src/HOL/Statespace/state_fun.ML

tuned proof

(*  Title:      HOL/Statespace/state_fun.ML
    Author:     Norbert Schirmer, TU Muenchen
*)

signature STATE_FUN =
sig
  val lookupN : string
  val updateN : string

  val mk_constr : theory -> typ -> term
  val mk_destr : theory -> typ -> term

  val lookup_simproc : simproc
  val update_simproc : simproc
  val ex_lookup_eq_simproc : simproc
  val ex_lookup_ss : simpset
  val lazy_conj_simproc : simproc
  val string_eq_simp_tac : int -> tactic

  val setup : theory -> theory
end;

structure StateFun: STATE_FUN =
struct

val lookupN = @{const_name StateFun.lookup};
val updateN = @{const_name StateFun.update};

val sel_name = HOLogic.dest_string;

fun mk_name i t =
  (case try sel_name t of
    SOME name => name
  | NONE =>
      (case t of
        Free (x, _) => x
      | Const (x, _) => x
      | _ => "x" ^ string_of_int i));

local

val conj1_False = @{thm conj1_False};
val conj2_False = @{thm conj2_False};
val conj_True = @{thm conj_True};
val conj_cong = @{thm conj_cong};

fun isFalse (Const (@{const_name False}, _)) = true
  | isFalse _ = false;

fun isTrue (Const (@{const_name True}, _)) = true
  | isTrue _ = false;

in

val lazy_conj_simproc =
  Simplifier.simproc_global @{theory HOL} "lazy_conj_simp" ["P & Q"]
    (fn thy => fn ss => fn t =>
      (case t of (Const (@{const_name HOL.conj},_) $ P $ Q) =>
        let
          val P_P' = Simplifier.rewrite ss (cterm_of thy P);
          val P' = P_P' |> prop_of |> Logic.dest_equals |> #2;
        in
          if isFalse P' then SOME (conj1_False OF [P_P'])
          else
            let
              val Q_Q' = Simplifier.rewrite ss (cterm_of thy Q);
              val Q' = Q_Q' |> prop_of |> Logic.dest_equals |> #2;
            in
              if isFalse Q' then SOME (conj2_False OF [Q_Q'])
              else if isTrue P' andalso isTrue Q' then SOME (conj_True OF [P_P', Q_Q'])
              else if P aconv P' andalso Q aconv Q' then NONE
              else SOME (conj_cong OF [P_P', Q_Q'])
            end
         end
      | _ => NONE));

val string_eq_simp_tac = simp_tac (HOL_basic_ss
  addsimps (@{thms list.inject} @ @{thms char.inject}
    @ @{thms list.distinct} @ @{thms char.distinct} @ @{thms simp_thms})
  addsimprocs [lazy_conj_simproc]
  |> Simplifier.add_cong @{thm block_conj_cong});

end;

val lookup_ss = (HOL_basic_ss
  addsimps (@{thms list.inject} @ @{thms char.inject}
    @ @{thms list.distinct} @ @{thms char.distinct} @ @{thms simp_thms}
    @ [@{thm StateFun.lookup_update_id_same}, @{thm StateFun.id_id_cancel},
      @{thm StateFun.lookup_update_same}, @{thm StateFun.lookup_update_other}])
  addsimprocs [lazy_conj_simproc]
  addSolver StateSpace.distinctNameSolver
  |> fold Simplifier.add_cong @{thms block_conj_cong});

val ex_lookup_ss = HOL_ss addsimps @{thms StateFun.ex_id};


structure Data = Generic_Data
(
  type T = simpset * simpset * bool;  (*lookup simpset, ex_lookup simpset, are simprocs installed*)
  val empty = (empty_ss, empty_ss, false);
  val extend = I;
  fun merge ((ss1, ex_ss1, b1), (ss2, ex_ss2, b2)) =
    (merge_ss (ss1, ss2), merge_ss (ex_ss1, ex_ss2), b1 orelse b2);
);

val init_state_fun_data =
  Context.theory_map (Data.put (lookup_ss, ex_lookup_ss, false));

