(*  Title:      Tools/induct.ML

    Author:     Markus Wenzel, TU Muenchen

Proof by cases, induction, and coinduction.

*)

signature INDUCT_ARGS =

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(Induct_Args: INDUCT_ARGS): 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 (Proof_Context.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 Induct_Args.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_global

    (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 Induct_Args.dest_def) (term_of ct) andalso

    not (is_some (Induct_Args.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 Induct_Args.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 Data = 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 = Data.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 (Data.map (f (name, thm)) x, thm) end;

fun del_att which =

  Thm.declaration_attribute (fn th => Data.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 = Data.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 (Proof_Context.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 Induct_Args.equal_def;

fun mark_constraints n ctxt = Conv.fconv_rule

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

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

val unmark_constraints = Conv.fconv_rule

  (Raw_Simplifier.rewrite true [Induct_Args.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 Induct_Args.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 = Induct_Args.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 = Proof_Context.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 @ [Induct_Args.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 =

  Raw_Simplifier.rewrite_term thy Induct_Args.atomize []

  #> Object_Logic.drop_judgment thy;

val atomize_cterm = Raw_Simplifier.rewrite true Induct_Args.atomize;

val atomize_tac = Simplifier.rewrite_goal_tac Induct_Args.atomize;

val inner_atomize_tac =

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

(* rulify *)

fun rulify_term thy =

  Raw_Simplifier.rewrite_term thy (Induct_Args.rulify @ conjunction_congs) [] #>

  Raw_Simplifier.rewrite_term thy Induct_Args.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 (Induct_Args.rulify @ conjunction_congs) THEN'

  Simplifier.rewrite_goal_tac Induct_Args.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 = Proof_Context.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_Trans.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 General.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 (Variable.revert_fixed 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 = Proof_Context.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 (Variable.revert_fixed 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 (Raw_Simplifier.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 = Proof_Context.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 (Proof_Context.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 = Proof_Context.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;