src/HOL/Tools/Nitpick/nitpick_scope.ML

 
   type dtype_spec = {
     typ: typ,
     card: int,
     co: bool,
-    precise: bool,
+    complete: bool,
+    concrete: bool,
     shallow: bool,
     constrs: constr_spec list}
 
   type scope = {
     ext_ctxt: extended_context,

     datatypes: dtype_spec list,
     ofs: int Typtab.table}
 
   val datatype_spec : dtype_spec list -> typ -> dtype_spec option
   val constr_spec : dtype_spec list -> styp -> constr_spec
-  val is_precise_type : dtype_spec list -> typ -> bool
-  val is_fully_comparable_type : dtype_spec list -> typ -> bool
+  val is_complete_type : dtype_spec list -> typ -> bool
+  val is_concrete_type : dtype_spec list -> typ -> bool
+  val is_exact_type : dtype_spec list -> typ -> bool
   val offset_of_type : int Typtab.table -> typ -> int
   val spec_of_type : scope -> typ -> int * int
   val pretties_for_scope : scope -> bool -> Pretty.T list
   val multiline_string_for_scope : scope -> string
   val scopes_equivalent : scope -> scope -> bool

 
 type dtype_spec = {
   typ: typ,
   card: int,
   co: bool,
-  precise: bool,
+  complete: bool,
+  concrete: bool,
   shallow: bool,
   constrs: constr_spec list}
 
 type scope = {
   ext_ctxt: extended_context,

                    constrs of
       SOME c => c
     | NONE => constr_spec dtypes x
 
 (* dtype_spec list -> typ -> bool *)
-fun is_precise_type dtypes (Type ("fun", Ts)) =
-    forall (is_precise_type dtypes) Ts
-  | is_precise_type dtypes (Type ("*", Ts)) = forall (is_precise_type dtypes) Ts
-  | is_precise_type dtypes T =
-    not (is_integer_type T) andalso #precise (the (datatype_spec dtypes T))
+fun is_complete_type dtypes (Type ("fun", [T1, T2])) =
+    is_concrete_type dtypes T1 andalso is_complete_type dtypes T2
+  | is_complete_type dtypes (Type ("*", Ts)) =
+    forall (is_complete_type dtypes) Ts
+  | is_complete_type dtypes T =
+    not (is_integer_type T) andalso #complete (the (datatype_spec dtypes T))
     handle Option.Option => true
-fun is_fully_comparable_type dtypes (Type ("fun", [T1, T2])) =
-    is_precise_type dtypes T1 andalso is_fully_comparable_type dtypes T2
-  | is_fully_comparable_type dtypes (Type ("*", Ts)) =
-    forall (is_fully_comparable_type dtypes) Ts
-  | is_fully_comparable_type _ _ = true
+and is_concrete_type dtypes (Type ("fun", [T1, T2])) =
+    is_complete_type dtypes T1 andalso is_concrete_type dtypes T2
+  | is_concrete_type dtypes (Type ("*", Ts)) =
+    forall (is_concrete_type dtypes) Ts
+  | is_concrete_type dtypes T =
+    #concrete (the (datatype_spec dtypes T)) handle Option.Option => true
+fun is_exact_type dtypes = is_complete_type dtypes andf is_concrete_type dtypes
 
 (* int Typtab.table -> typ -> int *)
 fun offset_of_type ofs T =
   case Typtab.lookup ofs T of
     SOME j0 => j0

 (* (string -> string) -> scope
    -> string list * string list * string list * string list * string list *)
 fun quintuple_for_scope quote ({ext_ctxt as {thy, ctxt, ...}, card_assigns,
                                 bisim_depth, datatypes, ...} : scope) =
   let
-    val (iter_asgns, card_asgns) =
+    val (iter_assigns, card_assigns) =
       card_assigns |> filter_out (curry (op =) @{typ bisim_iterator} o fst)
                    |> List.partition (is_fp_iterator_type o fst)
-    val (unimportant_card_asgns, important_card_asgns) =
-      card_asgns |> List.partition ((is_integer_type orf is_datatype thy) o fst)
+    val (unimportant_card_assigns, important_card_assigns) =
+      card_assigns |> List.partition ((is_integer_type orf is_datatype thy) o fst)
     val cards =
       map (fn (T, k) => quote (string_for_type ctxt T) ^ " = " ^
                         string_of_int k)
     fun maxes () =
       maps (map_filter

