theory QuotMain

imports QuotScript QuotList Prove

uses ("quotient_info.ML") 

     ("quotient.ML")

begin

ML {* Attrib.empty_binding *}

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::"'a \<Rightarrow> 'b"

where

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

definition

  REP::"'b \<Rightarrow> 'a"

where

  "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"

using ac bd

by (blast intro: R_trans R_sym)

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 *}

(* the auxiliary data for the quotient types *)

use "quotient_info.ML"

declare [[map list = (map, LIST_REL)]]

declare [[map * = (prod_fun, prod_rel)]]

declare [[map "fun" = (fun_map, FUN_REL)]]

ML {* maps_lookup @{theory} "List.list" *}

ML {* maps_lookup @{theory} "*" *}

ML {* maps_lookup @{theory} "fun" *}

(* definition of the quotient types *)

use "quotient.ML"

text {* FIXME: auxiliary function *}

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);

*}

section {* lifting of constants *}

ML {*

fun lookup_qenv qenv qty =

    SOME rty => SOME (qty, rty)

  | NONE => NONE)

*}

ML {*

(* calculates the aggregate abs and rep functions for a given type; 

   repF is for constants' arguments; absF is for constants;

   function types need to be treated specially, since repF and absF

   change

*)

datatype flag = absF | repF

fun negF absF = repF

  | negF repF = absF

fun get_fun flag qenv lthy ty =

let

  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 (maps_lookup (ProofContext.theory_of lthy) s) of

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

  end

  fun get_fun_fun fs_tys =

  let

    val (fs, tys) = split_list fs_tys

    val ([oty1, oty2], [nty1, nty2]) = split_list tys

    val oty = nty1 --> oty2

    val nty = oty1 --> nty2

    val ftys = map (op -->) tys

  in

    (list_comb (Const (@{const_name "fun_map"}, ftys ---> oty --> nty), fs), (oty, nty))

  end

  fun get_const flag (qty, rty) =

  let 

    val thy = ProofContext.theory_of lthy

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

  in

    case flag of

      absF => (Const (Sign.full_bname thy ("ABS_" ^ qty_name), rty --> qty), (rty, qty))

    | repF => (Const (Sign.full_bname thy ("REP_" ^ qty_name), qty --> rty), (qty, rty))

  end

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

in

  then (get_const flag (the (lookup_qenv qenv ty)))

  else (case ty of

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

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

        | Type ("fun" , [ty1, ty2]) => 

                 get_fun_fun [get_fun (negF flag) qenv lthy ty1, get_fun flag qenv lthy ty2]

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

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

       )

end

*}

ML {*

let

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

  val abs_fn  = (fst o get_fun absF qenv lthy) res_ty

in

end

*}

ML {*

fun diff (T, S) Ds =

  case (T, S) of

    (TVar v, TVar u) => if v = u then Ds else (T, S)::Ds 

  | (TFree x, TFree y) => if x = y then Ds else (T, S)::Ds

  | (Type (a, Ts), Type (b, Us)) => 

      if a = b then diffs (Ts, Us) Ds else (T, S)::Ds

  | _ => (T, S)::Ds

and diffs (T::Ts, U::Us) Ds = diffs (Ts, Us) (diff (T, U) Ds)

  | diffs ([], []) Ds = Ds

  | diffs _ _ = error "Unequal length of type arguments"

*}

ML {*

in

end

*}

ML {*

*}

ML {*

*}

ML {*

let

  val qty  = (Syntax.check_typ lthy o Syntax.parse_typ lthy) qtystr

  val prop = (Syntax.check_prop lthy o Syntax.parse_prop lthy) propstr

in

end

*}

ML {*

val _ = OuterSyntax.local_theory "quotient_def" "lifted definition of constants"

*}

section {* ATOMIZE *}

text {*

  Unabs_def converts a definition given as

    c \<equiv> %x. %y. f x y

  to a theorem of the form

    c x y \<equiv> f x y

  This function is needed to rewrite the right-hand

  side to the left-hand side.

