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
+ (Free (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 (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 = DatatypeProp.make_tnames (map body_type perm_types);
+ val perm_fun_def = PureThy.get_thm thy2 "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_global 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_global 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 ("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 "at_pt_inst";
+ val pt2 = PureThy.get_thm thy2 "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 = pt_inst_of thy2 a;
+ val pt2' = pt_inst RS pt2;
+ val pt2_ax = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
+ in List.take (map standard (split_conj_thm
+ (Goal.prove_global 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 ("List.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 _ => 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 "pt3";
+ val pt3_rev = PureThy.get_thm thy2 "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);
+ 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 = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
+ in List.take (map standard (split_conj_thm
+ (Goal.prove_global thy2 [] []
+ (augment_sort thy2 [pt_class_of thy2 a] (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 "cp1";
+ val dj_cp = PureThy.get_thm thy2 "dj_cp";
+ val pt_perm_compose = PureThy.get_thm thy2 "pt_perm_compose";
+ val pt_perm_compose_rev = PureThy.get_thm thy2 "pt_perm_compose_rev";
+ val dj_perm_perm_forget = PureThy.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;
+ val simps = simpset_of thy 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.certify_sort thy (cp_class :: pt_class);
+ val thms = split_conj_thm (Goal.prove_global 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 ("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
+ fold (fn (s, tvs) => fn thy => AxClass.prove_arity
+ (s, map (inter_sort thy sort o snd) tvs, [cp_class])
+ (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])
+ (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 |>
+ PureThy.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 = 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)));
+ 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'') = 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 (Free (List.nth (rep_set_names, k),
+ T --> HOLogic.boolT) $ free')) :: prems
+ | _ => prems,
+ snd (List.foldr mk_abs_fun (j', free) Ts) :: ts)
+ end;
+
+ val (_, _, prems, ts) = List.foldr mk_prem (1, 1, [], []) cargs;
+ 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 List.concat (apsnd ListPair.unzip (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,
+ (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) =
+ Inductive.add_inductive_global (serial_string ())
+ {quiet_mode = false, verbose = false, kind = Thm.internalK,
+ alt_name = Binding.name big_rep_name, coind = false, no_elim = true, no_ind = false,
+ skip_mono = true, fork_mono = false}
+ (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))
+ (rep_set_names' ~~ recTs'))
+ [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy3;
+
+ (**** Prove that representing set is closed under permutation ****)
+
+ val _ = warning "proving closure under permutation...";
+
+ val abs_perm = PureThy.get_thms thy4 "abs_perm";
+
+ 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_global thy4 [] []
+ (augment_sort thy4
+ (pt_class_of thy4 name :: map (cp_class_of thy4 name) (dt_atoms \ name))
+ (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 ("Nominal.perm", permT --> T --> T) $
+ Free ("pi", permT) $ Free (x, T)))
+ end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))
+ (fn _ => EVERY
+ [indtac rep_induct [] 1,
+ ALLGOALS (simp_tac (simpset_of thy4 addsimps
+ (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)), name') => fn thy =>
+ Typedef.add_typedef false (SOME (Binding.name name'))
+ (Binding.name name, map fst tvs, mx)
+ (Const ("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 (Name.variant (map fst tvs) "'a", HOLogic.typeS));
+ val pi = Free ("pi", permT);
+ val T = Type (Sign.intern_type thy name, map TFree tvs);
+ in apfst (pair r o hd)
+ (PureThy.add_defs_unchecked true [((Binding.name ("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 (rep_set_names'' ~~ recTs', length new_type_names) ~~
+ new_type_names);
+
+ val perm_defs = map snd typedefs;
+ val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs;
+ val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
+ val Rep_thms = map (collect_simp o #Rep 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.certify_sort thy
+ (pt_class :: map (cp_class_of thy atom) (dt_atoms \ atom))
+ in AxClass.prove_arity
+ (Sign.intern_type thy name,
+ map (inter_sort thy sort o snd) tvs, [pt_class])
+ (EVERY [Class.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
+ ("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.certify_sort thy
+ (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (dt_atoms \ atom1) @
+ (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else
+ pt_class_of thy atom2 :: map (cp_class_of thy atom2) (dt_atoms \ atom2)));
+ 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])
+ (EVERY [Class.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)
+ (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 ("Nominal.abs_fun", T --> U --> T -->
+ Type ("Nominal.noption", [U])) $ x $ t