                  o #constrs) datatypes
     fun iters () =
       map (fn (T, k) =>
               quote (Syntax.string_of_term ctxt
                          (Const (const_for_iterator_type T))) ^ " = " ^
-              string_of_int (k - 1)) iter_asgns
+              string_of_int (k - 1)) iter_assigns
     fun bisims () =
       if bisim_depth < 0 andalso forall (not o #co) datatypes then []
       else ["bisim_depth = " ^ string_of_int bisim_depth]
   in
     setmp_show_all_types
-        (fn () => (cards important_card_asgns, cards unimportant_card_asgns,
+        (fn () => (cards important_card_assigns, cards unimportant_card_assigns,
                    maxes (), iters (), bisims ())) ()
   end
 
 (* scope -> bool -> Pretty.T list *)
 fun pretties_for_scope scope verbose =

 fun project_row column (y, ks) = (y, [nth ks (Int.min (column, length ks - 1))])
 (* int -> block -> block *)
 fun project_block (column, block) = map (project_row column) block
 
 (* (''a * ''a -> bool) -> (''a option * int list) list -> ''a -> int list *)
-fun lookup_ints_assign eq asgns key =
-  case triple_lookup eq asgns key of
+fun lookup_ints_assign eq assigns key =
+  case triple_lookup eq assigns key of
     SOME ks => ks
   | NONE => raise ARG ("Nitpick_Scope.lookup_ints_assign", "")
 (* theory -> (typ option * int list) list -> typ -> int list *)
-fun lookup_type_ints_assign thy asgns T =
-  map (curry Int.max 1) (lookup_ints_assign (type_match thy) asgns T)
+fun lookup_type_ints_assign thy assigns T =
+  map (curry Int.max 1) (lookup_ints_assign (type_match thy) assigns T)
   handle ARG ("Nitpick_Scope.lookup_ints_assign", _) =>
          raise TYPE ("Nitpick_Scope.lookup_type_ints_assign", [T], [])
 (* theory -> (styp option * int list) list -> styp -> int list *)
-fun lookup_const_ints_assign thy asgns x =
-  lookup_ints_assign (const_match thy) asgns x
+fun lookup_const_ints_assign thy assigns x =
+  lookup_ints_assign (const_match thy) assigns x
   handle ARG ("Nitpick_Scope.lookup_ints_assign", _) =>
          raise TERM ("Nitpick_Scope.lookup_const_ints_assign", [Const x])
 
 (* theory -> (styp option * int list) list -> styp -> row option *)
-fun row_for_constr thy maxes_asgns constr =
-  SOME (Max constr, lookup_const_ints_assign thy maxes_asgns constr)
+fun row_for_constr thy maxes_assigns constr =
+  SOME (Max constr, lookup_const_ints_assign thy maxes_assigns constr)
   handle TERM ("lookup_const_ints_assign", _) => NONE
 
 (* extended_context -> (typ option * int list) list
    -> (styp option * int list) list -> (styp option * int list) list -> int list
    -> typ -> block *)
-fun block_for_type (ext_ctxt as {thy, ...}) cards_asgns maxes_asgns iters_asgns
-                   bisim_depths T =
+fun block_for_type (ext_ctxt as {thy, ...}) cards_assigns maxes_assigns
+                   iters_assigns bisim_depths T =
   if T = @{typ bisim_iterator} then
     [(Card T, map (fn k => Int.max (0, k) + 1) bisim_depths)]
   else if is_fp_iterator_type T then
     [(Card T, map (fn k => Int.max (0, k) + 1)
-                  (lookup_const_ints_assign thy iters_asgns
+                  (lookup_const_ints_assign thy iters_assigns
                                             (const_for_iterator_type T)))]
   else
-    (Card T, lookup_type_ints_assign thy cards_asgns T) ::
+    (Card T, lookup_type_ints_assign thy cards_assigns T) ::
     (case datatype_constrs ext_ctxt T of
        [_] => []
-     | constrs => map_filter (row_for_constr thy maxes_asgns) constrs)
+     | constrs => map_filter (row_for_constr thy maxes_assigns) constrs)
 
