src/HOL/Tools/datatype_aux.ML
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(*  Title:      HOL/Tools/datatype_aux.ML
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    Author:     Stefan Berghofer, TU Muenchen
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Auxiliary functions for defining datatypes.
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*)
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signature DATATYPE_AUX =
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sig
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  val quiet_mode : bool ref
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  val message : string -> unit
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  val add_path : bool -> string -> theory -> theory
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  val parent_path : bool -> theory -> theory
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  val store_thmss_atts : string -> string list -> attribute list list -> thm list list
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    -> theory -> thm list list * theory
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  val store_thmss : string -> string list -> thm list list -> theory -> thm list list * theory
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  val store_thms_atts : string -> string list -> attribute list list -> thm list
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    -> theory -> thm list * theory
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  val store_thms : string -> string list -> thm list -> theory -> thm list * theory
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  val split_conj_thm : thm -> thm list
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  val mk_conj : term list -> term
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  val mk_disj : term list -> term
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  val app_bnds : term -> int -> term
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  val cong_tac : int -> tactic
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  val indtac : thm -> string list -> int -> tactic
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  val exh_tac : (string -> thm) -> int -> tactic
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  datatype simproc_dist = FewConstrs of thm list
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                        | ManyConstrs of thm * simpset;
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  datatype dtyp =
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      DtTFree of string
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    | DtType of string * (dtyp list)
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    | DtRec of int;
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  type descr
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  type datatype_info
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  exception Datatype
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  exception Datatype_Empty of string
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  val name_of_typ : typ -> string
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  val dtyp_of_typ : (string * string list) list -> typ -> dtyp
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  val mk_Free : string -> typ -> int -> term
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  val is_rec_type : dtyp -> bool
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  val typ_of_dtyp : descr -> (string * sort) list -> dtyp -> typ
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  val dest_DtTFree : dtyp -> string
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  val dest_DtRec : dtyp -> int
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  val strip_dtyp : dtyp -> dtyp list * dtyp
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  val body_index : dtyp -> int
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  val mk_fun_dtyp : dtyp list -> dtyp -> dtyp
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  val get_nonrec_types : descr -> (string * sort) list -> typ list
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  val get_branching_types : descr -> (string * sort) list -> typ list
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  val get_arities : descr -> int list
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  val get_rec_types : descr -> (string * sort) list -> typ list
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  val check_nonempty : descr list -> unit
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  val unfold_datatypes : 
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    theory -> descr -> (string * sort) list -> datatype_info Symtab.table ->
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      descr -> int -> descr list * int
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end;
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structure DatatypeAux : DATATYPE_AUX =
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struct
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val quiet_mode = ref false;
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fun message s = if !quiet_mode then () else writeln s;
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fun add_path flat_names s = if flat_names then I else Sign.add_path s;
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fun parent_path flat_names = if flat_names then I else Sign.parent_path;
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(* store theorems in theory *)
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fun store_thmss_atts label tnames attss thmss =
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  fold_map (fn ((tname, atts), thms) =>
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    Sign.add_path tname
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    #> PureThy.add_thmss [((Binding.name label, thms), atts)]
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    #-> (fn thm::_ => Sign.parent_path #> pair thm)) (tnames ~~ attss ~~ thmss)
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  ##> Theory.checkpoint;
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fun store_thmss label tnames = store_thmss_atts label tnames (replicate (length tnames) []);
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fun store_thms_atts label tnames attss thmss =
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  fold_map (fn ((tname, atts), thms) =>
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    Sign.add_path tname
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    #> PureThy.add_thms [((Binding.name label, thms), atts)]
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    #-> (fn thm::_ => Sign.parent_path #> pair thm)) (tnames ~~ attss ~~ thmss)
