src/HOL/Nominal/nominal_inductive.ML
author wenzelm
Tue Feb 10 14:48:26 2015 +0100 (2015-02-10)
changeset 59498 50b60f501b05
parent 59058 a78612c67ec0
child 59582 0fbed69ff081
permissions -rw-r--r--
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
occasionally clarified use of context;
     1 (*  Title:      HOL/Nominal/nominal_inductive.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3 
     4 Infrastructure for proving equivariance and strong induction theorems
     5 for inductive predicates involving nominal datatypes.
     6 *)
     7 
     8 signature NOMINAL_INDUCTIVE =
     9 sig
    10   val prove_strong_ind: string -> (string * string list) list -> local_theory -> Proof.state
    11   val prove_eqvt: string -> string list -> local_theory -> local_theory
    12 end
    13 
    14 structure NominalInductive : NOMINAL_INDUCTIVE =
    15 struct
    16 
    17 val inductive_forall_def = @{thm induct_forall_def};
    18 val inductive_atomize = @{thms induct_atomize};
    19 val inductive_rulify = @{thms induct_rulify};
    20 
    21 fun rulify_term thy = Raw_Simplifier.rewrite_term thy inductive_rulify [];
    22 
    23 fun atomize_conv ctxt =
    24   Raw_Simplifier.rewrite_cterm (true, false, false) (K (K NONE))
    25     (put_simpset HOL_basic_ss ctxt addsimps inductive_atomize);
    26 fun atomize_intr ctxt = Conv.fconv_rule (Conv.prems_conv ~1 (atomize_conv ctxt));
    27 fun atomize_induct ctxt = Conv.fconv_rule (Conv.prems_conv ~1
    28   (Conv.params_conv ~1 (K (Conv.prems_conv ~1 (atomize_conv ctxt))) ctxt));
    29 
    30 fun preds_of ps t = inter (op = o apsnd dest_Free) ps (Term.add_frees t []);
    31 
    32 val fresh_prod = @{thm fresh_prod};
    33 
    34 val perm_bool = mk_meta_eq @{thm perm_bool_def};
    35 val perm_boolI = @{thm perm_boolI};
    36 val (_, [perm_boolI_pi, _]) = Drule.strip_comb (snd (Thm.dest_comb
    37   (Drule.strip_imp_concl (cprop_of perm_boolI))));
    38 
    39 fun mk_perm_bool pi th = th RS Drule.cterm_instantiate
    40   [(perm_boolI_pi, pi)] perm_boolI;
    41 
    42 fun mk_perm_bool_simproc names = Simplifier.simproc_global_i
    43   (theory_of_thm perm_bool) "perm_bool" [@{term "perm pi x"}] (fn ctxt =>
    44     fn Const (@{const_name Nominal.perm}, _) $ _ $ t =>
    45          if member (op =) names (the_default "" (try (head_of #> dest_Const #> fst) t))
    46          then SOME perm_bool else NONE
    47      | _ => NONE);
    48 
    49 fun transp ([] :: _) = []
    50   | transp xs = map hd xs :: transp (map tl xs);
    51 
    52 fun add_binders thy i (t as (_ $ _)) bs = (case strip_comb t of
    53       (Const (s, T), ts) => (case strip_type T of
    54         (Ts, Type (tname, _)) =>
    55           (case NominalDatatype.get_nominal_datatype thy tname of
    56              NONE => fold (add_binders thy i) ts bs
    57            | SOME {descr, index, ...} => (case AList.lookup op =
    58                  (#3 (the (AList.lookup op = descr index))) s of
    59                NONE => fold (add_binders thy i) ts bs
    60              | SOME cargs => fst (fold (fn (xs, x) => fn (bs', cargs') =>
    61                  let val (cargs1, (u, _) :: cargs2) = chop (length xs) cargs'
    62                  in (add_binders thy i u
    63                    (fold (fn (u, T) =>
    64                       if exists (fn j => j < i) (loose_bnos u) then I
    65                       else insert (op aconv o apply2 fst)
    66                         (incr_boundvars (~i) u, T)) cargs1 bs'), cargs2)
    67                  end) cargs (bs, ts ~~ Ts))))
    68       | _ => fold (add_binders thy i) ts bs)
    69     | (u, ts) => add_binders thy i u (fold (add_binders thy i) ts bs))
    70   | add_binders thy i (Abs (_, _, t)) bs = add_binders thy (i + 1) t bs
    71   | add_binders thy i _ bs = bs;
    72 
    73 fun split_conj f names (Const (@{const_name HOL.conj}, _) $ p $ q) _ = (case head_of p of
    74       Const (name, _) =>
    75         if member (op =) names name then SOME (f p q) else NONE
    76     | _ => NONE)
    77   | split_conj _ _ _ _ = NONE;
    78 
    79 fun strip_all [] t = t
    80   | strip_all (_ :: xs) (Const (@{const_name All}, _) $ Abs (s, T, t)) = strip_all xs t;
    81 
    82 (*********************************************************************)
    83 (* maps  R ... & (ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t))  *)
    84 (* or    ALL pi_1 ... pi_n z. P z (pi_1 o ... o pi_n o t)            *)
    85 (* to    R ... & id (ALL z. P z (pi_1 o ... o pi_n o t))             *)
    86 (* or    id (ALL z. P z (pi_1 o ... o pi_n o t))                     *)
    87 (*                                                                   *)
    88 (* where "id" protects the subformula from simplification            *)
    89 (*********************************************************************)
