src/HOL/Tools/Nitpick/nitpick_mono.ML
author wenzelm
Sat Jan 08 16:01:51 2011 +0100 (2011-01-08)
changeset 41471 54a58904a598
parent 41109 97bf008b9cfe
child 46081 8f6465f7021b
permissions -rw-r--r--
modernized structure Prop_Logic;
avoid ML structure aliases;
     1 (*  Title:      HOL/Tools/Nitpick/nitpick_mono.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2009, 2010
     4 
     5 Monotonicity inference for higher-order logic.
     6 *)
     7 
     8 signature NITPICK_MONO =
     9 sig
    10   type hol_context = Nitpick_HOL.hol_context
    11 
    12   val trace : bool Unsynchronized.ref
    13   val formulas_monotonic :
    14     hol_context -> bool -> typ -> term list * term list -> bool
    15 end;
    16 
    17 structure Nitpick_Mono : NITPICK_MONO =
    18 struct
    19 
    20 open Nitpick_Util
    21 open Nitpick_HOL
    22 
    23 datatype sign = Plus | Minus
    24 
    25 type var = int
    26 
    27 datatype annotation = Gen | New | Fls | Tru
    28 datatype annotation_atom = A of annotation | V of var
    29 
    30 type assign_literal = var * (sign * annotation)
    31 
    32 datatype mtyp =
    33   MAlpha |
    34   MFun of mtyp * annotation_atom * mtyp |
    35   MPair of mtyp * mtyp |
    36   MType of string * mtyp list |
    37   MRec of string * typ list
    38 
    39 type mdata =
    40   {hol_ctxt: hol_context,
    41    binarize: bool,
    42    alpha_T: typ,
    43    max_fresh: int Unsynchronized.ref,
    44    datatype_mcache: ((string * typ list) * mtyp) list Unsynchronized.ref,
    45    constr_mcache: (styp * mtyp) list Unsynchronized.ref}
    46 
    47 exception UNSOLVABLE of unit
    48 exception MTYPE of string * mtyp list * typ list
    49 
    50 val trace = Unsynchronized.ref false
    51 fun trace_msg msg = if !trace then tracing (msg ()) else ()
    52 
    53 fun string_for_sign Plus = "+"
    54   | string_for_sign Minus = "-"
    55 
    56 fun negate_sign Plus = Minus
    57   | negate_sign Minus = Plus
    58 
    59 val string_for_var = signed_string_of_int
    60 fun string_for_vars sep [] = "0\<^bsub>" ^ sep ^ "\<^esub>"
    61   | string_for_vars sep xs = space_implode sep (map string_for_var xs)
    62 fun subscript_string_for_vars sep xs =
    63   if null xs then "" else "\<^bsub>" ^ string_for_vars sep xs ^ "\<^esub>"
    64 
    65 fun string_for_annotation Gen = "G"
    66   | string_for_annotation New = "N"
    67   | string_for_annotation Fls = "F"
    68   | string_for_annotation Tru = "T"
    69 
    70 fun string_for_annotation_atom (A a) = string_for_annotation a
    71   | string_for_annotation_atom (V x) = string_for_var x
    72 
    73 fun string_for_assign_literal (x, (sn, a)) =
    74   string_for_var x ^ (case sn of Plus => " = " | Minus => " \<noteq> ") ^
    75   string_for_annotation a
    76 
    77 val bool_M = MType (@{type_name bool}, [])
    78 val dummy_M = MType (nitpick_prefix ^ "dummy", [])
    79 
    80 fun is_MRec (MRec _) = true
    81   | is_MRec _ = false
    82 fun dest_MFun (MFun z) = z
    83   | dest_MFun M = raise MTYPE ("Nitpick_Mono.dest_MFun", [M], [])
    84 
    85 val no_prec = 100
    86 
    87 fun precedence_of_mtype (MFun _) = 1
    88   | precedence_of_mtype (MPair _) = 2
    89   | precedence_of_mtype _ = no_prec
    90 
    91 val string_for_mtype =
    92   let
    93     fun aux outer_prec M =
    94       let
    95         val prec = precedence_of_mtype M
    96         val need_parens = (prec < outer_prec)
    97       in
    98         (if need_parens then "(" else "") ^
    99         (if M = dummy_M then
   100            "_"
   101          else case M of
   102              MAlpha => "\<alpha>"
   103            | MFun (M1, aa, M2) =>
   104              aux (prec + 1) M1 ^ " \<Rightarrow>\<^bsup>" ^
   105              string_for_annotation_atom aa ^ "\<^esup> " ^ aux prec M2
   106            | MPair (M1, M2) => aux (prec + 1) M1 ^ " \<times> " ^ aux prec M2
   107            | MType (s, []) =>
   108              if s = @{type_name prop} orelse s = @{type_name bool} then "o"