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  ##> Theory.checkpoint;
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fun store_thms label tnames = store_thms_atts label tnames (replicate (length tnames) []);
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(* split theorem thm_1 & ... & thm_n into n theorems *)
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fun split_conj_thm th =
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  ((th RS conjunct1)::(split_conj_thm (th RS conjunct2))) handle THM _ => [th];
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val mk_conj = foldr1 (HOLogic.mk_binop "op &");
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val mk_disj = foldr1 (HOLogic.mk_binop "op |");
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fun app_bnds t i = list_comb (t, map Bound (i - 1 downto 0));
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fun cong_tac i st = (case Logic.strip_assums_concl
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  (List.nth (prems_of st, i - 1)) of
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    _ $ (_ $ (f $ x) $ (g $ y)) =>
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      let
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        val cong' = Thm.lift_rule (Thm.cprem_of st i) cong;
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        val _ $ (_ $ (f' $ x') $ (g' $ y')) =
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          Logic.strip_assums_concl (prop_of cong');
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        val insts = map (pairself (cterm_of (Thm.theory_of_thm st)) o
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          apsnd (curry list_abs (Logic.strip_params (concl_of cong'))) o
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            apfst head_of) [(f', f), (g', g), (x', x), (y', y)]
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      in compose_tac (false, cterm_instantiate insts cong', 2) i st
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        handle THM _ => no_tac st
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      end
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  | _ => no_tac st);
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(* instantiate induction rule *)
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fun indtac indrule indnames i st =
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  let
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    val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
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    val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
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      (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
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    val getP = if can HOLogic.dest_imp (hd ts) then
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      (apfst SOME) o HOLogic.dest_imp else pair NONE;
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    val flt = if null indnames then I else
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      filter (fn Free (s, _) => s mem indnames | _ => false);
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    fun abstr (t1, t2) = (case t1 of
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        NONE => (case flt (OldTerm.term_frees t2) of
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            [Free (s, T)] => SOME (absfree (s, T, t2))
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          | _ => NONE)
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      | SOME (_ $ t') => SOME (Abs ("x", fastype_of t', abstract_over (t', t2))))
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    val cert = cterm_of (Thm.theory_of_thm st);
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    val insts = List.mapPartial (fn (t, u) => case abstr (getP u) of
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        NONE => NONE
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      | SOME u' => SOME (t |> getP |> snd |> head_of |> cert, cert u')) (ts ~~ ts');
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    val indrule' = cterm_instantiate insts indrule
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  in
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    rtac indrule' i st
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  end;
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(* perform exhaustive case analysis on last parameter of subgoal i *)
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fun exh_tac exh_thm_of i state =
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  let
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    val thy = Thm.theory_of_thm state;
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    val prem = nth (prems_of state) (i - 1);
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    val params = Logic.strip_params prem;
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    val (_, Type (tname, _)) = hd (rev params);
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    val exhaustion = Thm.lift_rule (Thm.cprem_of state i) (exh_thm_of tname);
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    val prem' = hd (prems_of exhaustion);
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    val _ $ (_ $ lhs $ _) = hd (rev (Logic.strip_assums_hyp prem'));
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    val exhaustion' = cterm_instantiate [(cterm_of thy (head_of lhs),
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      cterm_of thy (List.foldr (fn ((_, T), t) => Abs ("z", T, t))
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        (Bound 0) params))] exhaustion
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  in compose_tac (false, exhaustion', nprems_of exhaustion) i state
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  end;
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(* handling of distinctness theorems *)
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datatype simproc_dist = FewConstrs of thm list
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                      | ManyConstrs of thm * simpset;
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(********************** Internal description of datatypes *********************)
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datatype dtyp =
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    DtTFree of string
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  | DtType of string * (dtyp list)
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  | DtRec of int;
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(* information about datatypes *)