    90 
    91 fun inst_conj_all names ps pis (Const (@{const_name HOL.conj}, _) $ p $ q) _ =
    92       (case head_of p of
    93          Const (name, _) =>
    94            if member (op =) names name then SOME (HOLogic.mk_conj (p,
    95              Const (@{const_name Fun.id}, HOLogic.boolT --> HOLogic.boolT) $
    96                (subst_bounds (pis, strip_all pis q))))
    97            else NONE
    98        | _ => NONE)
    99   | inst_conj_all names ps pis t u =
   100       if member (op aconv) ps (head_of u) then
   101         SOME (Const (@{const_name Fun.id}, HOLogic.boolT --> HOLogic.boolT) $
   102           (subst_bounds (pis, strip_all pis t)))
   103       else NONE
   104   | inst_conj_all _ _ _ _ _ = NONE;
   105 
   106 fun inst_conj_all_tac ctxt k = EVERY
   107   [TRY (EVERY [etac conjE 1, rtac conjI 1, atac 1]),
   108    REPEAT_DETERM_N k (etac allE 1),
   109    simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm id_apply}]) 1];
   110 
   111 fun map_term f t u = (case f t u of
   112       NONE => map_term' f t u | x => x)
   113 and map_term' f (t $ u) (t' $ u') = (case (map_term f t t', map_term f u u') of
   114       (NONE, NONE) => NONE
   115     | (SOME t'', NONE) => SOME (t'' $ u)
   116     | (NONE, SOME u'') => SOME (t $ u'')
   117     | (SOME t'', SOME u'') => SOME (t'' $ u''))
   118   | map_term' f (Abs (s, T, t)) (Abs (s', T', t')) = (case map_term f t t' of
   119       NONE => NONE
   120     | SOME t'' => SOME (Abs (s, T, t'')))
   121   | map_term' _ _ _ = NONE;
   122 
   123 (*********************************************************************)
   124 (*         Prove  F[f t]  from  F[t],  where F is monotone           *)
   125 (*********************************************************************)
   126 
   127 fun map_thm ctxt f tac monos opt th =
   128   let
   129     val prop = prop_of th;
   130     fun prove t =
   131       Goal.prove ctxt [] [] t (fn _ =>
   132         EVERY [cut_facts_tac [th] 1, etac rev_mp 1,
   133           REPEAT_DETERM (FIRSTGOAL (resolve_tac ctxt monos)),
   134           REPEAT_DETERM (rtac impI 1 THEN (atac 1 ORELSE tac))])
   135   in Option.map prove (map_term f prop (the_default prop opt)) end;
   136 
   137 val eta_contract_cterm = Thm.dest_arg o Thm.cprop_of o Thm.eta_conversion;
   138 
   139 fun first_order_matchs pats objs = Thm.first_order_match
   140   (eta_contract_cterm (Conjunction.mk_conjunction_balanced pats),
   141    eta_contract_cterm (Conjunction.mk_conjunction_balanced objs));
   142 
   143 fun first_order_mrs ths th = ths MRS
   144   Thm.instantiate (first_order_matchs (cprems_of th) (map cprop_of ths)) th;
   145 
   146 fun prove_strong_ind s avoids ctxt =
   147   let
   148     val thy = Proof_Context.theory_of ctxt;
   149     val ({names, ...}, {raw_induct, intrs, elims, ...}) =
   150       Inductive.the_inductive ctxt (Sign.intern_const thy s);
   151     val ind_params = Inductive.params_of raw_induct;
   152     val raw_induct = atomize_induct ctxt raw_induct;
   153     val elims = map (atomize_induct ctxt) elims;
   154     val monos = Inductive.get_monos ctxt;
   155     val eqvt_thms = NominalThmDecls.get_eqvt_thms ctxt;
   156     val _ = (case subtract (op =) (fold (Term.add_const_names o Thm.prop_of) eqvt_thms []) names of
   157         [] => ()
   158       | xs => error ("Missing equivariance theorem for predicate(s): " ^
   159           commas_quote xs));
   160     val induct_cases = map (fst o fst) (fst (Rule_Cases.get (the
   161       (Induct.lookup_inductP ctxt (hd names)))));
   162     val ([raw_induct'], ctxt') = Variable.import_terms false [prop_of raw_induct] ctxt;
   163     val concls = raw_induct' |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop |>
   164       HOLogic.dest_conj |> map (HOLogic.dest_imp ##> strip_comb);
   165     val ps = map (fst o snd) concls;
   166 
   167     val _ = (case duplicates (op = o apply2 fst) avoids of
   168         [] => ()
   169       | xs => error ("Duplicate case names: " ^ commas_quote (map fst xs)));
   170     val _ = avoids |> forall (fn (a, xs) => null (duplicates (op =) xs) orelse
   171       error ("Duplicate variable names for case " ^ quote a));
   172     val _ = (case subtract (op =) induct_cases (map fst avoids) of
   173         [] => ()
   174       | xs => error ("No such case(s) in inductive definition: " ^ commas_quote xs));
   175     val avoids' = if null induct_cases then replicate (length intrs) ("", [])
   176       else map (fn name =>
   177         (name, the_default [] (AList.lookup op = avoids name))) induct_cases;
   178     fun mk_avoids params (name, ps) =
   179       let val k = length params - 1
   180       in map (fn x => case find_index (equal x o fst) params of
   181           ~1 => error ("No such variable in case " ^ quote name ^