+ end;
+
+ val (ty_idxs, _) = List.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 = 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, ("Nominal.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), List.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 = typ_of_dtyp descr'' sorts (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 = 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_bname thy) new_type_names;
+
+ fun make_constr_def tname T T' ((thy, defs, eqns),
+ (((cname_rep, _), (cname, cargs)), (cname', mx))) =
+ let
+ fun constr_arg ((dts, dt), (j, l_args, r_args)) =
+ let
+ val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts dt) i)
+ (dts ~~ (j upto j + length dts - 1))
+ val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)
+ in
+ (j + length dts + 1,
+ xs @ x :: l_args,
+ List.foldr mk_abs_fun
+ (case dt of
+ DtRec k => if k < length new_type_names then
+ Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts dt -->
+ typ_of_dtyp descr sorts dt) $ x
+ else error "nested recursion not (yet) supported"
+ | _ => x) xs :: r_args)
+ end
+
+ val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs;
+ 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_i [(Binding.name cname', constrT, mx)] |>
+ (PureThy.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 ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
+ (_, (_, _, 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')
+ ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
+ in
+ (parent_path (#flat_names config) 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) ~~
+ List.take (pdescr, length new_type_names) ~~
+ new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
+
+ val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs
+ val rep_inject_thms = map (#Rep_inject 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 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
+ Goal.prove_global thy8 [] []
+ (augment_sort thy8
+ (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))
+ (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 = DatatypeProp.make_distincts descr' sorts;
+ 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)) (k, t :: ts) =
+ let
+ val dist_thm = Goal.prove_global 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 (k, 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 List.concat (map (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 ("Nominal.perm", permT --> T --> T) $ pi $ t end;
+
+ fun constr_arg ((dts, dt), (j, l_args, r_args)) =
+ let
+ val Ts = map (typ_of_dtyp descr'' sorts) 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 descr'' sorts 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) = List.foldr constr_arg (1, [], []) dts;
+ val c = Const (cname, map fastype_of l_args ---> T)
+ in
+ Goal.prove_global thy8 [] []
+ (augment_sort thy8
+ (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))
+ (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 (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 = PureThy.get_thms thy8 "alpha";
+ val abs_fresh = PureThy.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) (dt_atoms \ s)) dt_atoms;
+
+ val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
+ let val T = 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
+ (Long_Name.append tname (Long_Name.base_name cname));
+
+ fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
+ let
+ val Ts_idx = map (typ_of_dtyp descr'' sorts) 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 descr'' sorts dt) (j + length dts);
+ val y = mk_Free "y" (typ_of_dtyp descr'' sorts dt) (j + length dts)
+ in
+ (j + length dts + 1,
+ xs @ (x :: args1), ys @ (y :: args2),
+ HOLogic.mk_eq
+ (List.foldr mk_abs_fun x xs, List.foldr mk_abs_fun y ys) :: eqs)
+ end;
+
+ val (_, args1, args2, eqs) = List.foldr make_inj (1, [], [], []) dts;
+ val Ts = map fastype_of args1;
+ val c = Const (cname, Ts ---> T)
+ in
+ Goal.prove_global 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 _ => 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)])
+ 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 List.concat (map (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 (typ_of_dtyp descr'' sorts) 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 descr'' sorts dt) (j + length dts)
+ in
+ (j + length dts + 1,
+ xs @ (x :: args1), List.foldr mk_abs_fun x xs :: args2)
+ end;
+
+ val (_, args1, args2) = List.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 = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;
+ val supp_thm = Goal.prove_global 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_name 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 :: finite_emptyI :: simp_thms @
+ abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
+ in
+ (supp_thm,
+ Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort
+ (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 (HOL_ss 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 ((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 $
+ (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $
+ (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 = Goal.prove_global 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' = map (fn r => r RS subst) Abs_inverse_thms;
+
+ val dt_induct_prop = DatatypeProp.make_ind descr' sorts;
+ val dt_induct = Goal.prove_global thy8 []
+ (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
+ (fn {prems, ...} => EVERY
