(*  Title:      Tools/induct.ML

     Author:     Markus Wenzel, TU Muenchen

 Proof by cases, induction, and coinduction.

 *)

 signature INDUCT_DATA =

 sig

   val cases_default: thm

   val atomize: thm list

   val rulify: thm list

   val rulify_fallback: thm list

   val equal_def: thm

   val dest_def: term -> (term * term) option

   val trivial_tac: int -> tactic

 end;

 signature INDUCT =

 sig

   (*rule declarations*)

   val vars_of: term -> term list

   val dest_rules: Proof.context ->

     {type_cases: (string * thm) list, pred_cases: (string * thm) list,

       type_induct: (string * thm) list, pred_induct: (string * thm) list,

       type_coinduct: (string * thm) list, pred_coinduct: (string * thm) list}

   val print_rules: Proof.context -> unit

   val lookup_casesT: Proof.context -> string -> thm option

   val lookup_casesP: Proof.context -> string -> thm option

   val lookup_inductT: Proof.context -> string -> thm option

   val lookup_inductP: Proof.context -> string -> thm option

   val lookup_coinductT: Proof.context -> string -> thm option

   val lookup_coinductP: Proof.context -> string -> thm option

   val find_casesT: Proof.context -> typ -> thm list

   val find_casesP: Proof.context -> term -> thm list

   val find_inductT: Proof.context -> typ -> thm list

   val find_inductP: Proof.context -> term -> thm list

   val find_coinductT: Proof.context -> typ -> thm list

   val find_coinductP: Proof.context -> term -> thm list

   val cases_type: string -> attribute

   val cases_pred: string -> attribute

   val cases_del: attribute

   val induct_type: string -> attribute

   val induct_pred: string -> attribute

   val induct_del: attribute

   val coinduct_type: string -> attribute

   val coinduct_pred: string -> attribute

   val coinduct_del: attribute

   val map_simpset: (simpset -> simpset) -> Context.generic -> Context.generic

   val induct_simp_add: attribute

   val induct_simp_del: attribute

   val no_simpN: string

   val casesN: string

   val inductN: string

   val coinductN: string

   val typeN: string

   val predN: string

   val setN: string

   (*proof methods*)

   val fix_tac: Proof.context -> int -> (string * typ) list -> int -> tactic

   val add_defs: (binding option * (term * bool)) option list -> Proof.context ->

     (term option list * thm list) * Proof.context

   val atomize_term: theory -> term -> term

   val atomize_cterm: conv

   val atomize_tac: int -> tactic

   val inner_atomize_tac: int -> tactic

   val rulified_term: thm -> theory * term

   val rulify_tac: int -> tactic

   val simplified_rule: Proof.context -> thm -> thm

   val simplify_tac: Proof.context -> int -> tactic

   val trivial_tac: int -> tactic

   val rotate_tac: int -> int -> int -> tactic

   val internalize: int -> thm -> thm

   val guess_instance: Proof.context -> thm -> int -> thm -> thm Seq.seq

   val cases_tac: Proof.context -> bool -> term option list list -> thm option ->

     thm list -> int -> cases_tactic

   val get_inductT: Proof.context -> term option list list -> thm list list

   val induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->

     (string * typ) list list -> term option list -> thm list option ->

     thm list -> int -> cases_tactic

   val coinduct_tac: Proof.context -> term option list -> term option list -> thm option ->

     thm list -> int -> cases_tactic

   val setup: theory -> theory

 end;

 functor Induct(Data: INDUCT_DATA): INDUCT =

 struct

 (** variables -- ordered left-to-right, preferring right **)

 fun vars_of tm =

   rev (distinct (op =) (Term.fold_aterms (fn (t as Var _) => cons t | _ => I) tm []));

 local

 val mk_var = Net.encode_type o #2 o Term.dest_Var;

 fun concl_var which thm = mk_var (which (vars_of (Thm.concl_of thm))) handle Empty =>

   raise THM ("No variables in conclusion of rule", 0, [thm]);

 in

 fun left_var_prem thm = mk_var (hd (vars_of (hd (Thm.prems_of thm)))) handle Empty =>

   raise THM ("No variables in major premise of rule", 0, [thm]);

 val left_var_concl = concl_var hd;

 val right_var_concl = concl_var List.last;

 end;

 (** constraint simplification **)

