(*  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 *)