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

    Author:     Cezary Kaliszyk and Christian Urban

Definition of a quotient type.

*)

signature QUOTIENT_TYPE =

sig

  val add_quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) * thm

    -> local_theory -> Quotient_Info.quotients * local_theory

  val quotient_type: ((string list * binding * mixfix) * (typ * term * bool)) list

    -> Proof.context -> Proof.state

  val quotient_type_cmd: ((((string list * binding) * mixfix) * string) * (bool * string)) list

    -> Proof.context -> Proof.state

end;

structure Quotient_Type: QUOTIENT_TYPE =

struct

(*** definition of quotient types ***)

val mem_def1 = @{lemma "y : Collect S ==> S y" by simp}

val mem_def2 = @{lemma "S y ==> y : Collect S" by simp}

(* constructs the term lambda (c::rty => bool). EX (x::rty). c = rel x *)

fun typedef_term rel rty lthy =

  let

    val [x, c] =

      [("x", rty), ("c", HOLogic.mk_setT rty)]

      |> Variable.variant_frees lthy [rel]

      |> map Free

  in

    HOLogic.Collect_const (HOLogic.mk_setT rty) $ (lambda c (HOLogic.exists_const rty $

        lambda x (HOLogic.mk_conj (rel $ x $ x,

        HOLogic.mk_eq (c, HOLogic.Collect_const rty $ (rel $ x))))))

  end

(* makes the new type definitions and proves non-emptyness *)

fun typedef_make (vs, qty_name, mx, rel, rty) equiv_thm lthy =

  let

    val typedef_tac =

      EVERY1 (map rtac [@{thm part_equivp_typedef}, equiv_thm])

  in

  (* FIXME: purely local typedef causes at the moment

     problems with type variables

    Typedef.add_typedef false NONE (qty_name, vs, mx)

      (typedef_term rel rty lthy) NONE typedef_tac lthy

  *)

  (* FIXME should really use local typedef here *)

    Local_Theory.background_theory_result

     (Typedef.add_typedef_global false NONE

       (qty_name, map (rpair dummyS) vs, mx)

         (typedef_term rel rty lthy)

           NONE typedef_tac) lthy

  end

(* tactic to prove the quot_type theorem for the new type *)

fun typedef_quot_type_tac equiv_thm ((_, typedef_info): Typedef.info) =

  let

    val rep_thm = #Rep typedef_info RS mem_def1

    val rep_inv = #Rep_inverse typedef_info

    val abs_inv = #Abs_inverse typedef_info

    val rep_inj = #Rep_inject typedef_info

  in

    (rtac @{thm quot_type.intro} THEN' RANGE [

      rtac equiv_thm,

      rtac rep_thm,

      rtac rep_inv,

      rtac abs_inv THEN' rtac mem_def2 THEN' atac,

      rtac rep_inj]) 1

  end

(* proves the quot_type theorem for the new type *)

fun typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy =

  let

    val quot_type_const = Const (@{const_name "quot_type"},

      fastype_of rel --> fastype_of abs --> fastype_of rep --> @{typ bool})

    val goal = HOLogic.mk_Trueprop (quot_type_const $ rel $ abs $ rep)

  in

    Goal.prove lthy [] [] goal

      (K (typedef_quot_type_tac equiv_thm typedef_info))

  end

(* main function for constructing a quotient type *)

fun add_quotient_type (((vs, qty_name, mx), (rty, rel, partial)), equiv_thm) lthy =

  let

    val part_equiv =

      if partial

      then equiv_thm

      else equiv_thm RS @{thm equivp_implies_part_equivp}

    (* generates the typedef *)

    val ((qty_full_name, typedef_info), lthy1) =

      typedef_make (vs, qty_name, mx, rel, rty) part_equiv lthy

    (* abs and rep functions from the typedef *)

    val Abs_ty = #abs_type (#1 typedef_info)

    val Rep_ty = #rep_type (#1 typedef_info)

    val Abs_name = #Abs_name (#1 typedef_info)

    val Rep_name = #Rep_name (#1 typedef_info)

    val Abs_const = Const (Abs_name, Rep_ty --> Abs_ty)

    val Rep_const = Const (Rep_name, Abs_ty --> Rep_ty)

    (* more useful abs and rep definitions *)

    val abs_const = Const (@{const_name quot_type.abs},

      (rty --> rty --> @{typ bool}) --> (Rep_ty --> Abs_ty) --> rty --> Abs_ty)

    val rep_const = Const (@{const_name quot_type.rep}, (Abs_ty --> Rep_ty) --> Abs_ty --> rty)

    val abs_trm = abs_const $ rel $ Abs_const

    val rep_trm = rep_const $ Rep_const

    val abs_name = Binding.prefix_name "abs_" qty_name

    val rep_name = Binding.prefix_name "rep_" qty_name

    val ((_, (_, abs_def)), lthy2) = lthy1

      |> Local_Theory.define ((abs_name, NoSyn), (Attrib.empty_binding, abs_trm))

    val ((_, (_, rep_def)), lthy3) = lthy2

      |> Local_Theory.define ((rep_name, NoSyn), (Attrib.empty_binding, rep_trm))

    (* quot_type theorem *)

    val quot_thm = typedef_quot_type_thm (rel, Abs_const, Rep_const, part_equiv, typedef_info) lthy3

    (* quotient theorem *)

    val quotient_thm_name = Binding.prefix_name "Quotient_" qty_name

    val quotient_thm =

      (quot_thm RS @{thm quot_type.Quotient})

      |> fold_rule [abs_def, rep_def]