 (* rearrange parameters and premises to allow application of one-point-rules *)

 fun swap_params_conv ctxt i j cv =

   let

     fun conv1 0 ctxt = Conv.forall_conv (cv o snd) ctxt

       | conv1 k ctxt =

           Conv.rewr_conv @{thm swap_params} then_conv

           Conv.forall_conv (conv1 (k-1) o snd) ctxt

     fun conv2 0 ctxt = conv1 j ctxt

       | conv2 k ctxt = Conv.forall_conv (conv2 (k-1) o snd) ctxt

   in conv2 i ctxt end;

 fun swap_prems_conv 0 = Conv.all_conv

   | swap_prems_conv i =

       Conv.implies_concl_conv (swap_prems_conv (i-1)) then_conv

       Conv.rewr_conv Drule.swap_prems_eq

 fun drop_judgment ctxt = Object_Logic.drop_judgment (ProofContext.theory_of ctxt);

 fun find_eq ctxt t =

   let

     val l = length (Logic.strip_params t);

     val Hs = Logic.strip_assums_hyp t;

     fun find (i, t) =

       case Data.dest_def (drop_judgment ctxt t) of

         SOME (Bound j, _) => SOME (i, j)

       | SOME (_, Bound j) => SOME (i, j)

       | _ => NONE

   in

     case get_first find (map_index I Hs) of

       NONE => NONE

     | SOME (0, 0) => NONE

     | SOME (i, j) => SOME (i, l-j-1, j)

   end;

 fun mk_swap_rrule ctxt ct = case find_eq ctxt (term_of ct) of

     NONE => NONE

   | SOME (i, k, j) => SOME (swap_params_conv ctxt k j (K (swap_prems_conv i)) ct);

 val rearrange_eqs_simproc = Simplifier.simproc

   (Thm.theory_of_thm Drule.swap_prems_eq) "rearrange_eqs" ["all t"]

   (fn thy => fn ss => fn t =>

      mk_swap_rrule (Simplifier.the_context ss) (cterm_of thy t))

 (* rotate k premises to the left by j, skipping over first j premises *)

 fun rotate_conv 0 j 0 = Conv.all_conv

   | rotate_conv 0 j k = swap_prems_conv j then_conv rotate_conv 1 j (k-1)

   | rotate_conv i j k = Conv.implies_concl_conv (rotate_conv (i-1) j k);

 fun rotate_tac j 0 = K all_tac

   | rotate_tac j k = SUBGOAL (fn (goal, i) => CONVERSION (rotate_conv

       j (length (Logic.strip_assums_hyp goal) - j - k) k) i);

 (* rulify operators around definition *)

 fun rulify_defs_conv ctxt ct =

   if exists_subterm (is_some o Data.dest_def) (term_of ct) andalso

     not (is_some (Data.dest_def (drop_judgment ctxt (term_of ct))))

   then

     (Conv.forall_conv (rulify_defs_conv o snd) ctxt else_conv

      Conv.implies_conv (Conv.try_conv (rulify_defs_conv ctxt))

        (Conv.try_conv (rulify_defs_conv ctxt)) else_conv

      Conv.first_conv (map Conv.rewr_conv Data.rulify) then_conv

        Conv.try_conv (rulify_defs_conv ctxt)) ct

   else Conv.no_conv ct;

 (** induct data **)

 (* rules *)

 type rules = (string * thm) Item_Net.T;

 fun init_rules index : rules =

   Item_Net.init

     (fn ((s1, th1), (s2, th2)) => s1 = s2 andalso Thm.eq_thm_prop (th1, th2))

     (single o index);

 fun filter_rules (rs: rules) th =

   filter (fn (_, th') => Thm.eq_thm_prop (th, th')) (Item_Net.content rs);

 fun lookup_rule (rs: rules) = AList.lookup (op =) (Item_Net.content rs);

 fun pretty_rules ctxt kind rs =

   let val thms = map snd (Item_Net.content rs)

   in Pretty.big_list kind (map (Display.pretty_thm ctxt) thms) end;

 (* context data *)

 structure InductData = Generic_Data

 (

   type T = (rules * rules) * (rules * rules) * (rules * rules) * simpset;

   val empty =

     ((init_rules (left_var_prem o #2), init_rules (Thm.major_prem_of o #2)),

      (init_rules (right_var_concl o #2), init_rules (Thm.major_prem_of o #2)),

      (init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)),

      empty_ss addsimprocs [rearrange_eqs_simproc] addsimps [Drule.norm_hhf_eq]);

   val extend = I;

   fun merge (((casesT1, casesP1), (inductT1, inductP1), (coinductT1, coinductP1), simpset1),

       ((casesT2, casesP2), (inductT2, inductP2), (coinductT2, coinductP2), simpset2)) =

     ((Item_Net.merge (casesT1, casesT2), Item_Net.merge (casesP1, casesP2)),

      (Item_Net.merge (inductT1, inductT2), Item_Net.merge (inductP1, inductP2)),

      (Item_Net.merge (coinductT1, coinductT2), Item_Net.merge (coinductP1, coinductP2)),

      merge_ss (simpset1, simpset2));