   182             " of inductive definition: " ^ quote x)
   183         | i => (Bound (k - i), snd (nth params i))) ps
   184       end;
   185 
   186     val prems = map (fn (prem, avoid) =>
   187       let
   188         val prems = map (incr_boundvars 1) (Logic.strip_assums_hyp prem);
   189         val concl = incr_boundvars 1 (Logic.strip_assums_concl prem);
   190         val params = Logic.strip_params prem
   191       in
   192         (params,
   193          fold (add_binders thy 0) (prems @ [concl]) [] @
   194            map (apfst (incr_boundvars 1)) (mk_avoids params avoid),
   195          prems, strip_comb (HOLogic.dest_Trueprop concl))
   196       end) (Logic.strip_imp_prems raw_induct' ~~ avoids');
   197 
   198     val atomTs = distinct op = (maps (map snd o #2) prems);
   199     val ind_sort = if null atomTs then @{sort type}
   200       else Sign.minimize_sort thy (Sign.certify_sort thy (map (fn T => Sign.intern_class thy
   201         ("fs_" ^ Long_Name.base_name (fst (dest_Type T)))) atomTs));
   202     val (fs_ctxt_tyname, _) = Name.variant "'n" (Variable.names_of ctxt');
   203     val ([fs_ctxt_name], ctxt'') = Variable.variant_fixes ["z"] ctxt';
   204     val fsT = TFree (fs_ctxt_tyname, ind_sort);
   205 
   206     val inductive_forall_def' = Drule.instantiate'
   207       [SOME (ctyp_of thy fsT)] [] inductive_forall_def;
   208 
   209     fun lift_pred' t (Free (s, T)) ts =
   210       list_comb (Free (s, fsT --> T), t :: ts);
   211     val lift_pred = lift_pred' (Bound 0);
   212 
   213     fun lift_prem (t as (f $ u)) =
   214           let val (p, ts) = strip_comb t
   215           in
   216             if member (op =) ps p then HOLogic.mk_induct_forall fsT $
   217               Abs ("z", fsT, lift_pred p (map (incr_boundvars 1) ts))
   218             else lift_prem f $ lift_prem u
   219           end
   220       | lift_prem (Abs (s, T, t)) = Abs (s, T, lift_prem t)
   221       | lift_prem t = t;
   222 
   223     fun mk_distinct [] = []
   224       | mk_distinct ((x, T) :: xs) = map_filter (fn (y, U) =>
   225           if T = U then SOME (HOLogic.mk_Trueprop
   226             (HOLogic.mk_not (HOLogic.eq_const T $ x $ y)))
   227           else NONE) xs @ mk_distinct xs;
   228 
   229     fun mk_fresh (x, T) = HOLogic.mk_Trueprop
   230       (NominalDatatype.fresh_const T fsT $ x $ Bound 0);
   231 
   232     val (prems', prems'') = split_list (map (fn (params, bvars, prems, (p, ts)) =>
   233       let
   234         val params' = params @ [("y", fsT)];
   235         val prem = Logic.list_implies
   236           (map mk_fresh bvars @ mk_distinct bvars @
   237            map (fn prem =>
   238              if null (preds_of ps prem) then prem
   239              else lift_prem prem) prems,
   240            HOLogic.mk_Trueprop (lift_pred p ts));
   241         val vs = map (Var o apfst (rpair 0)) (Term.rename_wrt_term prem params')
   242       in
   243         (Logic.list_all (params', prem), (rev vs, subst_bounds (vs, prem)))
   244       end) prems);
   245 
   246     val ind_vars =
   247       (Old_Datatype_Prop.indexify_names (replicate (length atomTs) "pi") ~~
   248        map NominalAtoms.mk_permT atomTs) @ [("z", fsT)];
   249     val ind_Ts = rev (map snd ind_vars);
   250 
   251     val concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   252       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
   253         HOLogic.list_all (ind_vars, lift_pred p
   254           (map (fold_rev (NominalDatatype.mk_perm ind_Ts)
   255             (map Bound (length atomTs downto 1))) ts)))) concls));
   256 
   257     val concl' = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   258       (map (fn (prem, (p, ts)) => HOLogic.mk_imp (prem,
   259         lift_pred' (Free (fs_ctxt_name, fsT)) p ts)) concls));
   260 
   261     val vc_compat = map (fn (params, bvars, prems, (p, ts)) =>
   262       map (fn q => Logic.list_all (params, incr_boundvars ~1 (Logic.list_implies
   263           (map_filter (fn prem =>
   264              if null (preds_of ps prem) then SOME prem
   265              else map_term (split_conj (K o I) names) prem prem) prems, q))))
   266         (mk_distinct bvars @
   267          maps (fn (t, T) => map (fn (u, U) => HOLogic.mk_Trueprop
   268            (NominalDatatype.fresh_const U T $ u $ t)) bvars)
   269              (ts ~~ binder_types (fastype_of p)))) prems;
   270 
   271     val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
   272     val pt2_atoms = map (fn aT => Global_Theory.get_thm thy
   273       ("pt_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "2")) atomTs;
   274     val eqvt_ss = simpset_of (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
   275       addsimps (eqvt_thms @ perm_pi_simp @ pt2_atoms)
   276       addsimprocs [mk_perm_bool_simproc [@{const_name Fun.id}],