 (* extended_context -> (typ option * int list) list
    -> (styp option * int list) list -> (styp option * int list) list -> int list
    -> typ list -> typ list -> block list *)
-fun blocks_for_types ext_ctxt cards_asgns maxes_asgns iters_asgns bisim_depths
-                     mono_Ts nonmono_Ts =
+fun blocks_for_types ext_ctxt cards_assigns maxes_assigns iters_assigns
+                     bisim_depths mono_Ts nonmono_Ts =
   let
     (* typ -> block *)
-    val block_for = block_for_type ext_ctxt cards_asgns maxes_asgns iters_asgns
-                                   bisim_depths
+    val block_for = block_for_type ext_ctxt cards_assigns maxes_assigns
+                                   iters_assigns bisim_depths
     val mono_block = maps block_for mono_Ts
     val nonmono_blocks = map block_for nonmono_Ts
   in mono_block :: nonmono_blocks end
 
 val sync_threshold = 5

 fun constr_max maxes x = the_default ~1 (AList.lookup (op =) maxes x)
 
 type scope_desc = (typ * int) list * (styp * int) list
 
 (* extended_context -> scope_desc -> typ * int -> bool *)
-fun is_surely_inconsistent_card_assign ext_ctxt (card_asgns, max_asgns) (T, k) =
+fun is_surely_inconsistent_card_assign ext_ctxt (card_assigns, max_assigns)
+                                       (T, k) =
   case datatype_constrs ext_ctxt T of
     [] => false
   | xs =>
     let
-      val precise_cards =
+      val exact_cards =
         map (Integer.prod
-             o map (bounded_precise_card_of_type ext_ctxt k 0 card_asgns)
+             o map (bounded_exact_card_of_type ext_ctxt k 0 card_assigns)
              o binder_types o snd) xs
-      val maxes = map (constr_max max_asgns) xs
+      val maxes = map (constr_max max_assigns) xs
       (* int -> int -> int *)
       fun effective_max 0 ~1 = k
         | effective_max 0 max = max
         | effective_max card ~1 = card
         | effective_max card max = Int.min (card, max)
-      val max = map2 effective_max precise_cards maxes |> Integer.sum
+      val max = map2 effective_max exact_cards maxes |> Integer.sum
       (* unit -> int *)
       fun doms_card () =
-        xs |> map (Integer.prod o map (bounded_card_of_type k ~1 card_asgns)
+        xs |> map (Integer.prod o map (bounded_card_of_type k ~1 card_assigns)
                    o binder_types o snd)
            |> Integer.sum
     in
       max < k
-      orelse (forall (not_equal 0) precise_cards andalso doms_card () < k)
+      orelse (forall (not_equal 0) exact_cards andalso doms_card () < k)
     end
     handle TYPE ("Nitpick_HOL.card_of_type", _, _) => false
 
 (* extended_context -> scope_desc -> bool *)
 fun is_surely_inconsistent_scope_description ext_ctxt
-                                             (desc as (card_asgns, _)) =
-  exists (is_surely_inconsistent_card_assign ext_ctxt desc) card_asgns
+                                             (desc as (card_assigns, _)) =
+  exists (is_surely_inconsistent_card_assign ext_ctxt desc) card_assigns
 
 (* extended_context -> scope_desc -> (typ * int) list option *)
-fun repair_card_assigns ext_ctxt (card_asgns, max_asgns) =
+fun repair_card_assigns ext_ctxt (card_assigns, max_assigns) =
   let
     (* (typ * int) list -> (typ * int) list -> (typ * int) list option *)
     fun aux seen [] = SOME seen
       | aux seen ((T, 0) :: _) = NONE
-      | aux seen ((T, k) :: asgns) =
+      | aux seen ((T, k) :: assigns) =
         (if is_surely_inconsistent_scope_description ext_ctxt
-                ((T, k) :: seen, max_asgns) then
+                ((T, k) :: seen, max_assigns) then
            raise SAME ()
          else
-           case aux ((T, k) :: seen) asgns of
-             SOME asgns => SOME asgns
+           case aux ((T, k) :: seen) assigns of
+             SOME assigns => SOME assigns
            | NONE => raise SAME ())
-        handle SAME () => aux seen ((T, k - 1) :: asgns)
-  in aux [] (rev card_asgns) end
+        handle SAME () => aux seen ((T, k - 1) :: assigns)
+  in aux [] (rev card_assigns) end
 