 );

 val get_local = InductData.get o Context.Proof;

 fun dest_rules ctxt =

   let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in

     {type_cases = Item_Net.content casesT,

      pred_cases = Item_Net.content casesP,

      type_induct = Item_Net.content inductT,

      pred_induct = Item_Net.content inductP,

      type_coinduct = Item_Net.content coinductT,

      pred_coinduct = Item_Net.content coinductP}

   end;

 fun print_rules ctxt =

   let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in

    [pretty_rules ctxt "coinduct type:" coinductT,

     pretty_rules ctxt "coinduct pred:" coinductP,

     pretty_rules ctxt "induct type:" inductT,

     pretty_rules ctxt "induct pred:" inductP,

     pretty_rules ctxt "cases type:" casesT,

     pretty_rules ctxt "cases pred:" casesP]

     |> Pretty.chunks |> Pretty.writeln

   end;

 val _ =

   Outer_Syntax.improper_command "print_induct_rules" "print induction and cases rules"

     Keyword.diag (Scan.succeed (Toplevel.no_timing o Toplevel.unknown_context o

       Toplevel.keep (print_rules o Toplevel.context_of)));

 (* access rules *)

 val lookup_casesT = lookup_rule o #1 o #1 o get_local;

 val lookup_casesP = lookup_rule o #2 o #1 o get_local;

 val lookup_inductT = lookup_rule o #1 o #2 o get_local;

 val lookup_inductP = lookup_rule o #2 o #2 o get_local;

 val lookup_coinductT = lookup_rule o #1 o #3 o get_local;

 val lookup_coinductP = lookup_rule o #2 o #3 o get_local;

 fun find_rules which how ctxt x =

   map snd (Item_Net.retrieve (which (get_local ctxt)) (how x));

 val find_casesT = find_rules (#1 o #1) Net.encode_type;

 val find_casesP = find_rules (#2 o #1) I;

 val find_inductT = find_rules (#1 o #2) Net.encode_type;

 val find_inductP = find_rules (#2 o #2) I;

 val find_coinductT = find_rules (#1 o #3) Net.encode_type;

 val find_coinductP = find_rules (#2 o #3) I;

 (** attributes **)

 local

 fun mk_att f g name arg =

   let val (x, thm) = g arg in (InductData.map (f (name, thm)) x, thm) end;

 fun del_att which = Thm.declaration_attribute (fn th => InductData.map (which (pairself (fn rs =>

   fold Item_Net.remove (filter_rules rs th) rs))));

 fun map1 f (x, y, z, s) = (f x, y, z, s);

 fun map2 f (x, y, z, s) = (x, f y, z, s);

 fun map3 f (x, y, z, s) = (x, y, f z, s);

 fun map4 f (x, y, z, s) = (x, y, z, f s);

 fun add_casesT rule x = map1 (apfst (Item_Net.update rule)) x;

 fun add_casesP rule x = map1 (apsnd (Item_Net.update rule)) x;

 fun add_inductT rule x = map2 (apfst (Item_Net.update rule)) x;

 fun add_inductP rule x = map2 (apsnd (Item_Net.update rule)) x;

 fun add_coinductT rule x = map3 (apfst (Item_Net.update rule)) x;

 fun add_coinductP rule x = map3 (apsnd (Item_Net.update rule)) x;

 val consumes0 = Rule_Cases.consumes_default 0;

 val consumes1 = Rule_Cases.consumes_default 1;

 in

 val cases_type = mk_att add_casesT consumes0;

 val cases_pred = mk_att add_casesP consumes1;

 val cases_del = del_att map1;

 val induct_type = mk_att add_inductT consumes0;

 val induct_pred = mk_att add_inductP consumes1;

 val induct_del = del_att map2;

 val coinduct_type = mk_att add_coinductT consumes0;

 val coinduct_pred = mk_att add_coinductP consumes1;

 val coinduct_del = del_att map3;

 fun map_simpset f = InductData.map (map4 f);

 fun induct_simp f =

   Thm.declaration_attribute (fn thm => fn context =>

       (map_simpset

         (Simplifier.with_context (Context.proof_of context) (fn ss => f (ss, [thm]))) context));

 val induct_simp_add = induct_simp (op addsimps);

 val induct_simp_del = induct_simp (op delsimps);

 end;