   277         NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun]);
   278     val fresh_bij = Global_Theory.get_thms thy "fresh_bij";
   279     val perm_bij = Global_Theory.get_thms thy "perm_bij";
   280     val fs_atoms = map (fn aT => Global_Theory.get_thm thy
   281       ("fs_" ^ Long_Name.base_name (fst (dest_Type aT)) ^ "1")) atomTs;
   282     val exists_fresh' = Global_Theory.get_thms thy "exists_fresh'";
   283     val fresh_atm = Global_Theory.get_thms thy "fresh_atm";
   284     val swap_simps = Global_Theory.get_thms thy "swap_simps";
   285     val perm_fresh_fresh = Global_Theory.get_thms thy "perm_fresh_fresh";
   286 
   287     fun obtain_fresh_name ts T (freshs1, freshs2, ctxt) =
   288       let
   289         (** protect terms to avoid that fresh_prod interferes with  **)
   290         (** pairs used in introduction rules of inductive predicate **)
   291         fun protect t =
   292           let val T = fastype_of t in Const (@{const_name Fun.id}, T --> T) $ t end;
   293         val p = foldr1 HOLogic.mk_prod (map protect ts @ freshs1);
   294         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
   295             (HOLogic.exists_const T $ Abs ("x", T,
   296               NominalDatatype.fresh_const T (fastype_of p) $
   297                 Bound 0 $ p)))
   298           (fn _ => EVERY
   299             [resolve_tac ctxt exists_fresh' 1,
   300              resolve_tac ctxt fs_atoms 1]);
   301         val (([(_, cx)], ths), ctxt') = Obtain.result
   302           (fn ctxt' => EVERY
   303             [etac exE 1,
   304              full_simp_tac (put_simpset HOL_ss ctxt' addsimps (fresh_prod :: fresh_atm)) 1,
   305              full_simp_tac (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]) 1,
   306              REPEAT (etac conjE 1)])
   307           [ex] ctxt
   308       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
   309 
   310     fun mk_ind_proof ctxt' thss =
   311       Goal.prove ctxt' [] prems' concl' (fn {prems = ihyps, context = ctxt} =>
   312         let val th = Goal.prove ctxt [] [] concl (fn {context, ...} =>
   313           rtac raw_induct 1 THEN
   314           EVERY (maps (fn ((((_, bvars, oprems, _), vc_compat_ths), ihyp), (vs, ihypt)) =>
   315             [REPEAT (rtac allI 1), simp_tac (put_simpset eqvt_ss context) 1,
   316              SUBPROOF (fn {prems = gprems, params, concl, context = ctxt', ...} =>
   317                let
   318                  val (params', (pis, z)) =
   319                    chop (length params - length atomTs - 1) (map (term_of o #2) params) ||>
   320                    split_last;
   321                  val bvars' = map
   322                    (fn (Bound i, T) => (nth params' (length params' - i), T)
   323                      | (t, T) => (t, T)) bvars;
   324                  val pi_bvars = map (fn (t, _) =>
   325                    fold_rev (NominalDatatype.mk_perm []) pis t) bvars';
   326                  val (P, ts) = strip_comb (HOLogic.dest_Trueprop (term_of concl));
   327                  val (freshs1, freshs2, ctxt'') = fold
   328                    (obtain_fresh_name (ts @ pi_bvars))
   329                    (map snd bvars') ([], [], ctxt');
   330                  val freshs2' = NominalDatatype.mk_not_sym freshs2;
   331                  val pis' = map NominalDatatype.perm_of_pair (pi_bvars ~~ freshs1);
   332                  fun concat_perm pi1 pi2 =
   333                    let val T = fastype_of pi1
   334                    in if T = fastype_of pi2 then
   335                        Const (@{const_name append}, T --> T --> T) $ pi1 $ pi2
   336                      else pi2
   337                    end;
   338                  val pis'' = fold (concat_perm #> map) pis' pis;
   339                  val env = Pattern.first_order_match thy (ihypt, prop_of ihyp)
   340                    (Vartab.empty, Vartab.empty);
   341                  val ihyp' = Thm.instantiate ([], map (apply2 (cterm_of thy))
   342                    (map (Envir.subst_term env) vs ~~
   343                     map (fold_rev (NominalDatatype.mk_perm [])
   344                       (rev pis' @ pis)) params' @ [z])) ihyp;
   345                  fun mk_pi th =
   346                    Simplifier.simplify (put_simpset HOL_basic_ss ctxt' addsimps [@{thm id_apply}]
   347                        addsimprocs [NominalDatatype.perm_simproc])
   348                      (Simplifier.simplify (put_simpset eqvt_ss ctxt')
   349                        (fold_rev (mk_perm_bool o cterm_of thy)
   350                          (rev pis' @ pis) th));
   351                  val (gprems1, gprems2) = split_list
   352                    (map (fn (th, t) =>
   353                       if null (preds_of ps t) then (SOME th, mk_pi th)
   354                       else