*}

ML {*

fun unabs_def ctxt def =

let

  val (lhs, rhs) = Thm.dest_equals (cprop_of def)

  val xs = strip_abs_vars (term_of rhs)

  val (_, ctxt') = Variable.add_fixes (map fst xs) ctxt

  val thy = ProofContext.theory_of ctxt'

  val cxs = map (cterm_of thy o Free) xs

  val new_lhs = Drule.list_comb (lhs, cxs)

  fun get_conv [] = Conv.rewr_conv def

    | get_conv (x::xs) = Conv.fun_conv (get_conv xs)

in

  get_conv xs new_lhs |>

  singleton (ProofContext.export ctxt' ctxt)

end

*}

lemma atomize_eqv[atomize]: 

  shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)" 

proof

  assume "A \<equiv> B" 

  then show "Trueprop A \<equiv> Trueprop B" by unfold

next

  assume *: "Trueprop A \<equiv> Trueprop B"

  have "A = B"

  proof (cases A)

    case True

    have "A" by fact

    then show "A = B" using * by simp

  next

    case False

    have "\<not>A" by fact

    then show "A = B" using * by auto

  qed

  then show "A \<equiv> B" by (rule eq_reflection)

qed

ML {*

fun atomize_thm thm =

let

  val thm' = Thm.freezeT (forall_intr_vars thm)

  val thm'' = ObjectLogic.atomize (cprop_of thm')

in

  @{thm Pure.equal_elim_rule1} OF [thm'', thm']

end

*}

ML {* atomize_thm @{thm list.induct} *}

section {* REGULARIZE *}

text {* tyRel takes a type and builds a relation that a quantifier over this

  type needs to respect. *}

ML {*

fun matches (ty1, ty2) =

  Type.raw_instance (ty1, Logic.varifyT ty2);

fun tyRel ty rty rel lthy =

  if matches (rty, ty)

  then rel

  else (case ty of

          Type (s, tys) =>

            let

              val tys_rel = map (fn ty => ty --> ty --> @{typ bool}) tys;

              val ty_out = ty --> ty --> @{typ bool};

              val tys_out = tys_rel ---> ty_out;

            in

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

               SOME (info) => list_comb (Const (#relfun info, tys_out), map (fn ty => tyRel ty rty rel lthy) tys)

             | NONE  => HOLogic.eq_const ty

            )

            end

        | _ => HOLogic.eq_const ty)

*}

definition

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

where

  "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"

(* TODO: Consider defining it with an "if"; sth like:

   Babs p m = \<lambda>x. if x \<in> p then m x else undefined

*)

ML {*

fun needs_lift (rty as Type (rty_s, _)) ty =

  case ty of

    Type (s, tys) =>

      (s = rty_s) orelse (exists (needs_lift rty) tys)

  | _ => false

*}

ML {*

fun mk_babs ty ty' = Const (@{const_name "Babs"}, [ty' --> @{typ bool}, ty] ---> ty)

fun mk_ball ty = Const (@{const_name "Ball"}, [ty, ty] ---> @{typ bool})

fun mk_bex ty = Const (@{const_name "Bex"}, [ty, ty] ---> @{typ bool})

fun mk_resp ty = Const (@{const_name Respects}, [[ty, ty] ---> @{typ bool}, ty] ---> @{typ bool})

*}

(* applies f to the subterm of an abstractions, otherwise to the given term *)

ML {*

fun apply_subt f trm =

  case trm of

    Abs (x, T, t) => 

       let 

         val (x', t') = Term.dest_abs (x, T, t)

       in

         Term.absfree (x', T, f t') 

       end

  | _ => f trm

*}

(* FIXME: if there are more than one quotient, then you have to look up the relation *)

ML {*

fun my_reg lthy rel rty trm =

  case trm of

    Abs (x, T, t) =>

       if (needs_lift rty T) then

         let

            val rrel = tyRel T rty rel lthy

         in

           (mk_babs (fastype_of trm) T) $ (mk_resp T $ rrel) $ (apply_subt (my_reg lthy rel rty) trm)

         end

       else

         Abs(x, T, (apply_subt (my_reg lthy rel rty) t))

  | Const (@{const_name "All"}, ty) $ (t as Abs (x, T, _)) =>

       let

          val ty1 = domain_type ty

          val ty2 = domain_type ty1

          val rrel = tyRel T rty rel lthy