val lookup_simproc =
  Simplifier.simproc_global @{theory} "lookup_simp" ["lookup d n (update d' c m v s)"]
    (fn thy => fn ss => fn t =>
      (case t of (Const (@{const_name StateFun.lookup}, lT) $ destr $ n $
                   (s as Const (@{const_name StateFun.update}, uT) $ _ $ _ $ _ $ _ $ _)) =>
        (let
          val (_::_::_::_::sT::_) = binder_types uT;
          val mi = maxidx_of_term t;
          fun mk_upds (Const (@{const_name StateFun.update}, uT) $ d' $ c $ m $ v $ s) =
                let
                  val (_ :: _ :: _ :: fT :: _ :: _) = binder_types uT;
                  val vT = domain_type fT;
                  val (s', cnt) = mk_upds s;
                  val (v', cnt') =
                    (case v of
                      Const (@{const_name K_statefun}, KT) $ v'' =>
                        (case v'' of
                          (Const (@{const_name StateFun.lookup}, _) $
                            (d as (Const (@{const_name Fun.id}, _))) $ n' $ _) =>
                              if d aconv c andalso n aconv m andalso m aconv n'
                              then (v,cnt) (* Keep value so that
                                              lookup_update_id_same can fire *)
                              else
                                (Const (@{const_name StateFun.K_statefun}, KT) $
                                  Var (("v", cnt), vT), cnt + 1)
                        | _ =>
                          (Const (@{const_name StateFun.K_statefun}, KT) $
                            Var (("v", cnt), vT), cnt + 1))
                     | _ => (v, cnt));
                in (Const (@{const_name StateFun.update}, uT) $ d' $ c $ m $ v' $ s', cnt') end
            | mk_upds s = (Var (("s", mi + 1), sT), mi + 2);

          val ct =
            cterm_of thy (Const (@{const_name StateFun.lookup}, lT) $ destr $ n $ fst (mk_upds s));
          val ctxt = Simplifier.the_context ss;
          val basic_ss = #1 (Data.get (Context.Proof ctxt));
          val ss' = Simplifier.context (Config.put simp_depth_limit 100 ctxt) basic_ss;
          val thm = Simplifier.rewrite ss' ct;
        in
          if (op aconv) (Logic.dest_equals (prop_of thm))
          then NONE
          else SOME thm
        end
        handle Option.Option => NONE)
      | _ => NONE ));


local

val meta_ext = @{thm StateFun.meta_ext};
val ss' = (HOL_ss addsimps
  (@{thm StateFun.update_apply} :: @{thm Fun.o_apply} :: @{thms list.inject} @ @{thms char.inject}
    @ @{thms list.distinct} @ @{thms char.distinct})
  addsimprocs [lazy_conj_simproc, StateSpace.distinct_simproc]
  |> fold Simplifier.add_cong @{thms block_conj_cong});

in

val update_simproc =
  Simplifier.simproc_global @{theory} "update_simp" ["update d c n v s"]
    (fn thy => fn ss => fn t =>
      (case t of
        ((upd as Const (@{const_name StateFun.update}, uT)) $ d $ c $ n $ v $ s) =>
          let
            val (_ :: _ :: _ :: _ :: sT :: _) = binder_types uT;
              (*"('v => 'a1) => ('a2 => 'v) => 'n => ('a1 => 'a2) => ('n => 'v) => ('n => 'v)"*)
            fun init_seed s = (Bound 0, Bound 0, [("s", sT)], [], false);

            fun mk_comp f fT g gT =
              let val T = domain_type fT --> range_type gT
              in (Const (@{const_name Fun.comp}, gT --> fT --> T) $ g $ f, T) end;

            fun mk_comps fs = foldl1 (fn ((f, fT), (g, gT)) => mk_comp g gT f fT) fs;

            fun append n c cT f fT d dT comps =
              (case AList.lookup (op aconv) comps n of
                SOME gTs => AList.update (op aconv) (n, [(c, cT), (f, fT), (d, dT)] @ gTs) comps
              | NONE => AList.update (op aconv) (n, [(c, cT), (f, fT), (d, dT)]) comps);

            fun split_list (x :: xs) = let val (xs', y) = split_last xs in (x, xs', y) end
              | split_list _ = error "StateFun.split_list";

            fun merge_upds n comps =
              let val ((c, cT), fs, (d, dT)) = split_list (the (AList.lookup (op aconv) comps n))
              in ((c, cT), fst (mk_comps fs), (d, dT)) end;

