berghofe@22313: (*  Title:      HOL/Nominal/nominal_inductive.ML
berghofe@22313:     Author:     Stefan Berghofer, TU Muenchen
berghofe@22313: 
berghofe@22530: Infrastructure for proving equivariance and strong induction theorems
berghofe@22530: for inductive predicates involving nominal datatypes.
berghofe@22313: *)
berghofe@22313: 
berghofe@22313: signature NOMINAL_INDUCTIVE =
berghofe@22313: sig
berghofe@30087:   val prove_strong_ind: string -> (string * string list) list -> local_theory -> Proof.state
berghofe@30087:   val prove_eqvt: string -> string list -> local_theory -> local_theory
berghofe@22313: end
berghofe@22313: 
berghofe@22313: structure NominalInductive : NOMINAL_INDUCTIVE =
berghofe@22313: struct
berghofe@22313: 
wenzelm@59940: val inductive_forall_def = @{thm HOL.induct_forall_def};
urbanc@33772: val inductive_atomize = @{thms induct_atomize};
urbanc@33772: val inductive_rulify = @{thms induct_rulify};
berghofe@24570: 
wenzelm@41228: fun rulify_term thy = Raw_Simplifier.rewrite_term thy inductive_rulify [];
berghofe@24570: 
wenzelm@51717: fun atomize_conv ctxt =
wenzelm@41228:   Raw_Simplifier.rewrite_cterm (true, false, false) (K (K NONE))
wenzelm@51717:     (put_simpset HOL_basic_ss ctxt addsimps inductive_atomize);
wenzelm@51717: fun atomize_intr ctxt = Conv.fconv_rule (Conv.prems_conv ~1 (atomize_conv ctxt));
wenzelm@24832: fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1
wenzelm@51717:   (Conv.params_conv ~1 (K (Conv.prems_conv ~1 (atomize_conv ctxt))) ctxt));
berghofe@24570: 
haftmann@44929: fun preds_of ps t = inter (op = o apsnd dest_Free) ps (Term.add_frees t []);
berghofe@30087: 
wenzelm@39159: val fresh_prod = @{thm fresh_prod};
berghofe@22530: 
haftmann@44689: val perm_bool = mk_meta_eq @{thm perm_bool_def};
wenzelm@39159: val perm_boolI = @{thm perm_boolI};
berghofe@22313: val (_, [perm_boolI_pi, _]) = Drule.strip_comb (snd (Thm.dest_comb
wenzelm@59582:   (Drule.strip_imp_concl (Thm.cprop_of perm_boolI))));
berghofe@22313: 
berghofe@25824: fun mk_perm_bool pi th = th RS Drule.cterm_instantiate
berghofe@25824:   [(perm_boolI_pi, pi)] perm_boolI;
berghofe@25824: 
wenzelm@38715: fun mk_perm_bool_simproc names = Simplifier.simproc_global_i
wenzelm@59582:   (Thm.theory_of_thm perm_bool) "perm_bool" [@{term "perm pi x"}] (fn ctxt =>
haftmann@44868:     fn Const (@{const_name Nominal.perm}, _) $ _ $ t =>
haftmann@36692:          if member (op =) names (the_default "" (try (head_of #> dest_Const #> fst) t))
berghofe@24570:          then SOME perm_bool else NONE
berghofe@24570:      | _ => NONE);
berghofe@24570: 
berghofe@22313: fun transp ([] :: _) = []
berghofe@22313:   | transp xs = map hd xs :: transp (map tl xs);
berghofe@22313: 
berghofe@22530: fun add_binders thy i (t as (_ $ _)) bs = (case strip_comb t of
berghofe@22530:       (Const (s, T), ts) => (case strip_type T of
berghofe@22530:         (Ts, Type (tname, _)) =>
haftmann@31938:           (case NominalDatatype.get_nominal_datatype thy tname of
berghofe@22530:              NONE => fold (add_binders thy i) ts bs
berghofe@22530:            | SOME {descr, index, ...} => (case AList.lookup op =
berghofe@22530:                  (#3 (the (AList.lookup op = descr index))) s of
berghofe@22530:                NONE => fold (add_binders thy i) ts bs
berghofe@22530:              | SOME cargs => fst (fold (fn (xs, x) => fn (bs', cargs') =>
berghofe@22530:                  let val (cargs1, (u, _) :: cargs2) = chop (length xs) cargs'
berghofe@22530:                  in (add_binders thy i u
berghofe@22530:                    (fold (fn (u, T) =>
berghofe@22530:                       if exists (fn j => j < i) (loose_bnos u) then I
wenzelm@59058:                       else insert (op aconv o apply2 fst)
berghofe@22530:                         (incr_boundvars (~i) u, T)) cargs1 bs'), cargs2)
berghofe@22530:                  end) cargs (bs, ts ~~ Ts))))
berghofe@22530:       | _ => fold (add_binders thy i) ts bs)
berghofe@22530:     | (u, ts) => add_binders thy i u (fold (add_binders thy i) ts bs))
berghofe@22530:   | add_binders thy i (Abs (_, _, t)) bs = add_binders thy (i + 1) t bs
berghofe@22530:   | add_binders thy i _ bs = bs;
berghofe@22530: 
haftmann@38795: fun split_conj f names (Const (@{const_name HOL.conj}, _) $ p $ q) _ = (case head_of p of
berghofe@24570:       Const (name, _) =>
haftmann@36692:         if member (op =) names name then SOME (f p q) else NONE
berghofe@24570:     | _ => NONE)
berghofe@24570:   | split_conj _ _ _ _ = NONE;
berghofe@24570: 
berghofe@24570: fun strip_all [] t = t
haftmann@38558:   | strip_all (_ :: xs) (Const (@{const_name All}, _) $ Abs (s, T, t)) = strip_all xs t;
berghofe@24570: 
berghofe@24570: (*********************************************************************)
berghofe@24570: (* maps  R ... & (ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t))  *)
berghofe@27722: (* or    ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t)            *)
berghofe@27722: (* to    R ... & id (ALL z. P z (pi_1 o ... o pi_n o t))             *)
berghofe@27722: (* or    id (ALL z. P z (pi_1 o ... o pi_n o t))                     *)
berghofe@24570: (*                                                                   *)
berghofe@24570: (* where "id" protects the subformula from simplification            *)
berghofe@24570: (*********************************************************************)
berghofe@24570: 
haftmann@38795: fun inst_conj_all names ps pis (Const (@{const_name HOL.conj}, _) $ p $ q) _ =
berghofe@24570:       (case head_of p of
berghofe@24570:          Const (name, _) =>
