(*  Title:      HOL/Tools/Transfer/transfer_bnf.ML

    Author:     Ondrej Kuncar, TU Muenchen

Setup for Transfer for types that are BNF.

*)

signature TRANSFER_BNF =

sig

  val base_name_of_bnf: BNF_Def.bnf -> binding

  val type_name_of_bnf: BNF_Def.bnf -> string

  val lookup_defined_pred_data: Proof.context -> string -> Transfer.pred_data

  val map_local_theory: (local_theory -> local_theory) -> theory -> theory

  val bnf_only_type_ctr: (BNF_Def.bnf -> 'a -> 'a) -> BNF_Def.bnf -> 'a -> 'a

end

structure Transfer_BNF : TRANSFER_BNF =

struct

open BNF_Util

open BNF_Def

open BNF_FP_Def_Sugar

(* util functions *)

fun base_name_of_bnf bnf = Binding.name (Binding.name_of (name_of_bnf bnf))

fun mk_Frees_free x Ts ctxt = Variable.variant_frees ctxt [] (mk_names (length Ts) x ~~ Ts) |> map Free

fun mk_Domainp P =

  let

    val PT = fastype_of P

    val argT = hd (binder_types PT)

  in

    Const (@{const_name Domainp}, PT --> argT --> HOLogic.boolT) $ P

  end

fun mk_pred pred_def args T =

  let

    val pred_name = pred_def |> prop_of |> HOLogic.dest_Trueprop |> fst o HOLogic.dest_eq 

      |> head_of |> fst o dest_Const

    val argsT = map fastype_of args

  in

    list_comb (Const (pred_name, argsT ---> (T --> HOLogic.boolT)), args)

  end

fun mk_eq_onp arg = 

  let

    val argT = domain_type (fastype_of arg)

  in

    Const (@{const_name eq_onp}, (argT --> HOLogic.boolT) --> argT --> argT --> HOLogic.boolT)

      $ arg

  end

fun subst_conv thm =

  Conv.top_sweep_conv (K (Conv.rewr_conv (safe_mk_meta_eq thm))) @{context}

fun type_name_of_bnf bnf = T_of_bnf bnf |> dest_Type |> fst

fun is_Type (Type _) = true

  | is_Type _ = false

fun map_local_theory f = Named_Target.theory_init #> f #> Local_Theory.exit_global

fun bnf_only_type_ctr f bnf = if is_Type (T_of_bnf bnf) then f bnf else I

fun bnf_of_fp_sugar (fp_sugar:fp_sugar) = nth (#bnfs (#fp_res fp_sugar)) (#fp_res_index fp_sugar)

fun fp_sugar_only_type_ctr f fp_sugar = 

  if is_Type (T_of_bnf (bnf_of_fp_sugar fp_sugar)) then f fp_sugar else I

(* relation constraints - bi_total & co. *)

fun mk_relation_constraint name arg =

  (Const (name, fastype_of arg --> HOLogic.boolT)) $ arg

fun side_constraint_tac bnf constr_defs ctxt i = 

  let

    val thms = constr_defs @ map mk_sym [rel_eq_of_bnf bnf, rel_conversep_of_bnf bnf,

      rel_OO_of_bnf bnf]

  in                

    (SELECT_GOAL (Local_Defs.unfold_tac ctxt thms) THEN' rtac (rel_mono_of_bnf bnf)

      THEN_ALL_NEW atac) i

  end

fun bi_constraint_tac constr_iff sided_constr_intros ctxt i = 

  (SELECT_GOAL (Local_Defs.unfold_tac ctxt [constr_iff]) THEN' 

    CONJ_WRAP' (fn thm => rtac thm THEN_ALL_NEW (REPEAT_DETERM o etac conjE THEN' atac)) sided_constr_intros) i

fun generate_relation_constraint_goal ctxt bnf constraint_def =

  let

    val constr_name = constraint_def |> prop_of |> HOLogic.dest_Trueprop |> fst o HOLogic.dest_eq

      |> head_of |> fst o dest_Const

    val live = live_of_bnf bnf

    val (((As, Bs), Ds), ctxt) = ctxt

      |> mk_TFrees live

      ||>> mk_TFrees live

      ||>> mk_TFrees (dead_of_bnf bnf)

    val relator = mk_rel_of_bnf Ds As Bs bnf

    val relsT = map2 mk_pred2T As Bs

    val (args, ctxt) = Ctr_Sugar_Util.mk_Frees "R" relsT ctxt

    val concl = HOLogic.mk_Trueprop (mk_relation_constraint constr_name (list_comb (relator, args)))

