(*  Title:      HOL/Tools/Quotient/quotient_def.ML

     Author:     Cezary Kaliszyk and Christian Urban

 Definitions for constants on quotient types.

 *)

 signature QUOTIENT_DEF =

 sig

   val add_quotient_def:

     ((binding * mixfix) * Attrib.binding) * (term * term) -> thm ->

     local_theory -> Quotient_Info.quotconsts * local_theory

   val quotient_def:

     (binding * typ option * mixfix) option * (Attrib.binding * (term * term)) ->

     local_theory -> Proof.state

   val quotient_def_cmd:

     (binding * string option * mixfix) option * (Attrib.binding * (string * string)) ->

     local_theory -> Proof.state

   val lift_raw_const: typ list -> (string * term * mixfix) -> local_theory ->

     Quotient_Info.quotconsts * local_theory

 end;

 structure Quotient_Def: QUOTIENT_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 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 = Quotient_Term.prove_quot_theorem 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 generate_code_cert ctxt def_thm rsp_thm (rty, qty) =

   let

     val quot_thm = Quotient_Term.prove_quot_theorem 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 = (snd o Thm.dest_comb o snd o Thm.dest_comb o cprop_of) 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 define_code_cert def_thm rsp_thm (rty, qty) lthy = 

   let

     val quot_thm = Quotient_Term.prove_quot_theorem lthy (get_body_types (rty, qty))

   in

     if Quotient_Type.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);

       in

         lthy

           |> (snd oo Local_Theory.note) ((Binding.empty, [add_abs_eqn_attrib]), [code_cert])

       end

     else

       lthy

   end

 fun define_code_eq 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) ((Binding.empty, [Code.add_default_eqn_attrib]), [simp_code_eq])

   end

 fun define_code def_thm rsp_thm (rty, qty) lthy =

   if body_type rty = body_type qty then 

     define_code_eq def_thm lthy

   else 

     define_code_cert def_thm rsp_thm (rty, qty) lthy

 (* The ML-interface for a quotient definition takes

    as argument:

     - an optional binding and mixfix annotation

     - attributes

     - the new constant as term

     - the rhs of the definition as term

     - respectfulness theorem for the rhs

    It stores the qconst_info in the quotconsts data slot.

    Restriction: At the moment the left- and right-hand

    side of the definition must be a constant.

 *)

 fun error_msg bind str =

   let

     val name = Binding.name_of bind

     val pos = Position.str_of (Binding.pos_of bind)

   in

     error ("Head of quotient_definition " ^

       quote str ^ " differs from declaration " ^ name ^ pos)

   end

 fun add_quotient_def ((var, (name, atts)), (lhs, rhs)) rsp_thm lthy =

   let

     val rty = fastype_of rhs

     val qty = fastype_of lhs

     val absrep_trm = 

       Quotient_Term.absrep_fun lthy Quotient_Term.AbsF (rty, qty) $ rhs

     val prop = Syntax.check_term lthy (Logic.mk_equals (lhs, absrep_trm))

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

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

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

       Local_Theory.define (var, ((Thm.def_binding_optional (#1 var) name, atts), newrhs)) lthy

     (* data storage *)

     val qconst_data = {qconst = trm, rconst = rhs, def = def_thm}

     fun get_rsp_thm_name (lhs_name, _) = Binding.suffix_name "_rsp" lhs_name

     val lthy'' = lthy'

       |> Local_Theory.declaration {syntax = false, pervasive = true}

         (fn phi =>

           (case Quotient_Info.transform_quotconsts phi qconst_data of

             qcinfo as {qconst = Const (c, _), ...} =>

               Quotient_Info.update_quotconsts c qcinfo

           | _ => I))

       |> (snd oo Local_Theory.note) 

         ((get_rsp_thm_name var, [Attrib.internal (K Quotient_Info.rsp_rules_add)]),

         [rsp_thm])

       |> define_code def_thm rsp_thm (rty, qty)

   in

     (qconst_data, lthy'')

   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 gen_quotient_def prep_vars prep_term (raw_var, (attr, (lhs_raw, rhs_raw))) lthy =

   let

     val (vars, ctxt) = prep_vars (the_list raw_var) lthy

     val T_opt = (case vars of [(_, SOME T, _)] => SOME T | _ => NONE)

     val lhs = prep_term T_opt ctxt lhs_raw

     val rhs = prep_term NONE ctxt rhs_raw

     val (lhs_str, lhs_ty) = dest_Free lhs handle TERM _ => error "Constant already defined."

     val _ = if null (strip_abs_vars rhs) then () else error "The definiens cannot be an abstraction"

     val _ = if is_Const rhs then () else warning "The definiens is not a constant"

     val var =

       (case vars of 

         [] => (Binding.name lhs_str, NoSyn)

       | [(binding, _, mx)] =>

           if Variable.check_name binding = lhs_str then (binding, mx)

           else error_msg binding lhs_str

       | _ => raise Match)

     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)

           | SOME _ => (case body_type (fastype_of lhs) of

             Type (typ_name, _) => ( SOME

               (#equiv_thm (the (Quotient_Info.lookup_quotients lthy typ_name)) 

                 RS @{thm Equiv_Relations.equivp_reflp} RS thm)

               handle _ => NONE)

             | _ => NONE

             )

           | _ => NONE

       end

     val rsp_rel = Quotient_Term.equiv_relation lthy (fastype_of rhs, lhs_ty)

     val internal_rsp_tm = HOLogic.mk_Trueprop (Syntax.check_term lthy (rsp_rel $ rhs $ 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 ctxt [] [] internal_rsp_tm 

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

       in

         snd (add_quotient_def ((var, attr), (lhs, 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 check_term' cnstr ctxt =

   Syntax.check_term ctxt o (case cnstr of SOME T => Type.constraint T | _ => I)

 fun read_term' cnstr ctxt =

   check_term' cnstr ctxt o Syntax.parse_term ctxt

 val quotient_def = gen_quotient_def Proof_Context.cert_vars check_term'

 val quotient_def_cmd = gen_quotient_def Proof_Context.read_vars read_term'

 (* a wrapper for automatically lifting a raw constant *)

 fun lift_raw_const qtys (qconst_name, rconst, mx) ctxt =

   let

     val rty = fastype_of rconst

     val qty = Quotient_Term.derive_qtyp ctxt qtys rty

     val lhs = Free (qconst_name, qty)

   in

     (*quotient_def (SOME (Binding.name qconst_name, NONE, mx), (Attrib.empty_binding, (lhs, rconst))) ctxt*)

     ({qconst = lhs, rconst = lhs, def = @{thm refl}}, ctxt)

   end

 (* parser and command *)

 val quotdef_parser =

   Scan.option Parse_Spec.constdecl --

     Parse.!!! (Parse_Spec.opt_thm_name ":" -- (Parse.term --| @{keyword "is"} -- Parse.term))

 val _ =

   Outer_Syntax.local_theory_to_proof @{command_spec "quotient_definition"}

     "definition for constants over the quotient type"

       (quotdef_parser >> quotient_def_cmd)

 end; (* structure *)