haftmann@36692:            if member (op =) names name then SOME (HOLogic.mk_conj (p,
haftmann@44868:              Const (@{const_name Fun.id}, HOLogic.boolT --> HOLogic.boolT) $
berghofe@24570:                (subst_bounds (pis, strip_all pis q))))
berghofe@24570:            else NONE
berghofe@24570:        | _ => NONE)
berghofe@24570:   | inst_conj_all names ps pis t u =
berghofe@24570:       if member (op aconv) ps (head_of u) then
haftmann@44868:         SOME (Const (@{const_name Fun.id}, HOLogic.boolT --> HOLogic.boolT) $
berghofe@24570:           (subst_bounds (pis, strip_all pis t)))
berghofe@24570:       else NONE
berghofe@24570:   | inst_conj_all _ _ _ _ _ = NONE;
berghofe@24570: 
wenzelm@51717: fun inst_conj_all_tac ctxt k = EVERY
berghofe@24570:   [TRY (EVERY [etac conjE 1, rtac conjI 1, atac 1]),
berghofe@24570:    REPEAT_DETERM_N k (etac allE 1),
wenzelm@51717:    simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm id_apply}]) 1];
berghofe@24570: 
berghofe@24570: fun map_term f t u = (case f t u of
berghofe@24570:       NONE => map_term' f t u | x => x)
berghofe@24570: and map_term' f (t $ u) (t' $ u') = (case (map_term f t t', map_term f u u') of
berghofe@24570:       (NONE, NONE) => NONE
berghofe@24570:     | (SOME t'', NONE) => SOME (t'' $ u)
berghofe@24570:     | (NONE, SOME u'') => SOME (t $ u'')
berghofe@24570:     | (SOME t'', SOME u'') => SOME (t'' $ u''))
berghofe@24570:   | map_term' f (Abs (s, T, t)) (Abs (s', T', t')) = (case map_term f t t' of
berghofe@24570:       NONE => NONE
berghofe@24570:     | SOME t'' => SOME (Abs (s, T, t'')))
berghofe@24570:   | map_term' _ _ _ = NONE;
berghofe@24570: 
berghofe@24570: (*********************************************************************)
berghofe@24570: (*         Prove  F[f t]  from  F[t],  where F is monotone           *)
berghofe@24570: (*********************************************************************)
berghofe@24570: 
berghofe@24570: fun map_thm ctxt f tac monos opt th =
berghofe@24570:   let
wenzelm@59582:     val prop = Thm.prop_of th;
berghofe@24570:     fun prove t =
berghofe@24570:       Goal.prove ctxt [] [] t (fn _ =>
berghofe@24570:         EVERY [cut_facts_tac [th] 1, etac rev_mp 1,
wenzelm@59498:           REPEAT_DETERM (FIRSTGOAL (resolve_tac ctxt monos)),
berghofe@24570:           REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))])
berghofe@24570:   in Option.map prove (map_term f prop (the_default prop opt)) end;
berghofe@24570: 
berghofe@27352: val eta_contract_cterm = Thm.dest_arg o Thm.cprop_of o Thm.eta_conversion;
berghofe@27352: 
berghofe@25824: fun first_order_matchs pats objs = Thm.first_order_match
berghofe@27352:   (eta_contract_cterm (Conjunction.mk_conjunction_balanced pats),
berghofe@27352:    eta_contract_cterm (Conjunction.mk_conjunction_balanced objs));
berghofe@25824: 
berghofe@25824: fun first_order_mrs ths th = ths MRS
wenzelm@59582:   Thm.instantiate (first_order_matchs (cprems_of th) (map Thm.cprop_of ths)) th;
berghofe@25824: 
berghofe@30087: fun prove_strong_ind s avoids ctxt =
berghofe@22313:   let
wenzelm@42361:     val thy = Proof_Context.theory_of ctxt;
berghofe@25824:     val ({names, ...}, {raw_induct, intrs, elims, ...}) =
haftmann@31723:       Inductive.the_inductive ctxt (Sign.intern_const thy s);
haftmann@31723:     val ind_params = Inductive.params_of raw_induct;
wenzelm@24832:     val raw_induct = atomize_induct ctxt raw_induct;
berghofe@25824:     val elims = map (atomize_induct ctxt) elims;
haftmann@31723:     val monos = Inductive.get_monos ctxt;
urbanc@24571:     val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;
haftmann@33040:     val _ = (case subtract (op =) (fold (Term.add_const_names o Thm.prop_of) eqvt_thms []) names of
berghofe@22788:         [] => ()
berghofe@22788:       | xs => error ("Missing equivariance theorem for predicate(s): " ^
berghofe@22788:           commas_quote xs));
nipkow@44045:     val induct_cases = map (fst o fst) (fst (Rule_Cases.get (the
wenzelm@24861:       (Induct.lookup_inductP ctxt (hd names)))));
wenzelm@59582:     val ([raw_induct'], ctxt') = Variable.import_terms false [Thm.prop_of raw_induct] ctxt;
berghofe@22530:     val concls = raw_induct' |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop |>
berghofe@22530:       HOLogic.dest_conj |> map (HOLogic.dest_imp ##> strip_comb);
berghofe@22530:     val ps = map (fst o snd) concls;
berghofe@22530: 
wenzelm@59058:     val _ = (case duplicates (op = o apply2 fst) avoids of
berghofe@22530:         [] => ()
berghofe@22530:       | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)));
wenzelm@57884:     val _ = avoids |> forall (fn (a, xs) => null (duplicates (op =) xs) orelse
wenzelm@57884:       error ("Duplicate variable names for case " ^ quote a));
haftmann@33040:     val _ = (case subtract (op =) induct_cases (map fst avoids) of
berghofe@22530:         [] => ()
berghofe@22530:       | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
berghofe@25824:     val avoids' = if null induct_cases then replicate (length intrs) ("", [])
berghofe@25824:       else map (fn name =>
berghofe@25824:         (name, the_default [] (AList.lookup op = avoids name))) induct_cases;
berghofe@22530:     fun mk_avoids params (name, ps) =
berghofe@22530:       let val k = length params - 1
berghofe@22530:       in map (fn x => case find_index (equal x o fst) params of
berghofe@22530:           ~1 => error ("No such variable in case " ^ quote name ^
berghofe@22530:             " of inductive definition: " ^ quote x)
berghofe@22530:         | i => (Bound (k - i), snd (nth params i))) ps
berghofe@22530:       end;