+ [rtac indrule_lemma' 1,
+ (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_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 "abs_supp";
+ val supp_atm = PureThy.get_thms thy8 "supp_atm";
+
+ val finite_supp_thms = map (fn atom =>
+ let val atomT = Type (atom, [])
+ in map standard (List.take
+ (split_conj_thm (Goal.prove_global thy8 [] []
+ (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
+ (HOLogic.mk_Trueprop
+ (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
+ Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
+ (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))
+ (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 ("fs_" ^ Long_Name.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];
+
+ (* 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 |>
+ PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>
+ PureThy.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>
+ Sign.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_eqvt_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 = fs_class_of thy atom;
+ val sort = 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])
+ (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 HOLogic.typeS
+ else Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms);
+ 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 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) = List.concat (map (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 = 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 = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
+ val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
+ val frees = tnames ~~ Ts;
+ val frees' = partition_cargs idxs frees;
+ val z = (Name.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))) (List.concat (map fst frees')) @
+ mk_fresh1 [] (List.concat (map fst frees')) @
+ mk_fresh2 [] frees'
+
+ 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 =>
+ fresh_const T fsT $ t $ u) i T)
+ (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
+ val tnames = DatatypeProp.make_tnames recTs;
+ val zs = Name.variant_list tnames (replicate (length descr'') "z");
+ 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 (Const ("Finite_Set.finite",
+ (snd (split_last (binder_types T)) --> HOLogic.boolT) -->
+ HOLogic.boolT) $ (f $ Free ("x", fsT'))))) 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');
+
+ 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 $
+ 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, _) =>
+ PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;
+ val pt2_atoms = map (fn Type (s, _) =>
+ PureThy.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;
+ val exists_fresh' = PureThy.get_thms thy9 "exists_fresh'";
+ val fs_atoms = PureThy.get_thms thy9 "fin_supp";
+ val abs_supp = PureThy.get_thms thy9 "abs_supp";
+ val perm_fresh_fresh = PureThy.get_thms thy9 "perm_fresh_fresh";
+ val calc_atm = PureThy.get_thms thy9 "calc_atm";
+ val fresh_atm = PureThy.get_thms thy9 "fresh_atm";
+ val fresh_left = PureThy.get_thms thy9 "fresh_left";
+ val perm_swap = PureThy.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 (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @
+ fin_set_supp @ ths)) 1]);
+ val (([cx], ths), ctxt') = Obtain.result
+ (fn _ => EVERY
+ [etac exE 1,
+ full_simp_tac (HOL_ss 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) (dt_atoms \ s)) 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 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 = HOL_ss 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 = HOL_ss addsimps abs_fun_eq1 ::
+ abs_perm @ calc_atm @ perm_swap;
+ val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @
+ fin_set_fresh @ calc_atm;
+ val ind_ss5 = HOL_basic_ss addsimps pt1_atoms;
+ val ind_ss6 = HOL_basic_ss addsimps flat perm_simps';
+ val th = Goal.prove context [] []
+ (augment_sort thy9 fs_cp_sort aux_ind_concl)
+ (fn {context = context1, ...} =>
+ EVERY (indtac 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 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 (HOL_ss 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 ("Nominal.fresh", _) $ _ $ _) =>
+ simp_tac ind_ss1' i
+ | _ $ (Const ("Not", _) $ _) =>
+ resolve_tac freshs2' i
+ | _ => asm_simp_tac (HOL_basic_ss addsimps
+ pt2_atoms addsimprocs [perm_simproc]) i)) 1])
+ val final = ProofContext.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 f
+ in (cterm_of thy9 f',
+ cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
+ end) fresh_fs) induct_aux;
+
+ val induct = Goal.prove_global thy9 []
+ (map (augment_sort thy9 fs_cp_sort) ind_prems)
+ (augment_sort thy9 fs_cp_sort ind_concl)
+ (fn {prems, ...} => EVERY
+ [rtac induct_aux' 1,
+ REPEAT (resolve_tac fs_atoms 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 |>
+ Sign.add_path big_name |>
+ PureThy.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>
+ PureThy.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>
+ PureThy.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];
+
+ (**** recursion combinator ****)
+
+ val _ = warning "defining recursion combinator ...";
+
+ val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;
+
+ val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts used;
+
+ val rec_sort = if null dt_atomTs then HOLogic.typeS else
+ 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 (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) = List.concat (map (fn y as (_, T) =>
