(*  Title:      HOL/BNF/Tools/bnf_fp_util.ML

     Author:     Dmitriy Traytel, TU Muenchen

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013

 Shared library for the datatype and codatatype constructions.

 *)

 signature BNF_FP_UTIL =

 sig

   datatype fp_kind = Least_FP | Greatest_FP

   type fp_result =

     {Ts: typ list,

      bnfs: BNF_Def.bnf list,

      ctors: term list,

      dtors: term list,

      xtor_un_folds: term list,

      xtor_co_recs: term list,

      xtor_co_induct: thm,

      xtor_strong_co_induct: thm,

      dtor_ctors: thm list,

      ctor_dtors: thm list,

      ctor_injects: thm list,

      xtor_map_thms: thm list,

      xtor_set_thmss: thm list list,

      xtor_rel_thms: thm list,

      xtor_un_fold_thms: thm list,

      xtor_co_rec_thms: thm list}

   val morph_fp_result: morphism -> fp_result -> fp_result

   val eq_fp_result: fp_result * fp_result -> bool

   val time: Timer.real_timer -> string -> Timer.real_timer

   val IITN: string

   val LevN: string

   val algN: string

   val behN: string

   val bisN: string

   val carTN: string

   val caseN: string

   val coN: string

   val coinductN: string

   val corecN: string

   val ctorN: string

   val ctor_dtorN: string

   val ctor_dtor_corecN: string

   val ctor_dtor_unfoldN: string

   val ctor_exhaustN: string

   val ctor_induct2N: string

   val ctor_inductN: string

   val ctor_injectN: string

   val ctor_foldN: string

   val ctor_fold_uniqueN: string

   val ctor_mapN: string

   val ctor_map_uniqueN: string

   val ctor_recN: string

   val ctor_rec_uniqueN: string

   val ctor_relN: string

   val ctor_set_inclN: string

   val ctor_set_set_inclN: string

   val disc_unfoldN: string

   val disc_unfold_iffN: string

   val disc_corecN: string

   val disc_corec_iffN: string

   val dtorN: string

   val dtor_coinductN: string

   val dtor_corecN: string

   val dtor_corec_uniqueN: string

   val dtor_ctorN: string

   val dtor_exhaustN: string

   val dtor_injectN: string

   val dtor_mapN: string

   val dtor_map_coinductN: string

   val dtor_map_strong_coinductN: string

   val dtor_map_uniqueN: string

   val dtor_relN: string

   val dtor_set_inclN: string

   val dtor_set_set_inclN: string

   val dtor_strong_coinductN: string

   val dtor_unfoldN: string

   val dtor_unfold_uniqueN: string

   val exhaustN: string

   val foldN: string

   val hsetN: string

   val hset_recN: string

   val inductN: string

   val injectN: string

   val isNodeN: string

   val lsbisN: string

   val mapN: string

   val map_uniqueN: string

   val min_algN: string

   val morN: string

   val nchotomyN: string

   val recN: string

   val rel_coinductN: string

   val rel_inductN: string

   val rel_injectN: string

   val rel_distinctN: string

   val rvN: string

   val sel_corecN: string

   val set_inclN: string

   val set_set_inclN: string

   val sel_unfoldN: string

   val setsN: string

   val simpsN: string

   val strTN: string

   val str_initN: string

   val strong_coinductN: string

   val sum_bdN: string

   val sum_bdTN: string

   val unfoldN: string

   val uniqueN: string

   (* TODO: Don't index set facts. Isabelle packages traditionally generate uniform names. *)