   355                         (map_thm ctxt' (split_conj (K o I) names)
   356                            (etac conjunct1 1) monos NONE th,
   357                          mk_pi (the (map_thm ctxt' (inst_conj_all names ps (rev pis''))
   358                            (inst_conj_all_tac ctxt' (length pis'')) monos (SOME t) th))))
   359                       (gprems ~~ oprems)) |>> map_filter I;
   360                  val vc_compat_ths' = map (fn th =>
   361                    let
   362                      val th' = first_order_mrs gprems1 th;
   363                      val (bop, lhs, rhs) = (case concl_of th' of
   364                          _ $ (fresh $ lhs $ rhs) =>
   365                            (fn t => fn u => fresh $ t $ u, lhs, rhs)
   366                        | _ $ (_ $ (_ $ lhs $ rhs)) =>
   367                            (curry (HOLogic.mk_not o HOLogic.mk_eq), lhs, rhs));
   368                      val th'' = Goal.prove ctxt'' [] [] (HOLogic.mk_Trueprop
   369                          (bop (fold_rev (NominalDatatype.mk_perm []) pis lhs)
   370                             (fold_rev (NominalDatatype.mk_perm []) pis rhs)))
   371                        (fn _ => simp_tac (put_simpset HOL_basic_ss ctxt'' addsimps
   372                           (fresh_bij @ perm_bij)) 1 THEN rtac th' 1)
   373                    in Simplifier.simplify (put_simpset eqvt_ss ctxt'' addsimps fresh_atm) th'' end)
   374                      vc_compat_ths;
   375                  val vc_compat_ths'' = NominalDatatype.mk_not_sym vc_compat_ths';
   376                  (** Since swap_simps simplifies (pi :: 'a prm) o (x :: 'b) to x **)
   377                  (** we have to pre-simplify the rewrite rules                   **)
   378                  val swap_simps_simpset = put_simpset HOL_ss ctxt'' addsimps swap_simps @
   379                     map (Simplifier.simplify (put_simpset HOL_ss ctxt'' addsimps swap_simps))
   380                       (vc_compat_ths'' @ freshs2');
   381                  val th = Goal.prove ctxt'' [] []
   382                    (HOLogic.mk_Trueprop (list_comb (P $ hd ts,
   383                      map (fold (NominalDatatype.mk_perm []) pis') (tl ts))))
   384                    (fn _ => EVERY ([simp_tac (put_simpset eqvt_ss ctxt'') 1, rtac ihyp' 1,
   385                      REPEAT_DETERM_N (nprems_of ihyp - length gprems)
   386                        (simp_tac swap_simps_simpset 1),
   387                      REPEAT_DETERM_N (length gprems)
   388                        (simp_tac (put_simpset HOL_basic_ss ctxt''
   389                           addsimps [inductive_forall_def']
   390                           addsimprocs [NominalDatatype.perm_simproc]) 1 THEN
   391                         resolve_tac ctxt'' gprems2 1)]));
   392                  val final = Goal.prove ctxt'' [] [] (term_of concl)
   393                    (fn _ => cut_facts_tac [th] 1 THEN full_simp_tac (put_simpset HOL_ss ctxt''
   394                      addsimps vc_compat_ths'' @ freshs2' @
   395                        perm_fresh_fresh @ fresh_atm) 1);
   396                  val final' = Proof_Context.export ctxt'' ctxt' [final];
   397                in resolve_tac ctxt' final' 1 end) context 1])
   398                  (prems ~~ thss ~~ ihyps ~~ prems'')))
   399         in
   400           cut_facts_tac [th] 1 THEN REPEAT (etac conjE 1) THEN
   401           REPEAT (REPEAT (resolve_tac ctxt [conjI, impI] 1) THEN
   402             etac impE 1 THEN atac 1 THEN REPEAT (etac @{thm allE_Nil} 1) THEN
   403             asm_full_simp_tac ctxt 1)
   404         end) |> singleton (Proof_Context.export ctxt' ctxt);
   405 
   406     (** strong case analysis rule **)
   407 
   408     val cases_prems = map (fn ((name, avoids), rule) =>
   409       let
   410         val ([rule'], ctxt') = Variable.import_terms false [prop_of rule] ctxt;
   411         val prem :: prems = Logic.strip_imp_prems rule';
   412         val concl = Logic.strip_imp_concl rule'
   413       in
   414         (prem,
   415          List.drop (snd (strip_comb (HOLogic.dest_Trueprop prem)), length ind_params),
   416          concl,
   417          fold_map (fn (prem, (_, avoid)) => fn ctxt =>
   418            let
   419              val prems = Logic.strip_assums_hyp prem;
   420              val params = Logic.strip_params prem;
   421              val bnds = fold (add_binders thy 0) prems [] @ mk_avoids params avoid;
   422              fun mk_subst (p as (s, T)) (i, j, ctxt, ps, qs, is, ts) =
   423                if member (op = o apsnd fst) bnds (Bound i) then
   424                  let
   425                    val ([s'], ctxt') = Variable.variant_fixes [s] ctxt;
   426                    val t = Free (s', T)
   427                  in (i + 1, j, ctxt', ps, (t, T) :: qs, i :: is, t :: ts) end
