(*  Title:      HOL/Codatatype/Tools/bnf_util.ML

    Author:     Dmitriy Traytel, TU Muenchen

    Copyright   2012

Library for bounded natural functors.

*)

signature BNF_UTIL =

sig

  val map3: ('a -> 'b -> 'c -> 'd) -> 'a list -> 'b list -> 'c list -> 'd list

  val map4: ('a -> 'b -> 'c -> 'd -> 'e) -> 'a list -> 'b list -> 'c list -> 'd list -> 'e list

  val map5: ('a -> 'b -> 'c -> 'd -> 'e -> 'f) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list

  val map6: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list

  val map7: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list

  val map8: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list -> 'i list

  val map9: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->

    'i list -> 'j list

  val map10: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->

    'i list -> 'j list -> 'k list

  val map11: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k -> 'l) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->

    'i list -> 'j list -> 'k list -> 'l list

  val map12: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k -> 'l -> 'm) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->

    'i list -> 'j list -> 'k list -> 'l list -> 'm list

  val fold_map2: ('a -> 'b -> 'c -> 'd * 'c) -> 'a list -> 'b list -> 'c -> 'd list * 'c

  val fold_map3: ('a -> 'b -> 'c -> 'd -> 'e * 'd) ->

    'a list -> 'b list -> 'c list -> 'd -> 'e list * 'd

  val fold_map4: ('a -> 'b -> 'c -> 'd -> 'e -> 'f * 'e) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e -> 'f list * 'e

  val fold_map5: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g * 'f) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f -> 'g list * 'f

  val fold_map6: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h * 'g) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g -> 'h list * 'g

  val fold_map7: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i * 'h) ->

    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h -> 'i list * 'h

  val interleave: 'a list -> 'a list -> 'a list

  val transpose: 'a list list -> 'a list list

  val seq_conds: (bool -> 'a -> 'b) -> int -> int -> 'a list -> 'b list

  val mk_fresh_names: Proof.context -> int -> string -> string list * Proof.context

  val mk_TFrees: int -> Proof.context -> typ list * Proof.context

  val mk_TFreess: int list -> Proof.context -> typ list list * Proof.context

  val mk_TFrees': sort list -> Proof.context -> typ list * Proof.context

  val mk_Frees: string -> typ list -> Proof.context -> term list * Proof.context

  val mk_Freess: string -> typ list list -> Proof.context -> term list list * Proof.context

  val mk_Freesss: string -> typ list list list -> Proof.context ->

    term list list list * Proof.context

  val mk_Freessss: string -> typ list list list list -> Proof.context ->

    term list list list list * Proof.context

  val mk_Frees': string -> typ list -> Proof.context ->

    (term list * (string * typ) list) * Proof.context

  val mk_Freess': string -> typ list list -> Proof.context ->

    (term list list * (string * typ) list list) * Proof.context

  val nonzero_string_of_int: int -> string

  val strip_typeN: int -> typ -> typ list * typ

  val mk_optionT: typ -> typ

  val mk_relT: typ * typ -> typ

  val dest_relT: typ -> typ * typ

  val mk_sumT: typ * typ -> typ

  val ctwo: term

  val fst_const: typ -> term

  val snd_const: typ -> term

  val Id_const: typ -> term

  val mk_Ball: term -> term -> term

  val mk_Bex: term -> term -> term

  val mk_Card_order: term -> term

  val mk_Field: term -> term

  val mk_Gr: term -> term -> term

  val mk_IfN: typ -> term list -> term list -> term

  val mk_Trueprop_eq: term * term -> term

  val mk_UNION: term -> term -> term

  val mk_Union: typ -> term

  val mk_card_binop: string -> (typ * typ -> typ) -> term -> term -> term

  val mk_card_of: term -> term

  val mk_card_order: term -> term

  val mk_ccexp: term -> term -> term

  val mk_cexp: term -> term -> term

  val mk_cinfinite: term -> term

  val mk_collect: term list -> typ -> term

  val mk_converse: term -> term

  val mk_cprod: term -> term -> term

  val mk_csum: term -> term -> term

  val mk_dir_image: term -> term -> term

  val mk_image: term -> term

  val mk_in: term list -> term list -> typ -> term

  val mk_ordLeq: term -> term -> term

  val mk_rel_comp: term * term -> term

  val mk_subset: term -> term -> term

  val mk_wpull: term -> term -> term -> term -> term -> (term * term) option -> term -> term -> term