   val mk_ctor_setN: int -> string

   val mk_dtor_setN: int -> string

   val mk_dtor_set_inductN: int -> string

   val mk_set_inductN: int -> string

   val datatype_word: fp_kind -> string

   val base_name_of_typ: typ -> string

   val mk_common_name: string list -> string

   val variant_types: string list -> sort list -> Proof.context ->

     (string * sort) list * Proof.context

   val variant_tfrees: string list -> Proof.context -> typ list * Proof.context

   val split_conj_thm: thm -> thm list

   val split_conj_prems: int -> thm -> thm

   val mk_sumTN: typ list -> typ

   val mk_sumTN_balanced: typ list -> typ

   val id_const: typ -> term

   val Inl_const: typ -> typ -> term

   val Inr_const: typ -> typ -> term

   val mk_Inl: typ -> term -> term

   val mk_Inr: typ -> term -> term

   val mk_InN: typ list -> term -> int -> term

   val mk_InN_balanced: typ -> int -> term -> int -> term

   val mk_sum_case: term * term -> term

   val mk_sum_caseN: term list -> term

   val mk_sum_caseN_balanced: term list -> term

   val dest_sumT: typ -> typ * typ

   val dest_sumTN: int -> typ -> typ list

   val dest_sumTN_balanced: int -> typ -> typ list

   val dest_tupleT: int -> typ -> typ list

   val mk_Field: term -> term

   val mk_If: term -> term -> term -> term

   val mk_union: term * term -> term

   val mk_sumEN: int -> thm

   val mk_sumEN_balanced: int -> thm

   val mk_sumEN_tupled_balanced: int list -> thm

   val mk_sum_casesN: int -> int -> thm

   val mk_sum_casesN_balanced: int -> int -> thm

   val fixpoint: ('a * 'a -> bool) -> ('a list -> 'a list) -> 'a list -> 'a list

   val fp_bnf: (binding list -> (string * sort) list -> typ list * typ list list ->

       BNF_Def.bnf list -> local_theory -> 'a) ->

     binding list -> (string * sort) list -> ((string * sort) * typ) list -> local_theory ->

     BNF_Def.bnf list * 'a

 end;

 structure BNF_FP_Util : BNF_FP_UTIL =

 struct

 open BNF_Comp

 open BNF_Def

 open BNF_Util

 datatype fp_kind = Least_FP | Greatest_FP;

 type fp_result =

   {Ts: typ list,

    bnfs: BNF_Def.bnf list,

    ctors: term list,

    dtors: term list,

    xtor_un_folds: term list,

    xtor_co_recs: term list,

    xtor_co_induct: thm,

    xtor_strong_co_induct: thm,

    dtor_ctors: thm list,

    ctor_dtors: thm list,

    ctor_injects: thm list,

    xtor_map_thms: thm list,

    xtor_set_thmss: thm list list,

    xtor_rel_thms: thm list,

    xtor_un_fold_thms: thm list,

    xtor_co_rec_thms: thm list};

 fun morph_fp_result phi {Ts, bnfs, ctors, dtors, xtor_un_folds, xtor_co_recs, xtor_co_induct,

     xtor_strong_co_induct, dtor_ctors, ctor_dtors, ctor_injects, xtor_map_thms, xtor_set_thmss,

     xtor_rel_thms, xtor_un_fold_thms, xtor_co_rec_thms} =

   {Ts = map (Morphism.typ phi) Ts,

    bnfs = map (morph_bnf phi) bnfs,

    ctors = map (Morphism.term phi) ctors,

    dtors = map (Morphism.term phi) dtors,

    xtor_un_folds = map (Morphism.term phi) xtor_un_folds,

    xtor_co_recs = map (Morphism.term phi) xtor_co_recs,

    xtor_co_induct = Morphism.thm phi xtor_co_induct,

    xtor_strong_co_induct = Morphism.thm phi xtor_strong_co_induct,

    dtor_ctors = map (Morphism.thm phi) dtor_ctors,

    ctor_dtors = map (Morphism.thm phi) ctor_dtors,

    ctor_injects = map (Morphism.thm phi) ctor_injects,

    xtor_map_thms = map (Morphism.thm phi) xtor_map_thms,

    xtor_set_thmss = map (map (Morphism.thm phi)) xtor_set_thmss,

    xtor_rel_thms = map (Morphism.thm phi) xtor_rel_thms,

    xtor_un_fold_thms = map (Morphism.thm phi) xtor_un_fold_thms,

    xtor_co_rec_thms = map (Morphism.thm phi) xtor_co_rec_thms};

 fun eq_fp_result ({bnfs = bnfs1, ...} : fp_result, {bnfs = bnfs2, ...} : fp_result) =

   eq_list eq_bnf (bnfs1, bnfs2);

 val timing = true;

 fun time timer msg = (if timing

   then warning (msg ^ ": " ^ ATP_Util.string_of_time (Timer.checkRealTimer timer))

   else (); Timer.startRealTimer ());

 val preN = "pre_"

 val rawN = "raw_"

 val coN = "co"

 val unN = "un"

 val algN = "alg"

 val IITN = "IITN"

 val foldN = "fold"

 val unfoldN = unN ^ foldN

 val uniqueN = "_unique"

 val simpsN = "simps"

 val ctorN = "ctor"

 val dtorN = "dtor"