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(* index, datatype name, type arguments, constructor name, types of constructor's arguments *)
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type descr = (int * (string * dtyp list * (string * dtyp list) list)) list;
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type datatype_info =
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  {index : int,
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   alt_names : string list option,
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   descr : descr,
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   sorts : (string * sort) list,
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   rec_names : string list,
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   rec_rewrites : thm list,
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   case_name : string,
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   case_rewrites : thm list,
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   induction : thm,
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   exhaustion : thm,
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   distinct : simproc_dist,
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   inject : thm list,
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   nchotomy : thm,
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   case_cong : thm,
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   weak_case_cong : thm};
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fun mk_Free s T i = Free (s ^ (string_of_int i), T);
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fun subst_DtTFree _ substs (T as (DtTFree name)) =
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      AList.lookup (op =) substs name |> the_default T
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  | subst_DtTFree i substs (DtType (name, ts)) =
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      DtType (name, map (subst_DtTFree i substs) ts)
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  | subst_DtTFree i _ (DtRec j) = DtRec (i + j);
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exception Datatype;
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exception Datatype_Empty of string;
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fun dest_DtTFree (DtTFree a) = a
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  | dest_DtTFree _ = raise Datatype;
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fun dest_DtRec (DtRec i) = i
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  | dest_DtRec _ = raise Datatype;
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fun is_rec_type (DtType (_, dts)) = exists is_rec_type dts
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  | is_rec_type (DtRec _) = true
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  | is_rec_type _ = false;
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fun strip_dtyp (DtType ("fun", [T, U])) = apfst (cons T) (strip_dtyp U)
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  | strip_dtyp T = ([], T);
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val body_index = dest_DtRec o snd o strip_dtyp;
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fun mk_fun_dtyp [] U = U
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  | mk_fun_dtyp (T :: Ts) U = DtType ("fun", [T, mk_fun_dtyp Ts U]);
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fun name_of_typ (Type (s, Ts)) =
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      let val s' = Sign.base_name s
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      in space_implode "_" (List.filter (not o equal "") (map name_of_typ Ts) @
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        [if Syntax.is_identifier s' then s' else "x"])
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      end
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  | name_of_typ _ = "";
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fun dtyp_of_typ _ (TFree (n, _)) = DtTFree n
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  | dtyp_of_typ _ (TVar _) = error "Illegal schematic type variable(s)"
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  | dtyp_of_typ new_dts (Type (tname, Ts)) =
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      (case AList.lookup (op =) new_dts tname of
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         NONE => DtType (tname, map (dtyp_of_typ new_dts) Ts)
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       | SOME vs => if map (try (fst o dest_TFree)) Ts = map SOME vs then
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             DtRec (find_index (curry op = tname o fst) new_dts)
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           else error ("Illegal occurrence of recursive type " ^ tname));
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fun typ_of_dtyp descr sorts (DtTFree a) = TFree (a, (the o AList.lookup (op =) sorts) a)
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  | typ_of_dtyp descr sorts (DtRec i) =
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      let val (s, ds, _) = (the o AList.lookup (op =) descr) i
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      in Type (s, map (typ_of_dtyp descr sorts) ds) end
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  | typ_of_dtyp descr sorts (DtType (s, ds)) =
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      Type (s, map (typ_of_dtyp descr sorts) ds);
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(* find all non-recursive types in datatype description *)
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fun get_nonrec_types descr sorts =
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  map (typ_of_dtyp descr sorts) (Library.foldl (fn (Ts, (_, (_, _, constrs))) =>
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    Library.foldl (fn (Ts', (_, cargs)) =>
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      filter_out is_rec_type cargs union Ts') (Ts, constrs)) ([], descr));
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(* get all recursive types in datatype description *)
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fun get_rec_types descr sorts = map (fn (_ , (s, ds, _)) =>
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  Type (s, map (typ_of_dtyp descr sorts) ds)) descr;
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(* get all branching types *)
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fun get_branching_types descr sorts =
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  map (typ_of_dtyp descr sorts) (fold (fn (_, (_, _, constrs)) =>
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    fold (fn (_, cargs) => fold (strip_dtyp #> fst #> fold (insert op =)) cargs)
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      constrs) descr []);
