theory QuotMain

imports QuotScript QuotList Prove

begin

locale QUOT_TYPE =

  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

  and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"

  and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"

  assumes equiv: "EQUIV R"

  and     rep_prop: "\<And>y. \<exists>x. Rep y = R x"

  and     rep_inverse: "\<And>x. Abs (Rep x) = x"

  and     abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"

  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"

begin

definition

  "ABS x \<equiv> Abs (R x)"

definition

  "REP a = Eps (Rep a)"

lemma lem9:

  shows "R (Eps (R x)) = R x"

proof -

  have a: "R x x" using equiv by (simp add: EQUIV_REFL_SYM_TRANS REFL_def)

  then have "R x (Eps (R x))" by (rule someI)

  then show "R (Eps (R x)) = R x"

    using equiv unfolding EQUIV_def by simp

qed

theorem thm10:

  shows "ABS (REP a) \<equiv> a"

  apply  (rule eq_reflection)

  unfolding ABS_def REP_def

proof -

  from rep_prop

  obtain x where eq: "Rep a = R x" by auto

  have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp

  also have "\<dots> = Abs (R x)" using lem9 by simp

  also have "\<dots> = Abs (Rep a)" using eq by simp

  also have "\<dots> = a" using rep_inverse by simp

  finally

  show "Abs (R (Eps (Rep a))) = a" by simp

qed

lemma REP_refl:

  shows "R (REP a) (REP a)"

unfolding REP_def

by (simp add: equiv[simplified EQUIV_def])

lemma lem7:

  shows "(R x = R y) = (Abs (R x) = Abs (R y))"

apply(rule iffI)

apply(simp)

apply(drule rep_inject[THEN iffD2])

apply(simp add: abs_inverse)

done

theorem thm11:

  shows "R r r' = (ABS r = ABS r')"

unfolding ABS_def

by (simp only: equiv[simplified EQUIV_def] lem7)

lemma REP_ABS_rsp:

  shows "R f (REP (ABS g)) = R f g"

  and   "R (REP (ABS g)) f = R g f"

by (simp_all add: thm10 thm11)

lemma QUOTIENT:

  "QUOTIENT R ABS REP"

apply(unfold QUOTIENT_def)

apply(simp add: thm10)

apply(simp add: REP_refl)

apply(subst thm11[symmetric])

apply(simp add: equiv[simplified EQUIV_def])

done

lemma R_trans:

  assumes ab: "R a b"

  and     bc: "R b c"

  shows "R a c"

proof -

  have tr: "TRANS R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

  moreover have ab: "R a b" by fact

  moreover have bc: "R b c" by fact

  ultimately show "R a c" unfolding TRANS_def by blast

qed

lemma R_sym:

  assumes ab: "R a b"

  shows "R b a"

proof -

  have re: "SYM R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

  then show "R b a" using ab unfolding SYM_def by blast

qed

lemma R_trans2:

  assumes ac: "R a c"

  and     bd: "R b d"

  shows "R a b = R c d"

proof

  assume "R a b"

  then have "R b a" using R_sym by blast

  then have "R b c" using ac R_trans by blast

  then have "R c b" using R_sym by blast

  then show "R c d" using bd R_trans by blast

next

  assume "R c d"

  then have "R a d" using ac R_trans by blast

  then have "R d a" using R_sym by blast

  then have "R b a" using bd R_trans by blast

  then show "R a b" using R_sym by blast

qed

lemma REPS_same:

  shows "R (REP a) (REP b) \<equiv> (a = b)"

proof -

  have "R (REP a) (REP b) = (a = b)"

  proof

    assume as: "R (REP a) (REP b)"

    from rep_prop

    obtain x y

      where eqs: "Rep a = R x" "Rep b = R y" by blast

    from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp

    then have "R x (Eps (R y))" using lem9 by simp

    then have "R (Eps (R y)) x" using R_sym by blast

    then have "R y x" using lem9 by simp

    then have "R x y" using R_sym by blast

    then have "ABS x = ABS y" using thm11 by simp

    then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp

    then show "a = b" using rep_inverse by simp

  next

    assume ab: "a = b"

    have "REFL R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp

    then show "R (REP a) (REP b)" unfolding REFL_def using ab by auto

  qed

  then show "R (REP a) (REP b) \<equiv> (a = b)" by simp

qed

end

section {* type definition for the quotient type *}

ML {*

(* constructs the term \<lambda>(c::ty \<Rightarrow> bool). \<exists>x. c = rel x *)

fun typedef_term rel ty lthy =

let

  val [x, c] = [("x", ty), ("c", ty --> @{typ bool})]

               |> Variable.variant_frees lthy [rel]

               |> map Free

in

  lambda c

    (HOLogic.exists_const ty $

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

end

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

let

  val typedef_tac =

     EVERY1 [rewrite_goal_tac @{thms mem_def},

             rtac @{thm exI},

             rtac @{thm exI},

             rtac @{thm refl}]