 val ctor_foldN = ctorN ^ "_" ^ foldN

 val dtor_unfoldN = dtorN ^ "_" ^ unfoldN

 val ctor_fold_uniqueN = ctor_foldN ^ uniqueN

 val dtor_unfold_uniqueN = dtor_unfoldN ^ uniqueN

 val ctor_dtor_unfoldN = ctorN ^ "_" ^ dtor_unfoldN

 val ctor_mapN = ctorN ^ "_" ^ mapN

 val dtor_mapN = dtorN ^ "_" ^ mapN

 val map_uniqueN = mapN ^ uniqueN

 val ctor_map_uniqueN = ctorN ^ "_" ^ map_uniqueN

 val dtor_map_uniqueN = dtorN ^ "_" ^ map_uniqueN

 val min_algN = "min_alg"

 val morN = "mor"

 val bisN = "bis"

 val lsbisN = "lsbis"

 val sum_bdTN = "sbdT"

 val sum_bdN = "sbd"

 val carTN = "carT"

 val strTN = "strT"

 val isNodeN = "isNode"

 val LevN = "Lev"

 val rvN = "recover"

 val behN = "beh"

 val setsN = "sets"

 val mk_ctor_setN = prefix (ctorN ^ "_") o mk_setN

 val mk_dtor_setN = prefix (dtorN ^ "_") o mk_setN

 fun mk_set_inductN i = mk_setN i ^ "_induct"

 val mk_dtor_set_inductN = prefix (dtorN ^ "_") o mk_set_inductN

 val str_initN = "str_init"

 val recN = "rec"

 val corecN = coN ^ recN

 val ctor_recN = ctorN ^ "_" ^ recN

 val ctor_rec_uniqueN = ctor_recN ^ uniqueN

 val dtor_corecN = dtorN ^ "_" ^ corecN

 val dtor_corec_uniqueN = dtor_corecN ^ uniqueN

 val ctor_dtor_corecN = ctorN ^ "_" ^ dtor_corecN

 val ctor_dtorN = ctorN ^ "_" ^ dtorN

 val dtor_ctorN = dtorN ^ "_" ^ ctorN

 val nchotomyN = "nchotomy"

 val injectN = "inject"

 val exhaustN = "exhaust"

 val ctor_injectN = ctorN ^ "_" ^ injectN

 val ctor_exhaustN = ctorN ^ "_" ^ exhaustN

 val dtor_injectN = dtorN ^ "_" ^ injectN

 val dtor_exhaustN = dtorN ^ "_" ^ exhaustN

 val ctor_relN = ctorN ^ "_" ^ relN

 val dtor_relN = dtorN ^ "_" ^ relN

 val inductN = "induct"

 val coinductN = coN ^ inductN

 val ctor_inductN = ctorN ^ "_" ^ inductN

 val ctor_induct2N = ctor_inductN ^ "2"

 val dtor_map_coinductN = dtor_mapN ^ "_" ^ coinductN

 val dtor_coinductN = dtorN ^ "_" ^ coinductN

 val strong_coinductN = "strong_" ^ coinductN

 val dtor_map_strong_coinductN = dtor_mapN ^ "_" ^ strong_coinductN

 val dtor_strong_coinductN = dtorN ^ "_" ^ strong_coinductN

 val hsetN = "Hset"

 val hset_recN = hsetN ^ "_rec"

 val set_inclN = "set_incl"

 val ctor_set_inclN = ctorN ^ "_" ^ set_inclN

 val dtor_set_inclN = dtorN ^ "_" ^ set_inclN

 val set_set_inclN = "set_set_incl"

 val ctor_set_set_inclN = ctorN ^ "_" ^ set_set_inclN

 val dtor_set_set_inclN = dtorN ^ "_" ^ set_set_inclN

 val caseN = "case"

 val discN = "disc"

 val disc_unfoldN = discN ^ "_" ^ unfoldN

 val disc_corecN = discN ^ "_" ^ corecN

 val iffN = "_iff"

 val disc_unfold_iffN = discN ^ "_" ^ unfoldN ^ iffN

 val disc_corec_iffN = discN ^ "_" ^ corecN ^ iffN

 val distinctN = "distinct"

 val rel_distinctN = relN ^ "_" ^ distinctN

 val injectN = "inject"

 val rel_injectN = relN ^ "_" ^ injectN

 val rel_coinductN = relN ^ "_" ^ coinductN

 val rel_inductN = relN ^ "_" ^ inductN

 val selN = "sel"

 val sel_unfoldN = selN ^ "_" ^ unfoldN

 val sel_corecN = selN ^ "_" ^ corecN

 fun datatype_word fp = (if fp = Greatest_FP then "co" else "") ^ "datatype";

 fun add_components_of_typ (Type (s, Ts)) =

     fold add_components_of_typ Ts #> cons (Long_Name.base_name s)

   | add_components_of_typ _ = I;

 fun base_name_of_typ T = space_implode "_" (add_components_of_typ T []);

 val mk_common_name = space_implode "_";

 fun dest_sumT (Type (@{type_name sum}, [T, T'])) = (T, T');

 fun dest_sumTN 1 T = [T]

   | dest_sumTN n (Type (@{type_name sum}, [T, T'])) = T :: dest_sumTN (n - 1) T';

 val dest_sumTN_balanced = Balanced_Tree.dest dest_sumT;

