src/HOL/Tools/list_to_set_comprehension.ML

removing dead code; tuned

(*  Title:      HOL/Tools/list_to_set_comprehension.ML
    Author:     Lukas Bulwahn, TU Muenchen

Simproc for rewriting list comprehensions applied to List.set to set
comprehension.
*)

signature LIST_TO_SET_COMPREHENSION =
sig
  val simproc : simpset -> cterm -> thm option
end;

structure List_to_Set_Comprehension : LIST_TO_SET_COMPREHENSION =
struct

(* conversion *)

fun all_exists_conv cv ctxt ct =
  case Thm.term_of ct of
    Const(@{const_name HOL.Ex}, _) $ Abs(_, _, _) =>
      Conv.arg_conv (Conv.abs_conv (all_exists_conv cv o #2) ctxt) ct
  | _ => cv ctxt ct

fun all_but_last_exists_conv cv ctxt ct =
  case Thm.term_of ct of
    Const (@{const_name HOL.Ex}, _) $ Abs (_, _, Const (@{const_name HOL.Ex}, _) $ _) =>
      Conv.arg_conv (Conv.abs_conv (all_but_last_exists_conv cv o #2) ctxt) ct
  | _ => cv ctxt ct

fun Collect_conv cv ctxt ct =
  (case Thm.term_of ct of
    Const (@{const_name Set.Collect}, _) $ Abs _ => Conv.arg_conv (Conv.abs_conv cv ctxt) ct
  | _ => raise CTERM ("Collect_conv", [ct]));

fun Trueprop_conv cv ct =
  (case Thm.term_of ct of
    Const (@{const_name Trueprop}, _) $ _ => Conv.arg_conv cv ct
  | _ => raise CTERM ("Trueprop_conv", [ct]));

fun eq_conv cv1 cv2 ct =
  (case Thm.term_of ct of
    Const (@{const_name HOL.eq}, _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv1) cv2 ct
  | _ => raise CTERM ("eq_conv", [ct]));

fun conj_conv cv1 cv2 ct =
  (case Thm.term_of ct of
    Const (@{const_name HOL.conj}, _) $ _ $ _ => Conv.combination_conv (Conv.arg_conv cv1) cv2 ct
  | _ => raise CTERM ("conj_conv", [ct]));

fun right_hand_set_comprehension_conv conv ctxt = Trueprop_conv (eq_conv Conv.all_conv
  (Collect_conv (all_exists_conv conv o #2) ctxt))

(* term abstraction of list comprehension patterns *)
 
datatype termlets = If | Case of (typ * int)

fun meta_eq th = th RS @{thm eq_reflection}

fun rewr_conv' th = Conv.rewr_conv (meta_eq th)