 (** attribute syntax **)

 val no_simpN = "no_simp";

 val casesN = "cases";

 val inductN = "induct";

 val coinductN = "coinduct";

 val typeN = "type";

 val predN = "pred";

 val setN = "set";

 local

 fun spec k arg =

   Scan.lift (Args.$$$ k -- Args.colon) |-- arg ||

   Scan.lift (Args.$$$ k) >> K "";

 fun attrib add_type add_pred del =

   spec typeN (Args.type_name false) >> add_type ||

   spec predN (Args.const false) >> add_pred ||

   spec setN (Args.const false) >> add_pred ||

   Scan.lift Args.del >> K del;

 in

 val attrib_setup =

   Attrib.setup @{binding cases} (attrib cases_type cases_pred cases_del)

     "declaration of cases rule" #>

   Attrib.setup @{binding induct} (attrib induct_type induct_pred induct_del)

     "declaration of induction rule" #>

   Attrib.setup @{binding coinduct} (attrib coinduct_type coinduct_pred coinduct_del)

     "declaration of coinduction rule" #>

   Attrib.setup @{binding induct_simp} (Attrib.add_del induct_simp_add induct_simp_del)

     "declaration of rules for simplifying induction or cases rules";

 end;

 (** method utils **)

 (* alignment *)

 fun align_left msg xs ys =

   let val m = length xs and n = length ys

   in if m < n then error msg else (take n xs ~~ ys) end;

 fun align_right msg xs ys =

   let val m = length xs and n = length ys

   in if m < n then error msg else (drop (m - n) xs ~~ ys) end;

 (* prep_inst *)

 fun prep_inst ctxt align tune (tm, ts) =

   let

     val cert = Thm.cterm_of (ProofContext.theory_of ctxt);

     fun prep_var (x, SOME t) =

           let

             val cx = cert x;

             val xT = #T (Thm.rep_cterm cx);

             val ct = cert (tune t);

             val tT = #T (Thm.rep_cterm ct);

           in

             if Type.could_unify (tT, xT) then SOME (cx, ct)

             else error (Pretty.string_of (Pretty.block

              [Pretty.str "Ill-typed instantiation:", Pretty.fbrk,

               Syntax.pretty_term ctxt (Thm.term_of ct), Pretty.str " ::", Pretty.brk 1,

               Syntax.pretty_typ ctxt tT]))

           end

       | prep_var (_, NONE) = NONE;

     val xs = vars_of tm;

   in

     align "Rule has fewer variables than instantiations given" xs ts

     |> map_filter prep_var

   end;

 (* trace_rules *)

 fun trace_rules _ kind [] = error ("Unable to figure out " ^ kind ^ " rule")

   | trace_rules ctxt _ rules = Method.trace ctxt rules;

 (* mark equality constraints in cases rule *)

 val equal_def' = Thm.symmetric Data.equal_def;

 fun mark_constraints n ctxt = Conv.fconv_rule

   (Conv.prems_conv (~1) (Conv.params_conv ~1 (K (Conv.prems_conv n

      (MetaSimplifier.rewrite false [equal_def']))) ctxt));

 val unmark_constraints = Conv.fconv_rule

   (MetaSimplifier.rewrite true [Data.equal_def]);

 (* simplify *)

 fun simplify_conv' ctxt =

   Simplifier.full_rewrite (Simplifier.context ctxt (#4 (get_local ctxt)));

 fun simplify_conv ctxt ct =

   if exists_subterm (is_some o Data.dest_def) (term_of ct) then

     (Conv.try_conv (rulify_defs_conv ctxt) then_conv simplify_conv' ctxt) ct

   else Conv.all_conv ct;

 fun gen_simplified_rule cv ctxt =

   Conv.fconv_rule (Conv.prems_conv ~1 (cv ctxt));

 val simplified_rule' = gen_simplified_rule simplify_conv';

 val simplified_rule = gen_simplified_rule simplify_conv;

 fun simplify_tac ctxt = CONVERSION (simplify_conv ctxt);

 val trivial_tac = Data.trivial_tac;

 (** cases method **)

 (*

   rule selection scheme:

           cases         - default case split

     `A t` cases ...     - predicate/set cases

           cases t       - type cases

     ...   cases ... r   - explicit rule

 *)

 local

 fun get_casesT ctxt ((SOME t :: _) :: _) = find_casesT ctxt (Term.fastype_of t)

   | get_casesT _ _ = [];

 fun get_casesP ctxt (fact :: _) = find_casesP ctxt (Thm.concl_of fact)

   | get_casesP _ _ = [];

 in

 fun cases_tac ctxt simp insts opt_rule facts =

   let

     val thy = ProofContext.theory_of ctxt;

     fun inst_rule r =

       (if null insts then r

        else (align_left "Rule has fewer premises than arguments given" (Thm.prems_of r) insts

          |> maps (prep_inst ctxt align_left I)

          |> Drule.cterm_instantiate) r) |>

       (if simp then mark_constraints (Rule_Cases.get_constraints r) ctxt else I) |>

       pair (Rule_Cases.get r);