 (* theory -> (typ * int) list -> typ * int -> typ * int *)
-fun repair_iterator_assign thy asgns (T as Type (s, Ts), k) =
+fun repair_iterator_assign thy assigns (T as Type (s, Ts), k) =
     (T, if T = @{typ bisim_iterator} then
-          let val co_cards = map snd (filter (is_codatatype thy o fst) asgns) in
-            Int.min (k, Integer.sum co_cards)
-          end
+          let
+            val co_cards = map snd (filter (is_codatatype thy o fst) assigns)
+          in Int.min (k, Integer.sum co_cards) end
         else if is_fp_iterator_type T then
           case Ts of
             [] => 1
-          | _ => bounded_card_of_type k ~1 asgns (foldr1 HOLogic.mk_prodT Ts)
+          | _ => bounded_card_of_type k ~1 assigns (foldr1 HOLogic.mk_prodT Ts)
         else
           k)
-  | repair_iterator_assign _ _ asgn = asgn
+  | repair_iterator_assign _ _ assign = assign
 
 (* row -> scope_desc -> scope_desc *)
-fun add_row_to_scope_descriptor (kind, ks) (card_asgns, max_asgns) =
+fun add_row_to_scope_descriptor (kind, ks) (card_assigns, max_assigns) =
   case kind of
-    Card T => ((T, the_single ks) :: card_asgns, max_asgns)
-  | Max x => (card_asgns, (x, the_single ks) :: max_asgns)
+    Card T => ((T, the_single ks) :: card_assigns, max_assigns)
+  | Max x => (card_assigns, (x, the_single ks) :: max_assigns)
 (* block -> scope_desc *)
 fun scope_descriptor_from_block block =
   fold_rev add_row_to_scope_descriptor block ([], [])
 (* extended_context -> block list -> int list -> scope_desc option *)
 fun scope_descriptor_from_combination (ext_ctxt as {thy, ...}) blocks columns =
   let
-    val (card_asgns, max_asgns) =
+    val (card_assigns, max_assigns) =
       maps project_block (columns ~~ blocks) |> scope_descriptor_from_block
-    val card_asgns = repair_card_assigns ext_ctxt (card_asgns, max_asgns) |> the
-  in
-    SOME (map (repair_iterator_assign thy card_asgns) card_asgns, max_asgns)
+    val card_assigns = repair_card_assigns ext_ctxt (card_assigns, max_assigns)
+                       |> the
+  in
+    SOME (map (repair_iterator_assign thy card_assigns) card_assigns,
+          max_assigns)
   end
   handle Option.Option => NONE
 
 (* theory -> (typ * int) list -> dtype_spec list -> int Typtab.table *)
-fun offset_table_for_card_assigns thy asgns dtypes =
+fun offset_table_for_card_assigns thy assigns dtypes =
   let
     (* int -> (int * int) list -> (typ * int) list -> int Typtab.table
        -> int Typtab.table *)
     fun aux next _ [] = Typtab.update_new (dummyT, next)
-      | aux next reusable ((T, k) :: asgns) =
+      | aux next reusable ((T, k) :: assigns) =
         if k = 1 orelse is_integer_type T then
-          aux next reusable asgns
+          aux next reusable assigns
         else if length (these (Option.map #constrs (datatype_spec dtypes T)))
                 > 1 then
-          Typtab.update_new (T, next) #> aux (next + k) reusable asgns
+          Typtab.update_new (T, next) #> aux (next + k) reusable assigns
         else
           case AList.lookup (op =) reusable k of
-            SOME j0 => Typtab.update_new (T, j0) #> aux next reusable asgns
+            SOME j0 => Typtab.update_new (T, j0) #> aux next reusable assigns
           | NONE => Typtab.update_new (T, next)
-                    #> aux (next + k) ((k, next) :: reusable) asgns
-  in aux 0 [] asgns Typtab.empty end
+                    #> aux (next + k) ((k, next) :: reusable) assigns
+  in aux 0 [] assigns Typtab.empty end
 