    (* name equivalence theorem *)

    val equiv_thm_name = Binding.suffix_name "_equivp" qty_name

    (* storing the quotients *)

    val quotients = {qtyp = Abs_ty, rtyp = rty, equiv_rel = rel, equiv_thm = equiv_thm}

    fun qinfo phi = Quotient_Info.transform_quotients phi quotients

    val lthy4 = lthy3

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

        (fn phi => Quotient_Info.update_quotients qty_full_name (qinfo phi))

      |> (snd oo Local_Theory.note)

        ((equiv_thm_name,

          if partial then [] else [Attrib.internal (K Quotient_Info.equiv_rules_add)]),

          [equiv_thm])

      |> (snd oo Local_Theory.note)

        ((quotient_thm_name, [Attrib.internal (K Quotient_Info.quotient_rules_add)]),

          [quotient_thm])

  in

    (quotients, lthy4)

  end

(* sanity checks for the quotient type specifications *)

fun sanity_check ((vs, qty_name, _), (rty, rel, _)) =

  let

    val rty_tfreesT = map fst (Term.add_tfreesT rty [])

    val rel_tfrees = map fst (Term.add_tfrees rel [])

    val rel_frees = map fst (Term.add_frees rel [])

    val rel_vars = Term.add_vars rel []

    val rel_tvars = Term.add_tvars rel []

    val qty_str = Binding.print qty_name ^ ": "

    val illegal_rel_vars =

      if null rel_vars andalso null rel_tvars then []

      else [qty_str ^ "illegal schematic variable(s) in the relation."]

    val dup_vs =

      (case duplicates (op =) vs of

        [] => []

      | dups => [qty_str ^ "duplicate type variable(s) on the lhs: " ^ commas_quote dups])

    val extra_rty_tfrees =

      (case subtract (op =) vs rty_tfreesT of

        [] => []

      | extras => [qty_str ^ "extra type variable(s) on the lhs: " ^ commas_quote extras])

    val extra_rel_tfrees =

      (case subtract (op =) vs rel_tfrees of

        [] => []

      | extras => [qty_str ^ "extra type variable(s) in the relation: " ^ commas_quote extras])

    val illegal_rel_frees =

      (case rel_frees of

        [] => []

      | xs => [qty_str ^ "illegal variable(s) in the relation: " ^ commas_quote xs])

    val errs = illegal_rel_vars @ dup_vs @ extra_rty_tfrees @ extra_rel_tfrees @ illegal_rel_frees

  in

    if null errs then () else error (cat_lines errs)

  end

(* check for existence of map functions *)

fun map_check ctxt (_, (rty, _, _)) =

  let

    fun map_check_aux rty warns =

      (case rty of

        Type (_, []) => warns

      | Type (s, _) => if is_some (Quotient_Info.lookup_quotmaps ctxt s) then warns else s :: warns

      | _ => warns)

    val warns = map_check_aux rty []

  in

    if null warns then ()

    else warning ("No map function defined for " ^ commas warns ^

      ". This will cause problems later on.")

  end

(*** interface and syntax setup ***)

(* the ML-interface takes a list of 5-tuples consisting of:

 - the name of the quotient type

 - its free type variables (first argument)

 - its mixfix annotation

 - the type to be quotient

 - the partial flag (a boolean)

 - the relation according to which the type is quotient

 it opens a proof-state in which one has to show that the

 relations are equivalence relations

*)

fun quotient_type quot_list lthy =

  let

    (* sanity check *)

    val _ = List.app sanity_check quot_list

    val _ = List.app (map_check lthy) quot_list

    fun mk_goal (rty, rel, partial) =

      let

        val equivp_ty = ([rty, rty] ---> @{typ bool}) --> @{typ bool}

        val const =

          if partial then @{const_name part_equivp} else @{const_name equivp}

      in

        HOLogic.mk_Trueprop (Const (const, equivp_ty) $ rel)

      end

    val goals = map (mk_goal o snd) quot_list

    fun after_qed [thms] = fold (snd oo add_quotient_type) (quot_list ~~ thms)

  in

    Proof.theorem NONE after_qed [map (rpair []) goals] lthy

  end

fun quotient_type_cmd specs lthy =

  let

    fun parse_spec ((((vs, qty_name), mx), rty_str), (partial, rel_str)) lthy =

      let

        val rty = Syntax.read_typ lthy rty_str

        val lthy1 = Variable.declare_typ rty lthy

        val rel =

          Syntax.parse_term lthy1 rel_str

          |> Type.constraint (rty --> rty --> @{typ bool})

          |> Syntax.check_term lthy1

        val lthy2 = Variable.declare_term rel lthy1

      in

        (((vs, qty_name, mx), (rty, rel, partial)), lthy2)

      end

    val (spec', lthy') = fold_map parse_spec specs lthy

  in

    quotient_type spec' lthy'

  end

val partial = Scan.optional (Parse.reserved "partial" -- Parse.$$$ ":" >> K true) false

val quotspec_parser =

  Parse.and_list1

    ((Parse.type_args -- Parse.binding) --

      Parse.opt_mixfix -- (Parse.$$$ "=" |-- Parse.typ) --

        (Parse.$$$ "/" |-- (partial -- Parse.term)))

val _ = Keyword.keyword "/"

val _ =

  Outer_Syntax.local_theory_to_proof "quotient_type"

    "quotient type definitions (require equivalence proofs)"

       Keyword.thy_goal (quotspec_parser >> quotient_type_cmd)

end;