(*  Title:      nominal_dt_rawperm.ML

    Author:     Cezary Kaliszyk

    Author:     Christian Urban

  Definitions of the raw permutations and

  proof that the raw datatypes are in the

  pt-class.

*)

signature NOMINAL_DT_RAWPERM =

sig

  val define_raw_perms: Datatype.descr -> (string * sort) list -> thm -> int -> theory -> 

    (term list * thm list * thm list) * theory

end

structure Nominal_Dt_RawPerm: NOMINAL_DT_RAWPERM =

struct

(* permutation function for one argument 

    - in case the argument is recursive it returns 

         permute_fn p arg

    - in case the argument is non-recursive it will return

         p o arg

*)

fun perm_arg permute_fn_frees p (arg_dty, arg) =

  if Datatype_Aux.is_rec_type arg_dty 

  then (nth permute_fn_frees (Datatype_Aux.body_index arg_dty)) $ p $ arg

  else mk_perm p arg

(* generates the equation for the permutation function for one constructor;

   i is the index of the corresponding datatype *)

fun perm_eq_constr dt_descr sorts permute_fn_frees i (cnstr_name, dts) =

let

  val p = Free ("p", @{typ perm})

  val arg_tys = map (Datatype_Aux.typ_of_dtyp dt_descr sorts) dts

  val arg_names = Name.variant_list ["p"] (Datatype_Prop.make_tnames arg_tys)

  val args = map Free (arg_names ~~ arg_tys)

  val cnstr = Const (cnstr_name, arg_tys ---> (nth_dtyp dt_descr sorts i))

  val lhs = (nth permute_fn_frees i) $ p $ list_comb (cnstr, args)

  val rhs = list_comb (cnstr, map (perm_arg permute_fn_frees p) (dts ~~ args))

  val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))

in

  (Attrib.empty_binding, eq)

end

(** proves the two pt-type class properties **)

fun prove_permute_zero lthy induct perm_defs perm_fns =

let

  val perm_types = map (body_type o fastype_of) perm_fns

  val perm_indnames = Datatype_Prop.make_tnames perm_types

  fun single_goal ((perm_fn, T), x) =

    HOLogic.mk_eq (perm_fn $ @{term "0::perm"} $ Free (x, T), Free (x, T))

  val goals =

    HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

      (map single_goal (perm_fns ~~ perm_types ~~ perm_indnames)))

  val simps = HOL_basic_ss addsimps (@{thm permute_zero} :: perm_defs)

  val tac = (Datatype_Aux.indtac induct perm_indnames 

             THEN_ALL_NEW asm_simp_tac simps) 1

in

  Goal.prove lthy perm_indnames [] goals (K tac)

  |> Datatype_Aux.split_conj_thm

end

fun prove_permute_plus lthy induct perm_defs perm_fns =

let

  val p = Free ("p", @{typ perm})

  val q = Free ("q", @{typ perm})

  val perm_types = map (body_type o fastype_of) perm_fns

  val perm_indnames = Datatype_Prop.make_tnames perm_types

  fun single_goal ((perm_fn, T), x) = HOLogic.mk_eq 

      (perm_fn $ (mk_plus p q) $ Free (x, T), perm_fn $ p $ (perm_fn $ q $ Free (x, T)))

  val goals =

    HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

      (map single_goal (perm_fns ~~ perm_types ~~ perm_indnames)))

  val simps = HOL_basic_ss addsimps (@{thm permute_plus} :: perm_defs)

  val tac = (Datatype_Aux.indtac induct perm_indnames

             THEN_ALL_NEW asm_simp_tac simps) 1

in

  Goal.prove lthy ("p" :: "q" :: perm_indnames) [] goals (K tac)

  |> Datatype_Aux.split_conj_thm 

end

(* user_dt_nos refers to the number of "un-unfolded" datatypes

   given by the user

*)

fun define_raw_perms dt_descr sorts induct_thm user_dt_nos thy =

let

  val all_full_tnames = map (fn (_, (n, _, _)) => n) dt_descr;

  val user_full_tnames = List.take (all_full_tnames, user_dt_nos);

  val perm_fn_names = prefix_dt_names dt_descr sorts "permute_"

  val perm_fn_types = map (fn (i, _) => perm_ty (nth_dtyp dt_descr sorts i)) dt_descr

  val perm_fn_frees = map Free (perm_fn_names ~~ perm_fn_types)

  fun perm_eq (i, (_, _, constrs)) = 

    map (perm_eq_constr dt_descr sorts perm_fn_frees i) constrs;

  val perm_eqs = maps perm_eq dt_descr;

  val lthy =

    Theory_Target.instantiation (user_full_tnames, [], @{sort pt}) thy;

  val ((perm_funs, perm_eq_thms), lthy') =

    Primrec.add_primrec

      (map (fn s => (Binding.name s, NONE, NoSyn)) perm_fn_names) perm_eqs lthy;

  val perm_zero_thms = prove_permute_zero lthy' induct_thm perm_eq_thms perm_funs

  val perm_plus_thms = prove_permute_plus lthy' induct_thm perm_eq_thms perm_funs

  val perm_zero_thms' = List.take (perm_zero_thms, user_dt_nos);

  val perm_plus_thms' = List.take (perm_plus_thms, user_dt_nos)

  val perms_name = space_implode "_" perm_fn_names

  val perms_zero_bind = Binding.name (perms_name ^ "_zero")

  val perms_plus_bind = Binding.name (perms_name ^ "_plus")

  fun tac _ (_, _, simps) =

    Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac simps)

  fun morphism phi (fvs, dfs, simps) =

    (map (Morphism.term phi) fvs, map (Morphism.thm phi) dfs, map (Morphism.thm phi) simps);

in

  lthy'

  |> snd o (Local_Theory.note ((perms_zero_bind, []), perm_zero_thms'))

  |> snd o (Local_Theory.note ((perms_plus_bind, []), perm_plus_thms'))

  |> Class_Target.prove_instantiation_exit_result morphism tac 

       (perm_funs, perm_eq_thms, perm_zero_thms' @ perm_plus_thms')

end

end (* structure *)