  val list_all_free: term list -> term -> term

  val list_exists_free: term list -> term -> term

  (*parameterized terms*)

  val mk_nthN: int -> term -> int -> term

  (*parameterized thms*)

  val mk_Un_upper: int -> int -> thm

  val mk_conjIN: int -> thm

  val mk_conjunctN: int -> int -> thm

  val conj_dests: int -> thm -> thm list

  val mk_disjIN: int -> int -> thm

  val mk_nthI: int -> int -> thm

  val mk_nth_conv: int -> int -> thm

  val mk_ordLeq_csum: int -> int -> thm -> thm

  val mk_UnIN: int -> int -> thm

  val ctrans: thm

  val o_apply: thm

  val mk_sym: thm -> thm

  val mk_trans: thm -> thm -> thm

  val mk_unabs_def: int -> thm -> thm

  val mk_permute: ''a list -> ''a list -> 'b list -> 'b list

  val find_indices: ''a list -> ''a list -> int list

  val certifyT: Proof.context -> typ -> ctyp

  val certify: Proof.context -> term -> cterm

  val parse_binding_colon: Token.T list -> binding * Token.T list

  val parse_opt_binding_colon: Token.T list -> binding * Token.T list

  val typedef: bool -> binding option -> binding * (string * sort) list * mixfix -> term ->

    (binding * binding) option -> tactic -> local_theory -> (string * Typedef.info) * local_theory

  val WRAP: ('a -> tactic) -> ('a -> tactic) -> 'a list -> tactic -> tactic

  val WRAP': ('a -> int -> tactic) -> ('a -> int -> tactic) -> 'a list -> (int -> tactic) -> int ->

    tactic

  val CONJ_WRAP_GEN: tactic -> ('a -> tactic) -> 'a list -> tactic

  val CONJ_WRAP_GEN': (int -> tactic) -> ('a -> int -> tactic) -> 'a list -> int -> tactic

  val CONJ_WRAP: ('a -> tactic) -> 'a list -> tactic

  val CONJ_WRAP': ('a -> int -> tactic) -> 'a list -> int -> tactic

end;

structure BNF_Util : BNF_UTIL =

struct

(* Library proper *)

fun map3 _ [] [] [] = []

  | map3 f (x1::x1s) (x2::x2s) (x3::x3s) = f x1 x2 x3 :: map3 f x1s x2s x3s

  | map3 _ _ _ _ = raise ListPair.UnequalLengths;

fun map4 _ [] [] [] [] = []

  | map4 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) = f x1 x2 x3 x4 :: map4 f x1s x2s x3s x4s

  | map4 _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map5 _ [] [] [] [] [] = []

  | map5 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) =

    f x1 x2 x3 x4 x5 :: map5 f x1s x2s x3s x4s x5s

  | map5 _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map6 _ [] [] [] [] [] [] = []

  | map6 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) =

    f x1 x2 x3 x4 x5 x6 :: map6 f x1s x2s x3s x4s x5s x6s

  | map6 _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map7 _ [] [] [] [] [] [] [] = []

  | map7 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) =

    f x1 x2 x3 x4 x5 x6 x7 :: map7 f x1s x2s x3s x4s x5s x6s x7s

  | map7 _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map8 _ [] [] [] [] [] [] [] [] = []

  | map8 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) (x8::x8s) =

    f x1 x2 x3 x4 x5 x6 x7 x8 :: map8 f x1s x2s x3s x4s x5s x6s x7s x8s

  | map8 _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map9 _ [] [] [] [] [] [] [] [] [] = []

  | map9 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)