+ List.mapPartial (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 ((rec_intr_ts, rec_prems, rec_prems',
+ rec_eq_prems, l), ((cname, cargs), idxs)) =
+ let
+ val Ts = map (typ_of_dtyp descr'' sorts) cargs;
+ val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
+ val frees' = partition_cargs idxs frees;
+ val binders = List.concat (map fst frees');
+ val atomTs = distinct op = (maps (map snd o fst) frees');
+ val recs = List.mapPartial
+ (fn ((_, 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, _) => List.nth (rec_result_Ts, i)) recs;
+ val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
+ (List.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 (List.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 ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
+ (Const ("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 (List.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 (List.concat prems2 @ prems3 @ prems1,
+ HOLogic.mk_Trueprop (rec_set $
+ list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
+ rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
+ rec_prems' @ map (fn fr => list_all_free (frees @ frees'',
+ Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
+ HOLogic.mk_Trueprop fr))) result_freshs,
+ rec_eq_prems @ [List.concat prems2 @ prems3],
+ l + 1)
+ end;
+
+ val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
+ Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
+ Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
+ (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets');
+
+ val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
+ thy10 |>
+ Inductive.add_inductive_global (serial_string ())
+ {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK,
+ alt_name = Binding.name big_rec_name, coind = false, no_elim = false, no_ind = false,
+ skip_mono = true, fork_mono = false}
+ (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) [] ||>
+ PureThy.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct");
+
+ (** equivariance **)
+
+ val fresh_bij = PureThy.get_thms thy11 "fresh_bij";
+ val perm_bij = PureThy.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 (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 => standard (th RS mp)) (split_conj_thm
+ (Goal.prove_global thy11 [] []
+ (augment_sort thy1 pt_cp_sort
+ (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
+ (fn _ => rtac rec_induct 1 THEN REPEAT
+ (simp_tac (Simplifier.theory_context thy11 HOL_basic_ss
+ addsimps flat perm_simps'
+ addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
+ (resolve_tac rec_intrs THEN_ALL_NEW
+ asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
+ val ths' = map (fn ((P, Q), th) =>
+ Goal.prove_global thy11 [] []
+ (augment_sort thy1 pt_cp_sort
+ (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))
+ (fn _ => dtac (Thm.instantiate ([],
+ [(cterm_of thy11 (Var (("pi", 0), permT)),
+ cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
+ NominalPermeq.perm_simp_tac HOL_ss 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 = PureThy.get_thm thy11 ("fs_" ^ name ^ "1");
+ val aset = HOLogic.mk_setT aT;
+ val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
+ val fins = map (fn (f, T) => HOLogic.mk_Trueprop
+ (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
+ (rec_fns ~~ rec_fn_Ts)
+ in
+ map (fn th => standard (th RS mp)) (split_conj_thm
+ (Goal.prove_global 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 ("Nominal.supp", U --> aset) $ y))
+ end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~
+ (1 upto length recTs))))))
+ (fn {prems = fins, ...} =>
+ (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
+ (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
+ end) dt_atomTs;
+
+ (** freshness **)
+
+ val finite_premss = map (fn aT =>
+ map (fn (f, T) => HOLogic.mk_Trueprop
+ (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
+ (Const ("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 = PureThy.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
+ standard (Goal.prove (ProofContext.init 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 ("Nominal.supp",
+ fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
+ supports_fresh) 1,
+ simp_tac (HOL_basic_ss addsimps
+ [supports_def, 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 (HOL_ss addsimps
+ (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
+ rtac rec_prem 1,
+ simp_tac (HOL_ss addsimps (fs_name ::
+ supp_prod :: finite_Un :: finite_prems)) 1,
+ simp_tac (HOL_ss addsimps (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 =
+ DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
+ val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
+ val rec_unique_frees' =
+ DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
+ val rec_unique_concls = map (fn ((x, U), R) =>
+ Const ("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 f, Abs ("z", HOLogic.unitT,
+ Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
+ fresh_fs @
+ map (fn (((P, T), (x, U)), Q) =>
+ (Var ((P, 0), Logic.varifyT (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 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 (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
+ val (([cx], ths), ctxt') = Obtain.result
+ (fn _ => EVERY
+ [etac exE 1,
+ full_simp_tac (HOL_ss 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 ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
+ (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