   109              else s
   110            | MType (s, Ms) => "(" ^ commas (map (aux 0) Ms) ^ ") " ^ s
   111            | MRec (s, _) => "[" ^ s ^ "]") ^
   112         (if need_parens then ")" else "")
   113       end
   114   in aux 0 end
   115 
   116 fun flatten_mtype (MPair (M1, M2)) = maps flatten_mtype [M1, M2]
   117   | flatten_mtype (MType (_, Ms)) = maps flatten_mtype Ms
   118   | flatten_mtype M = [M]
   119 
   120 fun initial_mdata hol_ctxt binarize alpha_T =
   121   ({hol_ctxt = hol_ctxt, binarize = binarize, alpha_T = alpha_T,
   122     max_fresh = Unsynchronized.ref 0, datatype_mcache = Unsynchronized.ref [],
   123     constr_mcache = Unsynchronized.ref []} : mdata)
   124 
   125 fun could_exist_alpha_subtype alpha_T (T as Type (_, Ts)) =
   126     T = alpha_T orelse (not (is_fp_iterator_type T) andalso
   127                         exists (could_exist_alpha_subtype alpha_T) Ts)
   128   | could_exist_alpha_subtype alpha_T T = (T = alpha_T)
   129 fun could_exist_alpha_sub_mtype _ (alpha_T as TFree _) T =
   130     could_exist_alpha_subtype alpha_T T
   131   | could_exist_alpha_sub_mtype ctxt alpha_T T =
   132     (T = alpha_T orelse is_datatype ctxt [(NONE, true)] T)
   133 
   134 fun exists_alpha_sub_mtype MAlpha = true
   135   | exists_alpha_sub_mtype (MFun (M1, _, M2)) =
   136     exists exists_alpha_sub_mtype [M1, M2]
   137   | exists_alpha_sub_mtype (MPair (M1, M2)) =
   138     exists exists_alpha_sub_mtype [M1, M2]
   139   | exists_alpha_sub_mtype (MType (_, Ms)) = exists exists_alpha_sub_mtype Ms
   140   | exists_alpha_sub_mtype (MRec _) = true
   141 
   142 fun exists_alpha_sub_mtype_fresh MAlpha = true
   143   | exists_alpha_sub_mtype_fresh (MFun (_, V _, _)) = true
   144   | exists_alpha_sub_mtype_fresh (MFun (_, _, M2)) =
   145     exists_alpha_sub_mtype_fresh M2
   146   | exists_alpha_sub_mtype_fresh (MPair (M1, M2)) =
   147     exists exists_alpha_sub_mtype_fresh [M1, M2]
   148   | exists_alpha_sub_mtype_fresh (MType (_, Ms)) =
   149     exists exists_alpha_sub_mtype_fresh Ms
   150   | exists_alpha_sub_mtype_fresh (MRec _) = true
   151 
   152 fun constr_mtype_for_binders z Ms =
   153   fold_rev (fn M => curry3 MFun M (A Gen)) Ms (MRec z)
   154 
   155 fun repair_mtype _ _ MAlpha = MAlpha
   156   | repair_mtype cache seen (MFun (M1, aa, M2)) =
   157     MFun (repair_mtype cache seen M1, aa, repair_mtype cache seen M2)
   158   | repair_mtype cache seen (MPair Mp) =
   159     MPair (pairself (repair_mtype cache seen) Mp)
   160   | repair_mtype cache seen (MType (s, Ms)) =
   161     MType (s, maps (flatten_mtype o repair_mtype cache seen) Ms)
   162   | repair_mtype cache seen (MRec (z as (s, _))) =
   163     case AList.lookup (op =) cache z |> the of
   164       MRec _ => MType (s, [])
   165     | M => if member (op =) seen M then MType (s, [])
   166            else repair_mtype cache (M :: seen) M
   167 
   168 fun repair_datatype_mcache cache =
   169   let
   170     fun repair_one (z, M) =
   171       Unsynchronized.change cache
   172           (AList.update (op =) (z, repair_mtype (!cache) [] M))
   173   in List.app repair_one (rev (!cache)) end
   174 
   175 fun repair_constr_mcache dtype_cache constr_mcache =
   176   let
   177     fun repair_one (x, M) =
   178       Unsynchronized.change constr_mcache
   179           (AList.update (op =) (x, repair_mtype dtype_cache [] M))
   180   in List.app repair_one (!constr_mcache) end
   181 
   182 fun is_fin_fun_supported_type @{typ prop} = true
   183   | is_fin_fun_supported_type @{typ bool} = true
   184   | is_fin_fun_supported_type (Type (@{type_name option}, _)) = true
   185   | is_fin_fun_supported_type _ = false
   186 
   187 (* TODO: clean this up *)
   188 fun fin_fun_body _ _ (t as @{term False}) = SOME t
   189   | fin_fun_body _ _ (t as Const (@{const_name None}, _)) = SOME t
   190   | fin_fun_body dom_T ran_T
   191                  ((t0 as Const (@{const_name If}, _))
   192                   $ (t1 as Const (@{const_name HOL.eq}, _) $ Bound 0 $ t1')