    val assms = map (HOLogic.mk_Trueprop o (mk_relation_constraint constr_name)) args

    val goal = Logic.list_implies (assms, concl)

  in

    (goal, ctxt)

  end

fun prove_relation_side_constraint ctxt bnf constraint_def =

  let

    val old_ctxt = ctxt

    val (goal, ctxt) = generate_relation_constraint_goal ctxt bnf constraint_def

    val thm = Goal.prove ctxt [] [] goal 

      (fn {context = ctxt, prems = _} => side_constraint_tac bnf [constraint_def] ctxt 1)

  in

    Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)

  end

fun prove_relation_bi_constraint ctxt bnf constraint_def side_constraints =

  let

    val old_ctxt = ctxt

    val (goal, ctxt) = generate_relation_constraint_goal ctxt bnf constraint_def

    val thm = Goal.prove ctxt [] [] goal 

      (fn {context = ctxt, prems = _} => bi_constraint_tac constraint_def side_constraints ctxt 1)

  in

    Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)

  end

val defs = [("left_total_rel", @{thm left_total_alt_def}), ("right_total_rel", @{thm right_total_alt_def}),

  ("left_unique_rel", @{thm left_unique_alt_def}), ("right_unique_rel", @{thm right_unique_alt_def})]

fun prove_relation_constraints bnf lthy =

  let

    val transfer_attr = @{attributes [transfer_rule]}

    val Tname = base_name_of_bnf bnf

    fun qualify suffix = Binding.qualified true suffix Tname

    val defs = map (apsnd (prove_relation_side_constraint lthy bnf)) defs

    val bi_total = prove_relation_bi_constraint lthy bnf @{thm bi_total_alt_def} 

      [snd (nth defs 0), snd (nth defs 1)]

    val bi_unique = prove_relation_bi_constraint lthy bnf @{thm bi_unique_alt_def} 

      [snd (nth defs 2), snd (nth defs 3)]

    val defs = ("bi_total_rel", bi_total) :: ("bi_unique_rel", bi_unique) :: defs

    val notes = maps (fn (name, thm) => [((qualify name, []), [([thm], transfer_attr)])]) defs

  in

    notes

  end

(* relator_eq *)

fun relator_eq bnf =

  [((Binding.empty, []), [([rel_eq_of_bnf bnf], @{attributes [relator_eq]})])]

(* predicator definition and Domainp and eq_onp theorem *)

fun define_pred bnf lthy =

  let

    fun mk_pred_name c = Binding.prefix_name "pred_" c

    val live = live_of_bnf bnf

    val Tname = base_name_of_bnf bnf

    val ((As, Ds), lthy) = lthy

      |> mk_TFrees live

      ||>> mk_TFrees (dead_of_bnf bnf)

    val T = mk_T_of_bnf Ds As bnf

    val sets = mk_sets_of_bnf (replicate live Ds) (replicate live As) bnf

    val argTs = map mk_pred1T As

    val args = mk_Frees_free "P" argTs lthy

    val conjs = map (fn (set, arg) => mk_Ball (set $ Bound 0) arg) (sets ~~ args)

    val rhs = Abs ("x", T, foldr1 HOLogic.mk_conj conjs)

    val pred_name = mk_pred_name Tname

    val headT = argTs ---> (T --> HOLogic.boolT)

    val head = Free (Binding.name_of pred_name, headT)

    val lhs = list_comb (head, args)

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

    val ((_, (_, pred_def)), lthy) = Specification.definition ((SOME (pred_name, SOME headT, NoSyn)), 

      ((Binding.empty, []), def)) lthy

  in

    (pred_def, lthy)

  end

fun Domainp_tac bnf pred_def ctxt i =

  let

    val n = live_of_bnf bnf

    val set_map's = set_map_of_bnf bnf

  in

    EVERY' [rtac ext, SELECT_GOAL (Local_Defs.unfold_tac ctxt [@{thm Domainp.simps}, 

        in_rel_of_bnf bnf, pred_def]), rtac iffI,

        REPEAT_DETERM o eresolve_tac [exE, conjE, CollectE], hyp_subst_tac ctxt,

        CONJ_WRAP' (fn set_map => EVERY' [rtac ballI, dtac (set_map RS equalityD1 RS set_mp),