     val ruleq =

       (case opt_rule of

         SOME r => Seq.single (inst_rule r)

       | NONE =>

           (get_casesP ctxt facts @ get_casesT ctxt insts @ [Data.cases_default])

           |> tap (trace_rules ctxt casesN)

           |> Seq.of_list |> Seq.maps (Seq.try inst_rule));

   in

     fn i => fn st =>

       ruleq

       |> Seq.maps (Rule_Cases.consume [] facts)

       |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>

         let val rule' =

           (if simp then simplified_rule' ctxt #> unmark_constraints else I) rule

         in

           CASES (Rule_Cases.make_common (thy,

               Thm.prop_of (Rule_Cases.internalize_params rule')) cases)

             ((Method.insert_tac more_facts THEN' Tactic.rtac rule' THEN_ALL_NEW

                 (if simp then TRY o trivial_tac else K all_tac)) i) st

         end)

   end;

 end;

 (** induct method **)

 val conjunction_congs = [@{thm Pure.all_conjunction}, @{thm imp_conjunction}];

 (* atomize *)

 fun atomize_term thy =

   MetaSimplifier.rewrite_term thy Data.atomize []

   #> Object_Logic.drop_judgment thy;

 val atomize_cterm = MetaSimplifier.rewrite true Data.atomize;

 val atomize_tac = Simplifier.rewrite_goal_tac Data.atomize;

 val inner_atomize_tac =

   Simplifier.rewrite_goal_tac (map Thm.symmetric conjunction_congs) THEN' atomize_tac;

 (* rulify *)

 fun rulify_term thy =

   MetaSimplifier.rewrite_term thy (Data.rulify @ conjunction_congs) [] #>

   MetaSimplifier.rewrite_term thy Data.rulify_fallback [];

 fun rulified_term thm =

   let

     val thy = Thm.theory_of_thm thm;

     val rulify = rulify_term thy;

     val (As, B) = Logic.strip_horn (Thm.prop_of thm);

   in (thy, Logic.list_implies (map rulify As, rulify B)) end;

 val rulify_tac =

   Simplifier.rewrite_goal_tac (Data.rulify @ conjunction_congs) THEN'

   Simplifier.rewrite_goal_tac Data.rulify_fallback THEN'

   Goal.conjunction_tac THEN_ALL_NEW

   (Simplifier.rewrite_goal_tac [@{thm Pure.conjunction_imp}] THEN' Goal.norm_hhf_tac);

 (* prepare rule *)

 fun rule_instance ctxt inst rule =

   Drule.cterm_instantiate (prep_inst ctxt align_left I (Thm.prop_of rule, inst)) rule;

 fun internalize k th =

   th |> Thm.permute_prems 0 k

   |> Conv.fconv_rule (Conv.concl_conv (Thm.nprems_of th - k) atomize_cterm);

 (* guess rule instantiation -- cannot handle pending goal parameters *)

 local

 fun dest_env thy env =

   let

     val cert = Thm.cterm_of thy;

     val certT = Thm.ctyp_of thy;

     val pairs = Vartab.dest (Envir.term_env env);

     val types = Vartab.dest (Envir.type_env env);

     val ts = map (cert o Envir.norm_term env o #2 o #2) pairs;

     val xs = map2 (curry (cert o Var)) (map #1 pairs) (map (#T o Thm.rep_cterm) ts);

   in (map (fn (xi, (S, T)) => (certT (TVar (xi, S)), certT T)) types, xs ~~ ts) end;

 in

 fun guess_instance ctxt rule i st =

   let

     val thy = ProofContext.theory_of ctxt;

     val maxidx = Thm.maxidx_of st;

     val goal = Thm.term_of (Thm.cprem_of st i);  (*exception Subscript*)

     val params = rev (Term.rename_wrt_term goal (Logic.strip_params goal));

   in

     if not (null params) then

       (warning ("Cannot determine rule instantiation due to pending parameter(s): " ^

         commas_quote (map (Syntax.string_of_term ctxt o Syntax.mark_boundT) params));

       Seq.single rule)

     else

       let

         val rule' = Thm.incr_indexes (maxidx + 1) rule;

         val concl = Logic.strip_assums_concl goal;

       in

         Unify.smash_unifiers thy [(Thm.concl_of rule', concl)] (Envir.empty (Thm.maxidx_of rule'))

         |> Seq.map (fn env => Drule.instantiate (dest_env thy env) rule')

       end

   end handle Subscript => Seq.empty;

 end;