 (* TODO: move something like this to "HOLogic"? *)

 fun dest_tupleT 0 @{typ unit} = []

   | dest_tupleT 1 T = [T]

   | dest_tupleT n (Type (@{type_name prod}, [T, T'])) = T :: dest_tupleT (n - 1) T';

 val mk_sumTN = Library.foldr1 mk_sumT;

 val mk_sumTN_balanced = Balanced_Tree.make mk_sumT;

 fun id_const T = Const (@{const_name id}, T --> T);

 fun Inl_const LT RT = Const (@{const_name Inl}, LT --> mk_sumT (LT, RT));

 fun mk_Inl RT t = Inl_const (fastype_of t) RT $ t;

 fun Inr_const LT RT = Const (@{const_name Inr}, RT --> mk_sumT (LT, RT));

 fun mk_Inr LT t = Inr_const LT (fastype_of t) $ t;

 fun mk_InN [_] t 1 = t

   | mk_InN (_ :: Ts) t 1 = mk_Inl (mk_sumTN Ts) t

   | mk_InN (LT :: Ts) t m = mk_Inr LT (mk_InN Ts t (m - 1))

   | mk_InN Ts t _ = raise (TYPE ("mk_InN", Ts, [t]));

 fun mk_InN_balanced sum_T n t k =

   let

     fun repair_types T (Const (s as @{const_name Inl}, _) $ t) = repair_inj_types T s fst t

       | repair_types T (Const (s as @{const_name Inr}, _) $ t) = repair_inj_types T s snd t

       | repair_types _ t = t

     and repair_inj_types T s get t =

       let val T' = get (dest_sumT T) in

         Const (s, T' --> T) $ repair_types T' t

       end;

   in

     Balanced_Tree.access {left = mk_Inl dummyT, right = mk_Inr dummyT, init = t} n k

     |> repair_types sum_T

   end;

 fun mk_sum_case (f, g) =

   let

     val fT = fastype_of f;

     val gT = fastype_of g;

   in

     Const (@{const_name sum_case},

       fT --> gT --> mk_sumT (domain_type fT, domain_type gT) --> range_type fT) $ f $ g

   end;

 val mk_sum_caseN = Library.foldr1 mk_sum_case;

 val mk_sum_caseN_balanced = Balanced_Tree.make mk_sum_case;

 fun mk_If p t f =

   let val T = fastype_of t;

   in Const (@{const_name If}, HOLogic.boolT --> T --> T --> T) $ p $ t $ f end;

 fun mk_Field r =

   let val T = fst (dest_relT (fastype_of r));

   in Const (@{const_name Field}, mk_relT (T, T) --> HOLogic.mk_setT T) $ r end;

 val mk_union = HOLogic.mk_binop @{const_name sup};

 (*dangerous; use with monotonic, converging functions only!*)

 fun fixpoint eq f X = if subset eq (f X, X) then X else fixpoint eq f (f X);

 fun variant_types ss Ss ctxt =

   let

     val (tfrees, _) =

       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);

     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;

   in (tfrees, ctxt') end;

 fun variant_tfrees ss =

   apfst (map TFree) o variant_types (map (prefix "'") ss) (replicate (length ss) HOLogic.typeS);

 (* stolen from "~~/src/HOL/Tools/Datatype/datatype_aux.ML" *)

 fun split_conj_thm th =

   ((th RS conjunct1) :: split_conj_thm (th RS conjunct2)) handle THM _ => [th];

 fun split_conj_prems limit th =

   let

     fun split n i th =

       if i = n then th else split n (i + 1) (conjI RSN (i, th)) handle THM _ => th;

   in split limit 1 th end;

 fun mk_sumEN 1 = @{thm one_pointE}

   | mk_sumEN 2 = @{thm sumE}

   | mk_sumEN n =

     (fold (fn i => fn thm => @{thm obj_sum_step} RSN (i, thm)) (2 upto n - 1) @{thm obj_sumE}) OF

       replicate n (impI RS allI);

 fun mk_obj_sumEN_balanced n =

   Balanced_Tree.make (fn (thm1, thm2) => thm1 RSN (1, thm2 RSN (2, @{thm obj_sumE_f})))