+
+ val rec_unique_thms = split_conj_thm (Goal.prove
+ (ProofContext.init thy11) (map fst rec_unique_frees)
+ (map (augment_sort thy11 fs_cp_sort)
+ (List.concat 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) (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 (HOL_ss addsimps [supp_prod, finite_Un]) 1])
+ (finite_thss ~~ finite_ctxt_ths) @
+ maps (fn ((_, idxss), elim) => maps (fn idxs =>
+ [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
+ REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
+ rtac 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
+ (HOL_ss addsimps List.concat distinct_thms)) 1] @
+ (if null idxs then [] else [hyp_subst_tac 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 = List.concat (map fst cargsl');
+ val (_, cargsr) = strip_comb rhs;
+ val cargsr' = partition_cargs idxs cargsr;
+ val boundsr = List.concat (map fst cargsr');
+ val (params1, _ :: params2) =
+ chop (length params div 2) (map term_of params);
+ val params' = params1 @ params2;
+ val rec_prems = filter (fn th => case prop_of th of
+ _ $ p => (case head_of p of
+ Const (s, _) => s mem rec_set_names
+ | _ => false)
+ | _ => false) prems';
+ val fresh_prems = filter (fn th => case prop_of th of
+ _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
+ | _ $ (Const ("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')
+ (List.concat (map 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_ss = HOL_ss addsimps
+ (finite_Diff :: List.concat 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_ss 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 (HOL_basic_ss 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 (HOL_ss addsimps
+ (calc_atm @ List.concat perm_simps' @
+ fresh_prems' @ freshs2' @ abs_perm @
+ alpha @ List.concat 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 (HOL_ss 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 = List.nth (List.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 (HOL_ss addsimps (fresh_prems' @ freshs2' @
+ perm_swap @ perm_fresh_fresh)) 1,
+ rtac th 1]))
+ in
+ Simplifier.simplify
+ (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
+ end) rec_prems2;
+
+ val ihs = filter (fn th => case prop_of th of
+ _ $ (Const ("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 (HOL_basic_ss addsimps
+ (th' :: perm_swap)) 1)
+ end) (rec_prems' ~~ ihs);
+
+ (** as # rs **)
+ val _ = warning "step 8: as # rs";
+ val rec_freshs = List.concat
+ (map (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 = List.concat (List.mapPartial (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 (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
+ map (fn th => rtac th 1)
+ (snd (List.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 (HOL_ss addsimps fresh_bij) 1 THEN
+ rtac th 1)
+ in
+ Goal.prove context'' [] []
+ (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))
+ (fn _ => EVERY
+ [cut_facts_tac [th'] 1,
+ full_simp_tac (Simplifier.theory_context thy11 HOL_ss
+ addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap
+ addsimprocs [NominalPermeq.perm_simproc_app]) 1,
+ full_simp_tac (HOL_ss 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
+ (HOL_ss addsimps (freshs2 @
+ fs_atoms @ fresh_atm @
+ List.concat (map 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 (Simplifier.context context'' HOL_ss
+ addsimps pi1_pi2_eqs @ rec_eqns
+ addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
+ TRY (simp_tac (HOL_ss 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 (HOL_basic_ss addsimps perm_fresh_fresh @
+ fresh_results @ fresh_results') 1);
+ val final' = ProofContext.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 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_i (map (fn ((name, T), T') =>
+ (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))
+ (reccomb_names ~~ recTs ~~ rec_result_Ts))
+ |> (PureThy.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 ("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' = List.concat 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 thy12 []
+ (map (augment_sort thy12 fs_cp_sort) prems')
+ (augment_sort thy12 fs_cp_sort concl')
+ (fn {prems, ...} => EVERY
+ [rewrite_goals_tac reccomb_defs,
+ rtac 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 ~~
+ DatatypeProp.make_primrecs new_type_names descr' sorts thy12);
+
+ val dt_infos = map (make_dt_info pdescr sorts induct reccomb_names rec_thms)
+ ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms);
+
+ (* FIXME: theorems are stored in database for testing only *)
+ val (_, thy13) = thy12 |>
+ PureThy.add_thmss
+ [((Binding.name "rec_equiv", List.concat rec_equiv_thms), []),
+ ((Binding.name "rec_equiv'", List.concat rec_equiv_thms'), []),
+ ((Binding.name "rec_fin_supp", List.concat rec_fin_supp_thms), []),
+ ((Binding.name "rec_fresh", List.concat rec_fresh_thms), []),
+ ((Binding.name "rec_unique", map standard rec_unique_thms), []),
+ ((Binding.name "recs", rec_thms), [])] ||>
+ Sign.parent_path ||>
+ map_nominal_datatypes (fold Symtab.update dt_infos);
+
+ in
+ thy13
+ end;
+
+val add_nominal_datatype = gen_add_nominal_datatype Datatype.read_typ;
+
+
+(* FIXME: The following stuff should be exported by Datatype *)
+
+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 Datatype.default_config names specs end;
+
+val _ =
+ OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
+ (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
+
+end;
+
+end