   193                   $ t2 $ t3) =
   194     (if loose_bvar1 (t1', 0) then
   195        NONE
   196      else case fin_fun_body dom_T ran_T t3 of
   197        NONE => NONE
   198      | SOME t3 =>
   199        SOME (t0 $ (Const (@{const_name is_unknown}, dom_T --> bool_T) $ t1')
   200                 $ (Const (@{const_name unknown}, ran_T)) $ (t0 $ t1 $ t2 $ t3)))
   201   | fin_fun_body _ _ _ = NONE
   202 
   203 (* FIXME: make sure well-annotated *)
   204 
   205 fun fresh_mfun_for_fun_type (mdata as {max_fresh, ...} : mdata) all_minus
   206                             T1 T2 =
   207   let
   208     val M1 = fresh_mtype_for_type mdata all_minus T1
   209     val M2 = fresh_mtype_for_type mdata all_minus T2
   210     val aa = if not all_minus andalso exists_alpha_sub_mtype_fresh M1 andalso
   211                 is_fin_fun_supported_type (body_type T2) then
   212                V (Unsynchronized.inc max_fresh)
   213              else
   214                A Gen
   215   in (M1, aa, M2) end
   216 and fresh_mtype_for_type (mdata as {hol_ctxt as {ctxt, ...}, binarize, alpha_T,
   217                                     datatype_mcache, constr_mcache, ...})
   218                          all_minus =
   219   let
   220     fun do_type T =
   221       if T = alpha_T then
   222         MAlpha
   223       else case T of
   224         Type (@{type_name fun}, [T1, T2]) =>
   225         MFun (fresh_mfun_for_fun_type mdata all_minus T1 T2)
   226       | Type (@{type_name prod}, [T1, T2]) => MPair (pairself do_type (T1, T2))
   227       | Type (z as (s, _)) =>
   228         if could_exist_alpha_sub_mtype ctxt alpha_T T then
   229           case AList.lookup (op =) (!datatype_mcache) z of
   230             SOME M => M
   231           | NONE =>
   232             let
   233               val _ = Unsynchronized.change datatype_mcache (cons (z, MRec z))
   234               val xs = binarized_and_boxed_datatype_constrs hol_ctxt binarize T
   235               val (all_Ms, constr_Ms) =
   236                 fold_rev (fn (_, T') => fn (all_Ms, constr_Ms) =>
   237                              let
   238                                val binder_Ms = map do_type (binder_types T')
   239                                val new_Ms = filter exists_alpha_sub_mtype_fresh
   240                                                    binder_Ms
   241                                val constr_M = constr_mtype_for_binders z
   242                                                                        binder_Ms
   243                              in
   244                                (union (op =) new_Ms all_Ms,
   245                                 constr_M :: constr_Ms)
   246                              end)
   247                          xs ([], [])
   248               val M = MType (s, all_Ms)
   249               val _ = Unsynchronized.change datatype_mcache
   250                           (AList.update (op =) (z, M))
   251               val _ = Unsynchronized.change constr_mcache
   252                           (append (xs ~~ constr_Ms))
   253             in
   254               if forall (not o is_MRec o snd) (!datatype_mcache) then
   255                 (repair_datatype_mcache datatype_mcache;
   256                  repair_constr_mcache (!datatype_mcache) constr_mcache;
   257                  AList.lookup (op =) (!datatype_mcache) z |> the)
   258               else
   259                 M
   260             end
   261         else
   262           MType (s, [])
   263       | _ => MType (simple_string_of_typ T, [])
   264   in do_type end
   265 
   266 val ground_and_sole_base_constrs = []
   267 (* FIXME: [@{const_name Nil}, @{const_name None}], cf. lists_empty *)
   268 
   269 fun prodM_factors (MPair (M1, M2)) = maps prodM_factors [M1, M2]
   270   | prodM_factors M = [M]
   271 fun curried_strip_mtype (MFun (M1, _, M2)) =
   272     curried_strip_mtype M2 |>> append (prodM_factors M1)
   273   | curried_strip_mtype M = ([], M)
