(*  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 *)