src/HOLCF/Tools/Domain/domain_theorems.ML

     |> Sign.parent_path
     |> pair (iso_rews @ when_rews @ con_rews @ sel_rews @ dis_rews @
         pat_rews @ dist_les @ dist_eqs)
 end; (* let *)
 
-fun prove_coinduction
+(******************************************************************************)
+(****************************** induction rules *******************************)
+(******************************************************************************)
+
+fun prove_induction
     (comp_dnam, eqs : eq list)
     (take_lemmas : thm list)
-    (thy : theory) : thm * theory =
+    (axs_reach : thm list)
+    (take_rews : thm list)
+    (thy : theory) =
 let
-
-val dnames = map (fst o fst) eqs;
-val comp_dname = Sign.full_bname thy comp_dnam;
-fun dc_take dn = %%:(dn^"_take");
-val x_name = idx_name dnames "x"; 
-val n_eqs = length eqs;
-
-val take_rews =
-    maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
-
-(* ----- define bisimulation predicate -------------------------------------- *)
-
-local
-  open HOLCF_Library
-  val dtypes  = map (Type o fst) eqs;
-  val relprod = mk_tupleT (map (fn tp => tp --> tp --> boolT) dtypes);
-  val bisim_bind = Binding.name (comp_dnam ^ "_bisim");
-  val bisim_type = relprod --> boolT;
-in
-  val (bisim_const, thy) =
-      Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
-end;
-
-local
-
-  fun legacy_infer_term thy t =
-      singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t);
-  fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t);
-  fun infer_props thy = map (apsnd (legacy_infer_prop thy));
-  fun add_defs_i x = PureThy.add_defs false (map Thm.no_attributes x);
-  fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;
-
-  val comp_dname = Sign.full_bname thy comp_dnam;
   val dnames = map (fst o fst) eqs;
+  val conss  = map  snd        eqs;
+  fun dc_take dn = %%:(dn^"_take");
   val x_name = idx_name dnames "x"; 
-
-  fun one_con (con, args) =
-    let
-      val nonrec_args = filter_out is_rec args;
-      val    rec_args = filter is_rec args;
-      val    recs_cnt = length rec_args;
-      val allargs     = nonrec_args @ rec_args
-                        @ map (upd_vname (fn s=> s^"'")) rec_args;
-      val allvns      = map vname allargs;
-      fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg;
-      val vns1        = map (vname_arg "" ) args;
-      val vns2        = map (vname_arg "'") args;
-      val allargs_cnt = length nonrec_args + 2*recs_cnt;
-      val rec_idxs    = (recs_cnt-1) downto 0;
-      val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg)
-                                             (allargs~~((allargs_cnt-1) downto 0)));
-      fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $ 
-                              Bound (2*recs_cnt-i) $ Bound (recs_cnt-i);
-      val capps =
-          List.foldr
-            mk_conj
-            (mk_conj(
-             Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1),
-             Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2)))
-            (mapn rel_app 1 rec_args);
-    in
-      List.foldr
-        mk_ex
-        (Library.foldr mk_conj
-                       (map (defined o Bound) nonlazy_idxs,capps)) allvns
-    end;
-  fun one_comp n (_,cons) =
-      mk_all (x_name(n+1),
-      mk_all (x_name(n+1)^"'",
-      mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0,
-      foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
-                      ::map one_con cons))));
-  val bisim_eqn =
-      %%:(comp_dname^"_bisim") ==
-         mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs));
-
-in
-  val ([ax_bisim_def], thy) =
-      thy
-        |> Sign.add_path comp_dnam
-        |> add_defs_infer [(Binding.name "bisim_def", bisim_eqn)]
-        ||> Sign.parent_path;
-end; (* local *)
-