   274 fun sel_mtype_from_constr_mtype s M =
   275   let
   276     val (arg_Ms, dataM) = curried_strip_mtype M
   277     val a = if member (op =) ground_and_sole_base_constrs
   278                              (constr_name_for_sel_like s) then
   279               Fls
   280             else
   281               Gen
   282   in
   283     MFun (dataM, A a,
   284           case sel_no_from_name s of ~1 => bool_M | n => nth arg_Ms n)
   285   end
   286 
   287 fun mtype_for_constr (mdata as {hol_ctxt = {ctxt, ...}, alpha_T, constr_mcache,
   288                                 ...}) (x as (_, T)) =
   289   if could_exist_alpha_sub_mtype ctxt alpha_T T then
   290     case AList.lookup (op =) (!constr_mcache) x of
   291       SOME M => M
   292     | NONE => if T = alpha_T then
   293                 let val M = fresh_mtype_for_type mdata false T in
   294                   (Unsynchronized.change constr_mcache (cons (x, M)); M)
   295                 end
   296               else
   297                 (fresh_mtype_for_type mdata false (body_type T);
   298                  AList.lookup (op =) (!constr_mcache) x |> the)
   299   else
   300     fresh_mtype_for_type mdata false T
   301 fun mtype_for_sel (mdata as {hol_ctxt, binarize, ...}) (x as (s, _)) =
   302   x |> binarized_and_boxed_constr_for_sel hol_ctxt binarize
   303     |> mtype_for_constr mdata |> sel_mtype_from_constr_mtype s
   304 
   305 fun resolve_annotation_atom asgs (V x) =
   306     x |> AList.lookup (op =) asgs |> Option.map A |> the_default (V x)
   307   | resolve_annotation_atom _ aa = aa
   308 fun resolve_mtype asgs =
   309   let
   310     fun aux MAlpha = MAlpha
   311       | aux (MFun (M1, aa, M2)) =
   312         MFun (aux M1, resolve_annotation_atom asgs aa, aux M2)
   313       | aux (MPair Mp) = MPair (pairself aux Mp)
   314       | aux (MType (s, Ms)) = MType (s, map aux Ms)
   315       | aux (MRec z) = MRec z
   316   in aux end
   317 
   318 datatype comp_op = Eq | Neq | Leq
   319 
   320 type comp = annotation_atom * annotation_atom * comp_op * var list
   321 type assign_clause = assign_literal list
   322 
   323 type constraint_set = comp list * assign_clause list
   324 
   325 fun string_for_comp_op Eq = "="
   326   | string_for_comp_op Neq = "\<noteq>"
   327   | string_for_comp_op Leq = "\<le>"
   328 
   329 fun string_for_comp (aa1, aa2, cmp, xs) =
   330   string_for_annotation_atom aa1 ^ " " ^ string_for_comp_op cmp ^
   331   subscript_string_for_vars " \<and> " xs ^ " " ^ string_for_annotation_atom aa2
   332 
   333 fun string_for_assign_clause NONE = "\<top>"
   334   | string_for_assign_clause (SOME []) = "\<bot>"
   335   | string_for_assign_clause (SOME asgs) =
   336     space_implode " \<or> " (map string_for_assign_literal asgs)
   337 
   338 fun add_assign_literal (x, (sn, a)) clauses =
   339   if exists (fn [(x', (sn', a'))] =>
   340                 x = x' andalso ((sn = sn' andalso a <> a') orelse
   341                                 (sn <> sn' andalso a = a'))
   342               | _ => false) clauses then
   343     NONE
   344   else
   345     SOME ([(x, (sn, a))] :: clauses)
   346 
   347 fun add_assign_disjunct _ NONE = NONE
   348   | add_assign_disjunct asg (SOME asgs) = SOME (insert (op =) asg asgs)
   349 
   350 fun add_assign_clause NONE = I
   351   | add_assign_clause (SOME clause) = insert (op =) clause
   352 
   353 fun annotation_comp Eq a1 a2 = (a1 = a2)
   354   | annotation_comp Neq a1 a2 = (a1 <> a2)
   355   | annotation_comp Leq a1 a2 = (a1 = a2 orelse a2 = Gen)
   356 
   357 fun sign_for_comp_op Eq = Plus
   358   | sign_for_comp_op Neq = Minus
   359   | sign_for_comp_op Leq = raise BAD ("sign_for_comp_op", "unexpected \"Leq\"")
   360 
   361 fun do_annotation_atom_comp Leq [] aa1 aa2 (cset as (comps, clauses)) =
   362     (case (aa1, aa2) of
   363        (A a1, A a2) => if annotation_comp Leq a1 a2 then SOME cset else NONE
   364      | _ => SOME (insert (op =) (aa1, aa2, Leq, []) comps, clauses))
   365   | do_annotation_atom_comp cmp [] aa1 aa2 (cset as (comps, clauses)) =