 (* int -> (typ * int) list -> typ -> int *)
-fun domain_card max card_asgns =
-  Integer.prod o map (bounded_card_of_type max max card_asgns) o binder_types
+fun domain_card max card_assigns =
+  Integer.prod o map (bounded_card_of_type max max card_assigns) o binder_types
 
 (* scope_desc -> bool -> int -> (int -> int) -> int -> int -> bool * styp
    -> constr_spec list -> constr_spec list *)
-fun add_constr_spec (card_asgns, max_asgns) co card sum_dom_cards num_self_recs
-                    num_non_self_recs (self_rec, x as (s, T)) constrs =
-  let
-    val max = constr_max max_asgns x
+fun add_constr_spec (card_assigns, max_assigns) co card sum_dom_cards
+                    num_self_recs num_non_self_recs (self_rec, x as (s, T))
+                    constrs =
+  let
+    val max = constr_max max_assigns x
     (* int -> int *)
     fun bound k = Int.min (card, (max >= 0 ? curry Int.min max) k)
     (* unit -> int *)
     fun next_delta () = if null constrs then 0 else #epsilon (hd constrs)
     val {delta, epsilon, exclusive, total} =

         let val delta = next_delta () in
           {delta = delta, epsilon = delta, exclusive = true, total = false}
         end
       else if not co andalso num_self_recs > 0 then
         if not self_rec andalso num_non_self_recs = 1
-           andalso domain_card 2 card_asgns T = 1 then
+           andalso domain_card 2 card_assigns T = 1 then
           {delta = 0, epsilon = 1, exclusive = (s = @{const_name Nil}),
            total = true}
         else if s = @{const_name Cons} then
           {delta = 1, epsilon = card, exclusive = true, total = false}
         else
           {delta = 0, epsilon = card, exclusive = false, total = false}
       else if card = sum_dom_cards (card + 1) then
         let val delta = next_delta () in
-          {delta = delta, epsilon = delta + domain_card card card_asgns T,
+          {delta = delta, epsilon = delta + domain_card card card_assigns T,
            exclusive = true, total = true}
         end
       else
         {delta = 0, epsilon = card,
          exclusive = (num_self_recs + num_non_self_recs = 1), total = false}
   in
     {const = x, delta = delta, epsilon = epsilon, exclusive = exclusive,
      explicit_max = max, total = total} :: constrs
   end
 
+(* extended_context -> (typ * int) list -> typ -> bool *)
+fun has_exact_card ext_ctxt card_assigns T =
+  let val card = card_of_type card_assigns T in
+    card = bounded_exact_card_of_type ext_ctxt (card + 1) 0 card_assigns T
+  end
+
 (* extended_context -> typ list -> scope_desc -> typ * int -> dtype_spec *)
 fun datatype_spec_from_scope_descriptor (ext_ctxt as {thy, ...}) shallow_dataTs
-                                        (desc as (card_asgns, _)) (T, card) =
+                                        (desc as (card_assigns, _)) (T, card) =
   let
     val shallow = member (op =) shallow_dataTs T
     val co = is_codatatype thy T
     val xs = boxed_datatype_constrs ext_ctxt T
     val self_recs = map (is_self_recursive_constr_type o snd) xs
     val (num_self_recs, num_non_self_recs) =
-      List.partition (curry (op =) true) self_recs |> pairself length
-    val precise = (card = bounded_precise_card_of_type ext_ctxt (card + 1) 0
-                                                       card_asgns T)
+      List.partition I self_recs |> pairself length
+    val complete = has_exact_card ext_ctxt card_assigns T
+    val concrete = xs |> maps (binder_types o snd) |> maps binder_types
+                      |> forall (has_exact_card ext_ctxt card_assigns)
     (* int -> int *)
     fun sum_dom_cards max =
-      map (domain_card max card_asgns o snd) xs |> Integer.sum
+      map (domain_card max card_assigns o snd) xs |> Integer.sum
     val constrs =
       fold_rev (add_constr_spec desc co card sum_dom_cards num_self_recs
                                 num_non_self_recs) (self_recs ~~ xs) []
   in
-    {typ = T, card = card, co = co, precise = precise, shallow = shallow,
-     constrs = constrs}
+    {typ = T, card = card, co = co, complete = complete, concrete = concrete,
+     shallow = shallow, constrs = constrs}
   end
 