          etac imageE, dtac set_rev_mp, atac, REPEAT_DETERM o eresolve_tac [CollectE, @{thm case_prodE}],

          hyp_subst_tac ctxt, rtac @{thm iffD2[OF arg_cong2[of _ _ _ _ Domainp, OF refl fst_conv]]},

          etac @{thm DomainPI}]) set_map's, 

        REPEAT_DETERM o etac conjE, REPEAT_DETERM o resolve_tac [exI, (refl RS conjI), rotate_prems 1 conjI], 

        rtac refl, rtac (box_equals OF [map_cong0_of_bnf bnf, map_comp_of_bnf bnf RS sym,

          map_id_of_bnf bnf]),

        REPEAT_DETERM_N n o (EVERY' [rtac @{thm box_equals[OF _ sym[OF o_apply] sym[OF id_apply]]},

          rtac @{thm fst_conv}]), rtac CollectI,

        CONJ_WRAP' (fn set_map => EVERY' [rtac (set_map RS @{thm ord_eq_le_trans}), 

          REPEAT_DETERM o resolve_tac [@{thm image_subsetI}, CollectI, @{thm case_prodI}],

          dtac (rotate_prems 1 bspec), atac, etac @{thm DomainpE}, etac @{thm someI}]) set_map's

      ] i

  end

fun prove_Domainp_rel ctxt bnf pred_def =

  let

    val live = live_of_bnf bnf

    val old_ctxt = ctxt

    val (((As, Bs), Ds), ctxt) = ctxt

      |> mk_TFrees live

      ||>> mk_TFrees live

      ||>> mk_TFrees (dead_of_bnf bnf)

    val relator = mk_rel_of_bnf Ds As Bs bnf

    val relsT = map2 mk_pred2T As Bs

    val T = mk_T_of_bnf Ds As bnf

    val (args, ctxt) = Ctr_Sugar_Util.mk_Frees "R" relsT ctxt

    val lhs = mk_Domainp (list_comb (relator, args))

    val rhs = mk_pred pred_def (map mk_Domainp args) T

    val goal = HOLogic.mk_eq (lhs, rhs) |> HOLogic.mk_Trueprop

    val thm = Goal.prove ctxt [] [] goal 

      (fn {context = ctxt, prems = _} => Domainp_tac bnf pred_def ctxt 1)

  in

    Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)

  end

fun pred_eq_onp_tac bnf pred_def ctxt i =

  (SELECT_GOAL (Local_Defs.unfold_tac ctxt [@{thm eq_onp_Grp}, 

    @{thm Ball_Collect}, pred_def]) THEN' CONVERSION (subst_conv (map_id0_of_bnf bnf RS sym))

  THEN' rtac (rel_Grp_of_bnf bnf)) i

fun prove_rel_eq_onp ctxt bnf pred_def =

  let

    val live = live_of_bnf bnf

    val old_ctxt = ctxt

    val ((As, Ds), ctxt) = ctxt

      |> mk_TFrees live

      ||>> mk_TFrees (dead_of_bnf bnf)

    val T = mk_T_of_bnf Ds As bnf

    val argTs = map mk_pred1T As

    val (args, ctxt) = mk_Frees "P" argTs ctxt

    val relator = mk_rel_of_bnf Ds As As bnf

    val lhs = list_comb (relator, map mk_eq_onp args)

    val rhs = mk_eq_onp (mk_pred pred_def args T)

    val goal = HOLogic.mk_eq (lhs, rhs) |> HOLogic.mk_Trueprop

    val thm = Goal.prove ctxt [] [] goal 

      (fn {context = ctxt, prems = _} => pred_eq_onp_tac bnf pred_def ctxt 1)

  in

    Drule.zero_var_indexes (singleton (Variable.export ctxt old_ctxt) thm)

  end

fun predicator bnf lthy =

  let

    val (pred_def, lthy) = define_pred bnf lthy

    val pred_def = Morphism.thm (Local_Theory.target_morphism lthy) pred_def

    val Domainp_rel = prove_Domainp_rel lthy bnf pred_def

    val rel_eq_onp = prove_rel_eq_onp lthy bnf pred_def

    fun qualify defname suffix = Binding.qualified true suffix defname

    val Domainp_rel_thm_name = qualify (base_name_of_bnf bnf) "Domainp_rel"

    val rel_eq_onp_thm_name = qualify (base_name_of_bnf bnf) "rel_eq_onp"

    val pred_data = {rel_eq_onp = rel_eq_onp}

    val type_name = type_name_of_bnf bnf

    val relator_domain_attr = @{attributes [relator_domain]}

    val notes = [((Domainp_rel_thm_name, []), [([Domainp_rel], relator_domain_attr)]),

