nominal2: QuotMain.thy@929bc55efff7

theory QuotMain
imports QuotScript QuotList Prove
uses ("quotient.ML")
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 {* Toplevel.theory *}

use "quotient.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" *}

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 {*

*}

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 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 (maps_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_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

  val thy = ProofContext.theory_of lthy

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

  fun mk_identity ty = Abs ("", 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 ("fun" , [ty1, ty2]) => 
                 get_fun_fun [get_fun (negF flag) rty qty lthy ty1, get_fun flag rty qty lthy ty2]
        | Type (s, tys) => get_fun_aux s (map (get_fun flag rty qty lthy) tys)
        | _ => raise ERROR ("no type variables")
       )
end
*}


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

  fun mk_fun_map (t1,t2) = Const (@{const_name "fun_map"}, dummyT) $ t1 $ t2

  val fns = Library.foldr (mk_fun_map) (rep_fns, abs_fn)
            |> Syntax.check_term lthy
in
  fns $ oconst
end
*}

ML {*
fun exchange_ty rty qty ty =
  if ty = rty
  then qty
  else
    (case ty of
       Type (s, tys) => Type (s, map (exchange_ty rty qty) tys)
      | _ => ty
    )
*}

ML {*
fun make_const_def nconst_bname oconst mx rty qty lthy =
let
  val oconst_ty = fastype_of oconst
  val nconst_ty = exchange_ty rty qty oconst_ty
  val nconst = Const (Binding.name_of nconst_bname, nconst_ty)
  val def_trm = get_const_def nconst oconst rty qty lthy
in
  define (nconst_bname, mx, def_trm) lthy
end
*}

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' = forall_intr_vars thm
  val thm'' = ObjectLogic.atomize (cprop_of thm')
in
  Thm.freezeT (Simplifier.rewrite_rule [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 tyRel ty rty rel lthy =
  if ty = rty 
  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 {*
(* trm \<Rightarrow> new_trm *)
fun regularise trm rty rel lthy =
  case trm of
    Abs (x, T, t) =>
      if (needs_lift rty T) then let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
        val lam_term = Term.lambda_name (x, v) rec_term;
        val sub_res_term = tyRel T rty rel lthy;
        val respects = Const (@{const_name Respects}, (fastype_of sub_res_term) --> T --> @{typ bool});
        val res_term = respects $ sub_res_term;
        val ty = fastype_of trm;
        val rabs = Const (@{const_name Babs}, (fastype_of res_term) --> ty --> ty);
        val rabs_term = (rabs $ res_term) $ lam_term;
      in
        rabs_term
      end else let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
      in
        Term.lambda_name (x, v) rec_term
      end
  | ((Const (@{const_name "All"}, at)) $ (Abs (x, T, t))) =>
      if (needs_lift rty T) then let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
        val lam_term = Term.lambda_name (x, v) rec_term;
        val sub_res_term = tyRel T rty rel lthy;
        val respects = Const (@{const_name Respects}, (fastype_of sub_res_term) --> T --> @{typ bool});
        val res_term = respects $ sub_res_term;
        val ty = fastype_of lam_term;
        val rall = Const (@{const_name Ball}, (fastype_of res_term) --> ty --> @{typ bool});
        val rall_term = (rall $ res_term) $ lam_term;
      in
        rall_term
      end else let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
        val lam_term = Term.lambda_name (x, v) rec_term
      in
        Const(@{const_name "All"}, at) $ lam_term
      end
  | ((Const (@{const_name "All"}, at)) $ P) =>
      let
        val (_, [al, _]) = dest_Type (fastype_of P);
        val ([x], lthy2) = Variable.variant_fixes [""] lthy;
        val v = (Free (x, al));
        val abs = Term.lambda_name (x, v) (P $ v);
      in regularise ((Const (@{const_name "All"}, at)) $ abs) rty rel lthy2 end
  | ((Const (@{const_name "Ex"}, at)) $ (Abs (x, T, t))) =>
      if (needs_lift rty T) then let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
        val lam_term = Term.lambda_name (x, v) rec_term;
        val sub_res_term = tyRel T rty rel lthy;
        val respects = Const (@{const_name Respects}, (fastype_of sub_res_term) --> T --> @{typ bool});
        val res_term = respects $ sub_res_term;
        val ty = fastype_of lam_term;
        val rall = Const (@{const_name Bex}, (fastype_of res_term) --> ty --> @{typ bool});
        val rall_term = (rall $ res_term) $ lam_term;
      in
        rall_term
      end else let
        val ([x'], lthy2) = Variable.variant_fixes [x] lthy;
        val v = Free (x', T);
        val t' = subst_bound (v, t);
        val rec_term = regularise t' rty rel lthy2;
        val lam_term = Term.lambda_name (x, v) rec_term
      in
        Const(@{const_name "Ex"}, at) $ lam_term
      end
  | ((Const (@{const_name "Ex"}, at)) $ P) =>
      let
        val (_, [al, _]) = dest_Type (fastype_of P);
        val ([x], lthy2) = Variable.variant_fixes [""] lthy;
        val v = (Free (x, al));
        val abs = Term.lambda_name (x, v) (P $ v);
      in regularise ((Const (@{const_name "Ex"}, at)) $ abs) rty rel lthy2 end
  | a $ b => (regularise a rty rel lthy) $ (regularise b rty rel lthy)
  | _ => trm