               (* mk_updterm returns
                *  - (orig-term-skeleton,simplified-term-skeleton, vars, b)
                *     where boolean b tells if a simplification has occurred.
                      "orig-term-skeleton = simplified-term-skeleton" is
                *     the desired simplification rule.
                * The algorithm first walks down the updates to the seed-state while
                * memorising the updates in the already-table. While walking up the
                * updates again, the optimised term is constructed.
                *)
            fun mk_updterm already
                (t as ((upd as Const (@{const_name StateFun.update}, uT)) $ d $ c $ n $ v $ s)) =
                  let
                    fun rest already = mk_updterm already;
                    val (dT :: cT :: nT :: vT :: sT :: _) = binder_types uT;
                      (*"('v => 'a1) => ('a2 => 'v) => 'n => ('a1 => 'a2) =>
                            ('n => 'v) => ('n => 'v)"*)
                  in
                    if member (op aconv) already n then
                      (case rest already s of
                        (trm, trm', vars, comps, _) =>
                          let
                            val i = length vars;
                            val kv = (mk_name i n, vT);
                            val kb = Bound i;
                            val comps' = append n c cT kb vT d dT comps;
                          in (upd $ d $ c $ n $ kb $ trm, trm', kv :: vars, comps',true) end)
                    else
                      (case rest (n :: already) s of
                        (trm, trm', vars, comps, b) =>
                          let
                            val i = length vars;
                            val kv = (mk_name i n, vT);
                            val kb = Bound i;
                            val comps' = append n c cT kb vT d dT comps;
                            val ((c', c'T), f', (d', d'T)) = merge_upds n comps';
                            val vT' = range_type d'T --> domain_type c'T;
                            val upd' =
                              Const (@{const_name StateFun.update},
                                d'T --> c'T --> nT --> vT' --> sT --> sT);
                          in
                            (upd $ d $ c $ n $ kb $ trm, upd' $ d' $ c' $ n $ f' $ trm', kv :: vars, comps', b)
                          end)
                  end
              | mk_updterm _ t = init_seed t;

            val ctxt = Simplifier.the_context ss |> Config.put simp_depth_limit 100;
            val ss1 = Simplifier.context ctxt ss';
            val ss2 = Simplifier.context ctxt (#1 (Data.get (Context.Proof ctxt)));
          in
            (case mk_updterm [] t of
              (trm, trm', vars, _, true) =>
                let
                  val eq1 =
                    Goal.prove ctxt [] []
                      (Logic.list_all (vars, Logic.mk_equals (trm, trm')))
                      (fn _ => rtac meta_ext 1 THEN simp_tac ss1 1);
                  val eq2 = Simplifier.asm_full_rewrite ss2 (Thm.dest_equals_rhs (cprop_of eq1));
                in SOME (Thm.transitive eq1 eq2) end
            | _ => NONE)
          end
      | _ => NONE));

end;


local

val swap_ex_eq = @{thm StateFun.swap_ex_eq};

fun is_selector thy T sel =
  let val (flds, more) = Record.get_recT_fields thy T
  in member (fn (s, (n, _)) => n = s) (more :: flds) sel end;

in

val ex_lookup_eq_simproc =
  Simplifier.simproc_global @{theory HOL} "ex_lookup_eq_simproc" ["Ex t"]
    (fn thy => fn ss => fn t =>
      let
        val ctxt = Simplifier.the_context ss |> Config.put simp_depth_limit 100;
        val ex_lookup_ss = #2 (Data.get (Context.Proof ctxt));
        val ss' = Simplifier.context ctxt ex_lookup_ss;
        fun prove prop =
          Goal.prove_global thy [] [] prop
            (fn _ => Record.split_simp_tac [] (K ~1) 1 THEN simp_tac ss' 1);

        fun mkeq (swap, Teq, lT, lo, d, n, x, s) i =
          let
            val (_ :: nT :: _) = binder_types lT;
            (*  ('v => 'a) => 'n => ('n => 'v) => 'a *)
            val x' = if not (Term.is_dependent x) then Bound 1 else raise TERM ("", [x]);
            val n' = if not (Term.is_dependent n) then Bound 2 else raise TERM ("", [n]);
            val sel' = lo $ d $ n' $ s;
          in (Const (@{const_name HOL.eq}, Teq) $ sel' $ x', hd (binder_types Teq), nT, swap) end;

        fun dest_state (s as Bound 0) = s
          | dest_state (s as (Const (sel, sT) $ Bound 0)) =
              if is_selector thy (domain_type sT) sel then s
              else raise TERM ("StateFun.ex_lookup_eq_simproc: not a record slector", [s])
          | dest_state s = raise TERM ("StateFun.ex_lookup_eq_simproc: not a record slector", [s]);

