(*  Title:      HOL/Tools/Lifting/lifting_def.ML

     Author:     Ondrej Kuncar

 Definitions for constants on quotient types.

 *)

 signature LIFTING_DEF =

 sig

   val add_lift_def:

     (binding * mixfix) -> typ -> term -> thm -> local_theory -> local_theory

   val lift_def_cmd:

     (binding * string option * mixfix) * string -> local_theory -> Proof.state

   val can_generate_code_cert: thm -> bool

 end;

 structure Lifting_Def: LIFTING_DEF =

 struct

 (** Interface and Syntax Setup **)

 (* Generation of the code certificate from the rsp theorem *)

 infix 0 MRSL

 fun ants MRSL thm = fold (fn rl => fn thm => rl RS thm) ants thm

 fun get_body_types (Type ("fun", [_, U]), Type ("fun", [_, V])) = get_body_types (U, V)

   | get_body_types (U, V)  = (U, V)

 fun get_binder_types (Type ("fun", [T, U]), Type ("fun", [V, W])) = (T, V) :: get_binder_types (U, W)

   | get_binder_types _ = []

 fun force_rty_type ctxt rty rhs = 

   let

     val thy = Proof_Context.theory_of ctxt

     val rhs_schematic = singleton (Variable.polymorphic ctxt) rhs

     val rty_schematic = fastype_of rhs_schematic

     val match = Sign.typ_match thy (rty_schematic, rty) Vartab.empty

   in

     Envir.subst_term_types match rhs_schematic

   end

 fun unabs_def ctxt def = 

   let

     val (_, rhs) = Thm.dest_equals (cprop_of def)

     fun dest_abs (Abs (var_name, T, _)) = (var_name, T)

       | dest_abs tm = raise TERM("get_abs_var",[tm])

     val (var_name, T) = dest_abs (term_of rhs)

     val (new_var_names, ctxt') = Variable.variant_fixes [var_name] ctxt

     val thy = Proof_Context.theory_of ctxt'

     val refl_thm = Thm.reflexive (cterm_of thy (Free (hd new_var_names, T)))

   in

     Thm.combination def refl_thm |>

     singleton (Proof_Context.export ctxt' ctxt)

   end

 fun unabs_all_def ctxt def = 

   let

     val (_, rhs) = Thm.dest_equals (cprop_of def)

     val xs = strip_abs_vars (term_of rhs)

   in  

     fold (K (unabs_def ctxt)) xs def

   end

 val map_fun_unfolded = 

   @{thm map_fun_def[abs_def]} |>

   unabs_def @{context} |>

   unabs_def @{context} |>

   Local_Defs.unfold @{context} [@{thm comp_def}]

 fun unfold_fun_maps ctm =

   let

     fun unfold_conv ctm =

       case (Thm.term_of ctm) of

         Const (@{const_name "map_fun"}, _) $ _ $ _ => 

           (Conv.arg_conv unfold_conv then_conv Conv.rewr_conv map_fun_unfolded) ctm

         | _ => Conv.all_conv ctm

     val try_beta_conv = Conv.try_conv (Thm.beta_conversion false)

   in

     (Conv.arg_conv (Conv.fun_conv unfold_conv then_conv try_beta_conv)) ctm

   end

 fun prove_rel ctxt rsp_thm (rty, qty) =

   let

     val ty_args = get_binder_types (rty, qty)

     fun disch_arg args_ty thm = 

       let

         val quot_thm = Lifting_Term.prove_quot_thm ctxt args_ty

       in

         [quot_thm, thm] MRSL @{thm apply_rsp''}

       end

   in

     fold disch_arg ty_args rsp_thm

   end

 exception CODE_CERT_GEN of string

 fun simplify_code_eq ctxt def_thm = 

   Local_Defs.unfold ctxt [@{thm o_def}, @{thm map_fun_def}, @{thm id_def}] def_thm

 fun can_generate_code_cert quot_thm  =

   case Lifting_Term.quot_thm_rel quot_thm of

     Const (@{const_name HOL.eq}, _) => true

     | Const (@{const_name invariant}, _) $ _  => true

     | _ => false

 fun generate_code_cert ctxt def_thm rsp_thm (rty, qty) =

   let

     val thy = Proof_Context.theory_of ctxt

     val quot_thm = Lifting_Term.prove_quot_thm ctxt (get_body_types (rty, qty))