*}

(* my version of regularise *)
(****************************)

(* some helper functions *)


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: assumes always the typ is qty! *)
(* FIXME: if there are more than one quotient, then you have to look up the relation *)
ML {*
fun my_reg rel trm =
  case trm of
    Abs (x, T, t) =>
       let 
          val ty1 = fastype_of trm
       in
         (mk_babs ty1 T) $ (mk_resp T $ rel) $ (apply_subt (my_reg rel) trm)    
       end
  | Const (@{const_name "All"}, ty) $ t =>
       let 
          val ty1 = domain_type ty
          val ty2 = domain_type ty1
       in
         (mk_ball ty1) $ (mk_resp ty2 $ rel) $ (apply_subt (my_reg rel) t)      
       end
  | Const (@{const_name "Ex"}, ty) $ t =>
       let 
          val ty1 = domain_type ty
          val ty2 = domain_type ty1
       in
         (mk_bex ty1) $ (mk_resp ty2 $ rel) $ (apply_subt (my_reg rel) t)    
       end
  | t1 $ t2 => (my_reg rel t1) $ (my_reg rel t2)
  | _ => trm
*}


(*fun prove_reg trm \<Rightarrow> thm (we might need some facts to do this)
  trm == new_trm
*)

text {* Assumes that the given theorem is atomized *}
ML {*
  fun build_regularize_goal thm rty rel lthy =
     Logic.mk_implies
       ((prop_of thm),
       (regularise (prop_of thm) rty rel lthy))
*}

section {* RepAbs injection *}

(* Needed to have a meta-equality *)
lemma id_def_sym: "(\<lambda>x. x) \<equiv> id"
by (simp add: id_def)

ML {*
fun build_repabs_term lthy thm constructors rty qty =
  let
    fun mk_rep tm =
      let
        val ty = exchange_ty rty qty (fastype_of tm)
      in fst (get_fun repF rty qty lthy ty) $ tm end

    fun mk_abs tm =
      let
        val ty = exchange_ty rty qty (fastype_of tm) in
      fst (get_fun absF rty qty lthy ty) $ tm end

    fun is_constructor (Const (x, _)) = member (op =) constructors x
      | is_constructor _ = false;

    fun build_aux lthy tm =
      case tm of
      Abs (a as (_, vty, _)) =>
      let
        val (vs, t) = Term.dest_abs a;
        val v = Free(vs, vty);
        val t' = lambda v (build_aux lthy t)
      in
      if (not (needs_lift rty (fastype_of tm))) then t'
      else mk_rep (mk_abs (
        if not (needs_lift rty vty) then t'
        else
          let
            val v' = mk_rep (mk_abs v);
            val t1 = Envir.beta_norm (t' $ v')
          in
            lambda v t1
          end
      ))
      end
    | x =>
      let
        val (opp, tms0) = Term.strip_comb tm
        val tms = map (build_aux lthy) tms0
        val ty = fastype_of tm
      in
        if (((fst (Term.dest_Const opp)) = @{const_name Respects}) handle _ => false)
          then (list_comb (opp, (hd tms0) :: (tl tms)))
      else if (is_constructor opp andalso needs_lift rty ty) then
          mk_rep (mk_abs (list_comb (opp,tms)))
        else if ((Term.is_Free opp) andalso (length tms > 0) andalso (needs_lift rty ty)) then
          mk_rep(mk_abs(list_comb(opp,tms)))
        else if tms = [] then opp
        else list_comb(opp, tms)
      end
  in
    MetaSimplifier.rewrite_term @{theory} @{thms id_def_sym} []
      (build_aux lthy (Thm.prop_of thm))
  end
*}

text {* Assumes that it is given a regularized theorem *}
ML {*
fun build_repabs_goal ctxt thm cons rty qty =
  Logic.mk_equals ((Thm.prop_of thm), (build_repabs_term ctxt thm cons rty qty))
*}

end
author	Christian Urban <urbanc@in.tum.de>
	Sun, 25 Oct 2009 23:44:41 +0100 (2009-10-25)
changeset 185	929bc55efff7
parent 182	c7eff9882bd8
child 187	f8fc085db38f
permissions	-rw-r--r--