 (* special renaming of rule parameters *)

 fun special_rename_params ctxt [[SOME (Free (z, Type (T, _)))]] [thm] =

       let

         val x = Name.clean (ProofContext.revert_skolem ctxt z);

         fun index i [] = []

           | index i (y :: ys) =

               if x = y then x ^ string_of_int i :: index (i + 1) ys

               else y :: index i ys;

         fun rename_params [] = []

           | rename_params ((y, Type (U, _)) :: ys) =

               (if U = T then x else y) :: rename_params ys

           | rename_params ((y, _) :: ys) = y :: rename_params ys;

         fun rename_asm A =

           let

             val xs = rename_params (Logic.strip_params A);

             val xs' =

               (case filter (fn x' => x' = x) xs of

                 [] => xs | [_] => xs | _ => index 1 xs);

           in Logic.list_rename_params (xs', A) end;

         fun rename_prop p =

           let val (As, C) = Logic.strip_horn p

           in Logic.list_implies (map rename_asm As, C) end;

         val cp' = cterm_fun rename_prop (Thm.cprop_of thm);

         val thm' = Thm.equal_elim (Thm.reflexive cp') thm;

       in [Rule_Cases.save thm thm'] end

   | special_rename_params _ _ ths = ths;

 (* fix_tac *)

 local

 fun goal_prefix k ((c as Const ("all", _)) $ Abs (a, T, B)) = c $ Abs (a, T, goal_prefix k B)

   | goal_prefix 0 _ = Term.dummy_pattern propT

   | goal_prefix k ((c as Const ("==>", _)) $ A $ B) = c $ A $ goal_prefix (k - 1) B

   | goal_prefix _ _ = Term.dummy_pattern propT;

 fun goal_params k (Const ("all", _) $ Abs (_, _, B)) = goal_params k B + 1

   | goal_params 0 _ = 0

   | goal_params k (Const ("==>", _) $ _ $ B) = goal_params (k - 1) B

   | goal_params _ _ = 0;

 fun meta_spec_tac ctxt n (x, T) = SUBGOAL (fn (goal, i) =>

   let

     val thy = ProofContext.theory_of ctxt;

     val cert = Thm.cterm_of thy;

     val v = Free (x, T);

     fun spec_rule prfx (xs, body) =

       @{thm Pure.meta_spec}

       |> Thm.rename_params_rule ([Name.clean (ProofContext.revert_skolem ctxt x)], 1)

       |> Thm.lift_rule (cert prfx)

       |> `(Thm.prop_of #> Logic.strip_assums_concl)

       |-> (fn pred $ arg =>

         Drule.cterm_instantiate

           [(cert (Term.head_of pred), cert (Logic.rlist_abs (xs, body))),

            (cert (Term.head_of arg), cert (Logic.rlist_abs (xs, v)))]);

     fun goal_concl k xs (Const ("all", _) $ Abs (a, T, B)) = goal_concl k ((a, T) :: xs) B

       | goal_concl 0 xs B =

           if not (Term.exists_subterm (fn t => t aconv v) B) then NONE

           else SOME (xs, Term.absfree (x, T, Term.incr_boundvars 1 B))

       | goal_concl k xs (Const ("==>", _) $ _ $ B) = goal_concl (k - 1) xs B

       | goal_concl _ _ _ = NONE;

   in

     (case goal_concl n [] goal of

       SOME concl =>

         (compose_tac (false, spec_rule (goal_prefix n goal) concl, 1) THEN' rtac asm_rl) i

     | NONE => all_tac)

   end);

 fun miniscope_tac p = CONVERSION o

   Conv.params_conv p (K (MetaSimplifier.rewrite true [Thm.symmetric Drule.norm_hhf_eq]));

 in

 fun fix_tac _ _ [] = K all_tac

   | fix_tac ctxt n xs = SUBGOAL (fn (goal, i) =>

      (EVERY' (map (meta_spec_tac ctxt n) xs) THEN'

       (miniscope_tac (goal_params n goal) ctxt)) i);

 end;

 (* add_defs *)

 fun add_defs def_insts =

   let

     fun add (SOME (_, (t, true))) ctxt = ((SOME t, []), ctxt)

       | add (SOME (SOME x, (t, _))) ctxt =

           let val ([(lhs, (_, th))], ctxt') =

             Local_Defs.add_defs [((x, NoSyn), (Thm.empty_binding, t))] ctxt

           in ((SOME lhs, [th]), ctxt') end

       | add (SOME (NONE, (t as Free _, _))) ctxt = ((SOME t, []), ctxt)

       | add (SOME (NONE, (t, _))) ctxt =

           let

             val ([s], _) = Name.variants ["x"] (Variable.names_of ctxt);

             val ([(lhs, (_, th))], ctxt') =

               Local_Defs.add_defs [((Binding.name s, NoSyn),

                 (Thm.empty_binding, t))] ctxt

           in ((SOME lhs, [th]), ctxt') end

       | add NONE ctxt = ((NONE, []), ctxt);

   in fold_map add def_insts #> apfst (split_list #> apsnd flat) end;