fun simproc ss redex =
  let
    val ctxt = Simplifier.the_context ss
    val thy = ProofContext.theory_of ctxt 
    val set_Nil_I = @{thm trans} OF [@{thm set.simps(1)}, @{thm empty_def}]
    val set_singleton = @{lemma "set [a] = {x. x = a}" by simp}
    val inst_Collect_mem_eq = @{lemma "set A = {x. x : set A}" by simp}
    val del_refl_eq = @{lemma "(t = t & P) == P" by simp} 
    fun mk_set T = Const (@{const_name List.set}, HOLogic.listT T --> HOLogic.mk_setT T)
    fun dest_set (Const (@{const_name List.set}, _) $ xs) = xs
    fun dest_singleton_list (Const (@{const_name List.Cons}, _)
      $ t $ (Const (@{const_name List.Nil}, _))) = t
      | dest_singleton_list t = raise TERM ("dest_singleton_list", [t])
    (* We check that one case returns a singleton list and all other cases
       return [], and return the index of the one singleton list case *) 
    fun possible_index_of_singleton_case cases =
      let  
        fun check (i, case_t) s =
          (case strip_abs_body case_t of
            (Const (@{const_name List.Nil}, _)) => s
          | t => (case s of NONE => SOME i | SOME s => NONE))
      in
        fold_index check cases NONE
      end
    (* returns (case_expr type index chosen_case) option  *)
    fun dest_case case_term =
      let
        val (case_const, args) = strip_comb case_term
      in
        case try dest_Const case_const of
          SOME (c, T) => (case Datatype_Data.info_of_case thy c of
            SOME _ => (case possible_index_of_singleton_case (fst (split_last args)) of
              SOME i => 
                let
                  val (Ts, _) = strip_type T
                  val T' = snd (split_last Ts)
                in SOME (snd (split_last args), T', i, nth args i) end
            | NONE => NONE)
          | NONE => NONE)
        | NONE => NONE
      end
    (* returns condition continuing term option *)
    fun dest_if (Const (@{const_name If}, _) $ cond $ then_t $ Const (@{const_name Nil}, _)) =
        SOME (cond, then_t)
      | dest_if _ = NONE
    fun tac _ [] =
      rtac set_singleton 1 ORELSE rtac inst_Collect_mem_eq 1
    | tac ctxt (If :: cont) =
      Splitter.split_tac [@{thm split_if}] 1
      THEN rtac @{thm conjI} 1
      THEN rtac @{thm impI} 1
      THEN Subgoal.FOCUS (fn {prems, context, ...} =>
        CONVERSION (right_hand_set_comprehension_conv (K
          (conj_conv (Conv.rewr_conv (snd (split_last prems) RS @{thm Eq_TrueI})) Conv.all_conv
           then_conv rewr_conv' @{thm simp_thms(22)})) context) 1) ctxt 1
      THEN tac ctxt cont
      THEN rtac @{thm impI} 1
      THEN Subgoal.FOCUS (fn {prems, context, ...} =>
          CONVERSION (right_hand_set_comprehension_conv (K
            (conj_conv (Conv.rewr_conv (snd (split_last prems) RS @{thm Eq_FalseI})) Conv.all_conv
             then_conv rewr_conv' @{thm simp_thms(24)})) context) 1) ctxt 1
      THEN rtac set_Nil_I 1
    | tac ctxt (Case (T, i) :: cont) =
      let
        val info = Datatype.the_info thy (fst (dest_Type T))
      in
        (* do case distinction *)
        Splitter.split_tac [#split info] 1
        THEN EVERY (map_index (fn (i', case_rewrite) =>
          (if i' < length (#case_rewrites info) - 1 then rtac @{thm conjI} 1 else all_tac)
          THEN REPEAT_DETERM (rtac @{thm allI} 1)
          THEN rtac @{thm impI} 1
          THEN (if i' = i then
            (* continue recursively *)
            Subgoal.FOCUS (fn {prems, context, ...} =>
              CONVERSION (Thm.eta_conversion then_conv right_hand_set_comprehension_conv (K
                  ((conj_conv 
                    (eq_conv Conv.all_conv (rewr_conv' (snd (split_last prems)))
                    then_conv (Conv.try_conv (Conv.rewrs_conv (map meta_eq (#inject info))))) Conv.all_conv)
                    then_conv (Conv.try_conv (Conv.rewr_conv del_refl_eq))
                    then_conv (Conv.try_conv (rewr_conv' @{thm conj_assoc})))) context
                then_conv (Trueprop_conv (eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt) =>
                  Conv.repeat_conv (all_but_last_exists_conv (K (rewr_conv' @{thm simp_thms(39)})) ctxt)) context)))) 1) ctxt 1
            THEN tac ctxt cont
          else
            Subgoal.FOCUS (fn {prems, context, ...} =>
              CONVERSION ((right_hand_set_comprehension_conv (K
                (conj_conv
                  ((eq_conv Conv.all_conv
                    (rewr_conv' (snd (split_last prems))))
                     then_conv (Conv.rewrs_conv (map (fn th => th RS @{thm Eq_FalseI}) (#distinct info)))) Conv.all_conv
                  then_conv (rewr_conv' @{thm simp_thms(24)}))) context)
               then_conv (Trueprop_conv (eq_conv Conv.all_conv (Collect_conv (fn (_, ctxt) =>
                   Conv.repeat_conv (Conv.bottom_conv (K (rewr_conv' @{thm simp_thms(36)})) ctxt)) context)))) 1) ctxt 1
            THEN rtac set_Nil_I 1)) (#case_rewrites info))
      end
    fun make_inner_eqs bound_vs Tis eqs t =
      case dest_case t of
        SOME (x, T, i, cont) =>
          let
            val (vs, body) = strip_abs (Pattern.eta_long (map snd bound_vs) cont)
            val x' = incr_boundvars (length vs) x
            val eqs' = map (incr_boundvars (length vs)) eqs
            val (constr_name, _) = nth (the (Datatype_Data.get_constrs thy (fst (dest_Type T)))) i
            val constr_t = list_comb (Const (constr_name, map snd vs ---> T), map Bound (((length vs) - 1) downto 0))
            val constr_eq = Const (@{const_name HOL.eq}, T --> T --> @{typ bool}) $ constr_t $ x'
          in
            make_inner_eqs (rev vs @ bound_vs) (Case (T, i) :: Tis) (constr_eq :: eqs') body
          end
      | NONE =>
        case dest_if t of
          SOME (condition, cont) => make_inner_eqs bound_vs (If :: Tis) (condition :: eqs) cont
        | NONE =>
          if eqs = [] then NONE (* no rewriting, nothing to be done *)
          else
            let
              val Type (@{type_name List.list}, [rT]) = fastype_of t
              val pat_eq =
                case try dest_singleton_list t of
                  SOME t' => Const (@{const_name HOL.eq}, rT --> rT --> @{typ bool})
                    $ Bound (length bound_vs) $ t'
                | NONE => Const (@{const_name Set.member}, rT --> HOLogic.mk_setT rT --> @{typ bool})
                  $ Bound (length bound_vs) $ (mk_set rT $ t)
              val reverse_bounds = curry subst_bounds
                ((map Bound ((length bound_vs - 1) downto 0)) @ [Bound (length bound_vs)])
              val eqs' = map reverse_bounds eqs
              val pat_eq' = reverse_bounds pat_eq
              val inner_t = fold (fn (v, T) => fn t => HOLogic.exists_const T $ absdummy (T, t))
                (rev bound_vs) (fold (curry HOLogic.mk_conj) eqs' pat_eq')
              val lhs = term_of redex
              val rhs = HOLogic.mk_Collect ("x", rT, inner_t)
              val rewrite_rule_t = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs))
            in
              SOME ((Goal.prove ctxt [] [] rewrite_rule_t (fn {context, ...} => tac context (rev Tis))) RS @{thm eq_reflection})
            end
  in
    make_inner_eqs [] [] [] (dest_set (term_of redex))
  end

end