(*  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

 end;

 structure Quotient_Def: QUOTIENT_DEF =

 struct

 (** Interface and Syntax Setup **)

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

 open Lifting_Util

 infix 0 MRSL

 (* 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.here (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 qualify defname suffix = Binding.name suffix

       |> Binding.qualify true defname

     val lhs_name = Binding.name_of (#1 var)

     val rsp_thm_name = qualify lhs_name "rsp"

     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) 

         ((rsp_thm_name, [Attrib.internal (K Quotient_Info.rsp_rules_add)]),

         [rsp_thm])

   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 (@{const_name 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 rel_fun_eq_eq_onp[THEN eq_reflection]}, @{thm rel_fun_eq_rel[THEN eq_reflection]}, 

             @{thm rel_fun_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 rel_fun}, _) $ _ $ _ => 

             (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 eq_onp_same_args[THEN eq_reflection]}))) 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 lthy (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 (@{const_name HOL.eq}, _) $ _ $ _) => SOME (@{thm refl} RS thm)

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

             Type (typ_name, _) =>

               try (fn () =>

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

                   RS @{thm Equiv_Relations.equivp_reflp} RS thm) ()

             | _ => 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 ctxt [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'

 (* 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 *)