      (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) =

    f x1 x2 x3 x4 x5 x6 x7 x8 x9 :: map9 f x1s x2s x3s x4s x5s x6s x7s x8s x9s

  | map9 _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map10 _ [] [] [] [] [] [] [] [] [] [] = []

  | map10 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)

      (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) =

    f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 :: map10 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s

  | map10 _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map11 _ [] [] [] [] [] [] [] [] [] [] [] = []

  | map11 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)

      (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) (x11::x11s) =

    f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 :: map11 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s x11s

  | map11 _ _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun map12 _ [] [] [] [] [] [] [] [] [] [] [] [] = []

  | map12 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s)

      (x6::x6s) (x7::x7s) (x8::x8s) (x9::x9s) (x10::x10s) (x11::x11s) (x12::x12s) =

    f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 ::

      map12 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s x11s x12s

  | map12 _ _ _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map2 _ [] [] acc = ([], acc)

  | fold_map2 f (x1::x1s) (x2::x2s) acc =

    let

      val (x, acc') = f x1 x2 acc;

      val (xs, acc'') = fold_map2 f x1s x2s acc';

    in (x :: xs, acc'') end

  | fold_map2 _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map3 _ [] [] [] acc = ([], acc)

  | fold_map3 f (x1::x1s) (x2::x2s) (x3::x3s) acc =

    let

      val (x, acc') = f x1 x2 x3 acc;

      val (xs, acc'') = fold_map3 f x1s x2s x3s acc';

    in (x :: xs, acc'') end

  | fold_map3 _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map4 _ [] [] [] [] acc = ([], acc)

  | fold_map4 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) acc =

    let

      val (x, acc') = f x1 x2 x3 x4 acc;

      val (xs, acc'') = fold_map4 f x1s x2s x3s x4s acc';

    in (x :: xs, acc'') end

  | fold_map4 _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map5 _ [] [] [] [] [] acc = ([], acc)

  | fold_map5 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) acc =

    let

      val (x, acc') = f x1 x2 x3 x4 x5 acc;

      val (xs, acc'') = fold_map5 f x1s x2s x3s x4s x5s acc';

    in (x :: xs, acc'') end

  | fold_map5 _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map6 _ [] [] [] [] [] [] acc = ([], acc)

  | fold_map6 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) acc =

    let

      val (x, acc') = f x1 x2 x3 x4 x5 x6 acc;

      val (xs, acc'') = fold_map6 f x1s x2s x3s x4s x5s x6s acc';

    in (x :: xs, acc'') end

  | fold_map6 _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map7 _ [] [] [] [] [] [] [] acc = ([], acc)

  | fold_map7 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) acc =

    let

      val (x, acc') = f x1 x2 x3 x4 x5 x6 x7 acc;

      val (xs, acc'') = fold_map7 f x1s x2s x3s x4s x5s x6s x7s acc';

    in (x :: xs, acc'') end

  | fold_map7 _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

(*stolen from ~~/src/HOL/Tools/SMT/smt_utils.ML*)

fun certify ctxt = Thm.cterm_of (Proof_Context.theory_of ctxt);

fun certifyT ctxt = Thm.ctyp_of (Proof_Context.theory_of ctxt);

val parse_binding_colon = Parse.binding --| @{keyword ":"};

val parse_opt_binding_colon = Scan.optional parse_binding_colon Binding.empty;

(*TODO: is this really different from Typedef.add_typedef_global?*)

fun typedef def opt_name typ set opt_morphs tac lthy =

  let

    val ((name, info), (lthy, lthy_old)) =

      lthy

      |> Typedef.add_typedef def opt_name typ set opt_morphs tac

      ||> `Local_Theory.restore;

    val phi = Proof_Context.export_morphism lthy_old lthy;

  in

    ((name, Typedef.transform_info phi info), lthy)

  end;

