(*  Title:      Tools/induct.ML

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

Proof by cases and induction.

*)

signature INDUCT_DATA =

sig

  val cases_default: thm

  val atomize: thm list

  val rulify: thm list

  val rulify_fallback: thm list

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 induct_type: string -> attribute

  val induct_pred: string -> attribute

  val coinduct_type: string -> attribute

  val coinduct_pred: string -> attribute

  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: (string option * term) option list -> Proof.context ->

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

  val atomize_term: theory -> term -> term

  val atomize_tac: int -> tactic

  val inner_atomize_tac: int -> tactic

  val rulified_term: thm -> theory * term

  val rulify_tac: int -> tactic

  val internalize: int -> thm -> thm

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

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

    thm list -> int -> cases_tactic

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

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

    cases_tactic

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

    thm option -> thm list -> int -> cases_tactic

  val setup: theory -> theory

end;

functor InductFun(Data: INDUCT_DATA): INDUCT =

struct

(** misc utils **)

(* encode_type -- for indexing purposes *)

fun encode_type (Type (c, Ts)) = Term.list_comb (Const (c, dummyT), map encode_type Ts)

  | encode_type (TFree (a, _)) = Free (a, dummyT)

  | encode_type (TVar (a, _)) = Var (a, dummyT);

(* 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 = 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;

(** induct data **)

(* rules *)

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

val init_rules =

  NetRules.init (fn ((s1: string, th1), (s2, th2)) => s1 = s2 andalso

    Thm.eq_thm_prop (th1, th2));

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

fun pretty_rules ctxt kind rs =

  let val thms = map snd (NetRules.rules rs)

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

(* context data *)

structure Induct = GenericDataFun

(

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

  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)));

  val extend = I;

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

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

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

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

      (NetRules.merge (coinductT1, coinductT2), NetRules.merge (coinductP1, coinductP2)));

);

val get_local = Induct.get o Context.Proof;

fun dest_rules ctxt =

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

    {type_cases = NetRules.rules casesT,

     pred_cases = NetRules.rules casesP,

     type_induct = NetRules.rules inductT,

     pred_induct = NetRules.rules inductP,

     type_coinduct = NetRules.rules coinductT,

     pred_coinduct = NetRules.rules 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 _ =

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

    OuterKeyword.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 (NetRules.retrieve (which (get_local ctxt)) (how x));

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

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

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

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

val find_coinductT = find_rules (#1 o #3) 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 (Induct.map (f (name, thm)) x, thm) end;

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

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

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

fun add_casesT rule x = map1 (apfst (NetRules.insert rule)) x;

fun add_casesP rule x = map1 (apsnd (NetRules.insert rule)) x;

fun add_inductT rule x = map2 (apfst (NetRules.insert rule)) x;

fun add_inductP rule x = map2 (apsnd (NetRules.insert rule)) x;

fun add_coinductT rule x = map3 (apfst (NetRules.insert rule)) x;

fun add_coinductP rule x = map3 (apsnd (NetRules.insert rule)) x;

fun consumes0 x = RuleCases.consumes_default 0 x;

fun consumes1 x = RuleCases.consumes_default 1 x;

in

val cases_type = mk_att add_casesT consumes0;

val cases_pred = mk_att add_casesP consumes1;

val induct_type = mk_att add_inductT consumes0;

val induct_pred = mk_att add_inductP consumes1;

val coinduct_type = mk_att add_coinductT consumes0;

val coinduct_pred = mk_att add_coinductP consumes1;

end;

(** attribute syntax **)

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 = Attrib.syntax

 (spec typeN Args.tyname >> add_type ||

  spec predN Args.const >> add_pred ||

  spec setN Args.const >> add_pred);

val cases_att = attrib cases_type cases_pred;

val induct_att = attrib induct_type induct_pred;

val coinduct_att = attrib coinduct_type coinduct_pred;

in

val attrib_setup = Attrib.add_attributes

 [(casesN, cases_att, "declaration of cases rule for type or predicate/set"),

  (inductN, induct_att, "declaration of induction rule for type or predicate/set"),

  (coinductN, coinduct_att, "declaration of coinduction rule for type or predicate/set")];

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 (Library.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 (Library.drop (m - n, xs) ~~ ys) end;

(* prep_inst *)

fun prep_inst thy align tune (tm, ts) =

  let

    val cert = Thm.cterm_of thy;

    fun prep_var (x, SOME t) =

          let

            val cx = cert x;

            val {T = xT, thy, ...} = Thm.rep_cterm cx;

            val ct = cert (tune t);

          in

            if Type.could_unify (#T (Thm.rep_cterm ct), xT) then SOME (cx, ct)

            else error (Pretty.string_of (Pretty.block

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

              Display.pretty_cterm ct, Pretty.str " ::", Pretty.brk 1,

              Display.pretty_ctyp (#T (Thm.crep_cterm ct))]))

          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;

(* make_cases *)

fun make_cases is_open rule =

  RuleCases.make_common is_open (Thm.theory_of_thm rule, Thm.prop_of rule);

fun warn_open true = legacy_feature "open rule cases in proof method"

  | warn_open false = ();

(** 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 is_open insts opt_rule facts =

  let

    val _ = warn_open is_open;

    val thy = ProofContext.theory_of ctxt;

    val cert = Thm.cterm_of thy;

    fun inst_rule r =

      if null insts then `RuleCases.get r

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

        |> maps (prep_inst thy align_left I)

        |> Drule.cterm_instantiate) r |> pair (RuleCases.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 (RuleCases.consume [] facts)

