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

     Author:     Ondrej Kuncar, TU Muenchen

 Setup for Lifting for types that are BNF.

 *)

 signature LIFTING_BNF =

 sig

 end

 structure Lifting_BNF : LIFTING_BNF =

 struct

 open BNF_Util

 open BNF_Def

 open Transfer_BNF

 (* Quotient map theorem *)

 fun Quotient_tac bnf ctxt i =

   let

     val rel_Grp = rel_Grp_of_bnf bnf

     fun get_lhs thm = thm |> concl_of |> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst

     val vars = get_lhs rel_Grp |> strip_comb |> snd |> map_filter (try (strip_comb #> snd #> hd))

     val UNIVs = map (fn var => HOLogic.mk_UNIV (var |> dest_Var |> snd |> dest_Type |> snd |> hd)) vars

     val inst = map2 (curry(pairself (certify ctxt))) vars UNIVs

     val rel_Grp_UNIV_sym = rel_Grp |> Drule.instantiate_normalize ([], inst) 

       |> Local_Defs.unfold ctxt @{thms subset_UNIV[THEN eqTrueI] UNIV_def[symmetric] simp_thms(21)}

       |> (fn thm => thm RS sym)

     val rel_mono = rel_mono_of_bnf bnf

     val rel_conversep_sym = rel_conversep_of_bnf bnf RS sym

   in

     EVERY' [SELECT_GOAL (Local_Defs.unfold_tac ctxt [@{thm Quotient_alt_def5}]), 

       REPEAT_DETERM o (etac conjE), rtac conjI, SELECT_GOAL (Local_Defs.unfold_tac ctxt [rel_Grp_UNIV_sym]),

       rtac rel_mono THEN_ALL_NEW atac, rtac conjI, SELECT_GOAL (Local_Defs.unfold_tac ctxt

         [rel_conversep_sym, rel_Grp_UNIV_sym]), rtac rel_mono THEN_ALL_NEW atac,

       SELECT_GOAL (Local_Defs.unfold_tac ctxt [rel_conversep_sym, rel_OO_of_bnf bnf RS sym]),

       hyp_subst_tac ctxt, rtac refl] i

   end

 fun mk_Quotient args =

   let

     val argTs = map fastype_of args

   in

     list_comb (Const (@{const_name Quotient}, argTs ---> HOLogic.boolT), args)

   end

 fun prove_Quotient_map bnf ctxt =

   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 argTss = map2 (fn a => fn b => [mk_pred2T a a, a --> b, b --> a,mk_pred2T a b]) As Bs

     val ((argss, argss'), ctxt) = fold_map2 mk_Frees ["R", "Abs", "Rep", "T"] (transpose argTss) ctxt

       |>> `transpose

     val assms = map (mk_Quotient #> HOLogic.mk_Trueprop) argss

     val R_rel = list_comb (mk_rel_of_bnf Ds As As bnf, nth argss' 0)

     val Abs_map = list_comb (mk_map_of_bnf Ds As Bs bnf, nth argss' 1)

     val Rep_map = list_comb (mk_map_of_bnf Ds Bs As bnf, nth argss' 2)

     val T_rel = list_comb (mk_rel_of_bnf Ds As Bs bnf, nth argss' 3)

     val concl = mk_Quotient [R_rel, Abs_map, Rep_map, T_rel] |> HOLogic.mk_Trueprop

     val goal = Logic.list_implies (assms, concl)

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

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

   in

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

   end

 fun Quotient_map bnf ctxt =

   let

     val Quotient = prove_Quotient_map bnf ctxt

     fun qualify defname suffix = Binding.qualified true suffix defname

     val Quotient_thm_name = qualify (base_name_of_bnf bnf) "Quotient"

     val notes = [((Quotient_thm_name, []), [([Quotient], @{attributes [quot_map]})])]

   in

     notes

   end

 (* relator_eq_onp  *)

 fun relator_eq_onp bnf ctxt =

   let

     val pred_data = lookup_defined_pred_data ctxt (type_name_of_bnf bnf)

   in

     [((Binding.empty, []), [([Transfer.rel_eq_onp pred_data], @{attributes [relator_eq_onp]})])]    

   end

 (* relator_mono  *)

 fun relator_mono bnf =

   [((Binding.empty, []), [([rel_mono_of_bnf bnf], @{attributes [relator_mono]})])]    

 (* relator_distr  *)

 fun relator_distr bnf =

   [((Binding.empty, []), [([rel_OO_of_bnf bnf RS sym], @{attributes [relator_distr]})])]

 (* interpretation *)

 fun lifting_bnf_interpretation bnf lthy =

   if dead_of_bnf bnf > 0 then lthy

   else

     let

       val notes = relator_eq_onp bnf lthy @ Quotient_map bnf lthy @ relator_mono bnf

         @ relator_distr bnf

     in

       snd (Local_Theory.notes notes lthy)

     end

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

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

 end