 (* induct_tac *)

 (*

   rule selection scheme:

     `A x` induct ...     - predicate/set induction

           induct x       - type induction

     ...   induct ... r   - explicit rule

 *)

 fun get_inductT ctxt insts =

   fold_rev (map_product cons) (insts |> map

       ((fn [] => NONE | ts => List.last ts) #>

         (fn NONE => TVar (("'a", 0), []) | SOME t => Term.fastype_of t) #>

         find_inductT ctxt)) [[]]

   |> filter_out (forall Rule_Cases.is_inner_rule);

 fun get_inductP ctxt (fact :: _) = map single (find_inductP ctxt (Thm.concl_of fact))

   | get_inductP _ _ = [];

 fun induct_tac ctxt simp def_insts arbitrary taking opt_rule facts =

   let

     val thy = ProofContext.theory_of ctxt;

     val ((insts, defs), defs_ctxt) = fold_map add_defs def_insts ctxt |>> split_list;

     val atomized_defs = map (map (Conv.fconv_rule atomize_cterm)) defs;

     fun inst_rule (concls, r) =

       (if null insts then `Rule_Cases.get r

        else (align_left "Rule has fewer conclusions than arguments given"

           (map Logic.strip_imp_concl (Logic.dest_conjunctions (Thm.concl_of r))) insts

         |> maps (prep_inst ctxt align_right (atomize_term thy))

         |> Drule.cterm_instantiate) r |> pair (Rule_Cases.get r))

       |> (fn ((cases, consumes), th) => (((cases, concls), consumes), th));

     val ruleq =

       (case opt_rule of

         SOME rs => Seq.single (inst_rule (Rule_Cases.strict_mutual_rule ctxt rs))

       | NONE =>

           (get_inductP ctxt facts @

             map (special_rename_params defs_ctxt insts) (get_inductT ctxt insts))

           |> map_filter (Rule_Cases.mutual_rule ctxt)

           |> tap (trace_rules ctxt inductN o map #2)

           |> Seq.of_list |> Seq.maps (Seq.try inst_rule));

     fun rule_cases ctxt rule =

       let val rule' = (if simp then simplified_rule ctxt else I)

         (Rule_Cases.internalize_params rule);

       in Rule_Cases.make_nested (Thm.prop_of rule') (rulified_term rule') end;

   in

     (fn i => fn st =>

       ruleq

       |> Seq.maps (Rule_Cases.consume (flat defs) facts)

       |> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>

         (PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>

           (CONJUNCTS (ALLGOALS

             let

               val adefs = nth_list atomized_defs (j - 1);

               val frees = fold (Term.add_frees o prop_of) adefs [];

               val xs = nth_list arbitrary (j - 1);

               val k = nth concls (j - 1) + more_consumes

             in

               Method.insert_tac (more_facts @ adefs) THEN'

                 (if simp then

                    rotate_tac k (length adefs) THEN'

                    fix_tac defs_ctxt k

                      (List.partition (member op = frees) xs |> op @)

                  else

                    fix_tac defs_ctxt k xs)

              end)

           THEN' inner_atomize_tac) j))

         THEN' atomize_tac) i st |> Seq.maps (fn st' =>

             guess_instance ctxt (internalize more_consumes rule) i st'

             |> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))

             |> Seq.maps (fn rule' =>

               CASES (rule_cases ctxt rule' cases)

                 (Tactic.rtac rule' i THEN

                   PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st'))))

     THEN_ALL_NEW_CASES

       ((if simp then simplify_tac ctxt THEN' (TRY o trivial_tac)

         else K all_tac)

        THEN_ALL_NEW rulify_tac)

   end;

 (** coinduct method **)

 (*

   rule selection scheme:

     goal "A x" coinduct ...   - predicate/set coinduction

                coinduct x     - type coinduction

                coinduct ... r - explicit rule

 *)

 local

 fun get_coinductT ctxt (SOME t :: _) = find_coinductT ctxt (Term.fastype_of t)

   | get_coinductT _ _ = [];

 fun get_coinductP ctxt goal = find_coinductP ctxt (Logic.strip_assums_concl goal);

 fun main_prop_of th =

   if Rule_Cases.get_consumes th > 0 then Thm.major_prem_of th else Thm.concl_of th;

 in

 fun coinduct_tac ctxt inst taking opt_rule facts =

   let

     val thy = ProofContext.theory_of ctxt;