(*Tactical WRAP surrounds a static given tactic (core) with two deterministic chains of tactics*)

fun WRAP gen_before gen_after xs core_tac =

  fold_rev (fn x => fn tac => gen_before x THEN tac THEN gen_after x) xs core_tac;

fun WRAP' gen_before gen_after xs core_tac =

  fold_rev (fn x => fn tac => gen_before x THEN' tac THEN' gen_after x) xs core_tac;

fun CONJ_WRAP_GEN conj_tac gen_tac xs =

  let val (butlast, last) = split_last xs;

  in WRAP (fn thm => conj_tac THEN gen_tac thm) (K all_tac) butlast (gen_tac last) end;

fun CONJ_WRAP_GEN' conj_tac gen_tac xs =

  let val (butlast, last) = split_last xs;

  in WRAP' (fn thm => conj_tac THEN' gen_tac thm) (K (K all_tac)) butlast (gen_tac last) end;

(*not eta-converted because of monotype restriction*)

fun CONJ_WRAP gen_tac = CONJ_WRAP_GEN (rtac conjI 1) gen_tac;

fun CONJ_WRAP' gen_tac = CONJ_WRAP_GEN' (rtac conjI) gen_tac;

(* Term construction *)

(** Fresh variables **)

fun nonzero_string_of_int 0 = ""

  | nonzero_string_of_int n = string_of_int n;

val mk_TFrees' = apfst (map TFree) oo Variable.invent_types;

fun mk_TFrees n = mk_TFrees' (replicate n HOLogic.typeS);

val mk_TFreess = fold_map mk_TFrees;

fun mk_names n x = if n = 1 then [x] else map (fn i => x ^ string_of_int i) (1 upto n);

fun mk_fresh_names ctxt = (fn xs => Variable.variant_fixes xs ctxt) oo mk_names;

fun mk_Frees x Ts ctxt = mk_fresh_names ctxt (length Ts) x |>> (fn xs => map2 (curry Free) xs Ts);

fun mk_Freess x Tss = fold_map2 mk_Frees (mk_names (length Tss) x) Tss;

fun mk_Freesss x Tsss = fold_map2 mk_Freess (mk_names (length Tsss) x) Tsss;

fun mk_Freessss x Tssss = fold_map2 mk_Freesss (mk_names (length Tssss) x) Tssss;

fun mk_Frees' x Ts ctxt = mk_fresh_names ctxt (length Ts) x |>> (fn xs => `(map Free) (xs ~~ Ts));

fun mk_Freess' x Tss = fold_map2 mk_Frees' (mk_names (length Tss) x) Tss #>> split_list;

(** Types **)

fun strip_typeN 0 T = ([], T)

  | strip_typeN n (Type (@{type_name fun}, [T, T'])) = strip_typeN (n - 1) T' |>> cons T

  | strip_typeN n T = raise TYPE ("strip_typeN", [T], []);

fun mk_optionT T = Type (@{type_name option}, [T]);

val mk_relT = HOLogic.mk_setT o HOLogic.mk_prodT;

val dest_relT = HOLogic.dest_prodT o HOLogic.dest_setT;

fun mk_sumT (LT, RT) = Type (@{type_name Sum_Type.sum}, [LT, RT]);

fun mk_partial_funT (ranT, domT) = domT --> mk_optionT ranT;

(** Constants **)

fun fst_const T = Const (@{const_name fst}, T --> fst (HOLogic.dest_prodT T));

fun snd_const T = Const (@{const_name snd}, T --> snd (HOLogic.dest_prodT T));

fun Id_const T = Const (@{const_name Id}, mk_relT (T, T));