berghofe@22530: 
berghofe@22530:     val prems = map (fn (prem, avoid) =>
berghofe@22530:       let
berghofe@22530:         val prems = map (incr_boundvars 1) (Logic.strip_assums_hyp prem);
berghofe@22530:         val concl = incr_boundvars 1 (Logic.strip_assums_concl prem);
berghofe@22530:         val params = Logic.strip_params prem
berghofe@22530:       in
berghofe@22530:         (params,
berghofe@22530:          fold (add_binders thy 0) (prems @ [concl]) [] @
berghofe@22530:            map (apfst (incr_boundvars 1)) (mk_avoids params avoid),
berghofe@22530:          prems, strip_comb (HOLogic.dest_Trueprop concl))
berghofe@22530:       end) (Logic.strip_imp_prems raw_induct' ~~ avoids');
berghofe@22530: 
berghofe@22530:     val atomTs = distinct op = (maps (map snd o #2) prems);
wenzelm@56253:     val ind_sort = if null atomTs then @{sort type}
wenzelm@36428:       else Sign.minimize_sort thy (Sign.certify_sort thy (map (fn T => Sign.intern_class thy
wenzelm@36428:         ("fs_" ^ Long_Name.base_name (fst (dest_Type T)))) atomTs));
wenzelm@43326:     val (fs_ctxt_tyname, _) = Name.variant "'n" (Variable.names_of ctxt');
berghofe@30087:     val ([fs_ctxt_name], ctxt'') = Variable.variant_fixes ["z"] ctxt';
berghofe@22530:     val fsT = TFree (fs_ctxt_tyname, ind_sort);
berghofe@22530: 
berghofe@24570:     val inductive_forall_def' = Drule.instantiate'
wenzelm@59621:       [SOME (Thm.global_ctyp_of thy fsT)] [] inductive_forall_def;
berghofe@24570: 
berghofe@22530:     fun lift_pred' t (Free (s, T)) ts =
berghofe@22530:       list_comb (Free (s, fsT --> T), t :: ts);
berghofe@22530:     val lift_pred = lift_pred' (Bound 0);
berghofe@22530: 
berghofe@24570:     fun lift_prem (t as (f $ u)) =
berghofe@22530:           let val (p, ts) = strip_comb t
berghofe@22530:           in
haftmann@44868:             if member (op =) ps p then HOLogic.mk_induct_forall fsT $
haftmann@44868:               Abs ("z", fsT, lift_pred p (map (incr_boundvars 1) ts))
berghofe@22530:             else lift_prem f $ lift_prem u
berghofe@22530:           end
berghofe@22530:       | lift_prem (Abs (s, T, t)) = Abs (s, T, lift_prem t)
berghofe@22530:       | lift_prem t = t;
berghofe@22530: 
berghofe@22530:     fun mk_distinct [] = []
wenzelm@32952:       | mk_distinct ((x, T) :: xs) = map_filter (fn (y, U) =>
berghofe@22530:           if T = U then SOME (HOLogic.mk_Trueprop
berghofe@22530:             (HOLogic.mk_not (HOLogic.eq_const T $ x $ y)))
berghofe@22530:           else NONE) xs @ mk_distinct xs;
berghofe@22530: 
berghofe@22530:     fun mk_fresh (x, T) = HOLogic.mk_Trueprop
haftmann@31938:       (NominalDatatype.fresh_const T fsT $ x $ Bound 0);
berghofe@22530: 
berghofe@22530:     val (prems', prems'') = split_list (map (fn (params, bvars, prems, (p, ts)) =>
berghofe@22530:       let
berghofe@22530:         val params' = params @ [("y", fsT)];
berghofe@22530:         val prem = Logic.list_implies
berghofe@22530:           (map mk_fresh bvars @ mk_distinct bvars @
berghofe@22530:            map (fn prem =>
berghofe@30087:              if null (preds_of ps prem) then prem
berghofe@22530:              else lift_prem prem) prems,
berghofe@22530:            HOLogic.mk_Trueprop (lift_pred p ts));
wenzelm@29276:         val vs = map (Var o apfst (rpair 0)) (Term.rename_wrt_term prem params')
berghofe@22530:       in
wenzelm@46218:         (Logic.list_all (params', prem), (rev vs, subst_bounds (vs, prem)))
berghofe@22530:       end) prems);
berghofe@22530: 
berghofe@22530:     val ind_vars =
blanchet@58112:       (Old_Datatype_Prop.indexify_names (replicate (length atomTs) "pi") ~~
berghofe@22530:        map NominalAtoms.mk_permT atomTs) @ [("z", fsT)];
berghofe@22530:     val ind_Ts = rev (map snd ind_vars);
berghofe@22530: 
berghofe@22530:     val concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@22530:       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
berghofe@22530:         HOLogic.list_all (ind_vars, lift_pred p
haftmann@31938:           (map (fold_rev (NominalDatatype.mk_perm ind_Ts)
berghofe@22530:             (map Bound (length atomTs downto 1))) ts)))) concls));
berghofe@22530: 
berghofe@22530:     val concl' = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@22530:       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
berghofe@22530:         lift_pred' (Free (fs_ctxt_name, fsT)) p ts)) concls));
berghofe@22530: 
berghofe@22530:     val vc_compat = map (fn (params, bvars, prems, (p, ts)) =>
wenzelm@46218:       map (fn q => Logic.list_all (params, incr_boundvars ~1 (Logic.list_implies
wenzelm@32952:           (map_filter (fn prem =>
berghofe@30087:              if null (preds_of ps prem) then SOME prem
berghofe@24570:              else map_term (split_conj (K o I) names) prem prem) prems, q))))
berghofe@22530:         (mk_distinct bvars @
berghofe@22530:          maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop
haftmann@31938:            (NominalDatatype.fresh_const U T $ u $ t)) bvars)
berghofe@22530:              (ts ~~ binder_types (fastype_of p)))) prems;
berghofe@22530: 
wenzelm@39557:     val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
wenzelm@39557:     val pt2_atoms = map (fn aT => Global_Theory.get_thm thy
wenzelm@30364:       ("pt_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "2")) atomTs;
wenzelm@54895:     val eqvt_ss = simpset_of (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
berghofe@27352:       addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)
haftmann@44868:       addsimprocs [mk_perm_bool_simproc [@{const_name Fun.id}],