   428                else (i + 1, j + 1, ctxt, p :: ps, qs, is, Bound j :: ts);
   429              val (_, _, ctxt', ps, qs, is, ts) = fold_rev mk_subst params
   430                (0, 0, ctxt, [], [], [], [])
   431            in
   432              ((ps, qs, is, map (curry subst_bounds (rev ts)) prems), ctxt')
   433            end) (prems ~~ avoids) ctxt')
   434       end)
   435         (Inductive.partition_rules' raw_induct (intrs ~~ avoids') ~~
   436          elims);
   437 
   438     val cases_prems' =
   439       map (fn (prem, args, concl, (prems, _)) =>
   440         let
   441           fun mk_prem (ps, [], _, prems) =
   442                 Logic.list_all (ps, Logic.list_implies (prems, concl))
   443             | mk_prem (ps, qs, _, prems) =
   444                 Logic.list_all (ps, Logic.mk_implies
   445                   (Logic.list_implies
   446                     (mk_distinct qs @
   447                      maps (fn (t, T) => map (fn u => HOLogic.mk_Trueprop
   448                       (NominalDatatype.fresh_const T (fastype_of u) $ t $ u))
   449                         args) qs,
   450                      HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   451                        (map HOLogic.dest_Trueprop prems))),
   452                    concl))
   453           in map mk_prem prems end) cases_prems;
   454 
   455     val cases_eqvt_simpset = put_simpset HOL_ss (Proof_Context.init_global thy)
   456       addsimps eqvt_thms @ swap_simps @ perm_pi_simp
   457       addsimprocs [NominalPermeq.perm_simproc_app,
   458         NominalPermeq.perm_simproc_fun];
   459 
   460     val simp_fresh_atm = map
   461       (Simplifier.simplify (put_simpset HOL_basic_ss (Proof_Context.init_global thy)
   462         addsimps fresh_atm));
   463 
   464     fun mk_cases_proof ((((name, thss), elim), (prem, args, concl, (prems, ctxt'))),
   465         prems') =
   466       (name, Goal.prove ctxt' [] (prem :: prems') concl
   467         (fn {prems = hyp :: hyps, context = ctxt1} =>
   468         EVERY (rtac (hyp RS elim) 1 ::
   469           map (fn (((_, vc_compat_ths), case_hyp), (_, qs, is, _)) =>
   470             SUBPROOF (fn {prems = case_hyps, params, context = ctxt2, concl, ...} =>
   471               if null qs then
   472                 rtac (first_order_mrs case_hyps case_hyp) 1
   473               else
   474                 let
   475                   val params' = map (term_of o #2 o nth (rev params)) is;
   476                   val tab = params' ~~ map fst qs;
   477                   val (hyps1, hyps2) = chop (length args) case_hyps;
   478                   (* turns a = t and [x1 # t, ..., xn # t] *)
   479                   (* into [x1 # a, ..., xn # a]            *)
   480                   fun inst_fresh th' ths =
   481                     let val (ths1, ths2) = chop (length qs) ths
   482                     in
   483                       (map (fn th =>
   484                          let
   485                            val (cf, ct) =
   486                              Thm.dest_comb (Thm.dest_arg (cprop_of th));
   487                            val arg_cong' = Drule.instantiate'
   488                              [SOME (ctyp_of_term ct)]
   489                              [NONE, SOME ct, SOME cf] (arg_cong RS iffD2);
   490                            val inst = Thm.first_order_match (ct,
   491                              Thm.dest_arg (Thm.dest_arg (cprop_of th')))
   492                          in [th', th] MRS Thm.instantiate inst arg_cong'
   493                          end) ths1,
   494                        ths2)
   495                     end;
   496                   val (vc_compat_ths1, vc_compat_ths2) =
   497                     chop (length vc_compat_ths - length args * length qs)
   498                       (map (first_order_mrs hyps2) vc_compat_ths);
   499                   val vc_compat_ths' =
   500                     NominalDatatype.mk_not_sym vc_compat_ths1 @
   501                     flat (fst (fold_map inst_fresh hyps1 vc_compat_ths2));
   502                   val (freshs1, freshs2, ctxt3) = fold
   503                     (obtain_fresh_name (args @ map fst qs @ params'))
   504                     (map snd qs) ([], [], ctxt2);
   505                   val freshs2' = NominalDatatype.mk_not_sym freshs2;
   506                   val pis = map (NominalDatatype.perm_of_pair)
   507                     ((freshs1 ~~ map fst qs) @ (params' ~~ freshs1));
   508                   val mk_pis = fold_rev mk_perm_bool (map (cterm_of thy) pis);
   509                   val obj = cterm_of thy (foldr1 HOLogic.mk_conj (map (map_aterms
   510                      (fn x as Free _ =>
   511                            if member (op =) args x then x
   512                            else (case AList.lookup op = tab x of
