src/HOL/BNF/Tools/bnf_util.ML

express weak pullback property of bnfs only in terms of the relator

(*  Title:      HOL/BNF/Tools/bnf_util.ML
    Author:     Dmitriy Traytel, TU Muenchen
    Copyright   2012

Library for bounded natural functors.
*)

signature BNF_UTIL =
sig
  include CTR_SUGAR_UTIL

  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 map13: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k -> 'l -> 'm -> 'n) ->
    '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 -> 'n list
  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 fold_map8: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j * 'i) ->
    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list -> 'i ->
    'j list * 'i
  val fold_map9: ('a -> 'b -> 'c -> 'd -> 'e -> 'f -> 'g -> 'h -> 'i -> 'j -> 'k * 'j) ->
    'a list -> 'b list -> 'c list -> 'd list -> 'e list -> 'f list -> 'g list -> 'h list ->
    'i list -> 'j -> 'k list * 'j
  val split_list4: ('a * 'b * 'c * 'd) list -> 'a list * 'b list * 'c list * 'd list
  val split_list5: ('a * 'b * 'c * 'd * 'e) list -> 'a list * 'b list * 'c list * 'd list * 'e list
  val find_indices: ('b * 'a -> bool) -> 'a list -> 'b list -> int list

  val mk_TFreess: int list -> Proof.context -> typ 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 nonzero_string_of_int: int -> string
  val retype_free: typ -> term -> term

  val binder_fun_types: typ -> typ list
  val body_fun_type: typ -> typ
  val num_binder_types: typ -> int
  val strip_fun_type: typ -> typ list * typ
  val strip_typeN: int -> typ -> typ list * typ

  val mk_pred2T: typ -> typ -> typ
  val mk_relT: typ * typ -> typ
  val dest_relT: typ -> typ * typ
  val dest_pred2T: 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_Grp: 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_cexp: term -> term -> term
  val mk_cinfinite: term -> term
  val mk_collect: term list -> typ -> term
  val mk_converse: term -> term
  val mk_conversep: term -> term
  val mk_cprod: term -> term -> term
  val mk_csum: term -> term -> term
  val mk_dir_image: term -> term -> term
  val mk_fun_rel: term -> term -> term
  val mk_image: term -> term
  val mk_in: term list -> term list -> typ -> term
  val mk_leq: term -> term -> term
  val mk_ordLeq: term -> term -> term
  val mk_rel_comp: term * term -> term
  val mk_rel_compp: term * 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_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 Pair_eqD: thm
  val Pair_eqI: thm
  val ctrans: thm
  val id_apply: thm
  val meta_mp: thm
  val meta_spec: thm
  val o_apply: thm
  val set_mp: thm
  val set_rev_mp: thm
  val subset_UNIV: thm
  val mk_sym: thm -> thm
  val mk_trans: thm -> thm -> thm

  val is_triv_implies: thm -> bool
  val is_refl: thm -> bool
  val is_concl_refl: thm -> bool
  val no_refl: thm list -> thm list
  val no_reflexive: thm list -> thm list

  val fold_thms: Proof.context -> thm list -> thm -> thm

  val parse_binding_colon: binding parser
  val parse_opt_binding_colon: binding parser
  val parse_type_args_named_constrained: (binding option * (string * string option)) list parser
  val parse_map_rel_bindings: (binding * binding) parser

  val typedef: binding * (string * sort) list * mixfix -> term ->
    (binding * binding) option -> tactic -> local_theory -> (string * Typedef.info) * local_theory
end;

structure BNF_Util : BNF_UTIL =
struct

open Ctr_Sugar_Util

(* Library proper *)

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 map13 _ [] [] [] [] [] [] [] [] [] [] [] [] [] = []
  | map13 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) (x13::x13s) =
    f x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ::
      map13 f x1s x2s x3s x4s x5s x6s x7s x8s x9s x10s x11s x12s x13s
  | map13 _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 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;

fun fold_map8 _ [] [] [] [] [] [] [] [] acc = ([], acc)
  | fold_map8 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) (x8::x8s)
      acc =
    let
      val (x, acc') = f x1 x2 x3 x4 x5 x6 x7 x8 acc;
      val (xs, acc'') = fold_map8 f x1s x2s x3s x4s x5s x6s x7s x8s acc';
    in (x :: xs, acc'') end
  | fold_map8 _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun fold_map9 _ [] [] [] [] [] [] [] [] [] acc = ([], acc)
  | fold_map9 f (x1::x1s) (x2::x2s) (x3::x3s) (x4::x4s) (x5::x5s) (x6::x6s) (x7::x7s) (x8::x8s)
      (x9::x9s) acc =
    let
      val (x, acc') = f x1 x2 x3 x4 x5 x6 x7 x8 x9 acc;
      val (xs, acc'') = fold_map9 f x1s x2s x3s x4s x5s x6s x7s x8s x9s acc';
    in (x :: xs, acc'') end
  | fold_map9 _ _ _ _ _ _ _ _ _ _ _ = raise ListPair.UnequalLengths;

fun split_list4 [] = ([], [], [], [])
  | split_list4 ((x1, x2, x3, x4) :: xs) =
    let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;

fun split_list5 [] = ([], [], [], [], [])
  | split_list5 ((x1, x2, x3, x4, x5) :: xs) =
    let val (xs1, xs2, xs3, xs4, xs5) = split_list5 xs;
    in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4, x5 :: xs5) end;

val parse_binding_colon = parse_binding --| @{keyword ":"};
val parse_opt_binding_colon = Scan.optional parse_binding_colon Binding.empty;

val parse_type_arg_constrained =
  Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);

val parse_type_arg_named_constrained =
   (Parse.minus --| @{keyword ":"} >> K NONE || parse_opt_binding_colon >> SOME) --
   parse_type_arg_constrained;

val parse_type_args_named_constrained =
  parse_type_arg_constrained >> (single o pair (SOME Binding.empty)) ||
  @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  Scan.succeed [];

val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;

val no_map_rel = (Binding.empty, Binding.empty);

fun extract_map_rel ("map", b) = apfst (K b)
  | extract_map_rel ("rel", b) = apsnd (K b)
  | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");

val parse_map_rel_bindings =
  @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
    >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  Scan.succeed no_map_rel;