(** Operators **)

val mk_Trueprop_eq = HOLogic.mk_Trueprop o HOLogic.mk_eq;

fun mk_IfN _ _ [t] = t

  | mk_IfN T (c :: cs) (t :: ts) =

    Const (@{const_name If}, HOLogic.boolT --> T --> T --> T) $ c $ t $ mk_IfN T cs ts;

fun mk_converse R =

  let

    val RT = dest_relT (fastype_of R);

    val RST = mk_relT (snd RT, fst RT);

  in Const (@{const_name converse}, fastype_of R --> RST) $ R end;

fun mk_rel_comp (R, S) =

  let

    val RT = fastype_of R;

    val ST = fastype_of S;

    val RST = mk_relT (fst (dest_relT RT), snd (dest_relT ST));

  in Const (@{const_name relcomp}, RT --> ST --> RST) $ R $ S end;

fun mk_Gr A f =

  let val ((AT, BT), FT) = `dest_funT (fastype_of f);

  in Const (@{const_name Gr}, HOLogic.mk_setT AT --> FT --> mk_relT (AT, BT)) $ A $ f end;

fun mk_image f =

  let val (T, U) = dest_funT (fastype_of f);

  in Const (@{const_name image},

    (T --> U) --> (HOLogic.mk_setT T) --> (HOLogic.mk_setT U)) $ f end;

fun mk_Ball X f =

  Const (@{const_name Ball}, fastype_of X --> fastype_of f --> HOLogic.boolT) $ X $ f;

fun mk_Bex X f =

  Const (@{const_name Bex}, fastype_of X --> fastype_of f --> HOLogic.boolT) $ X $ f;

fun mk_UNION X f =

  let val (T, U) = dest_funT (fastype_of f);

  in Const (@{const_name SUPR}, fastype_of X --> (T --> U) --> U) $ X $ f end;

fun mk_Union T =

  Const (@{const_name Sup}, HOLogic.mk_setT (HOLogic.mk_setT T) --> HOLogic.mk_setT T);

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;

fun mk_card_order bd =

  let

    val T = fastype_of bd;

    val AT = fst (dest_relT T);

  in

    Const (@{const_name card_order_on}, HOLogic.mk_setT AT --> T --> HOLogic.boolT) $

      (HOLogic.mk_UNIV AT) $ bd

  end;

fun mk_Card_order bd =

  let

    val T = fastype_of bd;

    val AT = fst (dest_relT T);

  in

    Const (@{const_name card_order_on}, HOLogic.mk_setT AT --> T --> HOLogic.boolT) $

      mk_Field bd $ bd

  end;

fun mk_cinfinite bd =

  Const (@{const_name cinfinite}, fastype_of bd --> HOLogic.boolT) $ bd;

fun mk_ordLeq t1 t2 =

  HOLogic.mk_mem (HOLogic.mk_prod (t1, t2),

    Const (@{const_name ordLeq}, mk_relT (fastype_of t1, fastype_of t2)));

fun mk_card_of A =

  let

    val AT = fastype_of A;

    val T = HOLogic.dest_setT AT;

  in

    Const (@{const_name card_of}, AT --> mk_relT (T, T)) $ A

  end;

fun mk_dir_image r f =

  let val (T, U) = dest_funT (fastype_of f);

  in Const (@{const_name dir_image}, mk_relT (T, T) --> (T --> U) --> mk_relT (U, U)) $ r $ f end;

(*FIXME: "x"?*)

(*(nth sets i) must be of type "T --> 'ai set"*)

fun mk_in As sets T =

  let

    fun in_single set A =

      let val AT = fastype_of A;

      in Const (@{const_name less_eq},

        AT --> AT --> HOLogic.boolT) $ (set $ Free ("x", T)) $ A end;

  in

    if length sets > 0

    then HOLogic.mk_Collect ("x", T, foldr1 (HOLogic.mk_conj) (map2 in_single sets As))

    else HOLogic.mk_UNIV T

  end;

fun mk_wpull A B1 B2 f1 f2 pseudo p1 p2 =

  let

    val AT = fastype_of A;

    val BT1 = fastype_of B1;

    val BT2 = fastype_of B2;