   513                              SOME y => y
   514                            | NONE => fold_rev (NominalDatatype.mk_perm []) pis x)
   515                        | x => x) o HOLogic.dest_Trueprop o prop_of) case_hyps));
   516                   val inst = Thm.first_order_match (Thm.dest_arg
   517                     (Drule.strip_imp_concl (hd (cprems_of case_hyp))), obj);
   518                   val th = Goal.prove ctxt3 [] [] (term_of concl)
   519                     (fn {context = ctxt4, ...} =>
   520                        rtac (Thm.instantiate inst case_hyp) 1 THEN
   521                        SUBPROOF (fn {prems = fresh_hyps, ...} =>
   522                          let
   523                            val fresh_hyps' = NominalDatatype.mk_not_sym fresh_hyps;
   524                            val case_simpset = cases_eqvt_simpset addsimps freshs2' @
   525                              simp_fresh_atm (vc_compat_ths' @ fresh_hyps');
   526                            val fresh_fresh_simpset = case_simpset addsimps perm_fresh_fresh;
   527                            val hyps1' = map
   528                              (mk_pis #> Simplifier.simplify fresh_fresh_simpset) hyps1;
   529                            val hyps2' = map
   530                              (mk_pis #> Simplifier.simplify case_simpset) hyps2;
   531                            val case_hyps' = hyps1' @ hyps2'
   532                          in
   533                            simp_tac case_simpset 1 THEN
   534                            REPEAT_DETERM (TRY (rtac conjI 1) THEN
   535                              resolve_tac ctxt4 case_hyps' 1)
   536                          end) ctxt4 1)
   537                   val final = Proof_Context.export ctxt3 ctxt2 [th]
   538                 in resolve_tac ctxt2 final 1 end) ctxt1 1)
   539                   (thss ~~ hyps ~~ prems))) |>
   540                   singleton (Proof_Context.export ctxt' ctxt))
   541 
   542   in
   543     ctxt'' |>
   544     Proof.theorem NONE (fn thss => fn ctxt => (* FIXME ctxt/ctxt' should be called lthy/lthy' *)
   545       let
   546         val rec_name = space_implode "_" (map Long_Name.base_name names);
   547         val rec_qualified = Binding.qualify false rec_name;
   548         val ind_case_names = Rule_Cases.case_names induct_cases;
   549         val induct_cases' = Inductive.partition_rules' raw_induct
   550           (intrs ~~ induct_cases); 
   551         val thss' = map (map (atomize_intr ctxt)) thss;
   552         val thsss = Inductive.partition_rules' raw_induct (intrs ~~ thss');
   553         val strong_raw_induct =
   554           mk_ind_proof ctxt thss' |> Inductive.rulify ctxt;
   555         val strong_cases = map (mk_cases_proof ##> Inductive.rulify ctxt)
   556           (thsss ~~ elims ~~ cases_prems ~~ cases_prems');
   557         val strong_induct_atts =
   558           map (Attrib.internal o K)
   559             [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of strong_raw_induct))];
   560         val strong_induct =
   561           if length names > 1 then strong_raw_induct
   562           else strong_raw_induct RSN (2, rev_mp);
   563         val ((_, [strong_induct']), ctxt') = ctxt |> Local_Theory.note
   564           ((rec_qualified (Binding.name "strong_induct"), strong_induct_atts), [strong_induct]);
   565         val strong_inducts =
   566           Project_Rule.projects ctxt (1 upto length names) strong_induct';
   567       in
   568         ctxt' |>
   569         Local_Theory.notes
   570           [((rec_qualified (Binding.name "strong_inducts"), []),
   571             strong_inducts |> map (fn th => ([th],
   572               [Attrib.internal (K ind_case_names),
   573                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th)))])))] |> snd |>
   574         Local_Theory.notes (map (fn ((name, elim), (_, cases)) =>
   575             ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "strong_cases"),
   576               [Attrib.internal (K (Rule_Cases.case_names (map snd cases))),
   577                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim)))]), [([elim], [])]))
   578           (strong_cases ~~ induct_cases')) |> snd
   579       end)
   580       (map (map (rulify_term thy #> rpair [])) vc_compat)
   581   end;
   582 
   583 fun prove_eqvt s xatoms ctxt =  (* FIXME ctxt should be called lthy *)
   584   let
   585     val thy = Proof_Context.theory_of ctxt;
   586     val ({names, ...}, {raw_induct, intrs, elims, ...}) =
   587       Inductive.the_inductive ctxt (Sign.intern_const thy s);
   588     val raw_induct = atomize_induct ctxt raw_induct;
   589     val elims = map (atomize_induct ctxt) elims;
   590     val intrs = map (atomize_intr ctxt) intrs;
   591     val monos = Inductive.get_monos ctxt;
   592     val intrs' = Inductive.unpartition_rules intrs