   366     (case (aa1, aa2) of
   367        (A a1, A a2) => if annotation_comp cmp a1 a2 then SOME cset else NONE
   368      | (V x1, A a2) =>
   369        clauses |> add_assign_literal (x1, (sign_for_comp_op cmp, a2))
   370                |> Option.map (pair comps)
   371      | (A _, V _) => do_annotation_atom_comp cmp [] aa2 aa1 cset
   372      | (V _, V _) => SOME (insert (op =) (aa1, aa2, cmp, []) comps, clauses))
   373   | do_annotation_atom_comp cmp xs aa1 aa2 (comps, clauses) =
   374     SOME (insert (op =) (aa1, aa2, cmp, xs) comps, clauses)
   375 
   376 fun add_annotation_atom_comp cmp xs aa1 aa2 (comps, clauses) =
   377   (trace_msg (fn () => "*** Add " ^ string_for_comp (aa1, aa2, cmp, xs));
   378    case do_annotation_atom_comp cmp xs aa1 aa2 (comps, clauses) of
   379      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   380    | SOME cset => cset)
   381 
   382 fun do_mtype_comp _ _ _ _ NONE = NONE
   383   | do_mtype_comp _ _ MAlpha MAlpha cset = cset
   384   | do_mtype_comp Eq xs (MFun (M11, aa1, M12)) (MFun (M21, aa2, M22))
   385                   (SOME cset) =
   386     cset |> do_annotation_atom_comp Eq xs aa1 aa2
   387          |> do_mtype_comp Eq xs M11 M21 |> do_mtype_comp Eq xs M12 M22
   388   | do_mtype_comp Leq xs (MFun (M11, aa1, M12)) (MFun (M21, aa2, M22))
   389                   (SOME cset) =
   390     (if exists_alpha_sub_mtype M11 then
   391        cset |> do_annotation_atom_comp Leq xs aa1 aa2
   392             |> do_mtype_comp Leq xs M21 M11
   393             |> (case aa2 of
   394                   A Gen => I
   395                 | A _ => do_mtype_comp Leq xs M11 M21
   396                 | V x => do_mtype_comp Leq (x :: xs) M11 M21)
   397      else
   398        SOME cset)
   399     |> do_mtype_comp Leq xs M12 M22
   400   | do_mtype_comp cmp xs (M1 as MPair (M11, M12)) (M2 as MPair (M21, M22))
   401                   cset =
   402     (cset |> fold (uncurry (do_mtype_comp cmp xs)) [(M11, M21), (M12, M22)]
   403      handle ListPair.UnequalLengths =>
   404             raise MTYPE ("Nitpick_Mono.do_mtype_comp", [M1, M2], []))
   405   | do_mtype_comp _ _ (MType _) (MType _) cset =
   406     cset (* no need to compare them thanks to the cache *)
   407   | do_mtype_comp cmp _ M1 M2 _ =
   408     raise MTYPE ("Nitpick_Mono.do_mtype_comp (" ^ string_for_comp_op cmp ^ ")",
   409                  [M1, M2], [])
   410 
   411 fun add_mtype_comp cmp M1 M2 cset =
   412   (trace_msg (fn () => "*** Add " ^ string_for_mtype M1 ^ " " ^
   413                        string_for_comp_op cmp ^ " " ^ string_for_mtype M2);
   414    case SOME cset |> do_mtype_comp cmp [] M1 M2 of
   415      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   416    | SOME cset => cset)
   417 
   418 val add_mtypes_equal = add_mtype_comp Eq
   419 val add_is_sub_mtype = add_mtype_comp Leq
   420 
   421 fun do_notin_mtype_fv _ _ _ NONE = NONE
   422   | do_notin_mtype_fv Minus _ MAlpha cset = cset
   423   | do_notin_mtype_fv Plus [] MAlpha _ = NONE
   424   | do_notin_mtype_fv Plus [asg] MAlpha (SOME clauses) =
   425     clauses |> add_assign_literal asg
   426   | do_notin_mtype_fv Plus unless MAlpha (SOME clauses) =
   427     SOME (insert (op =) unless clauses)
   428   | do_notin_mtype_fv sn unless (MFun (M1, A a, M2)) cset =
   429     cset |> (if a <> Gen andalso sn = Plus then do_notin_mtype_fv Plus unless M1
   430              else I)
   431          |> (if a = Gen orelse sn = Plus then do_notin_mtype_fv Minus unless M1
   432              else I)
   433          |> do_notin_mtype_fv sn unless M2
   434   | do_notin_mtype_fv Plus unless (MFun (M1, V x, M2)) cset =
   435     cset |> (case add_assign_disjunct (x, (Plus, Gen)) (SOME unless) of
   436                NONE => I
   437              | SOME unless' => do_notin_mtype_fv Plus unless' M1)
   438          |> do_notin_mtype_fv Minus unless M1
   439          |> do_notin_mtype_fv Plus unless M2
   440   | do_notin_mtype_fv Minus unless (MFun (M1, V x, M2)) cset =