-(* ----- theorem concerning coinduction ------------------------------------- *)
-
-local
+  val P_name = idx_name dnames "P";
   val pg = pg' thy;
-  val xs = mapn (fn n => K (x_name n)) 1 dnames;
-  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
-  val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
-  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
-  val _ = trace " Proving coind_lemma...";
-  val coind_lemma =
-    let
-      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
-      fun mk_eqn n dn =
-        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
-        (dc_take dn $ %:"n" ` bnd_arg n 1);
-      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
-      val goal =
-        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
-          Library.foldr mk_all2 (xs,
-            Library.foldr mk_imp (mapn mk_prj 0 dnames,
-              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
-      fun x_tacs ctxt n x = [
-        rotate_tac (n+1) 1,
-        etac all2E 1,
-        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
-        TRY (safe_tac HOL_cs),
-        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
-      fun tacs ctxt = [
-        rtac impI 1,
-        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
-        simp_tac take_ss 1,
-        safe_tac HOL_cs] @
-        flat (mapn (x_tacs ctxt) 0 xs);
-    in pg [ax_bisim_def] goal tacs end;
-in
-  val _ = trace " Proving coind...";
-  val coind = 
-    let
-      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
-      fun mk_eqn x = %:x === %:(x^"'");
-      val goal =
-        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
-          Logic.list_implies (mapn mk_prj 0 xs,
-            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
-      val tacs =
-        TRY (safe_tac HOL_cs) ::
-        maps (fn take_lemma => [
-          rtac take_lemma 1,
-          cut_facts_tac [coind_lemma] 1,
-          fast_tac HOL_cs 1])
-        take_lemmas;
-    in pg [] goal (K tacs) end;
-end; (* local *)
-
-in
-  (coind, thy)
-end;
-
-fun comp_theorems (comp_dnam, eqs: eq list) thy =
-let
-val map_tab = Domain_Take_Proofs.get_map_tab thy;
-
-val dnames = map (fst o fst) eqs;
-val conss  = map  snd        eqs;
-val comp_dname = Sign.full_bname thy comp_dnam;
-
-val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
-
-val pg = pg' thy;
-
-(* ----- getting the composite axiom and definitions ------------------------ *)
-
-local
-  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
-in
-  val axs_take_def   = map (ga "take_def"  ) dnames;
-  val axs_chain_take = map (ga "chain_take") dnames;
-  val axs_lub_take   = map (ga "lub_take"  ) dnames;
-  val axs_finite_def = map (ga "finite_def") dnames;
-end;
-
-local
-  fun gt  s dn = PureThy.get_thm  thy (dn ^ "." ^ s);
-  fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
-in
-  val cases = map (gt  "casedist" ) dnames;
-  val con_rews  = maps (gts "con_rews" ) dnames;
-end;
-
-fun dc_take dn = %%:(dn^"_take");
-val x_name = idx_name dnames "x"; 
-val P_name = idx_name dnames "P";
-val n_eqs = length eqs;
-
-(* ----- theorems concerning finite approximation and finite induction ------ *)
-
-val take_rews =
-    maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
-
-local
+
+  local
+    fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
+    fun gts s dn = PureThy.get_thms thy (dn ^ "." ^ s);
+  in
+    val axs_chain_take = map (ga "chain_take") dnames;
+    val axs_finite_def = map (ga "finite_def") dnames;
+    val cases = map (ga  "casedist" ) dnames;
+    val con_rews  = maps (gts "con_rews" ) dnames;
+  end;
+
   fun one_con p (con, args) =
     let
       val P_names = map P_name (1 upto (length dnames));
       val vns = Name.variant_list P_names (map vname args);
       val nonlazy_vns = map snd (filter_out (is_lazy o fst) (args ~~ vns));