wenzelm@51717:         NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun]);
wenzelm@39557:     val fresh_bij = Global_Theory.get_thms thy "fresh_bij";
wenzelm@39557:     val perm_bij = Global_Theory.get_thms thy "perm_bij";
wenzelm@39557:     val fs_atoms = map (fn aT => Global_Theory.get_thm thy
wenzelm@30364:       ("fs_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "1")) atomTs;
wenzelm@39557:     val exists_fresh' = Global_Theory.get_thms thy "exists_fresh'";
wenzelm@39557:     val fresh_atm = Global_Theory.get_thms thy "fresh_atm";
wenzelm@39557:     val swap_simps = Global_Theory.get_thms thy "swap_simps";
wenzelm@39557:     val perm_fresh_fresh = Global_Theory.get_thms thy "perm_fresh_fresh";
berghofe@22530: 
berghofe@22530:     fun obtain_fresh_name ts T (freshs1, freshs2, ctxt) =
berghofe@22530:       let
berghofe@27722:         (** protect terms to avoid that fresh_prod interferes with  **)
berghofe@22530:         (** pairs used in introduction rules of inductive predicate **)
berghofe@22530:         fun protect t =
haftmann@44868:           let val T = fastype_of t in Const (@{const_name Fun.id}, T --> T) $ t end;
berghofe@22530:         val p = foldr1 HOLogic.mk_prod (map protect ts @ freshs1);
berghofe@22530:         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
berghofe@22530:             (HOLogic.exists_const T $ Abs ("x", T,
haftmann@31938:               NominalDatatype.fresh_const T (fastype_of p) $
berghofe@22530:                 Bound 0 $ p)))
berghofe@22530:           (fn _ => EVERY
wenzelm@59498:             [resolve_tac ctxt exists_fresh' 1,
wenzelm@59498:              resolve_tac ctxt fs_atoms 1]);
wenzelm@32202:         val (([(_, cx)], ths), ctxt') = Obtain.result
wenzelm@51717:           (fn ctxt' => EVERY
berghofe@22530:             [etac exE 1,
wenzelm@51717:              full_simp_tac (put_simpset HOL_ss ctxt' addsimps (fresh_prod :: fresh_atm)) 1,
wenzelm@51717:              full_simp_tac (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]) 1,
berghofe@22530:              REPEAT (etac conjE 1)])
berghofe@22530:           [ex] ctxt
wenzelm@59582:       in (freshs1 @ [Thm.term_of cx], freshs2 @ ths, ctxt') end;
berghofe@22530: 
berghofe@30087:     fun mk_ind_proof ctxt' thss =
berghofe@30087:       Goal.prove ctxt' [] prems' concl' (fn {prems = ihyps, context = ctxt} =>
berghofe@22530:         let val th = Goal.prove ctxt [] [] concl (fn {context, ...} =>
berghofe@22530:           rtac raw_induct 1 THEN
berghofe@22530:           EVERY (maps (fn ((((_, bvars, oprems, _), vc_compat_ths), ihyp), (vs, ihypt)) =>
wenzelm@51717:             [REPEAT (rtac allI 1), simp_tac (put_simpset eqvt_ss context) 1,
berghofe@22530:              SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>
berghofe@22530:                let
berghofe@22530:                  val (params', (pis, z)) =
wenzelm@59582:                    chop (length params - length atomTs - 1) (map (Thm.term_of o #2) params) ||>
berghofe@22530:                    split_last;
berghofe@22530:                  val bvars' = map
berghofe@22530:                    (fn (Bound i, T) => (nth params' (length params' - i), T)
berghofe@22530:                      | (t, T) => (t, T)) bvars;
berghofe@22530:                  val pi_bvars = map (fn (t, _) =>
haftmann@31938:                    fold_rev (NominalDatatype.mk_perm []) pis t) bvars';
wenzelm@59582:                  val (P, ts) = strip_comb (HOLogic.dest_Trueprop (Thm.term_of concl));
berghofe@22530:                  val (freshs1, freshs2, ctxt'') = fold
berghofe@22530:                    (obtain_fresh_name (ts @ pi_bvars))
berghofe@22530:                    (map snd bvars') ([], [], ctxt');
haftmann@31938:                  val freshs2' = NominalDatatype.mk_not_sym freshs2;
haftmann@31938:                  val pis' = map NominalDatatype.perm_of_pair (pi_bvars ~~ freshs1);
berghofe@24570:                  fun concat_perm pi1 pi2 =
berghofe@24570:                    let val T = fastype_of pi1
berghofe@24570:                    in if T = fastype_of pi2 then
wenzelm@56253:                        Const (@{const_name append}, T --> T --> T) $ pi1 $ pi2
berghofe@24570:                      else pi2
berghofe@24570:                    end;
berghofe@24570:                  val pis'' = fold (concat_perm #> map) pis' pis;
wenzelm@59582:                  val env = Pattern.first_order_match thy (ihypt, Thm.prop_of ihyp)
berghofe@22530:                    (Vartab.empty, Vartab.empty);
wenzelm@59621:                  val ihyp' = Thm.instantiate ([], map (apply2 (Thm.global_cterm_of thy))
wenzelm@32035:                    (map (Envir.subst_term env) vs ~~
haftmann@31938:                     map (fold_rev (NominalDatatype.mk_perm [])
berghofe@22530:                       (rev pis' @ pis)) params' @ [z])) ihyp;
berghofe@24570:                  fun mk_pi th =
wenzelm@54983:                    Simplifier.simplify (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]
haftmann@31938:                        addsimprocs [NominalDatatype.perm_simproc])
wenzelm@54983:                      (Simplifier.simplify (put_simpset eqvt_ss ctxt')
wenzelm@59621:                        (fold_rev (mk_perm_bool o Thm.global_cterm_of thy)
berghofe@25824:                          (rev pis' @ pis) th));
berghofe@24570:                  val (gprems1, gprems2) = split_list
berghofe@24570:                    (map (fn (th, t) =>
berghofe@30087:                       if null (preds_of ps t) then (SOME th, mk_pi th)
berghofe@24570:                       else
wenzelm@54983:                         (map_thm ctxt' (split_conj (K o I) names)