    val FT1 = fastype_of f1;

    val FT2 = fastype_of f2;

    val PT1 = fastype_of p1;

    val PT2 = fastype_of p2;

    val T1 = HOLogic.dest_setT BT1;

    val T2 = HOLogic.dest_setT BT2;

    val domP = domain_type PT1;

    val ranF = range_type FT1;

    val _ = if is_some pseudo orelse

               (HOLogic.dest_setT AT = domP andalso

               domain_type FT1 = T1 andalso

               domain_type FT2 = T2 andalso

               domain_type PT2 = domP andalso

               range_type PT1 = T1 andalso

               range_type PT2 = T2 andalso

               range_type FT2 = ranF)

      then () else raise TYPE ("mk_wpull", [BT1, BT2, FT1, FT2, PT1, PT2], []);

  in

    (case pseudo of

      NONE => Const (@{const_name wpull},

        AT --> BT1 --> BT2 --> FT1 --> FT2 --> PT1 --> PT2 --> HOLogic.boolT) $

        A $ B1 $ B2 $ f1 $ f2 $ p1 $ p2

    | SOME (e1, e2) => Const (@{const_name wppull},

        AT --> BT1 --> BT2 --> FT1 --> FT2 --> fastype_of e1 --> fastype_of e2 -->

          PT1 --> PT2 --> HOLogic.boolT) $

        A $ B1 $ B2 $ f1 $ f2 $ e1 $ e2 $ p1 $ p2)

  end;

fun mk_subset t1 t2 =

  Const (@{const_name less_eq}, (fastype_of t1) --> (fastype_of t2) --> HOLogic.boolT) $ t1 $ t2;

fun mk_card_binop binop typop t1 t2 =

  let

    val (T1, relT1) = `(fst o dest_relT) (fastype_of t1);

    val (T2, relT2) = `(fst o dest_relT) (fastype_of t2);

  in

    Const (binop, relT1 --> relT2 --> mk_relT (typop (T1, T2), typop (T1, T2))) $ t1 $ t2

  end;

val mk_csum = mk_card_binop @{const_name csum} mk_sumT;

val mk_cprod = mk_card_binop @{const_name cprod} HOLogic.mk_prodT;

val mk_cexp = mk_card_binop @{const_name cexp} mk_partial_funT;

val mk_ccexp = mk_card_binop @{const_name ccexp} mk_partial_funT;

val ctwo = @{term ctwo};

fun mk_collect xs defT =

  let val T = (case xs of [] => defT | (x::_) => fastype_of x);

  in Const (@{const_name collect}, HOLogic.mk_setT T --> T) $ (HOLogic.mk_set T xs) end;

fun mk_permute src dest xs = map (nth xs o (fn x => find_index ((curry op =) x) src)) dest;

val list_all_free =

  fold_rev (fn free => fn P =>

    let val (x, T) = Term.dest_Free free;

    in HOLogic.all_const T $ Term.absfree (x, T) P end);

val list_exists_free =

  fold_rev (fn free => fn P =>

    let val (x, T) = Term.dest_Free free;

    in HOLogic.exists_const T $ Term.absfree (x, T) P end);

fun find_indices xs ys = map_filter I

  (map_index (fn (i, y) => if member (op =) xs y then SOME i else NONE) ys);

fun mk_trans thm1 thm2 = trans OF [thm1, thm2];

fun mk_sym thm = sym OF [thm];