   in
     val n__eqs = mapn (fn n => fn (_,cons) => (n,cons)) 0 eqs;
     val is_emptys = map warn n__eqs;
     val is_finite = forall (not o lazy_rec_to [] false) n__eqs;
   end;
-in (* local *)
   val _ = trace " Proving finite_ind...";
   val finite_ind =
     let
       fun concf n dn = %:(P_name n) $ (dc_take dn $ %:"n" `%(x_name n));
       val goal = ind_term concf;

           tacs1 @ maps cases_tacs (conss ~~ cases)
         end;
     in pg'' thy [] goal tacf
        handle ERROR _ => (warning "Proof of finite_ind failed."; TrueI)
     end;
-
-  val _ = trace " Proving take_lemmas...";
-  val take_lemmas =
-    let
-      fun take_lemma (ax_chain_take, ax_lub_take) =
-        Drule.export_without_context
-        (@{thm lub_ID_take_lemma} OF [ax_chain_take, ax_lub_take]);
-    in map take_lemma (axs_chain_take ~~ axs_lub_take) end;
-
-  val axs_reach =
-    let
-      fun reach (ax_chain_take, ax_lub_take) =
-        Drule.export_without_context
-        (@{thm lub_ID_reach} OF [ax_chain_take, ax_lub_take]);
-    in map reach (axs_chain_take ~~ axs_lub_take) end;
 
 (* ----- theorems concerning finiteness and induction ----------------------- *)
 
   val global_ctxt = ProofContext.init thy;
 

       handle THM _ =>
              (warning "Induction proofs failed (THM raised)."; ([], TrueI))
            | ERROR _ =>
              (warning "Cannot prove induction rule"; ([], TrueI));
 