     val fun_rel = prove_rel ctxt rsp_thm (rty, qty)

     val abs_rep_thm = [quot_thm, fun_rel] MRSL @{thm Quotient_rep_abs}

     val abs_rep_eq = 

       case (HOLogic.dest_Trueprop o prop_of) fun_rel of

         Const (@{const_name HOL.eq}, _) $ _ $ _ => abs_rep_thm

         | Const (@{const_name invariant}, _) $ _ $ _ $ _ => abs_rep_thm RS @{thm invariant_to_eq}

         | _ => raise CODE_CERT_GEN "relation is neither equality nor invariant"

     val unfolded_def = Conv.fconv_rule unfold_fun_maps def_thm

     val unabs_def = unabs_all_def ctxt unfolded_def

     val rep = (cterm_of thy o Lifting_Term.quot_thm_rep) quot_thm

     val rep_refl = Thm.reflexive rep RS @{thm meta_eq_to_obj_eq}

     val repped_eq = [rep_refl, unabs_def RS @{thm meta_eq_to_obj_eq}] MRSL @{thm cong}

     val code_cert = [repped_eq, abs_rep_eq] MRSL @{thm trans}

   in

     simplify_code_eq ctxt code_cert

   end

 fun is_abstype ctxt typ =

   let

     val thy = Proof_Context.theory_of ctxt

     val type_name = (fst o dest_Type) typ

   in

     (snd oo Code.get_type) thy type_name

   end

 fun define_code_cert code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy = 

   let

     val (rty_body, qty_body) = get_body_types (rty, qty)

     val quot_thm = Lifting_Term.prove_quot_thm lthy (rty_body, qty_body)

   in

     if can_generate_code_cert quot_thm then

       let

         val code_cert = generate_code_cert lthy def_thm rsp_thm (rty, qty)

         val add_abs_eqn_attribute = 

           Thm.declaration_attribute (fn thm => Context.mapping (Code.add_abs_eqn thm) I)

         val add_abs_eqn_attrib = Attrib.internal (K add_abs_eqn_attribute);

         val lthy' = 

           (snd oo Local_Theory.note) ((code_eqn_thm_name, []), [code_cert]) lthy

       in

         if is_abstype lthy qty_body then

           (snd oo Local_Theory.note) ((Binding.empty, [add_abs_eqn_attrib]), [code_cert]) lthy'

         else

           lthy'

       end

     else

       lthy

   end

 fun define_code_eq code_eqn_thm_name def_thm lthy =

   let

     val unfolded_def = Conv.fconv_rule unfold_fun_maps def_thm

     val code_eq = unabs_all_def lthy unfolded_def

     val simp_code_eq = simplify_code_eq lthy code_eq

   in

     lthy

       |> (snd oo Local_Theory.note) ((code_eqn_thm_name, [Code.add_default_eqn_attrib]), [simp_code_eq])

   end

 fun define_code code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy =

   if body_type rty = body_type qty then 

     define_code_eq code_eqn_thm_name def_thm lthy

   else 

     define_code_cert code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy

 fun add_lift_def var qty rhs rsp_thm lthy =

   let

     val rty = fastype_of rhs

     val quotient_thm = Lifting_Term.prove_quot_thm lthy (rty, qty)

     val absrep_trm =  Lifting_Term.quot_thm_abs quotient_thm

     val rty_forced = (domain_type o fastype_of) absrep_trm

     val forced_rhs = force_rty_type lthy rty_forced rhs

     val lhs = Free (Binding.print (#1 var), qty)

     val prop = Logic.mk_equals (lhs, absrep_trm $ forced_rhs)

     val (_, prop') = Local_Defs.cert_def lthy prop

     val (_, newrhs) = Local_Defs.abs_def prop'

