src/HOL/Tools/Quotient/quotient_def.ML
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(*  Title:      HOL/Tools/Quotient/quotient_def.ML
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    Author:     Cezary Kaliszyk and Christian Urban
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Definitions for constants on quotient types.
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*)
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signature QUOTIENT_DEF =
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sig
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  val add_quotient_def:
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    ((binding * mixfix) * Attrib.binding) * (term * term) -> thm ->
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    local_theory -> Quotient_Info.quotconsts * local_theory
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  val quotient_def:
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    (binding * typ option * mixfix) option * (Attrib.binding * (term * term)) ->
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    local_theory -> Proof.state
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  val quotient_def_cmd:
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    (binding * string option * mixfix) option * (Attrib.binding * (string * string)) ->
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    local_theory -> Proof.state
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end;
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structure Quotient_Def: QUOTIENT_DEF =
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struct
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(** Interface and Syntax Setup **)
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(* Generation of the code certificate from the rsp theorem *)
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open Lifting_Util
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infix 0 MRSL
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(* The ML-interface for a quotient definition takes
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   as argument:
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    - an optional binding and mixfix annotation
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    - attributes
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    - the new constant as term
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    - the rhs of the definition as term
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    - respectfulness theorem for the rhs
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   It stores the qconst_info in the quotconsts data slot.
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   Restriction: At the moment the left- and right-hand
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   side of the definition must be a constant.
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*)
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fun error_msg bind str =
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  let
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    val name = Binding.name_of bind
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    val pos = Position.here (Binding.pos_of bind)
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  in
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    error ("Head of quotient_definition " ^
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      quote str ^ " differs from declaration " ^ name ^ pos)
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  end
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fun add_quotient_def ((var, (name, atts)), (lhs, rhs)) rsp_thm lthy =
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  let
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    val rty = fastype_of rhs
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    val qty = fastype_of lhs
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    val absrep_trm =
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      Quotient_Term.absrep_fun lthy Quotient_Term.AbsF (rty, qty) $ rhs
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    val prop = Syntax.check_term lthy (Logic.mk_equals (lhs, absrep_trm))
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    val (_, prop') = Local_Defs.cert_def lthy (K []) prop
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    val (_, newrhs) = Local_Defs.abs_def prop'
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    val ((qconst, (_ , def)), lthy') =
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      Local_Theory.define (var, ((Thm.def_binding_optional (#1 var) name, atts), newrhs)) lthy
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    fun qconst_data phi =
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      Quotient_Info.transform_quotconsts phi {qconst = qconst, rconst = rhs, def = def}
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    fun qualify defname suffix = Binding.name suffix
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      |> Binding.qualify true defname
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    val lhs_name = Binding.name_of (#1 var)
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    val rsp_thm_name = qualify lhs_name "rsp"
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    val lthy'' = lthy'
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      |> Local_Theory.declaration {syntax = false, pervasive = true, pos = \<^here>}
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        (fn phi =>
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          (case qconst_data phi of
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            qcinfo as {qconst = Const (c, _), ...} =>
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              Quotient_Info.update_quotconsts (c, qcinfo)
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          | _ => I))
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      |> (snd oo Local_Theory.note)
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        ((rsp_thm_name, @{attributes [quot_respect]}), [rsp_thm])
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  in
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    (qconst_data Morphism.identity, lthy'')
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  end
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fun mk_readable_rsp_thm_eq tm ctxt =
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  let
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    val ctm = Thm.cterm_of ctxt tm
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    fun abs_conv2 cv = Conv.abs_conv (Conv.abs_conv (cv o #2) o #2) ctxt
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    fun erase_quants ctxt' ctm' =
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      case (Thm.term_of ctm') of
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        Const (\<^const_name>\<open>HOL.eq\<close>, _) $ _ $ _ => Conv.all_conv ctm'
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        | _ => (Conv.binder_conv (erase_quants o #2) ctxt' then_conv
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          Conv.rewr_conv @{thm fun_eq_iff[symmetric, THEN eq_reflection]}) ctm'
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    val norm_fun_eq = abs_conv2 erase_quants then_conv Thm.eta_conversion
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    fun simp_arrows_conv ctm =
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      let
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        val unfold_conv = Conv.rewrs_conv
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          [@{thm rel_fun_eq_eq_onp[THEN eq_reflection]}, @{thm rel_fun_eq_rel[THEN eq_reflection]},
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            @{thm rel_fun_def[THEN eq_reflection]}]
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        val left_conv = simp_arrows_conv then_conv Conv.try_conv norm_fun_eq
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        fun binop_conv2 cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
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      in
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        case (Thm.term_of ctm) of
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          Const (\<^const_name>\<open>rel_fun\<close>, _) $ _ $ _ =>