-end; (* local *)
-
-val (coind, thy) = prove_coinduction (comp_dnam, eqs) take_lemmas thy;
-
 val inducts = Project_Rule.projections (ProofContext.init thy) ind;
 fun ind_rule (dname, rule) = ((Binding.empty, [rule]), [Induct.induct_type dname]);
 val induct_failed = (Thm.prop_of ind = Thm.prop_of TrueI);
+
+in thy |> Sign.add_path comp_dnam
+       |> snd o PureThy.add_thmss [
+           ((Binding.name "finites"    , finites     ), []),
+           ((Binding.name "finite_ind" , [finite_ind]), []),
+           ((Binding.name "ind"        , [ind]       ), [])]
+       |> (if induct_failed then I
+           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
+       |> Sign.parent_path
+end; (* prove_induction *)
+
+(******************************************************************************)
+(************************ bisimulation and coinduction ************************)
+(******************************************************************************)
+
+fun prove_coinduction
+    (comp_dnam, eqs : eq list)
+    (take_lemmas : thm list)
+    (thy : theory) : thm * theory =
+let
+
+val dnames = map (fst o fst) eqs;
+val comp_dname = Sign.full_bname thy comp_dnam;
+fun dc_take dn = %%:(dn^"_take");
+val x_name = idx_name dnames "x"; 
+val n_eqs = length eqs;
+
+val take_rews =
+    maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
+
+(* ----- define bisimulation predicate -------------------------------------- *)
+
+local
+  open HOLCF_Library
+  val dtypes  = map (Type o fst) eqs;
+  val relprod = mk_tupleT (map (fn tp => tp --> tp --> boolT) dtypes);
+  val bisim_bind = Binding.name (comp_dnam ^ "_bisim");
+  val bisim_type = relprod --> boolT;
+in
+  val (bisim_const, thy) =
+      Sign.declare_const ((bisim_bind, bisim_type), NoSyn) thy;
+end;
+
+local
+
+  fun legacy_infer_term thy t =
+      singleton (Syntax.check_terms (ProofContext.init thy)) (Sign.intern_term thy t);
+  fun legacy_infer_prop thy t = legacy_infer_term thy (TypeInfer.constrain propT t);
+  fun infer_props thy = map (apsnd (legacy_infer_prop thy));
+  fun add_defs_i x = PureThy.add_defs false (map Thm.no_attributes x);
+  fun add_defs_infer defs thy = add_defs_i (infer_props thy defs) thy;
+
+  val comp_dname = Sign.full_bname thy comp_dnam;
+  val dnames = map (fst o fst) eqs;
+  val x_name = idx_name dnames "x"; 
+
+  fun one_con (con, args) =
+    let
+      val nonrec_args = filter_out is_rec args;
+      val    rec_args = filter is_rec args;
+      val    recs_cnt = length rec_args;
+      val allargs     = nonrec_args @ rec_args
+                        @ map (upd_vname (fn s=> s^"'")) rec_args;
+      val allvns      = map vname allargs;
+      fun vname_arg s arg = if is_rec arg then vname arg^s else vname arg;
+      val vns1        = map (vname_arg "" ) args;
+      val vns2        = map (vname_arg "'") args;
+      val allargs_cnt = length nonrec_args + 2*recs_cnt;
+      val rec_idxs    = (recs_cnt-1) downto 0;
+      val nonlazy_idxs = map snd (filter_out (fn (arg,_) => is_lazy arg)
+                                             (allargs~~((allargs_cnt-1) downto 0)));
+      fun rel_app i ra = proj (Bound(allargs_cnt+2)) eqs (rec_of ra) $ 
+                              Bound (2*recs_cnt-i) $ Bound (recs_cnt-i);
+      val capps =
+          List.foldr
+            mk_conj
+            (mk_conj(
+             Bound(allargs_cnt+1)===list_ccomb(%%:con,map (bound_arg allvns) vns1),
+             Bound(allargs_cnt+0)===list_ccomb(%%:con,map (bound_arg allvns) vns2)))
+            (mapn rel_app 1 rec_args);
+    in
+      List.foldr
+        mk_ex
+        (Library.foldr mk_conj
+                       (map (defined o Bound) nonlazy_idxs,capps)) allvns
+    end;
+  fun one_comp n (_,cons) =
+      mk_all (x_name(n+1),
+      mk_all (x_name(n+1)^"'",
+      mk_imp (proj (Bound 2) eqs n $ Bound 1 $ Bound 0,
+      foldr1 mk_disj (mk_conj(Bound 1 === UU,Bound 0 === UU)
+                      ::map one_con cons))));
+  val bisim_eqn =
+      %%:(comp_dname^"_bisim") ==
+         mk_lam("R", foldr1 mk_conj (mapn one_comp 0 eqs));
+
+in
+  val ([ax_bisim_def], thy) =
+      thy
+        |> Sign.add_path comp_dnam
+        |> add_defs_infer [(Binding.name "bisim_def", bisim_eqn)]
+        ||> Sign.parent_path;
+end; (* local *)
+
+(* ----- theorem concerning coinduction ------------------------------------- *)
+
+local
+  val pg = pg' thy;
+  val xs = mapn (fn n => K (x_name n)) 1 dnames;