berghofe@24570:                            (etac conjunct1 1) monos NONE th,
wenzelm@54983:                          mk_pi (the (map_thm ctxt' (inst_conj_all names ps (rev pis''))
wenzelm@54983:                            (inst_conj_all_tac ctxt' (length pis'')) monos (SOME t) th))))
wenzelm@32952:                       (gprems ~~ oprems)) |>> map_filter I;
berghofe@22530:                  val vc_compat_ths' = map (fn th =>
berghofe@22530:                    let
berghofe@25824:                      val th' = first_order_mrs gprems1 th;
wenzelm@59582:                      val (bop, lhs, rhs) = (case Thm.concl_of th' of
berghofe@22530:                          _ $ (fresh $ lhs $ rhs) =>
berghofe@22530:                            (fn t => fn u => fresh $ t $ u, lhs, rhs)
berghofe@22530:                        | _ $ (_ $ (_ $ lhs $ rhs)) =>
berghofe@22530:                            (curry (HOLogic.mk_not o HOLogic.mk_eq), lhs, rhs));
berghofe@22530:                      val th'' = Goal.prove ctxt'' [] [] (HOLogic.mk_Trueprop
haftmann@31938:                          (bop (fold_rev (NominalDatatype.mk_perm []) pis lhs)
haftmann@31938:                             (fold_rev (NominalDatatype.mk_perm []) pis rhs)))
wenzelm@51717:                        (fn _ => simp_tac (put_simpset HOL_basic_ss ctxt'' addsimps
berghofe@22530:                           (fresh_bij @ perm_bij)) 1 THEN rtac th' 1)
wenzelm@51717:                    in Simplifier.simplify (put_simpset eqvt_ss ctxt'' addsimps fresh_atm) th'' end)
berghofe@22530:                      vc_compat_ths;
haftmann@31938:                  val vc_compat_ths'' = NominalDatatype.mk_not_sym vc_compat_ths';
berghofe@27449:                  (** Since swap_simps simplifies (pi :: 'a prm) o (x :: 'b) to x **)
berghofe@27449:                  (** we have to pre-simplify the rewrite rules                   **)
wenzelm@51717:                  val swap_simps_simpset = put_simpset HOL_ss ctxt'' addsimps swap_simps @
wenzelm@51717:                     map (Simplifier.simplify (put_simpset HOL_ss ctxt'' addsimps swap_simps))
berghofe@22530:                       (vc_compat_ths'' @ freshs2');
berghofe@22530:                  val th = Goal.prove ctxt'' [] []
berghofe@22530:                    (HOLogic.mk_Trueprop (list_comb (P $ hd ts,
haftmann@31938:                      map (fold (NominalDatatype.mk_perm []) pis') (tl ts))))
wenzelm@51717:                    (fn _ => EVERY ([simp_tac (put_simpset eqvt_ss ctxt'') 1, rtac ihyp' 1,
wenzelm@59582:                      REPEAT_DETERM_N (Thm.nprems_of ihyp - length gprems)
wenzelm@51717:                        (simp_tac swap_simps_simpset 1),
berghofe@22530:                      REPEAT_DETERM_N (length gprems)
wenzelm@51717:                        (simp_tac (put_simpset HOL_basic_ss ctxt''
berghofe@27847:                           addsimps [inductive_forall_def']
haftmann@31938:                           addsimprocs [NominalDatatype.perm_simproc]) 1 THEN
wenzelm@59498:                         resolve_tac ctxt'' gprems2 1)]));
wenzelm@59582:                  val final = Goal.prove ctxt'' [] [] (Thm.term_of concl)
wenzelm@51717:                    (fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (put_simpset HOL_ss ctxt''
berghofe@22530:                      addsimps vc_compat_ths'' @ freshs2' @
berghofe@22530:                        perm_fresh_fresh @ fresh_atm) 1);
wenzelm@42361:                  val final' = Proof_Context.export ctxt'' ctxt' [final];
wenzelm@59498:                in resolve_tac ctxt' final' 1 end) context 1])
berghofe@22530:                  (prems ~~ thss ~~ ihyps ~~ prems'')))
berghofe@22530:         in
berghofe@22530:           cut_facts_tac [th] 1 THEN REPEAT (etac conjE 1) THEN
wenzelm@59498:           REPEAT (REPEAT (resolve_tac ctxt [conjI, impI] 1) THEN
wenzelm@27228:             etac impE 1 THEN atac 1 THEN REPEAT (etac @{thm allE_Nil} 1) THEN
wenzelm@51717:             asm_full_simp_tac ctxt 1)
wenzelm@42361:         end) |> singleton (Proof_Context.export ctxt' ctxt);
berghofe@22530: 
berghofe@25824:     (** strong case analysis rule **)
berghofe@25824: 
berghofe@25824:     val cases_prems = map (fn ((name, avoids), rule) =>
berghofe@25824:       let
wenzelm@59582:         val ([rule'], ctxt') = Variable.import_terms false [Thm.prop_of rule] ctxt;
berghofe@30087:         val prem :: prems = Logic.strip_imp_prems rule';
berghofe@30087:         val concl = Logic.strip_imp_concl rule'
berghofe@25824:       in
berghofe@25824:         (prem,
berghofe@25824:          List.drop (snd (strip_comb (HOLogic.dest_Trueprop prem)), length ind_params),
berghofe@25824:          concl,
berghofe@30087:          fold_map (fn (prem, (_, avoid)) => fn ctxt =>
berghofe@25824:            let
berghofe@25824:              val prems = Logic.strip_assums_hyp prem;
berghofe@25824:              val params = Logic.strip_params prem;
berghofe@25824:              val bnds = fold (add_binders thy 0) prems [] @ mk_avoids params avoid;
berghofe@30087:              fun mk_subst (p as (s, T)) (i, j, ctxt, ps, qs, is, ts) =
berghofe@25824:                if member (op = o apsnd fst) bnds (Bound i) then
berghofe@25824:                  let
berghofe@30087:                    val ([s'], ctxt') = Variable.variant_fixes [s] ctxt;
berghofe@25824:                    val t = Free (s', T)
berghofe@30087:                  in (i + 1, j, ctxt', ps, (t, T) :: qs, i :: is, t :: ts) end
berghofe@30087:                else (i + 1, j + 1, ctxt, p :: ps, qs, is, Bound j :: ts);
berghofe@30087:              val (_, _, ctxt', ps, qs, is, ts) = fold_rev mk_subst params