   441     cset |> (case fold (fn a => add_assign_disjunct (x, (Plus, a))) [Fls, Tru]
   442                        (SOME unless) of
   443                NONE => I
   444              | SOME unless' => do_notin_mtype_fv Plus unless' M1)
   445          |> do_notin_mtype_fv Minus unless M2
   446   | do_notin_mtype_fv sn unless (MPair (M1, M2)) cset =
   447     cset |> fold (do_notin_mtype_fv sn unless) [M1, M2]
   448   | do_notin_mtype_fv sn unless (MType (_, Ms)) cset =
   449     cset |> fold (do_notin_mtype_fv sn unless) Ms
   450  | do_notin_mtype_fv _ _ M _ =
   451    raise MTYPE ("Nitpick_Mono.do_notin_mtype_fv", [M], [])
   452 
   453 fun add_notin_mtype_fv sn unless M (comps, clauses) =
   454   (trace_msg (fn () => "*** Add " ^ string_for_mtype M ^ " is " ^
   455                        (case sn of Minus => "concrete" | Plus => "complete"));
   456    case SOME clauses |> do_notin_mtype_fv sn unless M of
   457      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   458    | SOME clauses => (comps, clauses))
   459 
   460 fun add_mtype_is_concrete x y z = add_notin_mtype_fv Minus x y z
   461 fun add_mtype_is_complete x y z = add_notin_mtype_fv Plus x y z
   462 
   463 val bool_table =
   464   [(Gen, (false, false)),
   465    (New, (false, true)),
   466    (Fls, (true, false)),
   467    (Tru, (true, true))]
   468 
   469 fun fst_var n = 2 * n
   470 fun snd_var n = 2 * n + 1
   471 
   472 val bools_from_annotation = AList.lookup (op =) bool_table #> the
   473 val annotation_from_bools = AList.find (op =) bool_table #> the_single
   474 
   475 fun prop_for_bool b = if b then Prop_Logic.True else Prop_Logic.False
   476 fun prop_for_bool_var_equality (v1, v2) =
   477   Prop_Logic.SAnd (Prop_Logic.SOr (Prop_Logic.BoolVar v1, Prop_Logic.SNot (Prop_Logic.BoolVar v2)),
   478            Prop_Logic.SOr (Prop_Logic.SNot (Prop_Logic.BoolVar v1), Prop_Logic.BoolVar v2))
   479 fun prop_for_assign (x, a) =
   480   let val (b1, b2) = bools_from_annotation a in
   481     Prop_Logic.SAnd (Prop_Logic.BoolVar (fst_var x) |> not b1 ? Prop_Logic.SNot,
   482              Prop_Logic.BoolVar (snd_var x) |> not b2 ? Prop_Logic.SNot)
   483   end
   484 fun prop_for_assign_literal (x, (Plus, a)) = prop_for_assign (x, a)
   485   | prop_for_assign_literal (x, (Minus, a)) = Prop_Logic.SNot (prop_for_assign (x, a))
   486 fun prop_for_atom_assign (A a', a) = prop_for_bool (a = a')
   487   | prop_for_atom_assign (V x, a) = prop_for_assign_literal (x, (Plus, a))
   488 fun prop_for_atom_equality (aa1, A a2) = prop_for_atom_assign (aa1, a2)
   489   | prop_for_atom_equality (A a1, aa2) = prop_for_atom_assign (aa2, a1)
   490   | prop_for_atom_equality (V x1, V x2) =
   491     Prop_Logic.SAnd (prop_for_bool_var_equality (pairself fst_var (x1, x2)),
   492              prop_for_bool_var_equality (pairself snd_var (x1, x2)))
   493 val prop_for_assign_clause = Prop_Logic.exists o map prop_for_assign_literal
   494 fun prop_for_exists_var_assign_literal xs a =
   495   Prop_Logic.exists (map (fn x => prop_for_assign_literal (x, (Plus, a))) xs)
   496 fun prop_for_comp (aa1, aa2, Eq, []) =
   497     Prop_Logic.SAnd (prop_for_comp (aa1, aa2, Leq, []),
   498              prop_for_comp (aa2, aa1, Leq, []))
   499   | prop_for_comp (aa1, aa2, Neq, []) =
   500     Prop_Logic.SNot (prop_for_comp (aa1, aa2, Eq, []))
   501   | prop_for_comp (aa1, aa2, Leq, []) =
   502     Prop_Logic.SOr (prop_for_atom_equality (aa1, aa2), prop_for_atom_assign (aa2, Gen))
   503   | prop_for_comp (aa1, aa2, cmp, xs) =
   504     Prop_Logic.SOr (prop_for_exists_var_assign_literal xs Gen,
   505             prop_for_comp (aa1, aa2, cmp, []))
   506 
   507 fun extract_assigns max_var assigns asgs =
   508   fold (fn x => fn accum =>
   509            if AList.defined (op =) asgs x then
   510              accum
   511            else case (fst_var x, snd_var x) |> pairself assigns of
   512              (NONE, NONE) => accum
   513            | bp => (x, annotation_from_bools (pairself (the_default false) bp))