(*TODO: is this really different from Typedef.add_typedef_global?*)
fun typedef (b, Ts, mx) set opt_morphs tac lthy =
  let
    (*Work around loss of qualification in "typedef" axioms by replicating it in the name*)
    val b' = fold_rev Binding.prefix_name (map (suffix "_" o fst) (#2 (Binding.dest b))) b;
    val ((name, info), (lthy, lthy_old)) =
      lthy
      |> Typedef.add_typedef (b', Ts, mx) 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;



(* Term construction *)

(** Fresh variables **)

fun nonzero_string_of_int 0 = ""
  | nonzero_string_of_int n = string_of_int n;

val mk_TFreess = fold_map mk_TFrees;

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 retype_free T (Free (s, _)) = Free (s, T)
  | retype_free _ t = raise TERM ("retype_free", [t]);


(** Types **)

(*stolen from ~~/src/HOL/Tools/Nitpick/nitpick_hol.ML*)
fun num_binder_types (Type (@{type_name fun}, [_, T2])) =
    1 + num_binder_types T2
  | num_binder_types _ = 0

(*maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T)*)
fun strip_typeN 0 T = ([], T)
  | strip_typeN n (Type (@{type_name fun}, [T, T'])) = strip_typeN (n - 1) T' |>> cons T
  | strip_typeN _ T = raise TYPE ("strip_typeN", [T], []);

(*maps [T1,...,Tn]--->T-->U to ([T1,T2,...,Tn], T-->U), where U is not a function type*)
fun strip_fun_type T = strip_typeN (num_binder_types T - 1) T;

val binder_fun_types = fst o strip_fun_type;
val body_fun_type = snd o strip_fun_type;

fun mk_pred2T T U = mk_predT [T, U];
val mk_relT = HOLogic.mk_setT o HOLogic.mk_prodT;
val dest_relT = HOLogic.dest_prodT o HOLogic.dest_setT;
val dest_pred2T = apsnd Term.domain_type o Term.dest_funT;
fun mk_sumT (LT, RT) = Type (@{type_name Sum_Type.sum}, [LT, RT]);


(** 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 **)

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_conversep R =
  let
    val RT = dest_pred2T (fastype_of R);
    val RST = mk_pred2T (snd RT) (fst RT);
  in Const (@{const_name conversep}, fastype_of R --> RST) $ R end;

fun mk_rel_compp (R, S) =
  let
    val RT = fastype_of R;
    val ST = fastype_of S;
    val RST = mk_pred2T (fst (dest_pred2T RT)) (snd (dest_pred2T ST));
  in Const (@{const_name relcompp}, RT --> ST --> RST) $ R $ S end;

fun mk_Grp A f =
  let val ((AT, BT), FT) = `dest_funT (fastype_of f);
  in Const (@{const_name Grp}, HOLogic.mk_setT AT --> FT --> mk_pred2T 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;

fun mk_fun_rel R S =
  let
    val ((RA, RB), RT) = `dest_pred2T (fastype_of R);
    val ((SA, SB), ST) = `dest_pred2T (fastype_of S);
  in Const (@{const_name fun_rel}, RT --> ST --> mk_pred2T (RA --> SA) (RB --> SB)) $ R $ S 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_leq 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} (op --> o swap);
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 find_indices eq xs ys = map_filter I
  (map_index (fn (i, y) => if member eq xs y then SOME i else NONE) ys);

fun mk_trans thm1 thm2 = trans OF [thm1, thm2];
fun mk_sym thm = thm RS sym;

(*TODO: antiquote heavily used theorems once*)
val Pair_eqD = @{thm iffD1[OF Pair_eq]};
val Pair_eqI = @{thm iffD2[OF Pair_eq]};
val ctrans = @{thm ordLeq_transitive};
val id_apply = @{thm id_apply};
val meta_mp = @{thm meta_mp};
val meta_spec = @{thm meta_spec};
val o_apply = @{thm o_apply};
val set_mp = @{thm set_mp};
val set_rev_mp = @{thm set_rev_mp};
val subset_UNIV = @{thm subset_UNIV};

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 fstI} else @{thm sndI}
    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_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 is_triv_implies thm =
  op aconv (Logic.dest_implies (Thm.prop_of thm))
  handle TERM _ => false;

fun is_refl_prop t =
  op aconv (HOLogic.dest_eq (HOLogic.dest_Trueprop t))
  handle TERM _ => false;

val is_refl = is_refl_prop o Thm.prop_of;
val is_concl_refl = is_refl_prop o Logic.strip_imp_concl o Thm.prop_of;

val no_refl = filter_out is_refl;
val no_reflexive = filter_out Thm.is_reflexive;

fun fold_thms ctxt thms = Local_Defs.fold ctxt (distinct Thm.eq_thm_prop thms);

end;