(*TODO: antiquote heavily used theorems once*)

val ctrans = @{thm ordLeq_transitive};

val o_apply = @{thm o_apply};

fun mk_nthN 1 t 1 = t

  | mk_nthN _ t 1 = HOLogic.mk_fst t

  | mk_nthN 2 t 2 = HOLogic.mk_snd t

  | mk_nthN n t m = mk_nthN (n - 1) (HOLogic.mk_snd t) (m - 1);

fun mk_nth_conv n m =

  let

    fun thm b = if b then @{thm fst_snd} else @{thm snd_snd}

    fun mk_nth_conv _ 1 1 = refl

      | mk_nth_conv _ _ 1 = @{thm fst_conv}

      | mk_nth_conv _ 2 2 = @{thm snd_conv}

      | mk_nth_conv b _ 2 = @{thm snd_conv} RS thm b

      | mk_nth_conv b n m = mk_nth_conv false (n - 1) (m - 1) RS thm b;

  in mk_nth_conv (not (m = n)) n m end;

fun mk_nthI 1 1 = @{thm TrueE[OF TrueI]}

  | mk_nthI n m = fold (curry op RS) (replicate (m - 1) @{thm sndI})

    (if m = n then @{thm TrueE[OF TrueI]} else @{thm fstI});

fun mk_conjunctN 1 1 = @{thm TrueE[OF TrueI]}

  | mk_conjunctN _ 1 = conjunct1

  | mk_conjunctN 2 2 = conjunct2

  | mk_conjunctN n m = conjunct2 RS (mk_conjunctN (n - 1) (m - 1));

fun conj_dests n thm = map (fn k => thm RS mk_conjunctN n k) (1 upto n);

fun mk_conjIN 1 = @{thm TrueE[OF TrueI]}

  | mk_conjIN n = mk_conjIN (n - 1) RSN (2, conjI);

fun mk_disjIN 1 1 = @{thm TrueE[OF TrueI]}

  | mk_disjIN _ 1 = disjI1

  | mk_disjIN 2 2 = disjI2

  | mk_disjIN n m = (mk_disjIN (n - 1) (m - 1)) RS disjI2;

fun mk_ordLeq_csum 1 1 thm = thm

  | mk_ordLeq_csum _ 1 thm = @{thm ordLeq_transitive} OF [thm, @{thm ordLeq_csum1}]

  | mk_ordLeq_csum 2 2 thm = @{thm ordLeq_transitive} OF [thm, @{thm ordLeq_csum2}]

  | mk_ordLeq_csum n m thm = @{thm ordLeq_transitive} OF

    [mk_ordLeq_csum (n - 1) (m - 1) thm, @{thm ordLeq_csum2[OF Card_order_csum]}];

local

  fun mk_Un_upper' 0 = subset_refl

    | mk_Un_upper' 1 = @{thm Un_upper1}

    | mk_Un_upper' k = Library.foldr (op RS o swap)

      (replicate (k - 1) @{thm subset_trans[OF Un_upper1]}, @{thm Un_upper1});

in

  fun mk_Un_upper 1 1 = subset_refl

    | mk_Un_upper n 1 = mk_Un_upper' (n - 2) RS @{thm subset_trans[OF Un_upper1]}

    | mk_Un_upper n m = mk_Un_upper' (n - m) RS @{thm subset_trans[OF Un_upper2]};

end;

local

  fun mk_UnIN' 0 = @{thm UnI2}

    | mk_UnIN' m = mk_UnIN' (m - 1) RS @{thm UnI1};

in

  fun mk_UnIN 1 1 = @{thm TrueE[OF TrueI]}

    | mk_UnIN n 1 = Library.foldr1 (op RS o swap) (replicate (n - 1) @{thm UnI1})

    | mk_UnIN n m = mk_UnIN' (n - m)

end;

fun interleave xs ys = flat (map2 (fn x => fn y => [x, y]) xs ys);

fun transpose [] = []

  | transpose ([] :: xss) = transpose xss

  | transpose xss = map hd xss :: transpose (map tl xss);

fun seq_conds f n k xs =

  if k = n then

    map (f false) (take (k - 1) xs)

  else

    let val (negs, pos) = split_last (take k xs) in

      map (f false) negs @ [f true pos]

    end;

fun mk_unabs_def 0 thm = thm

  | mk_unabs_def n thm = mk_unabs_def (n - 1) thm RS @{thm spec[OF iffD1[OF fun_eq_iff]]};

end;