   514                    :: accum)
   515        (max_var downto 1) asgs
   516 
   517 fun print_problem comps clauses =
   518   trace_msg (fn () => "*** Problem:\n" ^
   519                       cat_lines (map string_for_comp comps @
   520                                  map (string_for_assign_clause o SOME) clauses))
   521 
   522 fun print_solution asgs =
   523   trace_msg (fn () => "*** Solution:\n" ^
   524       (asgs
   525        |> map swap
   526        |> AList.group (op =)
   527        |> map (fn (a, xs) => string_for_annotation a ^ ": " ^
   528                              string_for_vars ", " (sort int_ord xs))
   529        |> space_implode "\n"))
   530 
   531 (* The ML solver timeout should correspond more or less to the overhead of
   532    invoking an external prover. *)
   533 val ml_solver_timeout = SOME (seconds 0.02)
   534 
   535 fun solve tac_timeout max_var (comps, clauses) =
   536   let
   537     val asgs =
   538       map_filter (fn [(x, (Plus, a))] => SOME (x, a) | _ => NONE) clauses
   539     fun do_assigns assigns =
   540       SOME (extract_assigns max_var assigns asgs |> tap print_solution)
   541     val _ = print_problem comps clauses
   542     val prop =
   543       Prop_Logic.all (map prop_for_comp comps @ map prop_for_assign_clause clauses)
   544   in
   545     if Prop_Logic.eval (K false) prop then
   546       do_assigns (K (SOME false))
   547     else if Prop_Logic.eval (K true) prop then
   548       do_assigns (K (SOME true))
   549     else
   550       let
   551         (* use the first ML solver (to avoid startup overhead) *)
   552         val (ml_solvers, nonml_solvers) =
   553           !SatSolver.solvers
   554           |> List.partition (member (op =) ["dptsat", "dpll"] o fst)
   555         val res =
   556           if null nonml_solvers then
   557             time_limit tac_timeout (snd (hd ml_solvers)) prop
   558           else
   559             time_limit ml_solver_timeout (snd (hd ml_solvers)) prop
   560             handle TimeLimit.TimeOut =>
   561                    time_limit tac_timeout (SatSolver.invoke_solver "auto") prop
   562       in
   563         case res of
   564           SatSolver.SATISFIABLE assigns => do_assigns assigns
   565         | _ => (trace_msg (K "*** Unsolvable"); NONE)
   566       end
   567       handle TimeLimit.TimeOut => (trace_msg (K "*** Timed out"); NONE)
   568   end
   569 
   570 type mcontext =
   571   {bound_Ts: typ list,
   572    bound_Ms: mtyp list,
   573    frame: (int * annotation_atom) list,
   574    frees: (styp * mtyp) list,
   575    consts: (styp * mtyp) list}
   576 
   577 fun string_for_bound ctxt Ms (j, aa) =
   578   Syntax.string_of_term ctxt (Bound (length Ms - j - 1)) ^ " :\<^bsup>" ^
   579   string_for_annotation_atom aa ^ "\<^esup> " ^
   580   string_for_mtype (nth Ms (length Ms - j - 1))
   581 fun string_for_free relevant_frees ((s, _), M) =
   582   if member (op =) relevant_frees s then SOME (s ^ " : " ^ string_for_mtype M)
   583   else NONE
   584 fun string_for_mcontext ctxt t ({bound_Ms, frame, frees, ...} : mcontext) =
   585   (map_filter (string_for_free (Term.add_free_names t [])) frees @
   586    map (string_for_bound ctxt bound_Ms) frame)
   587   |> commas |> enclose "[" "]"
   588 
   589 val initial_gamma =
   590   {bound_Ts = [], bound_Ms = [], frame = [], frees = [], consts = []}
   591 
   592 fun push_bound aa T M {bound_Ts, bound_Ms, frame, frees, consts} =
   593   {bound_Ts = T :: bound_Ts, bound_Ms = M :: bound_Ms,
   594    frame = frame @ [(length bound_Ts, aa)], frees = frees, consts = consts}
   595 fun pop_bound {bound_Ts, bound_Ms, frame, frees, consts} =
   596   {bound_Ts = tl bound_Ts, bound_Ms = tl bound_Ms,
   597    frame = frame |> filter_out (fn (j, _) => j = length bound_Ts - 1),
   598    frees = frees, consts = consts}
   599   handle List.Empty => initial_gamma (* FIXME: needed? *)
   600 
   601 fun set_frame frame ({bound_Ts, bound_Ms, frees, consts, ...} : mcontext) =