     fun inst_rule r =

       if null inst then `Rule_Cases.get r

       else Drule.cterm_instantiate (prep_inst ctxt align_right I (main_prop_of r, inst)) r

         |> pair (Rule_Cases.get r);

     fun ruleq goal =

       (case opt_rule of

         SOME r => Seq.single (inst_rule r)

       | NONE =>

           (get_coinductP ctxt goal @ get_coinductT ctxt inst)

           |> tap (trace_rules ctxt coinductN)

           |> Seq.of_list |> Seq.maps (Seq.try inst_rule));

   in

     SUBGOAL_CASES (fn (goal, i) => fn st =>

       ruleq goal

       |> Seq.maps (Rule_Cases.consume [] facts)

       |> Seq.maps (fn ((cases, (_, more_facts)), rule) =>

         guess_instance ctxt rule i st

         |> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))

         |> Seq.maps (fn rule' =>

           CASES (Rule_Cases.make_common (thy,

               Thm.prop_of (Rule_Cases.internalize_params rule')) cases)

             (Method.insert_tac more_facts i THEN Tactic.rtac rule' i) st)))

   end;

 end;

 (** concrete syntax **)

 val arbitraryN = "arbitrary";

 val takingN = "taking";

 val ruleN = "rule";

 local

 fun single_rule [rule] = rule

   | single_rule _ = error "Single rule expected";

 fun named_rule k arg get =

   Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--

     (fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>

       (case get (Context.proof_of context) name of SOME x => x

       | NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));

 fun rule get_type get_pred =

   named_rule typeN (Args.type_name false) get_type ||

   named_rule predN (Args.const false) get_pred ||

   named_rule setN (Args.const false) get_pred ||

   Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;

 val cases_rule = rule lookup_casesT lookup_casesP >> single_rule;

 val induct_rule = rule lookup_inductT lookup_inductP;

 val coinduct_rule = rule lookup_coinductT lookup_coinductP >> single_rule;

 val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.term >> SOME;

 val inst' = Scan.lift (Args.$$$ "_") >> K NONE ||

   Args.term >> (SOME o rpair false) ||

   Scan.lift (Args.$$$ "(") |-- (Args.term >> (SOME o rpair true)) --|

     Scan.lift (Args.$$$ ")");

 val def_inst =

   ((Scan.lift (Args.binding --| (Args.$$$ "\<equiv>" || Args.$$$ "==")) >> SOME)

       -- (Args.term >> rpair false)) >> SOME ||

     inst' >> Option.map (pair NONE);

 val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>

   error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));

 fun unless_more_args scan = Scan.unless (Scan.lift

   ((Args.$$$ arbitraryN || Args.$$$ takingN || Args.$$$ typeN ||

     Args.$$$ predN || Args.$$$ setN || Args.$$$ ruleN) -- Args.colon)) scan;

 val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--

   Parse.and_list1' (Scan.repeat (unless_more_args free))) [];

 val taking = Scan.optional (Scan.lift (Args.$$$ takingN -- Args.colon) |--

   Scan.repeat1 (unless_more_args inst)) [];

 in

 val cases_setup =

   Method.setup @{binding cases}

     (Args.mode no_simpN --

       (Parse.and_list' (Scan.repeat (unless_more_args inst)) -- Scan.option cases_rule) >>

       (fn (no_simp, (insts, opt_rule)) => fn ctxt =>

         METHOD_CASES (fn facts => Seq.DETERM (HEADGOAL

           (cases_tac ctxt (not no_simp) insts opt_rule facts)))))

     "case analysis on types or predicates/sets";

 val induct_setup =

   Method.setup @{binding induct}

     (Args.mode no_simpN -- (Parse.and_list' (Scan.repeat (unless_more_args def_inst)) --

       (arbitrary -- taking -- Scan.option induct_rule)) >>

       (fn (no_simp, (insts, ((arbitrary, taking), opt_rule))) => fn ctxt =>

         RAW_METHOD_CASES (fn facts =>

           Seq.DETERM

             (HEADGOAL (induct_tac ctxt (not no_simp) insts arbitrary taking opt_rule facts)))))

     "induction on types or predicates/sets";

 val coinduct_setup =

   Method.setup @{binding coinduct}

     (Scan.repeat (unless_more_args inst) -- taking -- Scan.option coinduct_rule >>

       (fn ((insts, taking), opt_rule) => fn ctxt =>

         RAW_METHOD_CASES (fn facts =>

           Seq.DETERM (HEADGOAL (coinduct_tac ctxt insts taking opt_rule facts)))))

     "coinduction on types or predicates/sets";

 end;

 (** theory setup **)

 val setup = attrib_setup #> cases_setup  #> induct_setup #> coinduct_setup;

 end;