     val ((_, (_ , def_thm)), lthy') = 

       Local_Theory.define (var, ((Thm.def_binding (#1 var), []), newrhs)) lthy

     val transfer_thm = [quotient_thm, rsp_thm, def_thm] MRSL @{thm Quotient_to_transfer}

         |> Raw_Simplifier.rewrite_rule (Transfer.get_relator_eq lthy')

     fun qualify defname suffix = Binding.qualified true suffix defname

     val lhs_name = (#1 var)

     val rsp_thm_name = qualify lhs_name "rsp"

     val code_eqn_thm_name = qualify lhs_name "rep_eq"

     val transfer_thm_name = qualify lhs_name "transfer"

     val transfer_attr = Attrib.internal (K Transfer.transfer_add)

   in

     lthy'

       |> (snd oo Local_Theory.note) ((rsp_thm_name, []), [rsp_thm])

       |> (snd oo Local_Theory.note) ((transfer_thm_name, [transfer_attr]), [transfer_thm])

       |> define_code code_eqn_thm_name def_thm rsp_thm (rty_forced, qty)

   end

 fun mk_readable_rsp_thm_eq tm lthy =

   let

     val ctm = cterm_of (Proof_Context.theory_of lthy) tm

     fun norm_fun_eq ctm = 

       let

         fun abs_conv2 cv = Conv.abs_conv (K (Conv.abs_conv (K cv) lthy)) lthy

         fun erase_quants ctm' =

           case (Thm.term_of ctm') of

             Const ("HOL.eq", _) $ _ $ _ => Conv.all_conv ctm'

             | _ => (Conv.binder_conv (K erase_quants) lthy then_conv 

               Conv.rewr_conv @{thm fun_eq_iff[symmetric, THEN eq_reflection]}) ctm'

       in

         (abs_conv2 erase_quants then_conv Thm.eta_conversion) ctm

       end

     fun simp_arrows_conv ctm =

       let

         val unfold_conv = Conv.rewrs_conv 

           [@{thm fun_rel_eq_invariant[THEN eq_reflection]}, @{thm fun_rel_eq_rel[THEN eq_reflection]}, 

             @{thm fun_rel_def[THEN eq_reflection]}]

         val left_conv = simp_arrows_conv then_conv Conv.try_conv norm_fun_eq

         fun binop_conv2 cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2

       in

         case (Thm.term_of ctm) of

           Const (@{const_name "fun_rel"}, _) $ _ $ _ => 

             (binop_conv2  left_conv simp_arrows_conv then_conv unfold_conv) ctm

           | _ => Conv.all_conv ctm

       end

     val unfold_ret_val_invs = Conv.bottom_conv 

       (K (Conv.try_conv (Conv.rewr_conv @{thm invariant_same_args}))) lthy 

     val simp_conv = Conv.arg_conv (Conv.fun2_conv simp_arrows_conv)

     val univq_conv = Conv.rewr_conv @{thm HOL.all_simps(6)[symmetric, THEN eq_reflection]}

     val univq_prenex_conv = Conv.top_conv (K (Conv.try_conv univq_conv)) lthy

     val beta_conv = Thm.beta_conversion true

     val eq_thm = 

       (simp_conv then_conv univq_prenex_conv then_conv beta_conv then_conv unfold_ret_val_invs) ctm

   in

     Object_Logic.rulify(eq_thm RS Drule.equal_elim_rule2)

   end

 fun lift_def_cmd (raw_var, rhs_raw) lthy =

   let

     val ((binding, SOME qty, mx), lthy') = yield_singleton Proof_Context.read_vars raw_var lthy 

     val rhs = (Syntax.check_term lthy' o Syntax.parse_term lthy') rhs_raw

     fun try_to_prove_refl thm = 

       let

         val lhs_eq =

           thm

           |> prop_of

           |> Logic.dest_implies

           |> fst

           |> strip_all_body

           |> try HOLogic.dest_Trueprop

       in

         case lhs_eq of

           SOME (Const ("HOL.eq", _) $ _ $ _) => SOME (@{thm refl} RS thm)

