(*  Title:      nominal_dt_alpha.ML

    Author:     Cezary Kaliszyk

    Author:     Christian Urban

  Definitions of the alpha relations.

*)

signature NOMINAL_DT_ALPHA =

sig

  val define_raw_alpha: Datatype_Aux.descr -> (string * sort) list -> bn_info ->

    bclause list list list -> term list -> Proof.context -> 

end

structure Nominal_Dt_Alpha: NOMINAL_DT_ALPHA =

struct

(* construct the compound terms for prod_fv and prod_alpha *)

fun mk_prod_fv (t1, t2) =

let

  val ty1 = fastype_of t1

  val ty2 = fastype_of t2 

  val resT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2) --> @{typ "atom set"}

in

  Const (@{const_name "prod_fv"}, [ty1, ty2] ---> resT) $ t1 $ t2

end

fun mk_prod_alpha (t1, t2) =

let

  val ty1 = fastype_of t1

  val ty2 = fastype_of t2 

  val prodT = HOLogic.mk_prodT (domain_type ty1, domain_type ty2)

  val resT = [prodT, prodT] ---> @{typ "bool"}

in

  Const (@{const_name "prod_alpha"}, [ty1, ty2] ---> resT) $ t1 $ t2

end

(* generates the compound binder terms *)

fun mk_binders lthy bmode args bodies = 

let  

  fun bind_set lthy args (NONE, i) = setify lthy (nth args i)

    | bind_set _ args (SOME bn, i) = bn $ (nth args i)

  fun bind_lst lthy args (NONE, i) = listify lthy (nth args i)

    | bind_lst _ args (SOME bn, i) = bn $ (nth args i)

  val (combine_fn, bind_fn) =

    case bmode of

      Lst => (mk_append, bind_lst) 

    | Set => (mk_union,  bind_set)

    | Res => (mk_union,  bind_set)

in

  foldl1 combine_fn (map (bind_fn lthy args) bodies)

end

(* produces the term for an alpha with abstraction *)

fun mk_alpha_term bmode fv alpha args args' binders binders' =

let

  val (alpha_name, binder_ty) = 

    case bmode of

      Lst => (@{const_name "alpha_lst"}, @{typ "atom list"})

    | Set => (@{const_name "alpha_gen"}, @{typ "atom set"})

    | Res => (@{const_name "alpha_res"}, @{typ "atom set"})

  val ty = fastype_of args

  val pair_ty = HOLogic.mk_prodT (binder_ty, ty)

  val alpha_ty = [ty, ty] ---> @{typ "bool"}

  val fv_ty = ty --> @{typ "atom set"}

  val pair_lhs = HOLogic.mk_prod (binders, args)

  val pair_rhs = HOLogic.mk_prod (binders', args')

in

  HOLogic.exists_const @{typ perm} $ Abs ("p", @{typ perm},

    Const (alpha_name, [pair_ty, alpha_ty, fv_ty, @{typ "perm"}, pair_ty] ---> @{typ bool}) 

      $ pair_lhs $ alpha $ fv $ (Bound 0) $ pair_rhs)

end

(* for non-recursive binders we have to produce alpha_bn premises *)

fun mk_alpha_bn_prem alpha_bn_map args args' bodies binder = 

  case binder of

    (NONE, _) => []

  | (SOME bn, i) =>

     if member (op=) bodies i then [] 

     else [the (AList.lookup (op=) alpha_bn_map bn) $ (nth args i) $ (nth args' i)]

(* generat the premises for an alpha rule; mk_frees is used

   if no binders are present *)

fun mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause =

let

  fun mk_frees i =

    let

      val arg = nth args i

      val arg' = nth args' i

      val ty = fastype_of arg

    in

      if nth is_rec i

      then fst (the (AList.lookup (op=) alpha_map ty)) $ arg $ arg'

      else HOLogic.mk_eq (arg, arg')

    end

  fun mk_alpha_fv i = 

    let

      val ty = fastype_of (nth args i)

    in

      case AList.lookup (op=) alpha_map ty of

        NONE => (HOLogic.eq_const ty, supp_const ty) 

      | SOME (alpha, fv) => (alpha, fv) 

    end  

in

  case bclause of

    BC (_, [], bodies) => map (HOLogic.mk_Trueprop o mk_frees) bodies 

  | BC (bmode, binders, bodies) => 

    let

      val (alphas, fvs) = split_list (map mk_alpha_fv bodies)

      val comp_fv = foldl1 mk_prod_fv fvs

      val comp_alpha = foldl1 mk_prod_alpha alphas

      val comp_args = foldl1 HOLogic.mk_prod (map (nth args) bodies)

      val comp_args' = foldl1 HOLogic.mk_prod (map (nth args') bodies)

      val comp_binders = mk_binders lthy bmode args binders

      val comp_binders' = mk_binders lthy bmode args' binders

      val alpha_prem = 

        mk_alpha_term bmode comp_fv comp_alpha comp_args comp_args' comp_binders comp_binders'

      val alpha_bn_prems = flat (map (mk_alpha_bn_prem alpha_bn_map args args' bodies) binders)

    in

      map HOLogic.mk_Trueprop (alpha_prem::alpha_bn_prems)

    end

end

(* produces the introduction rule for an alpha rule *)

fun mk_alpha_intros lthy alpha_map alpha_bn_map (constr, ty, arg_tys, is_rec) bclauses = 

let

  val arg_names = Datatype_Prop.make_tnames arg_tys

  val arg_names' = Name.variant_list arg_names arg_names

  val args = map Free (arg_names ~~ arg_tys)

  val args' = map Free (arg_names' ~~ arg_tys)

  val alpha = fst (the (AList.lookup (op=) alpha_map ty))

  val concl = HOLogic.mk_Trueprop (alpha $ list_comb (constr, args) $ list_comb (constr, args'))

  val prems = map (mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args')) bclauses

in

  Library.foldr Logic.mk_implies (flat prems, concl)

end

(* produces the premise of an alpha-bn rule; we only need to

   treat the case special where the binding clause is empty;