        fun dest_sel_eq
              (Const (@{const_name HOL.eq}, Teq) $
                ((lo as (Const (@{const_name StateFun.lookup}, lT))) $ d $ n $ s) $ X) =
              (false, Teq, lT, lo, d, n, X, dest_state s)
          | dest_sel_eq
              (Const (@{const_name HOL.eq}, Teq) $ X $
                ((lo as (Const (@{const_name StateFun.lookup}, lT))) $ d $ n $ s)) =
              (true, Teq, lT, lo, d, n, X, dest_state s)
          | dest_sel_eq _ = raise TERM ("", []);
      in
        (case t of
          Const (@{const_name Ex}, Tex) $ Abs (s, T, t) =>
            (let
              val (eq, eT, nT, swap) = mkeq (dest_sel_eq t) 0;
              val prop =
                Logic.list_all ([("n", nT), ("x", eT)],
                  Logic.mk_equals (Const (@{const_name Ex}, Tex) $ Abs (s, T, eq), @{term True}));
              val thm = Drule.export_without_context (prove prop);
              val thm' = if swap then swap_ex_eq OF [thm] else thm
            in SOME thm' end handle TERM _ => NONE)
        | _ => NONE)
      end handle Option.Option => NONE);

end;

val val_sfx = "V";
val val_prfx = "StateFun."
fun deco base_prfx s = val_prfx ^ (base_prfx ^ suffix val_sfx s);

fun mkUpper str =
  (case String.explode str of
    [] => ""
  | c::cs => String.implode (Char.toUpper c :: cs));

fun mkName (Type (T,args)) = implode (map mkName args) ^ mkUpper (Long_Name.base_name T)
  | mkName (TFree (x,_)) = mkUpper (Long_Name.base_name x)
  | mkName (TVar ((x,_),_)) = mkUpper (Long_Name.base_name x);

fun is_datatype thy = is_some o Datatype.get_info thy;

fun mk_map "List.list" = Syntax.const "List.map"
  | mk_map n = Syntax.const ("StateFun.map_" ^ Long_Name.base_name n);

fun gen_constr_destr comp prfx thy (Type (T, [])) =
      Syntax.const (deco prfx (mkUpper (Long_Name.base_name T)))
  | gen_constr_destr comp prfx thy (T as Type ("fun",_)) =
      let val (argTs, rangeT) = strip_type T;
      in
        comp
          (Syntax.const (deco prfx (implode (map mkName argTs) ^ "Fun")))
          (fold (fn x => fn y => x $ y)
            (replicate (length argTs) (Syntax.const "StateFun.map_fun"))
            (gen_constr_destr comp prfx thy rangeT))
      end
  | gen_constr_destr comp prfx thy (T' as Type (T, argTs)) =
      if is_datatype thy T
      then (* datatype args are recursively embedded into val *)
        (case argTs of
          [argT] =>
            comp
              ((Syntax.const (deco prfx (mkUpper (Long_Name.base_name T)))))
              ((mk_map T $ gen_constr_destr comp prfx thy argT))
        | _ => raise (TYPE ("StateFun.gen_constr_destr", [T'], [])))
      else (* type args are not recursively embedded into val *)
        Syntax.const (deco prfx (implode (map mkName argTs) ^ mkUpper (Long_Name.base_name T)))
  | gen_constr_destr thy _ _ T = raise (TYPE ("StateFun.gen_constr_destr", [T], []));

val mk_constr = gen_constr_destr (fn a => fn b => Syntax.const @{const_name Fun.comp} $ a $ b) "";
val mk_destr = gen_constr_destr (fn a => fn b => Syntax.const @{const_name Fun.comp} $ b $ a) "the_";


val statefun_simp_attr = Thm.declaration_attribute (fn thm => fn ctxt =>
  let
    val (lookup_ss, ex_lookup_ss, simprocs_active) = Data.get ctxt;
    val (lookup_ss', ex_lookup_ss') =
      (case concl_of thm of
        (_ $ ((Const (@{const_name Ex}, _) $ _))) => (lookup_ss, ex_lookup_ss addsimps [thm])
      | _ => (lookup_ss addsimps [thm], ex_lookup_ss));
    fun activate_simprocs ctxt =
      if simprocs_active then ctxt
      else Simplifier.map_ss (fn ss => ss addsimprocs [lookup_simproc, update_simproc]) ctxt;
  in
    ctxt
    |> activate_simprocs
    |> Data.put (lookup_ss', ex_lookup_ss', true)
  end);

val setup =
  init_state_fun_data #>
  Attrib.setup @{binding statefun_simp} (Scan.succeed statefun_simp_attr)
    "simplification in statespaces";

end;