   593       (map (fn (((s, ths), (_, k)), th) =>
   594            (s, ths ~~ Inductive.infer_intro_vars th k ths))
   595          (Inductive.partition_rules raw_induct intrs ~~
   596           Inductive.arities_of raw_induct ~~ elims));
   597     val k = length (Inductive.params_of raw_induct);
   598     val atoms' = NominalAtoms.atoms_of thy;
   599     val atoms =
   600       if null xatoms then atoms' else
   601       let val atoms = map (Sign.intern_type thy) xatoms
   602       in
   603         (case duplicates op = atoms of
   604              [] => ()
   605            | xs => error ("Duplicate atoms: " ^ commas xs);
   606          case subtract (op =) atoms' atoms of
   607              [] => ()
   608            | xs => error ("No such atoms: " ^ commas xs);
   609          atoms)
   610       end;
   611     val perm_pi_simp = Global_Theory.get_thms thy "perm_pi_simp";
   612     val (([t], [pi]), ctxt') = ctxt |>
   613       Variable.import_terms false [concl_of raw_induct] ||>>
   614       Variable.variant_fixes ["pi"];
   615     val eqvt_simpset = put_simpset HOL_basic_ss ctxt' addsimps
   616       (NominalThmDecls.get_eqvt_thms ctxt' @ perm_pi_simp) addsimprocs
   617       [mk_perm_bool_simproc names,
   618        NominalPermeq.perm_simproc_app, NominalPermeq.perm_simproc_fun];
   619     val ps = map (fst o HOLogic.dest_imp)
   620       (HOLogic.dest_conj (HOLogic.dest_Trueprop t));
   621     fun eqvt_tac pi (intr, vs) st =
   622       let
   623         fun eqvt_err s =
   624           let val ([t], ctxt'') = Variable.import_terms true [prop_of intr] ctxt'
   625           in error ("Could not prove equivariance for introduction rule\n" ^
   626             Syntax.string_of_term ctxt'' t ^ "\n" ^ s)
   627           end;
   628         val res = SUBPROOF (fn {context = ctxt'', prems, params, ...} =>
   629           let
   630             val prems' = map (fn th => the_default th (map_thm ctxt''
   631               (split_conj (K I) names) (etac conjunct2 1) monos NONE th)) prems;
   632             val prems'' = map (fn th => Simplifier.simplify eqvt_simpset
   633               (mk_perm_bool (cterm_of thy pi) th)) prems';
   634             val intr' = Drule.cterm_instantiate (map (cterm_of thy) vs ~~
   635                map (cterm_of thy o NominalDatatype.mk_perm [] pi o term_of o #2) params)
   636                intr
   637           in (rtac intr' THEN_ALL_NEW (TRY o resolve_tac ctxt'' prems'')) 1
   638           end) ctxt' 1 st
   639       in
   640         case (Seq.pull res handle THM (s, _, _) => eqvt_err s) of
   641           NONE => eqvt_err ("Rule does not match goal\n" ^
   642             Syntax.string_of_term ctxt' (hd (prems_of st)))
   643         | SOME (th, _) => Seq.single th
   644       end;
   645     val thss = map (fn atom =>
   646       let val pi' = Free (pi, NominalAtoms.mk_permT (Type (atom, [])))
   647       in map (fn th => zero_var_indexes (th RS mp))
   648         (Old_Datatype_Aux.split_conj_thm (Goal.prove ctxt' [] []
   649           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map (fn p =>
   650             let
   651               val (h, ts) = strip_comb p;
   652               val (ts1, ts2) = chop k ts
   653             in
   654               HOLogic.mk_imp (p, list_comb (h, ts1 @
   655                 map (NominalDatatype.mk_perm [] pi') ts2))
   656             end) ps)))
   657           (fn _ => EVERY (rtac raw_induct 1 :: map (fn intr_vs =>
   658               full_simp_tac eqvt_simpset 1 THEN
   659               eqvt_tac pi' intr_vs) intrs')) |>
   660           singleton (Proof_Context.export ctxt' ctxt)))
   661       end) atoms
   662   in
   663     ctxt |>
   664     Local_Theory.notes (map (fn (name, ths) =>
   665         ((Binding.qualified_name (Long_Name.qualify (Long_Name.base_name name) "eqvt"),
   666           [Attrib.internal (K NominalThmDecls.eqvt_add)]), [(ths, [])]))
   667       (names ~~ transp thss)) |> snd
   668   end;
   669 
   670 
   671 (* outer syntax *)
   672 
   673 val _ =
   674   Outer_Syntax.local_theory_to_proof @{command_spec "nominal_inductive"}
   675     "prove equivariance and strong induction theorem for inductive predicate involving nominal datatypes"
   676     (Parse.xname -- Scan.optional (@{keyword "avoids"} |-- Parse.and_list1 (Parse.name --
   677       (@{keyword ":"} |-- Scan.repeat1 Parse.name))) [] >> (fn (name, avoids) =>
   678         prove_strong_ind name avoids));
   679 
   680 val _ =
   681   Outer_Syntax.local_theory @{command_spec "equivariance"}
   682     "prove equivariance for inductive predicate involving nominal datatypes"
   683     (Parse.xname -- Scan.optional (@{keyword "["} |-- Parse.list1 Parse.name --| @{keyword "]"}) [] >>
   684       (fn (name, atoms) => prove_eqvt name atoms));
   685 
   686 end