in

  LocalTheory.theory_result

    (Typedef.add_typedef false NONE

         (typedef_term rel ty 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

  val rep_inv = #Rep_inverse typedef_info

  val abs_inv = #Abs_inverse typedef_info

  val rep_inj = #Rep_inject typedef_info

  val ss = HOL_basic_ss addsimps @{thms mem_def}

  val rep_thm_simpd = Simplifier.asm_full_simplify ss rep_thm

  val abs_inv_simpd = Simplifier.asm_full_simplify ss abs_inv

in

  EVERY1 [rtac @{thm QUOT_TYPE.intro},

          rtac equiv_thm,

          rtac rep_thm_simpd,

          rtac rep_inv,

          rtac abs_inv_simpd, rtac @{thm exI}, rtac @{thm refl},

          rtac rep_inj]

end

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

*}

text {* two wrappers for define and note *}

ML {*

fun make_def (name, mx, rhs) lthy =

let

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

     LocalTheory.define Thm.internalK ((name, mx), (Attrib.empty_binding, rhs)) lthy

in

  ((rhs, thm), lthy')

end

*}

ML {*

fun note_thm (name, thm) lthy =

let

  val ((_,[thm']), lthy') = LocalTheory.note Thm.theoremK ((name, []), [thm]) lthy

in

  (thm', lthy')

end

*}

ML {*

val no_vars = Thm.rule_attribute (fn context => fn th =>

  let

    val ctxt = Variable.set_body false (Context.proof_of context);

    val ((_, [th']), _) = Variable.import true [th] ctxt;

  in th' end);

*}

ML {*

let

  (* generates typedef *)

  (* abs and rep functions *)

  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 (abs_name, rep_ty --> abs_ty)

  val rep = Const (rep_name, abs_ty --> rep_ty)

  (* 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)

  val REP_trm = Syntax.check_term lthy1 (REP_const $ rep)

  val ABS_name = Binding.prefix_name "ABS_" qty_name

  val REP_name = Binding.prefix_name "REP_" qty_name

  val (((ABS, ABS_def), (REP, REP_def)), lthy2) =

         lthy1 |> make_def (ABS_name, NoSyn, ABS_trm)

               ||>> make_def (REP_name, NoSyn, REP_trm)

  (* quot_type theorem *)

  val quot_thm = typedef_quot_type_thm (rel, abs, rep, equiv_thm, typedef_info) lthy2

  val quot_thm_name = Binding.prefix_name "QUOT_TYPE_" qty_name

  (* quotient theorem *)

  val quotient_thm = typedef_quotient_thm (rel, ABS, REP, ABS_def, REP_def, quot_thm) lthy2

  val quotient_thm_name = Binding.prefix_name "QUOTIENT_" qty_name

  (* interpretation *)

  val bindd = ((Binding.make ("", Position.none)), ([]: Attrib.src list))

  val ((_, [eqn1pre]), lthy3) = Variable.import true [ABS_def] lthy2;

  val eqn1i = Thm.prop_of (symmetric eqn1pre)

  val ((_, [eqn2pre]), lthy4) = Variable.import true [REP_def] lthy3;

  val eqn2i = Thm.prop_of (symmetric eqn2pre)

  val exp_morphism = ProofContext.export_morphism lthy4 (ProofContext.init (ProofContext.theory_of lthy4));

  val exp_term = Morphism.term exp_morphism;

  val exp = Morphism.thm exp_morphism;

  val mthd = Method.SIMPLE_METHOD ((rtac quot_thm 1) THEN

    ALLGOALS (simp_tac (HOL_basic_ss addsimps [(symmetric (exp ABS_def)), (symmetric (exp REP_def))])))

  val mthdt = Method.Basic (fn _ => mthd)

  val bymt = Proof.global_terminal_proof (mthdt, NONE)

  val exp_i = [(@{const_name QUOT_TYPE}, ((("QUOT_TYPE_I_" ^ (Binding.name_of qty_name)), true),

    Expression.Named [

     ("R", rel),

     ("Abs", abs),

     ("Rep", rep)

    ]))]

in

  lthy4

  |> note_thm (quot_thm_name, quot_thm)

  ||>> note_thm (quotient_thm_name, quotient_thm)

  ||> LocalTheory.theory (fn thy =>

      let

        val global_eqns = map exp_term [eqn2i, eqn1i];