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

        CASES (make_cases is_open rule cases)

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

  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 []

  #> ObjectLogic.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 thy inst rule =

  Drule.cterm_instantiate (prep_inst thy 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 as Envir.Envir {iTs, ...}) =

  let

    val cert = Thm.cterm_of thy;

    val certT = Thm.ctyp_of thy;

    val pairs = Envir.alist_of 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)) (Vartab.dest iTs), xs ~~ ts) end;

in

fun guess_instance rule i st =

  let

    val {thy, maxidx, ...} = Thm.rep_thm st;

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

    val params = rev (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 (Sign.string_of_term thy 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 (#maxidx (Thm.rep_thm 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 = 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 List.filter (equal 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 [RuleCases.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 certT = Thm.ctyp_of thy;

    val v = Free (x, T);

    fun spec_rule prfx (xs, body) =

      @{thm Pure.meta_spec}

      |> Thm.rename_params_rule ([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.forall_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 (SOME x, t)) ctxt =

          let val ([(lhs, (_, th))], ctxt') = LocalDefs.add_defs [((x, NoSyn), (("", []), t))] ctxt

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

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

      | 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

*)

local

fun get_inductT ctxt insts =

  fold_rev multiply (insts |> map_filter (fn [] => NONE | ts => List.last ts)

    |> map (find_inductT ctxt o Term.fastype_of)) [[]]

  |> filter_out (forall PureThy.is_internal);

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

  | get_inductP _ _ = [];

in

fun induct_tac ctxt is_open def_insts arbitrary taking opt_rule facts =

  let

    val _ = warn_open is_open;

    val thy = ProofContext.theory_of ctxt;

    val cert = Thm.cterm_of thy;

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

    val atomized_defs = map (map (Conv.fconv_rule ObjectLogic.atomize)) defs;

    fun inst_rule (concls, r) =

      (if null insts then `RuleCases.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 thy align_right (atomize_term thy))

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

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

    val ruleq =

      (case opt_rule of

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

      | NONE =>

          (get_inductP ctxt facts @

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

          |> map_filter (RuleCases.mutual_rule ctxt)

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

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

    fun rule_cases rule =

      RuleCases.make_nested is_open (Thm.prop_of rule) (rulified_term rule);

  in

    (fn i => fn st =>

      ruleq

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

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

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

          (CONJUNCTS (ALLGOALS

            (Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1))

              THEN' fix_tac defs_ctxt

                (nth concls (j - 1) + more_consumes)

                (nth_list arbitrary (j - 1))))

          THEN' inner_atomize_tac) j))

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

            guess_instance (internalize more_consumes rule) i st'

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

            |> Seq.maps (fn rule' =>

              CASES (rule_cases rule' cases)

                (Tactic.rtac rule' i THEN

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

    THEN_ALL_NEW_CASES rulify_tac

  end;

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 RuleCases.get_consumes th > 0 then Thm.major_prem_of th else Thm.concl_of th;

in

fun coinduct_tac ctxt is_open inst taking opt_rule facts =

  let

    val _ = warn_open is_open;

    val thy = ProofContext.theory_of ctxt;

    val cert = Thm.cterm_of thy;

    fun inst_rule r =

      if null inst then `RuleCases.get r

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

        |> pair (RuleCases.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 (RuleCases.consume [] facts)

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

        guess_instance rule i st

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

        |> Seq.maps (fn rule' =>

          CASES (make_cases is_open rule' cases)

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

  end;

end;

(** concrete syntax **)

val openN = "open";

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.tyname get_type ||

  named_rule predN Args.const get_pred ||

  named_rule setN Args.const 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 def_inst =

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

      -- Args.term) >> SOME ||

    inst >> Option.map (pair NONE);

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

  error ("Bad free variable: " ^ Syntax.string_of_term (Context.proof_of context) 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) |--

  Args.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

fun cases_meth src =

  Method.syntax (Args.mode openN --

    (Args.and_list (Scan.repeat (unless_more_args inst)) -- Scan.option cases_rule)) src

  #> (fn ((is_open, (insts, opt_rule)), ctxt) =>

    Method.METHOD_CASES (fn facts =>

      Seq.DETERM (HEADGOAL (cases_tac ctxt is_open insts opt_rule facts))));

fun induct_meth src =

  Method.syntax (Args.mode openN --

    (Args.and_list (Scan.repeat (unless_more_args def_inst)) --

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

  #> (fn ((is_open, (insts, ((arbitrary, taking), opt_rule))), ctxt) =>

    Method.RAW_METHOD_CASES (fn facts =>

      Seq.DETERM (HEADGOAL (induct_tac ctxt is_open insts arbitrary taking opt_rule facts))));

fun coinduct_meth src =

  Method.syntax (Args.mode openN --

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

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

    Method.RAW_METHOD_CASES (fn facts =>

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

end;

(** theory setup **)

val setup =

  attrib_setup #>

  Method.add_methods

    [(casesN, cases_meth, "case analysis on types or predicates/sets"),

     (inductN, induct_meth, "induction on types or predicates/sets"),

     (coinductN, coinduct_meth, "coinduction on types or predicates/sets")];

end;