berghofe@30087:                (0, 0, ctxt, [], [], [], [])
berghofe@25824:            in
berghofe@30087:              ((ps, qs, is, map (curry subst_bounds (rev ts)) prems), ctxt')
berghofe@30087:            end) (prems ~~ avoids) ctxt')
berghofe@25824:       end)
haftmann@31723:         (Inductive.partition_rules' raw_induct (intrs ~~ avoids') ~~
berghofe@30087:          elims);
berghofe@25824: 
berghofe@25824:     val cases_prems' =
berghofe@30087:       map (fn (prem, args, concl, (prems, _)) =>
berghofe@25824:         let
berghofe@25824:           fun mk_prem (ps, [], _, prems) =
wenzelm@46218:                 Logic.list_all (ps, Logic.list_implies (prems, concl))
berghofe@25824:             | mk_prem (ps, qs, _, prems) =
wenzelm@46218:                 Logic.list_all (ps, Logic.mk_implies
berghofe@25824:                   (Logic.list_implies
berghofe@25824:                     (mk_distinct qs @
berghofe@25824:                      maps (fn (t, T) => map (fn u => HOLogic.mk_Trueprop
haftmann@31938:                       (NominalDatatype.fresh_const T (fastype_of u) $ t $ u))
berghofe@25824:                         args) qs,
berghofe@25824:                      HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
berghofe@25824:                        (map HOLogic.dest_Trueprop prems))),
berghofe@25824:                    concl))
berghofe@25824:           in map mk_prem prems end) cases_prems;
berghofe@25824: 
wenzelm@54895:     val cases_eqvt_simpset = put_simpset HOL_ss (Proof_Context.init_global thy)
berghofe@27449:       addsimps eqvt_thms @ swap_simps @ perm_pi_simp
berghofe@27352:       addsimprocs [NominalPermeq.perm_simproc_app,
berghofe@27352:         NominalPermeq.perm_simproc_fun];
berghofe@27352: 
berghofe@27352:     val simp_fresh_atm = map
wenzelm@54895:       (Simplifier.simplify (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
wenzelm@54895:         addsimps fresh_atm));
berghofe@25824: 
berghofe@30087:     fun mk_cases_proof ((((name, thss), elim), (prem, args, concl, (prems, ctxt'))),
berghofe@25824:         prems') =
berghofe@30087:       (name, Goal.prove ctxt' [] (prem :: prems') concl
wenzelm@26711:         (fn {prems = hyp :: hyps, context = ctxt1} =>
berghofe@25824:         EVERY (rtac (hyp RS elim) 1 ::
berghofe@25824:           map (fn (((_, vc_compat_ths), case_hyp), (_, qs, is, _)) =>
berghofe@25824:             SUBPROOF (fn {prems = case_hyps, params, context = ctxt2, concl, ...} =>
berghofe@25824:               if null qs then
berghofe@25824:                 rtac (first_order_mrs case_hyps case_hyp) 1
berghofe@25824:               else
berghofe@25824:                 let
wenzelm@59582:                   val params' = map (Thm.term_of o #2 o nth (rev params)) is;
berghofe@25824:                   val tab = params' ~~ map fst qs;
berghofe@25824:                   val (hyps1, hyps2) = chop (length args) case_hyps;
berghofe@25824:                   (* turns a = t and [x1 # t, ..., xn # t] *)
berghofe@25824:                   (* into [x1 # a, ..., xn # a]            *)
berghofe@25824:                   fun inst_fresh th' ths =
berghofe@25824:                     let val (ths1, ths2) = chop (length qs) ths
berghofe@25824:                     in
berghofe@25824:                       (map (fn th =>
berghofe@25824:                          let
berghofe@25824:                            val (cf, ct) =
wenzelm@59582:                              Thm.dest_comb (Thm.dest_arg (Thm.cprop_of th));
berghofe@25824:                            val arg_cong' = Drule.instantiate'
wenzelm@59586:                              [SOME (Thm.ctyp_of_cterm ct)]
berghofe@25824:                              [NONE, SOME ct, SOME cf] (arg_cong RS iffD2);
berghofe@25824:                            val inst = Thm.first_order_match (ct,
wenzelm@59582:                              Thm.dest_arg (Thm.dest_arg (Thm.cprop_of th')))
berghofe@25824:                          in [th', th] MRS Thm.instantiate inst arg_cong'
berghofe@25824:                          end) ths1,
berghofe@25824:                        ths2)
berghofe@25824:                     end;
berghofe@25824:                   val (vc_compat_ths1, vc_compat_ths2) =
berghofe@25824:                     chop (length vc_compat_ths - length args * length qs)
berghofe@25824:                       (map (first_order_mrs hyps2) vc_compat_ths);
berghofe@25824:                   val vc_compat_ths' =
haftmann@31938:                     NominalDatatype.mk_not_sym vc_compat_ths1 @
berghofe@25824:                     flat (fst (fold_map inst_fresh hyps1 vc_compat_ths2));
berghofe@25824:                   val (freshs1, freshs2, ctxt3) = fold
berghofe@25824:                     (obtain_fresh_name (args @ map fst qs @ params'))
berghofe@25824:                     (map snd qs) ([], [], ctxt2);
haftmann@31938:                   val freshs2' = NominalDatatype.mk_not_sym freshs2;
haftmann@31938:                   val pis = map (NominalDatatype.perm_of_pair)
berghofe@25824:                     ((freshs1 ~~ map fst qs) @ (params' ~~ freshs1));
wenzelm@59621:                   val mk_pis = fold_rev mk_perm_bool (map (Thm.global_cterm_of thy) pis);
wenzelm@59621:                   val obj = Thm.global_cterm_of thy (foldr1 HOLogic.mk_conj (map (map_aterms
berghofe@25824:                      (fn x as Free _ =>
haftmann@36692:                            if member (op =) args x then x
berghofe@25824:                            else (case AList.lookup op = tab x of
berghofe@25824:                              SOME y => y