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            (binop_conv2 left_conv simp_arrows_conv then_conv unfold_conv) ctm
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          | _ => Conv.all_conv ctm
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      end
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    val unfold_ret_val_invs = Conv.bottom_conv
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      (K (Conv.try_conv (Conv.rewr_conv @{thm eq_onp_same_args[THEN eq_reflection]}))) ctxt
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    val simp_conv = Conv.arg_conv (Conv.fun2_conv simp_arrows_conv)
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    val univq_conv = Conv.rewr_conv @{thm HOL.all_simps(6)[symmetric, THEN eq_reflection]}
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    val univq_prenex_conv = Conv.top_conv (K (Conv.try_conv univq_conv)) ctxt
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    val beta_conv = Thm.beta_conversion true
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    val eq_thm =
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      (simp_conv then_conv univq_prenex_conv then_conv beta_conv then_conv unfold_ret_val_invs) ctm
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  in
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    Object_Logic.rulify ctxt (eq_thm RS Drule.equal_elim_rule2)
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  end
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fun gen_quotient_def prep_var parse_term (raw_var, (attr, (raw_lhs, raw_rhs))) lthy =
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  let
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    val (opt_var, ctxt) =
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      (case raw_var of
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        NONE => (NONE, lthy)
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      | SOME var => prep_var var lthy |>> SOME)
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    val lhs_constraints = (case opt_var of SOME (_, SOME T, _) => [T] | _ => [])
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    fun prep_term Ts = parse_term ctxt #> fold Type.constraint Ts #> Syntax.check_term ctxt;
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    val lhs = prep_term lhs_constraints raw_lhs
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    val rhs = prep_term [] raw_rhs
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    val (lhs_str, lhs_ty) = dest_Free lhs handle TERM _ => error "Constant already defined"
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    val _ = if null (strip_abs_vars rhs) then () else error "The definiens cannot be an abstraction"
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    val _ = if is_Const rhs then () else warning "The definiens is not a constant"
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    val var =
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      (case opt_var of
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        NONE => (Binding.name lhs_str, NoSyn)
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      | SOME (binding, _, mx) =>
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          if Variable.check_name binding = lhs_str then (binding, mx)
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          else error_msg binding lhs_str);
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    fun try_to_prove_refl thm =
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      let
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        val lhs_eq =
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          thm
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          |> Thm.prop_of
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          |> Logic.dest_implies
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          |> fst
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          |> strip_all_body
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          |> try HOLogic.dest_Trueprop
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      in
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        case lhs_eq of
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          SOME (Const (\<^const_name>\<open>HOL.eq\<close>, _) $ _ $ _) => SOME (@{thm refl} RS thm)
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          | SOME _ => (case body_type (fastype_of lhs) of
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            Type (typ_name, _) =>
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              \<^try>\<open>
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                #equiv_thm (the (Quotient_Info.lookup_quotients lthy typ_name))
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                  RS @{thm Equiv_Relations.equivp_reflp} RS thm\<close>
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            | _ => NONE
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            )
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          | _ => NONE
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      end
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    val rsp_rel = Quotient_Term.equiv_relation lthy (fastype_of rhs, lhs_ty)
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    val internal_rsp_tm = HOLogic.mk_Trueprop (Syntax.check_term lthy (rsp_rel $ rhs $ rhs))
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    val readable_rsp_thm_eq = mk_readable_rsp_thm_eq internal_rsp_tm lthy
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    val maybe_proven_rsp_thm = try_to_prove_refl readable_rsp_thm_eq
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    val (readable_rsp_tm, _) = Logic.dest_implies (Thm.prop_of readable_rsp_thm_eq)
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    fun after_qed thm_list lthy =
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      let
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        val internal_rsp_thm =
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          case thm_list of
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            [] => the maybe_proven_rsp_thm
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          | [[thm]] => Goal.prove ctxt [] [] internal_rsp_tm
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            (fn _ =>
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              resolve_tac ctxt [readable_rsp_thm_eq] 1 THEN
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              Proof_Context.fact_tac ctxt [thm] 1)
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      in
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        snd (add_quotient_def ((var, attr), (lhs, rhs)) internal_rsp_thm lthy)
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      end
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  in
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    case maybe_proven_rsp_thm of
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      SOME _ => Proof.theorem NONE after_qed [] lthy
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      | NONE =>  Proof.theorem NONE after_qed [[(readable_rsp_tm,[])]] lthy
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  end
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val quotient_def = gen_quotient_def Proof_Context.cert_var (K I)
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val quotient_def_cmd = gen_quotient_def Proof_Context.read_var Syntax.parse_term
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(* command syntax *)
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val _ =
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  Outer_Syntax.local_theory_to_proof \<^command_keyword>\<open>quotient_definition\<close>
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    "definition for constants over the quotient type"
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      (Scan.option Parse_Spec.constdecl --
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        Parse.!!! (Parse_Spec.opt_thm_name ":" -- (Parse.term -- (\<^keyword>\<open>is\<close> |-- Parse.term)))
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        >> quotient_def_cmd);
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end;