(*  Author:     Christian Urban and Makarius

The nominal induct proof method.

*)

structure NominalInduct:

sig

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

    (string * typ) list -> (string * typ) list list -> thm list -> thm list -> int -> Rule_Cases.cases_tactic

  val nominal_induct_method: (Proof.context -> Proof.method) context_parser

end =

struct

(* proper tuples -- nested left *)

fun tupleT Ts = HOLogic.unitT |> fold (fn T => fn U => HOLogic.mk_prodT (U, T)) Ts;

fun tuple ts = HOLogic.unit |> fold (fn t => fn u => HOLogic.mk_prod (u, t)) ts;

fun tuple_fun Ts (xi, T) =

  Library.funpow (length Ts) HOLogic.mk_split

    (Var (xi, (HOLogic.unitT :: Ts) ---> Term.range_type T));

val split_all_tuples =

  Simplifier.full_simplify (HOL_basic_ss addsimps

    @{thms split_conv split_paired_all unit_all_eq1} @

    @{thms fresh_Unit_elim fresh_Pair_elim} @

    @{thms fresh_star_Unit_elim fresh_star_Pair_elim fresh_star_Un_elim} @

    @{thms fresh_star_insert_elim fresh_star_empty_elim})

(* prepare rule *)

fun inst_mutual_rule ctxt insts avoiding rules =

  let

    val (nconcls, joined_rule) = Rule_Cases.strict_mutual_rule ctxt rules;

    val concls = Logic.dest_conjunctions (Thm.concl_of joined_rule);

    val (cases, consumes) = Rule_Cases.get joined_rule;

    val l = length rules;

    val _ =

      if length insts = l then ()

      else error ("Bad number of instantiations for " ^ string_of_int l ^ " rules");

    fun subst inst concl =

      let

        val vars = Induct.vars_of concl;

        val m = length vars and n = length inst;

        val _ = if m >= n + 2 then () else error "Too few variables in conclusion of rule";

        val P :: x :: ys = vars;

        val zs = drop (m - n - 2) ys;

      in

        (P, tuple_fun (map #2 avoiding) (Term.dest_Var P)) ::

        (x, tuple (map Free avoiding)) ::

        map_filter (fn (z, SOME t) => SOME (z, t) | _ => NONE) (zs ~~ inst)

      end;

     val substs =

       map2 subst insts concls |> flat |> distinct (op =)

       |> map (pairself (Thm.cterm_of (Proof_Context.theory_of ctxt)));

  in 

    (((cases, nconcls), consumes), Drule.cterm_instantiate substs joined_rule) 

  end;

fun rename_params_rule internal xs rule =

  let

    val tune =

      if internal then Name.internal

      else fn x => the_default x (try Name.dest_internal x);

    val n = length xs;

    fun rename prem =

      let

        val ps = Logic.strip_params prem;

        val p = length ps;

        val ys =

          if p < n then []

          else map (tune o #1) (take (p - n) ps) @ xs;

      in Logic.list_rename_params ys prem end;

    fun rename_prems prop =

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

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

  in Thm.equal_elim (Thm.reflexive (Drule.cterm_fun rename_prems (Thm.cprop_of rule))) rule end;

(* nominal_induct_tac *)

fun nominal_induct_tac ctxt simp def_insts avoiding fixings rules facts =

  let

    val thy = Proof_Context.theory_of ctxt;

    val cert = Thm.cterm_of thy;

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

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

    val finish_rule =

      split_all_tuples

      #> rename_params_rule true

        (map (Name.clean o Variable.revert_fixed defs_ctxt o fst) avoiding);

    fun rule_cases ctxt r =

      let val r' = if simp then Induct.simplified_rule ctxt r else r

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

  in

    (fn i => fn st =>

      rules

      |> inst_mutual_rule ctxt insts avoiding

      |> 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 fixings (j - 1);

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

            in

              Method.insert_tac (more_facts @ adefs) THEN'

                (if simp then

                   Induct.rotate_tac k (length adefs) THEN'

                   Induct.arbitrary_tac defs_ctxt k

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

                 else

                   Induct.arbitrary_tac defs_ctxt k xs)

            end)

          THEN' Induct.inner_atomize_tac) j))

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

            Induct.guess_instance ctxt

              (finish_rule (Induct.internalize more_consumes rule)) i st'

            |> Seq.maps (fn rule' =>

              CASES (rule_cases ctxt rule' cases)

                (Tactic.rtac (rename_params_rule false [] rule') i THEN

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

    THEN_ALL_NEW_CASES

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

        else K all_tac)

       THEN_ALL_NEW Induct.rulify_tac)

  end;

(* concrete syntax *)

local

val avoidingN = "avoiding";

val fixingN = "arbitrary";  (* to be consistent with induct; hopefully this changes again *)

val ruleN = "rule";

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.$$$ avoidingN || Args.$$$ fixingN || Args.$$$ ruleN) -- Args.colon)) scan;

val avoiding = Scan.optional (Scan.lift (Args.$$$ avoidingN -- Args.colon) |--

  Scan.repeat (unless_more_args free)) [];

val fixing = Scan.optional (Scan.lift (Args.$$$ fixingN -- Args.colon) |--

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

val rule_spec = Scan.lift (Args.$$$ "rule" -- Args.colon) |-- Attrib.thms;

in

val nominal_induct_method =

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

  avoiding -- fixing -- rule_spec) >>

  (fn (no_simp, (((x, y), z), w)) => fn ctxt =>

    RAW_METHOD_CASES (fn facts =>

      HEADGOAL (nominal_induct_tac ctxt (not no_simp) x y z w facts)));

end

end;