           | _ => NONE

       end

     val rsp_rel = Lifting_Term.equiv_relation lthy' (fastype_of rhs, qty)

     val rty_forced = (domain_type o fastype_of) rsp_rel;

     val forced_rhs = force_rty_type lthy' rty_forced rhs;

     val internal_rsp_tm = HOLogic.mk_Trueprop (rsp_rel $ forced_rhs $ forced_rhs)

     val readable_rsp_thm_eq = mk_readable_rsp_thm_eq internal_rsp_tm lthy'

     val maybe_proven_rsp_thm = try_to_prove_refl readable_rsp_thm_eq

     val (readable_rsp_tm, _) = Logic.dest_implies (prop_of readable_rsp_thm_eq)

     fun after_qed thm_list lthy = 

       let

         val internal_rsp_thm =

           case thm_list of

             [] => the maybe_proven_rsp_thm

           | [[thm]] => Goal.prove lthy [] [] internal_rsp_tm 

             (fn _ => rtac readable_rsp_thm_eq 1 THEN Proof_Context.fact_tac [thm] 1)

       in

         add_lift_def (binding, mx) qty rhs internal_rsp_thm lthy

       end

   in

     case maybe_proven_rsp_thm of

       SOME _ => Proof.theorem NONE after_qed [] lthy'

       | NONE =>  Proof.theorem NONE after_qed [[(readable_rsp_tm,[])]] lthy'

   end

 fun quot_thm_err ctxt (rty, qty) pretty_msg =

   let

     val error_msg = cat_lines

        ["Lifting failed for the following types:",

         Pretty.string_of (Pretty.block

          [Pretty.str "Raw type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty]),

         Pretty.string_of (Pretty.block

          [Pretty.str "Abstract type:", Pretty.brk 2, Syntax.pretty_typ ctxt qty]),

         "",

         (Pretty.string_of (Pretty.block

          [Pretty.str "Reason:", Pretty.brk 2, pretty_msg]))]

   in

     error error_msg

   end

 fun check_rty_err ctxt (rty_schematic, rty_forced) (raw_var, rhs_raw) =

   let

     val (_, ctxt') = yield_singleton Proof_Context.read_vars raw_var ctxt 

     val rhs = (Syntax.check_term ctxt' o Syntax.parse_term ctxt') rhs_raw

     val error_msg = cat_lines

        ["Lifting failed for the following term:",

         Pretty.string_of (Pretty.block

          [Pretty.str "Term:", Pretty.brk 2, Syntax.pretty_term ctxt rhs]),

         Pretty.string_of (Pretty.block

          [Pretty.str "Type:", Pretty.brk 2, Syntax.pretty_typ ctxt rty_schematic]),

         "",

         (Pretty.string_of (Pretty.block

          [Pretty.str "Reason:", 

           Pretty.brk 2, 

           Pretty.str "The type of the term cannot be instancied to",

           Pretty.brk 1,

           Pretty.quote (Syntax.pretty_typ ctxt rty_forced),

           Pretty.str "."]))]

     in

       error error_msg

     end

 fun lift_def_cmd_with_err_handling (raw_var, rhs_raw) lthy =

   (lift_def_cmd (raw_var, rhs_raw) lthy

     handle Lifting_Term.QUOT_THM (rty, qty, msg) => quot_thm_err lthy (rty, qty) msg)

     handle Lifting_Term.CHECK_RTY (rty_schematic, rty_forced) => 

       check_rty_err lthy (rty_schematic, rty_forced) (raw_var, rhs_raw)

 (* parser and command *)

 val liftdef_parser =

   ((Parse.binding -- (@{keyword "::"} |-- (Parse.typ >> SOME) -- Parse.opt_mixfix')) >> Parse.triple2)

     --| @{keyword "is"} -- Parse.term

 val _ =

   Outer_Syntax.local_theory_to_proof @{command_spec "lift_definition"}

     "definition for constants over the quotient type"

       (liftdef_parser >> lift_def_cmd_with_err_handling)

 end; (* structure *)