      ((rel_eq_onp_thm_name, []), [([rel_eq_onp], [])])]

    val lthy = Local_Theory.declaration {syntax = false, pervasive = true}

      (fn phi => Transfer.update_pred_data type_name (Transfer.morph_pred_data phi pred_data)) lthy

  in

    (notes, lthy)

  end

(* BNF interpretation *)

fun transfer_bnf_interpretation bnf lthy =

  let

    val constr_notes = if dead_of_bnf bnf > 0 then []

      else prove_relation_constraints bnf lthy

    val relator_eq_notes = if dead_of_bnf bnf > 0 then [] 

      else relator_eq bnf

    val (pred_notes, lthy) = predicator bnf lthy

  in

    snd (Local_Theory.notes (constr_notes @ relator_eq_notes @ pred_notes) lthy)

  end

val _ = Context.>> (Context.map_theory (bnf_interpretation

  (bnf_only_type_ctr (fn bnf => map_local_theory (transfer_bnf_interpretation bnf)))))

(* simplification rules for the predicator *)

fun lookup_defined_pred_data lthy name =

  case (Transfer.lookup_pred_data lthy name) of

    SOME data => data

    | NONE => (error "lookup_pred_data: something went utterly wrong")

fun prove_pred_inject lthy (fp_sugar:fp_sugar) =

  let

    val involved_types = distinct op= (

        map type_name_of_bnf (#nested_bnfs fp_sugar) 

      @ map type_name_of_bnf (#nesting_bnfs fp_sugar)

      @ map type_name_of_bnf (#bnfs (#fp_res fp_sugar)))

    val eq_onps = map (Transfer.rel_eq_onp o lookup_defined_pred_data lthy) involved_types

    val live = live_of_bnf (bnf_of_fp_sugar fp_sugar)

    val old_lthy = lthy

    val (As, lthy) = mk_TFrees live lthy

    val predTs = map mk_pred1T As

    val (preds, lthy) = mk_Frees "P" predTs lthy

    val args = map mk_eq_onp preds

    val cTs = map (SOME o certifyT lthy) (maps (replicate 2) As)

    val cts = map (SOME o certify lthy) args

    fun get_rhs thm = thm |> concl_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> snd

    fun is_eqn thm = can get_rhs thm

    fun rel2pred_massage thm =

      let

        fun pred_eq_onp_conj 0 = error "not defined"

          | pred_eq_onp_conj 1 = @{thm eq_onp_same_args}

          | pred_eq_onp_conj n = 

            let

              val conj_cong = @{thm arg_cong2[of _ _ _ _ "op \<and>"]}

              val start = @{thm eq_onp_same_args} RS conj_cong

            in

              @{thm eq_onp_same_args} RS (funpow (n - 2) (fn thm => start RS thm) start)

            end

        val n = if is_eqn thm then thm |> get_rhs |> HOLogic.dest_conj |> length else 0

      in

        thm

        |> Drule.instantiate' cTs cts

        |> Local_Defs.unfold lthy eq_onps

        |> (fn thm => if n > 0 then @{thm box_equals} 

              OF [thm, @{thm eq_onp_same_args}, pred_eq_onp_conj n]

            else thm RS (@{thm eq_onp_same_args} RS iffD1))

      end

    val rel_injects = #rel_injects fp_sugar

  in

    rel_injects

    |> map rel2pred_massage

    |> Variable.export lthy old_lthy

    |> map Drule.zero_var_indexes

  end

(* fp_sugar interpretation *)

fun transfer_fp_sugar_interpretation fp_sugar lthy =

  let

    val pred_injects = prove_pred_inject lthy fp_sugar

    fun qualify defname suffix = Binding.qualified true suffix defname

    val pred_inject_thm_name = qualify (base_name_of_bnf (bnf_of_fp_sugar fp_sugar)) "pred_inject"

    val simp_attrs = @{attributes [simp]}

  in

    snd (Local_Theory.note ((pred_inject_thm_name, simp_attrs), pred_injects) lthy)

  end

val _ = Context.>> (Context.map_theory (fp_sugar_interpretation

  (fp_sugar_only_type_ctr (fn fp_sugar => map_local_theory (transfer_fp_sugar_interpretation fp_sugar)))))

end