haftmann@31938:                            | NONE => fold_rev (NominalDatatype.mk_perm []) pis x)
wenzelm@59582:                        | x => x) o HOLogic.dest_Trueprop o Thm.prop_of) case_hyps));
berghofe@25824:                   val inst = Thm.first_order_match (Thm.dest_arg
berghofe@25824:                     (Drule.strip_imp_concl (hd (cprems_of case_hyp))), obj);
wenzelm@59582:                   val th = Goal.prove ctxt3 [] [] (Thm.term_of concl)
berghofe@25824:                     (fn {context = ctxt4, ...} =>
berghofe@25824:                        rtac (Thm.instantiate inst case_hyp) 1 THEN
berghofe@25824:                        SUBPROOF (fn {prems = fresh_hyps, ...} =>
berghofe@25824:                          let
haftmann@31938:                            val fresh_hyps' = NominalDatatype.mk_not_sym fresh_hyps;
wenzelm@51717:                            val case_simpset = cases_eqvt_simpset addsimps freshs2' @
berghofe@27352:                              simp_fresh_atm (vc_compat_ths' @ fresh_hyps');
wenzelm@51717:                            val fresh_fresh_simpset = case_simpset addsimps perm_fresh_fresh;
berghofe@25824:                            val hyps1' = map
wenzelm@51717:                              (mk_pis #> Simplifier.simplify fresh_fresh_simpset) hyps1;
berghofe@25824:                            val hyps2' = map
wenzelm@51717:                              (mk_pis #> Simplifier.simplify case_simpset) hyps2;
berghofe@27449:                            val case_hyps' = hyps1' @ hyps2'
berghofe@25824:                          in
wenzelm@51717:                            simp_tac case_simpset 1 THEN
berghofe@25824:                            REPEAT_DETERM (TRY (rtac conjI 1) THEN
wenzelm@59498:                              resolve_tac ctxt4 case_hyps' 1)
berghofe@25824:                          end) ctxt4 1)
wenzelm@42361:                   val final = Proof_Context.export ctxt3 ctxt2 [th]
wenzelm@59498:                 in resolve_tac ctxt2 final 1 end) ctxt1 1)
berghofe@30087:                   (thss ~~ hyps ~~ prems))) |>
wenzelm@42361:                   singleton (Proof_Context.export ctxt' ctxt))
berghofe@25824: 
berghofe@22530:   in
berghofe@30087:     ctxt'' |>
wenzelm@50771:     Proof.theorem NONE (fn thss => fn ctxt => (* FIXME ctxt/ctxt' should be called lthy/lthy' *)
berghofe@22530:       let
wenzelm@30364:         val rec_name = space_implode "_" (map Long_Name.base_name names);
wenzelm@30223:         val rec_qualified = Binding.qualify false rec_name;
wenzelm@33368:         val ind_case_names = Rule_Cases.case_names induct_cases;
haftmann@31723:         val induct_cases' = Inductive.partition_rules' raw_induct
berghofe@25824:           (intrs ~~ induct_cases); 
wenzelm@51717:         val thss' = map (map (atomize_intr ctxt)) thss;
haftmann@31723:         val thsss = Inductive.partition_rules' raw_induct (intrs ~~ thss');
berghofe@24570:         val strong_raw_induct =
wenzelm@51717:           mk_ind_proof ctxt thss' |> Inductive.rulify ctxt;
wenzelm@51717:         val strong_cases = map (mk_cases_proof ##> Inductive.rulify ctxt)
berghofe@25824:           (thsss ~~ elims ~~ cases_prems ~~ cases_prems');
wenzelm@50771:         val strong_induct_atts =
wenzelm@50771:           map (Attrib.internal o K)
wenzelm@50771:             [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of strong_raw_induct))];
berghofe@22530:         val strong_induct =
wenzelm@50771:           if length names > 1 then strong_raw_induct
wenzelm@50771:           else strong_raw_induct RSN (2, rev_mp);
wenzelm@33671:         val ((_, [strong_induct']), ctxt') = ctxt |> Local_Theory.note
wenzelm@50771:           ((rec_qualified (Binding.name "strong_induct"), strong_induct_atts), [strong_induct]);
berghofe@22530:         val strong_inducts =
wenzelm@33670:           Project_Rule.projects ctxt (1 upto length names) strong_induct';
berghofe@22530:       in
berghofe@30087:         ctxt' |>
wenzelm@50771:         Local_Theory.notes
wenzelm@50771:           [((rec_qualified (Binding.name "strong_inducts"), []),
wenzelm@50771:             strong_inducts |> map (fn th => ([th],
wenzelm@50771:               [Attrib.internal (K ind_case_names),
wenzelm@50771:                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th)))])))] |> snd |>
wenzelm@33671:         Local_Theory.notes (map (fn ((name, elim), (_, cases)) =>
wenzelm@30435:             ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "strong_cases"),
wenzelm@33368:               [Attrib.internal (K (Rule_Cases.case_names (map snd cases))),
wenzelm@50771:                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim)))]), [([elim], [])]))
berghofe@30087:           (strong_cases ~~ induct_cases')) |> snd
berghofe@30087:       end)
berghofe@24570:       (map (map (rulify_term thy #> rpair [])) vc_compat)
berghofe@22530:   end;
berghofe@22530: 
wenzelm@54991: fun prove_eqvt s xatoms ctxt =  (* FIXME ctxt should be called lthy *)
berghofe@22530:   let
wenzelm@42361:     val thy = Proof_Context.theory_of ctxt;
berghofe@22788:     val ({names, ...}, {raw_induct, intrs, elims, ...}) =
haftmann@31723:       Inductive.the_inductive ctxt (Sign.intern_const thy s);
wenzelm@24832:     val raw_induct = atomize_induct ctxt raw_induct;
wenzelm@24832:     val elims = map (atomize_induct ctxt) elims;
wenzelm@51717:     val intrs = map (atomize_intr ctxt) intrs;
haftmann@31723:     val monos = Inductive.get_monos ctxt;
haftmann@31723:     val intrs' = Inductive.unpartition_rules intrs
berghofe@22788:       (map (fn (((s, ths), (_, k)), th) =>