        (* Not sure if the following context should not be used *)

        val (global_eqns2, lthy5) = Variable.import_terms true global_eqns lthy4;

        val global_eqns3 = map (fn t => (bindd, t)) global_eqns2;

      in ProofContext.theory_of (bymt (Expression.interpretation (exp_i, []) global_eqns3 thy)) end)

end

*}

section {* various tests for quotient types*}

datatype trm =

  var  "nat"

| app  "trm" "trm"

| lam  "nat" "trm"

axiomatization

  RR :: "trm \<Rightarrow> trm \<Rightarrow> bool"

where

  r_eq: "EQUIV RR"

local_setup {*

*}

term Rep_qtrm

term REP_qtrm

term Abs_qtrm

term ABS_qtrm

thm QUOT_TYPE_qtrm

thm QUOTIENT_qtrm

(* Test interpretation *)

thm QUOT_TYPE_I_qtrm.thm11

thm QUOT_TYPE.thm11

print_theorems

thm Rep_qtrm

text {* another test *}

datatype 'a trm' =

  var'  "'a"

| app'  "'a trm'" "'a trm'"

| lam'  "'a" "'a trm'"

consts R' :: "'a trm' \<Rightarrow> 'a trm' \<Rightarrow> bool"

axioms r_eq': "EQUIV R'"

local_setup {*

*}

print_theorems

term ABS_qtrm'

term REP_qtrm'

thm QUOT_TYPE_qtrm'

thm QUOTIENT_qtrm'

thm Rep_qtrm'

text {* a test with lists of terms *}

datatype t =

  vr "string"

| ap "t list"

| lm "string" "t"

consts Rt :: "t \<Rightarrow> t \<Rightarrow> bool"

axioms t_eq: "EQUIV Rt"

local_setup {*

*}

section {* lifting of constants *}

text {* information about map-functions for type-constructor *}

ML {*

type typ_info = {mapfun: string}

local

  structure Data = TheoryDataFun

  (type T = typ_info Symtab.table

   val empty = Symtab.empty

   val copy = I

   val extend = I

   fun merge _ = Symtab.merge (K true))

in

  val lookup = Symtab.lookup o Data.get

  fun update k v = Data.map (Symtab.update (k, v))

end

*}

(* mapfuns for some standard types *)

setup {*

  update @{type_name "list"} {mapfun = @{const_name "map"}} #> 

  update @{type_name "*"}    {mapfun = @{const_name "prod_fun"}} #> 

  update @{type_name "fun"}  {mapfun = @{const_name "fun_map"}}

*}

ML {* lookup @{theory} @{type_name list} *}

ML {*

datatype flag = absF | repF

fun get_fun flag rty qty lthy ty =

let

  val qty_name = Long_Name.base_name (fst (dest_Type qty))

  fun get_fun_aux s fs_tys =

  let

    val (fs, tys) = split_list fs_tys

    val (otys, ntys) = split_list tys

    val oty = Type (s, otys)

    val nty = Type (s, ntys)

    val ftys = map (op -->) tys

  in

   (case (lookup (ProofContext.theory_of lthy) s) of

      SOME info => (list_comb (Const (#mapfun info, ftys ---> oty --> nty), fs), (oty, nty))

    | NONE      => raise ERROR ("no map association for type " ^ s))

  end

  fun get_const absF = (Const ("QuotMain.ABS_" ^ qty_name, rty --> qty), (rty, qty))

    | get_const repF = (Const ("QuotMain.REP_" ^ qty_name, qty --> rty), (qty, rty))

  fun mk_identity ty = Abs ("x", ty, Bound 0)

in

  if ty = qty

  then (get_const flag)

  else (case ty of

          TFree _ => (mk_identity ty, (ty, ty))

        | Type (_, []) => (mk_identity ty, (ty, ty))

        | Type (s, tys) => get_fun_aux s (map (get_fun flag rty qty lthy) tys)

        | _ => raise ERROR ("no type variables")

       )

end

*}

ML {*

  get_fun repF @{typ t} @{typ qt} @{context} @{typ "t * nat"}

  |> fst

  |> Syntax.string_of_term @{context}

  |> writeln

*}

ML {*

  get_fun absF @{typ t} @{typ qt} @{context} @{typ "t * nat"}

  |> fst

  |> Syntax.string_of_term @{context}

  |> writeln

*}

text {* produces the definition for a lifted constant *}

ML {*

fun get_const_def nconst oconst rty qty lthy =

let

  val ty = fastype_of nconst

  val (arg_tys, res_ty) = strip_type ty

  val fresh_args = arg_tys |> map (pair "x")

                           |> Variable.variant_frees lthy [nconst, oconst]

                           |> map Free

  val rep_fns = map (fst o get_fun repF rty qty lthy) arg_tys

  val abs_fn  = (fst o get_fun absF rty qty lthy) res_ty

in

  map (op $) (rep_fns ~~ fresh_args)

  |> curry list_comb oconst

  |> curry (op $) abs_fn

  |> fold_rev lambda fresh_args

end

*}

ML {*

fun exchange_ty rty qty ty =

  if ty = rty

  then qty

  else

    (case ty of