+  fun bnd_arg n i = Bound(2*(n_eqs - n)-i-1);
+  val take_ss = HOL_ss addsimps (@{thm Rep_CFun_strict1} :: take_rews);
+  val sproj = prj (fn s => K("fst("^s^")")) (fn s => K("snd("^s^")"));
+  val _ = trace " Proving coind_lemma...";
+  val coind_lemma =
+    let
+      fun mk_prj n _ = proj (%:"R") eqs n $ bnd_arg n 0 $ bnd_arg n 1;
+      fun mk_eqn n dn =
+        (dc_take dn $ %:"n" ` bnd_arg n 0) ===
+        (dc_take dn $ %:"n" ` bnd_arg n 1);
+      fun mk_all2 (x,t) = mk_all (x, mk_all (x^"'", t));
+      val goal =
+        mk_trp (mk_imp (%%:(comp_dname^"_bisim") $ %:"R",
+          Library.foldr mk_all2 (xs,
+            Library.foldr mk_imp (mapn mk_prj 0 dnames,
+              foldr1 mk_conj (mapn mk_eqn 0 dnames)))));
+      fun x_tacs ctxt n x = [
+        rotate_tac (n+1) 1,
+        etac all2E 1,
+        eres_inst_tac ctxt [(("P", 1), sproj "R" eqs n^" "^x^" "^x^"'")] (mp RS disjE) 1,
+        TRY (safe_tac HOL_cs),
+        REPEAT (CHANGED (asm_simp_tac take_ss 1))];
+      fun tacs ctxt = [
+        rtac impI 1,
+        InductTacs.induct_tac ctxt [[SOME "n"]] 1,
+        simp_tac take_ss 1,
+        safe_tac HOL_cs] @
+        flat (mapn (x_tacs ctxt) 0 xs);
+    in pg [ax_bisim_def] goal tacs end;
+in
+  val _ = trace " Proving coind...";
+  val coind = 
+    let
+      fun mk_prj n x = mk_trp (proj (%:"R") eqs n $ %:x $ %:(x^"'"));
+      fun mk_eqn x = %:x === %:(x^"'");
+      val goal =
+        mk_trp (%%:(comp_dname^"_bisim") $ %:"R") ===>
+          Logic.list_implies (mapn mk_prj 0 xs,
+            mk_trp (foldr1 mk_conj (map mk_eqn xs)));
+      val tacs =
+        TRY (safe_tac HOL_cs) ::
+        maps (fn take_lemma => [
+          rtac take_lemma 1,
+          cut_facts_tac [coind_lemma] 1,
+          fast_tac HOL_cs 1])
+        take_lemmas;
+    in pg [] goal (K tacs) end;
+end; (* local *)
+
+in
+  (coind, thy)
+end;
+
+fun comp_theorems (comp_dnam, eqs: eq list) thy =
+let
+val map_tab = Domain_Take_Proofs.get_map_tab thy;
+
+val dnames = map (fst o fst) eqs;
+val comp_dname = Sign.full_bname thy comp_dnam;
+
+val _ = message ("Proving induction properties of domain "^comp_dname^" ...");
+
+(* ----- getting the composite axiom and definitions ------------------------ *)
+
+(* Test for indirect recursion *)
+local
+  fun indirect_arg arg =
+      rec_of arg = ~1 andalso Datatype_Aux.is_rec_type (dtyp_of arg);
+  fun indirect_con (_, args) = exists indirect_arg args;
+  fun indirect_eq (_, cons) = exists indirect_con cons;
+in
+  val is_indirect = exists indirect_eq eqs;
+  val _ = if is_indirect
+          then message "Definition uses indirect recursion."
+          else ();
+end;
+
+(* theorems about take *)
+
+local
+  fun ga s dn = PureThy.get_thm thy (dn ^ "." ^ s);
+  val axs_chain_take = map (ga "chain_take") dnames;
+  val axs_lub_take   = map (ga "lub_take"  ) dnames;
+in
+  val _ = trace " Proving take_lemmas...";
+  val take_lemmas =
+    let
+      fun take_lemma (ax_chain_take, ax_lub_take) =
+          Drule.export_without_context
+            (@{thm lub_ID_take_lemma} OF [ax_chain_take, ax_lub_take]);
+    in map take_lemma (axs_chain_take ~~ axs_lub_take) end;
+  val axs_reach =
+    let
+      fun reach (ax_chain_take, ax_lub_take) =
+          Drule.export_without_context
+            (@{thm lub_ID_reach} OF [ax_chain_take, ax_lub_take]);
+    in map reach (axs_chain_take ~~ axs_lub_take) end;
+end;
+
+val take_rews =
+    maps (fn dn => PureThy.get_thms thy (dn ^ ".take_rews")) dnames;
+
+(* prove induction rules, unless definition is indirect recursive *)
+val thy =
+    if is_indirect then thy else
+    prove_induction (comp_dnam, eqs) take_lemmas axs_reach take_rews thy;
+
+val (coind, thy) = prove_coinduction (comp_dnam, eqs) take_lemmas thy;
 
 in thy |> Sign.add_path comp_dnam
        |> snd o PureThy.add_thmss [
            ((Binding.name "take_lemmas", take_lemmas ), []),
            ((Binding.name "reach"      , axs_reach   ), []),
-           ((Binding.name "finites"    , finites     ), []),
-           ((Binding.name "finite_ind" , [finite_ind]), []),
-           ((Binding.name "ind"        , [ind]       ), []),
            ((Binding.name "coind"      , [coind]     ), [])]
-       |> (if induct_failed then I
-           else snd o PureThy.add_thmss (map ind_rule (dnames ~~ inducts)))
        |> Sign.parent_path |> pair take_rews
 end; (* let *)
 end; (* struct *)