haftmann@31723:            (s, ths ~~ Inductive.infer_intro_vars th k ths))
haftmann@31723:          (Inductive.partition_rules raw_induct intrs ~~
haftmann@31723:           Inductive.arities_of raw_induct ~~ elims));
haftmann@31723:     val k = length (Inductive.params_of raw_induct);
berghofe@22730:     val atoms' = NominalAtoms.atoms_of thy;
berghofe@22730:     val atoms =
berghofe@22730:       if null xatoms then atoms' else
berghofe@22730:       let val atoms = map (Sign.intern_type thy) xatoms
berghofe@22730:       in
berghofe@22730:         (case duplicates op = atoms of
berghofe@22730:              [] => ()
berghofe@22730:            | xs => error ("Duplicate atoms: " ^ commas xs);
haftmann@33040:          case subtract (op =) atoms' atoms of
berghofe@22730:              [] => ()
berghofe@22730:            | xs => error ("No such atoms: " ^ commas xs);
berghofe@22730:          atoms)
berghofe@22730:       end;
wenzelm@39557:     val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
berghofe@30087:     val (([t], [pi]), ctxt') = ctxt |>
wenzelm@59582:       Variable.import_terms false [Thm.concl_of raw_induct] ||>>
berghofe@30087:       Variable.variant_fixes ["pi"];
wenzelm@54991:     val eqvt_simpset = put_simpset HOL_basic_ss ctxt' addsimps
wenzelm@54991:       (NominalThmDecls.get_eqvt_thms ctxt' @ perm_pi_simp) addsimprocs
wenzelm@54991:       [mk_perm_bool_simproc names,
wenzelm@54991:        NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun];
berghofe@22313:     val ps = map (fst o HOLogic.dest_imp)
berghofe@22313:       (HOLogic.dest_conj (HOLogic.dest_Trueprop t));
wenzelm@54991:     fun eqvt_tac pi (intr, vs) st =
berghofe@22544:       let
berghofe@30087:         fun eqvt_err s =
wenzelm@59582:           let val ([t], ctxt'') = Variable.import_terms true [Thm.prop_of intr] ctxt'
berghofe@30087:           in error ("Could not prove equivariance for introduction rule\n" ^
wenzelm@54991:             Syntax.string_of_term ctxt'' t ^ "\n" ^ s)
berghofe@30087:           end;
wenzelm@54983:         val res = SUBPROOF (fn {context = ctxt'', prems, params, ...} =>
berghofe@22788:           let
wenzelm@54983:             val prems' = map (fn th => the_default th (map_thm ctxt''
berghofe@24570:               (split_conj (K I) names) (etac conjunct2 1) monos NONE th)) prems;
wenzelm@51717:             val prems'' = map (fn th => Simplifier.simplify eqvt_simpset
wenzelm@59621:               (mk_perm_bool (Thm.global_cterm_of thy pi) th)) prems';
wenzelm@59621:             val intr' = Drule.cterm_instantiate (map (Thm.global_cterm_of thy) vs ~~
wenzelm@59621:                map (Thm.global_cterm_of thy o NominalDatatype.mk_perm [] pi o Thm.term_of o #2) params)
berghofe@22788:                intr
wenzelm@59498:           in (rtac intr' THEN_ALL_NEW (TRY o resolve_tac ctxt'' prems'')) 1
berghofe@30087:           end) ctxt' 1 st
berghofe@22544:       in
berghofe@22544:         case (Seq.pull res handle THM (s, _, _) => eqvt_err s) of
berghofe@22544:           NONE => eqvt_err ("Rule does not match goal\n" ^
wenzelm@59582:             Syntax.string_of_term ctxt' (hd (Thm.prems_of st)))
berghofe@22544:         | SOME (th, _) => Seq.single th
berghofe@22544:       end;
berghofe@22313:     val thss = map (fn atom =>
berghofe@25824:       let val pi' = Free (pi, NominalAtoms.mk_permT (Type (atom, [])))
berghofe@22530:       in map (fn th => zero_var_indexes (th RS mp))
blanchet@58112:         (Old_Datatype_Aux.split_conj_thm (Goal.prove ctxt' [] []
berghofe@22313:           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn p =>
berghofe@30087:             let
berghofe@30087:               val (h, ts) = strip_comb p;
berghofe@30087:               val (ts1, ts2) = chop k ts
berghofe@30087:             in
berghofe@30087:               HOLogic.mk_imp (p, list_comb (h, ts1 @
haftmann@31938:                 map (NominalDatatype.mk_perm [] pi') ts2))
berghofe@30087:             end) ps)))
wenzelm@54991:           (fn _ => EVERY (rtac raw_induct 1 :: map (fn intr_vs =>
wenzelm@51717:               full_simp_tac eqvt_simpset 1 THEN
wenzelm@54991:               eqvt_tac pi' intr_vs) intrs')) |>
wenzelm@42361:           singleton (Proof_Context.export ctxt' ctxt)))
berghofe@22544:       end) atoms
berghofe@22544:   in
berghofe@30087:     ctxt |>
wenzelm@33671:     Local_Theory.notes (map (fn (name, ths) =>
wenzelm@30435:         ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "eqvt"),
berghofe@30087:           [Attrib.internal (K NominalThmDecls.eqvt_add)]), [(ths, [])]))
berghofe@30087:       (names ~~ transp thss)) |> snd
berghofe@22544:   end;
berghofe@22313: 
berghofe@22313: 
berghofe@22313: (* outer syntax *)
berghofe@22313: 
wenzelm@24867: val _ =
wenzelm@59936:   Outer_Syntax.local_theory_to_proof @{command_keyword nominal_inductive}
wenzelm@36960:     "prove equivariance and strong induction theorem for inductive predicate involving nominal datatypes"
wenzelm@46949:     (Parse.xname -- Scan.optional (@{keyword "avoids"} |-- Parse.and_list1 (Parse.name --
wenzelm@46949:       (@{keyword ":"} |-- Scan.repeat1 Parse.name))) [] >> (fn (name, avoids) =>
berghofe@30087:         prove_strong_ind name avoids));
berghofe@22313: 
wenzelm@24867: val _ =
wenzelm@59936:   Outer_Syntax.local_theory @{command_keyword equivariance}
wenzelm@46961:     "prove equivariance for inductive predicate involving nominal datatypes"
wenzelm@46949:     (Parse.xname -- Scan.optional (@{keyword "["} |-- Parse.list1 Parse.name --| @{keyword "]"}) [] >>
berghofe@30087:       (fn (name, atoms) => prove_eqvt name atoms));
berghofe@22530: 
berghofe@22313: end