signature QUOTIENT_TYPE =

sig

  exception LIFT_MATCH of string

  val quotient_type: ((binding * mixfix) * (typ * term)) list -> Proof.context -> Proof.state

  val quotient_type_cmd: (((bstring * mixfix) * string) * string) list -> Proof.context -> Proof.state

end;

structure Quotient_Type: QUOTIENT_TYPE =

struct

open Quotient_Info;

exception LIFT_MATCH of string

(* wrappers for define, note, Attrib.internal and theorem_i *)

fun define (name, mx, rhs) lthy =

let

  val ((rhs, (_ , thm)), lthy') =

     Local_Theory.define ((name, mx), (Attrib.empty_binding, rhs)) lthy

in

  ((rhs, thm), lthy')

end

fun note (name, thm, attrs) lthy =

let

  val ((_,[thm']), lthy') = Local_Theory.note ((name, attrs), [thm]) lthy

in

  (thm', lthy')

end

fun intern_attr at = Attrib.internal (K at)

fun theorem after_qed goals ctxt =

let

  val goals' = map (rpair []) goals

  fun after_qed' thms = after_qed (the_single thms)

in 

  Proof.theorem_i NONE after_qed' [goals'] ctxt

end

(* definition of quotient types *)

(********************************)

val mem_def1 = @{lemma "y : S ==> S y" by (simp add: mem_def)}

val mem_def2 = @{lemma "S y ==> y : S" by (simp add: mem_def)}

(* 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

  lambda c

    (HOLogic.exists_const rty $

       lambda x (HOLogic.mk_eq (c, (rel $ x))))

end

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

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

let

  val typedef_tac =

     EVERY1 [rtac @{thm exI},

             rtac mem_def2, 

             rtac @{thm exI},

             rtac @{thm refl}]

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

in

  Local_Theory.theory_result

    (Typedef.add_typedef false NONE

       (qty_name, tfrees, 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 = mem_def2 RS (#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,

          EVERY' [rtac abs_inv, rtac @{thm exI}, rtac @{thm refl}],

          rtac rep_inj]) 1

end

(* proves the Quot_Type theorem *)

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

let

  val quot_type_const = Const (@{const_name "Quot_Type"}, dummyT)

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

             |> Syntax.check_term lthy

in

  Goal.prove lthy [] [] goal

    (K (typedef_quot_type_tac equiv_thm typedef_info))

end

(* proves the quotient theorem *)

fun typedef_quotient_thm (rel, abs, rep, abs_def, rep_def, quot_type_thm) lthy =

let

  val quotient_const = Const (@{const_name "Quotient"}, dummyT)

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

             |> Syntax.check_term lthy

  val typedef_quotient_thm_tac =

    EVERY1 [K (rewrite_goals_tac [abs_def, rep_def]),

            rtac @{thm Quot_Type.Quotient},

            rtac quot_type_thm]

in

  Goal.prove lthy [] [] goal

    (K typedef_quotient_thm_tac)

end

(* main function for constructing a quotient type *)

fun mk_typedef_main (((qty_name, mx), (rty, rel)), equiv_thm) lthy =

let

  (* generates the typedef *)

  val ((qty_full_name, typedef_info), lthy1) = typedef_make (qty_name, mx, rel, rty) lthy

  (* abs and rep functions from the typedef *)

  val Abs_ty = #abs_type typedef_info

  val Rep_ty = #rep_type typedef_info

  val Abs_name = #Abs_name typedef_info

  val Rep_name = #Rep_name typedef_info

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

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

  (* more abstract abs and rep definitions *)

  val abs_const = Const (@{const_name "Quot_Type.abs"}, dummyT )

  val rep_const = Const (@{const_name "Quot_Type.rep"}, dummyT )

  val abs_trm = Syntax.check_term lthy1 (abs_const $ rel $ Abs_const)

  val rep_trm = Syntax.check_term lthy1 (rep_const $ Rep_const)

  val abs_name = Binding.prefix_name "abs_" qty_name

  val rep_name = Binding.prefix_name "rep_" qty_name

  val ((abs, abs_def), lthy2) = define (abs_name, NoSyn, abs_trm) lthy1

  val ((rep, rep_def), lthy3) = define (rep_name, NoSyn, rep_trm) lthy2

  (* quot_type theorem - needed below *)

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

  (* quotient theorem *)  

  val quotient_thm = typedef_quotient_thm (rel, abs, rep, abs_def, rep_def, quot_thm) lthy3

  val quotient_thm_name = Binding.prefix_name "Quotient_" qty_name

  (* name equivalence theorem *)

  val equiv_thm_name = Binding.suffix_name "_equivp" qty_name

  (* storing the quot-info *)

  (* FIXME: VarifyT should not be used - at the moment it allows matching against the types. *)

in

  lthy4

  |> note (quotient_thm_name, quotient_thm, [intern_attr quotient_rules_add])

  ||>> note (equiv_thm_name, equiv_thm, [intern_attr equiv_rules_add])

end

(* interface and syntax setup *)

(* the ML-interface takes a list of 4-tuples consisting of  *)

(*                                                          *)

(* - the name of the quotient type                          *)

(* - its mixfix annotation                                  *)

(* - the type to be quotient                                *)

(* - the relation according to which the type is quotient   *)

fun quotient_type quot_list lthy = 

let

  fun mk_goal (rty, rel) =

  let

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

  in 

    HOLogic.mk_Trueprop (Const (@{const_name equivp}, equivp_ty) $ rel)

  end

  val goals = map (mk_goal o snd) quot_list

  fun after_qed thms lthy =

    fold_map mk_typedef_main (quot_list ~~ thms) lthy |> snd

in

  theorem after_qed goals lthy

end

fun quotient_type_cmd spec lthy = 

let

  fun parse_spec (((qty_str, mx), rty_str), rel_str) =

  let

    val qty_name = Binding.name qty_str

    val rty = Syntax.read_typ lthy rty_str

    val rel = Syntax.read_term lthy rel_str 

  in

    ((qty_name, mx), (rty, rel))

  end

in

  quotient_type (map parse_spec spec) lthy

end

val quotspec_parser = 

    OuterParse.and_list1

     (OuterParse.short_ident -- OuterParse.opt_infix -- 

       (OuterParse.$$$ "=" |-- OuterParse.typ) -- 

         (OuterParse.$$$ "/" |-- OuterParse.term))

val _ = OuterKeyword.keyword "/"

val _ = 

    OuterSyntax.local_theory_to_proof "quotient_type" 

      "quotient type definitions (require equivalence proofs)"

         OuterKeyword.thy_goal (quotspec_parser >> quotient_type_cmd)

end; (* structure *)