 (* extended_context -> int -> typ list -> scope_desc -> scope *)
 fun scope_from_descriptor (ext_ctxt as {thy, ...}) sym_break shallow_dataTs
-                          (desc as (card_asgns, _)) =
+                          (desc as (card_assigns, _)) =
   let
     val datatypes =
       map (datatype_spec_from_scope_descriptor ext_ctxt shallow_dataTs desc)
-          (filter (is_datatype thy o fst) card_asgns)
-    val bisim_depth = card_of_type card_asgns @{typ bisim_iterator} - 1
-  in
-    {ext_ctxt = ext_ctxt, card_assigns = card_asgns, datatypes = datatypes,
+          (filter (is_datatype thy o fst) card_assigns)
+    val bisim_depth = card_of_type card_assigns @{typ bisim_iterator} - 1
+  in
+    {ext_ctxt = ext_ctxt, card_assigns = card_assigns, datatypes = datatypes,
      bisim_depth = bisim_depth,
      ofs = if sym_break <= 0 then Typtab.empty
-           else offset_table_for_card_assigns thy card_asgns datatypes}
+           else offset_table_for_card_assigns thy card_assigns datatypes}
   end
 
 (* theory -> typ list -> (typ option * int list) list
    -> (typ option * int list) list *)
 fun fix_cards_assigns_wrt_boxing _ _ [] = []
-  | fix_cards_assigns_wrt_boxing thy Ts ((SOME T, ks) :: cards_asgns) =
+  | fix_cards_assigns_wrt_boxing thy Ts ((SOME T, ks) :: cards_assigns) =
     (if is_fun_type T orelse is_pair_type T then
        Ts |> filter (curry (type_match thy o swap) T o unbox_type)
           |> map (rpair ks o SOME)
      else
-       [(SOME T, ks)]) @ fix_cards_assigns_wrt_boxing thy Ts cards_asgns
-  | fix_cards_assigns_wrt_boxing thy Ts ((NONE, ks) :: cards_asgns) =
-    (NONE, ks) :: fix_cards_assigns_wrt_boxing thy Ts cards_asgns
+       [(SOME T, ks)]) @ fix_cards_assigns_wrt_boxing thy Ts cards_assigns
+  | fix_cards_assigns_wrt_boxing thy Ts ((NONE, ks) :: cards_assigns) =
+    (NONE, ks) :: fix_cards_assigns_wrt_boxing thy Ts cards_assigns
 
 val max_scopes = 4096
 val distinct_threshold = 512
 
 (* extended_context -> int -> (typ option * int list) list
    -> (styp option * int list) list -> (styp option * int list) list -> int list
    -> typ list -> typ list -> typ list -> int * scope list *)
-fun all_scopes (ext_ctxt as {thy, ...}) sym_break cards_asgns maxes_asgns
-               iters_asgns bisim_depths mono_Ts nonmono_Ts shallow_dataTs =
-  let
-    val cards_asgns = fix_cards_assigns_wrt_boxing thy mono_Ts cards_asgns
-    val blocks = blocks_for_types ext_ctxt cards_asgns maxes_asgns iters_asgns
-                                  bisim_depths mono_Ts nonmono_Ts
+fun all_scopes (ext_ctxt as {thy, ...}) sym_break cards_assigns maxes_assigns
+               iters_assigns bisim_depths mono_Ts nonmono_Ts shallow_dataTs =
+  let
+    val cards_assigns = fix_cards_assigns_wrt_boxing thy mono_Ts cards_assigns
+    val blocks = blocks_for_types ext_ctxt cards_assigns maxes_assigns
+                                  iters_assigns bisim_depths mono_Ts nonmono_Ts
     val ranks = map rank_of_block blocks
     val all = all_combinations_ordered_smartly (map (rpair 0) ranks)
     val head = take max_scopes all
     val descs = map_filter (scope_descriptor_from_combination ext_ctxt blocks)
                            head