     (replicate n asm_rl);

 fun mk_sumEN_balanced' n all_impIs = mk_obj_sumEN_balanced n OF all_impIs RS @{thm obj_one_pointE};

 fun mk_sumEN_balanced 1 = @{thm one_pointE} (*optimization*)

   | mk_sumEN_balanced 2 = @{thm sumE} (*optimization*)

   | mk_sumEN_balanced n = mk_sumEN_balanced' n (replicate n (impI RS allI));

 fun mk_tupled_allIN 0 = @{thm unit_all_impI}

   | mk_tupled_allIN 1 = @{thm impI[THEN allI]}

   | mk_tupled_allIN 2 = @{thm prod_all_impI} (*optimization*)

   | mk_tupled_allIN n = mk_tupled_allIN (n - 1) RS @{thm prod_all_impI_step};

 fun mk_sumEN_tupled_balanced ms =

   let val n = length ms in

     if forall (curry (op =) 1) ms then mk_sumEN_balanced n

     else mk_sumEN_balanced' n (map mk_tupled_allIN ms)

   end;

 fun mk_sum_casesN 1 1 = refl

   | mk_sum_casesN _ 1 = @{thm sum.cases(1)}

   | mk_sum_casesN 2 2 = @{thm sum.cases(2)}

   | mk_sum_casesN n k = trans OF [@{thm sum_case_step(2)}, mk_sum_casesN (n - 1) (k - 1)];

 fun mk_sum_step base step thm =

   if Thm.eq_thm_prop (thm, refl) then base else trans OF [step, thm];

 fun mk_sum_casesN_balanced 1 1 = refl

   | mk_sum_casesN_balanced n k =

     Balanced_Tree.access {left = mk_sum_step @{thm sum.cases(1)} @{thm sum_case_step(1)},

       right = mk_sum_step @{thm sum.cases(2)} @{thm sum_case_step(2)}, init = refl} n k;

 fun fp_bnf construct_fp bs resBs eqs lthy =

   let

     val timer = time (Timer.startRealTimer ());

     val (lhss, rhss) = split_list eqs;

     (* FIXME: because of "@ lhss", the output could contain type variables that are not in the

        input; also, "fp_sort" should put the "resBs" first and in the order in which they appear *)

     fun fp_sort Ass =

       subtract (op =) lhss (filter (fn T => exists (fn Ts => member (op =) Ts T) Ass) resBs) @ lhss;

     fun raw_qualify b = Binding.qualify true (Binding.name_of (Binding.prefix_name rawN b));

     val ((bnfs, (deadss, livess)), (unfold_set, lthy)) = apfst (apsnd split_list o split_list)

       (fold_map2 (fn b => bnf_of_typ Smart_Inline (raw_qualify b) fp_sort) bs rhss

         (empty_unfolds, lthy));

     val name = mk_common_name (map Binding.name_of bs);

     fun qualify i =

       let val namei = name ^ nonzero_string_of_int i;

       in Binding.qualify true namei end;

     val Ass = map (map dest_TFree) livess;

     val resDs = fold (subtract (op =)) Ass resBs;

     val Ds = fold (fold Term.add_tfreesT) deadss [];

     val _ = (case Library.inter (op =) Ds lhss of [] => ()

       | A :: _ => error ("Inadmissible type recursion (cannot take fixed point of dead type \

         \variable " ^ quote (Syntax.string_of_typ lthy (TFree A)) ^ ")"));

     val timer = time (timer "Construction of BNFs");

     val ((kill_poss, _), (bnfs', (unfold_set', lthy'))) =

       normalize_bnfs qualify Ass Ds fp_sort bnfs unfold_set lthy;

     val Dss = map3 (append oo map o nth) livess kill_poss deadss;

     val ((pre_bnfs, deadss), lthy'') =

       fold_map3 (seal_bnf unfold_set') (map (Binding.prefix_name preN) bs) Dss bnfs' lthy'

       |>> split_list;

     val timer = time (timer "Normalization & sealing of BNFs");

     val res = construct_fp bs resBs (map TFree resDs, deadss) pre_bnfs lthy'';

     val timer = time (timer "FP construction in total");

   in

     timer; (pre_bnfs, res)

   end;

 end;