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fun get_arities descr = fold (fn (_, (_, _, constrs)) =>
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  fold (fn (_, cargs) => fold (insert op =) (map (length o fst o strip_dtyp)
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    (List.filter is_rec_type cargs))) constrs) descr [];
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(* nonemptiness check for datatypes *)
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fun check_nonempty descr =
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  let
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    val descr' = List.concat descr;
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    fun is_nonempty_dt is i =
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      let
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        val (_, _, constrs) = (the o AList.lookup (op =) descr') i;
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        fun arg_nonempty (_, DtRec i) = if i mem is then false
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              else is_nonempty_dt (i::is) i
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          | arg_nonempty _ = true;
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      in exists ((forall (arg_nonempty o strip_dtyp)) o snd) constrs
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      end
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  in assert_all (fn (i, _) => is_nonempty_dt [i] i) (hd descr)
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    (fn (_, (s, _, _)) => raise Datatype_Empty s)
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  end;
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(* unfold a list of mutually recursive datatype specifications *)
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(* all types of the form DtType (dt_name, [..., DtRec _, ...]) *)
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(* need to be unfolded                                         *)
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fun unfold_datatypes sign orig_descr sorts (dt_info : datatype_info Symtab.table) descr i =
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  let
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    fun typ_error T msg = error ("Non-admissible type expression\n" ^
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      Syntax.string_of_typ_global sign (typ_of_dtyp (orig_descr @ descr) sorts T) ^ "\n" ^ msg);
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    fun get_dt_descr T i tname dts =
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      (case Symtab.lookup dt_info tname of
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         NONE => typ_error T (tname ^ " is not a datatype - can't use it in\
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           \ nested recursion")
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       | (SOME {index, descr, ...}) =>
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           let val (_, vars, _) = (the o AList.lookup (op =) descr) index;
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               val subst = ((map dest_DtTFree vars) ~~ dts) handle Library.UnequalLengths =>
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                 typ_error T ("Type constructor " ^ tname ^ " used with wrong\
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                  \ number of arguments")
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           in (i + index, map (fn (j, (tn, args, cs)) => (i + j,
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             (tn, map (subst_DtTFree i subst) args,
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              map (apsnd (map (subst_DtTFree i subst))) cs))) descr)
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           end);
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    (* unfold a single constructor argument *)
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    fun unfold_arg ((i, Ts, descrs), T) =
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      if is_rec_type T then
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        let val (Us, U) = strip_dtyp T
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        in if exists is_rec_type Us then
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            typ_error T "Non-strictly positive recursive occurrence of type"
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          else (case U of
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              DtType (tname, dts) =>  
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                let
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                  val (index, descr) = get_dt_descr T i tname dts;
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                  val (descr', i') = unfold_datatypes sign orig_descr sorts
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                    dt_info descr (i + length descr)
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                in (i', Ts @ [mk_fun_dtyp Us (DtRec index)], descrs @ descr') end
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            | _ => (i, Ts @ [T], descrs))
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        end
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      else (i, Ts @ [T], descrs);
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    (* unfold a constructor *)
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    fun unfold_constr ((i, constrs, descrs), (cname, cargs)) =
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      let val (i', cargs', descrs') = Library.foldl unfold_arg ((i, [], descrs), cargs)
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      in (i', constrs @ [(cname, cargs')], descrs') end;
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    (* unfold a single datatype *)
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    fun unfold_datatype ((i, dtypes, descrs), (j, (tname, tvars, constrs))) =
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      let val (i', constrs', descrs') =
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        Library.foldl unfold_constr ((i, [], descrs), constrs)
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      in (i', dtypes @ [(j, (tname, tvars, constrs'))], descrs')
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      end;
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    val (i', descr', descrs) = Library.foldl unfold_datatype ((i, [],[]), descr);
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  in (descr' :: descrs, i') end;
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end;