   - if the body is not included in the bn_info, then we either

     produce an equation or an alpha-premise

   - if the body is included in the bn_info, then we create

     either a recursive call to alpha-bn, or no premise  *)

fun mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args') bclause =

let

  fun mk_alpha_bn_prem alpha_map alpha_bn_map bn_args (args, args') i = 

  let

    val arg = nth args i

    val arg' = nth args' i

    val ty = fastype_of arg

  in

    case AList.lookup (op=) bn_args i of

      NONE => (case (AList.lookup (op=) alpha_map ty) of

                 NONE =>  [HOLogic.mk_eq (arg, arg')]

               | SOME (alpha, _) => [alpha $ arg $ arg'])

    | SOME (NONE) => []

    | SOME (SOME bn) => [the (AList.lookup (op=) alpha_bn_map bn) $ arg $ arg']

  end  

in

  case bclause of

    BC (_, [], bodies) => 

      map HOLogic.mk_Trueprop 

        (flat (map (mk_alpha_bn_prem alpha_map alpha_bn_map bn_args (args, args')) bodies))

  | _ => mk_alpha_prems lthy alpha_map alpha_bn_map is_rec (args, args') bclause

end

fun mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map (bn_args, (constr, _, arg_tys, is_rec)) bclauses =

let

  val arg_names = Datatype_Prop.make_tnames arg_tys

  val arg_names' = Name.variant_list arg_names arg_names

  val args = map Free (arg_names ~~ arg_tys)

  val args' = map Free (arg_names' ~~ arg_tys)

  val alpha_bn = the (AList.lookup (op=) alpha_bn_map bn_trm)

  val concl = HOLogic.mk_Trueprop (alpha_bn $ list_comb (constr, args) $ list_comb (constr, args'))

  val prems = map (mk_alpha_bn lthy alpha_map alpha_bn_map bn_args is_rec (args, args')) bclauses

in

  Library.foldr Logic.mk_implies (flat prems, concl)

end

fun mk_alpha_bn_intros lthy alpha_map alpha_bn_map constrs_info bclausesss (bn_trm, bn_n, bn_argss) = 

let

  val nth_constrs_info = nth constrs_info bn_n

  val nth_bclausess = nth bclausesss bn_n

in

  map2 (mk_alpha_bn_intro lthy bn_trm alpha_map alpha_bn_map) (bn_argss ~~ nth_constrs_info) nth_bclausess

end

fun define_raw_alpha descr sorts bn_info bclausesss fvs lthy =

let

  val alpha_names = prefix_dt_names descr sorts "alpha_"

  val alpha_arg_tys = all_dtyps descr sorts

  val alpha_tys = map (fn ty => [ty, ty] ---> @{typ bool}) alpha_arg_tys

  val alpha_frees = map Free (alpha_names ~~ alpha_tys)

  val alpha_map = alpha_arg_tys ~~ (alpha_frees ~~ fvs)

  val (bns, bn_tys) = split_list (map (fn (bn, i, _) => (bn, i)) bn_info)

  val bn_names = map (fn bn => Long_Name.base_name (fst (dest_Const bn))) bns

  val alpha_bn_names = map (prefix "alpha_") bn_names

  val alpha_bn_arg_tys = map (fn i => nth_dtyp descr sorts i) bn_tys

  val alpha_bn_tys = map (fn ty => [ty, ty] ---> @{typ "bool"}) alpha_bn_arg_tys

  val alpha_bn_frees = map Free (alpha_bn_names ~~ alpha_bn_tys)

  val alpha_bn_map = bns ~~ alpha_bn_frees

  val constrs_info = all_dtyp_constrs_types descr sorts

  val alpha_intros = map2 (map2 (mk_alpha_intros lthy alpha_map alpha_bn_map)) constrs_info bclausesss 

  val alpha_bn_intros = map (mk_alpha_bn_intros lthy alpha_map alpha_bn_map constrs_info bclausesss) bn_info

  val all_alpha_names = map2 (fn s => fn ty => ((Binding.name s, ty), NoSyn))

    (alpha_names @ alpha_bn_names) (alpha_tys @ alpha_bn_tys)

  val all_alpha_intros = map (pair Attrib.empty_binding) (flat alpha_intros @ flat alpha_bn_intros)

  val (alphas, lthy') = Inductive.add_inductive_i

     {quiet_mode = true, verbose = false, alt_name = Binding.empty,

      coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}

     all_alpha_names [] all_alpha_intros [] lthy

  val alpha_induct_loc = #raw_induct alphas;

  val alpha_intros_loc = #intrs alphas;

  val alpha_cases_loc = #elims alphas;

  val phi = ProofContext.export_morphism lthy' lthy;

  val alpha_induct = Morphism.thm phi alpha_induct_loc;

  val alpha_intros = map (Morphism.thm phi) alpha_intros_loc

  val alpha_cases = map (Morphism.thm phi) alpha_cases_loc

in

end

end (* structure *)