   602   {bound_Ts = bound_Ts, bound_Ms = bound_Ms, frame = frame, frees = frees,
   603    consts = consts}
   604 
   605 fun add_comp_frame aa cmp = fold (add_annotation_atom_comp cmp [] aa o snd)
   606 
   607 fun add_bound_frame j frame =
   608   let
   609     val (new_frame, gen_frame) = List.partition (curry (op =) j o fst) frame
   610   in
   611     add_comp_frame (A New) Leq new_frame
   612     #> add_comp_frame (A Gen) Eq gen_frame
   613   end
   614 
   615 fun fresh_frame ({max_fresh, ...} : mdata) fls tru =
   616   map (apsnd (fn aa =>
   617                  case (aa, fls, tru) of
   618                    (A Fls, SOME a, _) => A a
   619                  | (A Tru, _, SOME a) => A a
   620                  | (A Gen, _, _) => A Gen
   621                  | _ => V (Unsynchronized.inc max_fresh)))
   622 
   623 fun conj_clauses res_aa aa1 aa2 =
   624   [[(aa1, (Neq, Tru)), (aa2, (Neq, Tru)), (res_aa, (Eq, Tru))],
   625    [(aa1, (Neq, Fls)), (res_aa, (Eq, Fls))],
   626    [(aa2, (Neq, Fls)), (res_aa, (Eq, Fls))],
   627    [(aa1, (Neq, Gen)), (aa2, (Eq, Fls)), (res_aa, (Eq, Gen))],
   628    [(aa1, (Neq, New)), (aa2, (Eq, Fls)), (res_aa, (Eq, Gen))],
   629    [(aa1, (Eq, Fls)), (aa2, (Neq, Gen)), (res_aa, (Eq, Gen))],
   630    [(aa1, (Eq, Fls)), (aa2, (Neq, New)), (res_aa, (Eq, Gen))]]
   631 
   632 fun disj_clauses res_aa aa1 aa2 =
   633   [[(aa1, (Neq, Tru)), (res_aa, (Eq, Tru))],
   634    [(aa2, (Neq, Tru)), (res_aa, (Eq, Tru))],
   635    [(aa1, (Neq, Fls)), (aa2, (Neq, Fls)), (res_aa, (Eq, Fls))],
   636    [(aa1, (Neq, Gen)), (aa2, (Eq, Tru)), (res_aa, (Eq, Gen))],
   637    [(aa1, (Neq, New)), (aa2, (Eq, Tru)), (res_aa, (Eq, Gen))],
   638    [(aa1, (Eq, Tru)), (aa2, (Neq, Gen)), (res_aa, (Eq, Gen))],
   639    [(aa1, (Eq, Tru)), (aa2, (Neq, New)), (res_aa, (Eq, Gen))]]
   640 
   641 fun imp_clauses res_aa aa1 aa2 =
   642   [[(aa1, (Neq, Fls)), (res_aa, (Eq, Tru))],
   643    [(aa2, (Neq, Tru)), (res_aa, (Eq, Tru))],
   644    [(aa1, (Neq, Tru)), (aa2, (Neq, Fls)), (res_aa, (Eq, Fls))],
   645    [(aa1, (Neq, Gen)), (aa2, (Eq, Tru)), (res_aa, (Eq, Gen))],
   646    [(aa1, (Neq, New)), (aa2, (Eq, Tru)), (res_aa, (Eq, Gen))],
   647    [(aa1, (Eq, Fls)), (aa2, (Neq, Gen)), (res_aa, (Eq, Gen))],
   648    [(aa1, (Eq, Fls)), (aa2, (Neq, New)), (res_aa, (Eq, Gen))]]
   649 
   650 val meta_conj_spec = ("\<and>", conj_clauses)
   651 val meta_imp_spec = ("\<implies>", imp_clauses)
   652 val conj_spec = ("\<and>", conj_clauses)
   653 val disj_spec = ("\<or>", disj_clauses)
   654 val imp_spec = ("\<implies>", imp_clauses)
   655 
   656 fun add_annotation_clause_from_quasi_clause _ NONE = NONE
   657   | add_annotation_clause_from_quasi_clause [] accum = accum
   658   | add_annotation_clause_from_quasi_clause ((aa, (cmp, a)) :: rest) accum =
   659     case aa of
   660       A a' => if annotation_comp cmp a' a then NONE
   661               else add_annotation_clause_from_quasi_clause rest accum
   662     | V x => add_annotation_clause_from_quasi_clause rest accum
   663              |> Option.map (cons (x, (sign_for_comp_op cmp, a)))
   664 
   665 fun assign_clause_from_quasi_clause unless =
   666   add_annotation_clause_from_quasi_clause unless (SOME [])
   667 
   668 fun add_connective_var conn mk_quasi_clauses res_aa aa1 aa2 =
   669   (trace_msg (fn () => "*** Add " ^ string_for_annotation_atom res_aa ^ " = " ^
   670                        string_for_annotation_atom aa1 ^ " " ^ conn ^ " " ^
   671                        string_for_annotation_atom aa2);
   672    fold (add_assign_clause o assign_clause_from_quasi_clause)
   673         (mk_quasi_clauses res_aa aa1 aa2))
   674 fun add_connective_frames conn mk_quasi_clauses res_frame frame1 frame2 =
   675   fold I (map3 (fn (_, res_aa) => fn (_, aa1) => fn (_, aa2) =>
   676                    add_connective_var conn mk_quasi_clauses res_aa aa1 aa2)
   677                res_frame frame1 frame2)
   678 
   679 fun kill_unused_in_frame is_in (accum as ({frame, ...} : mcontext, _)) =
   680   let val (used_frame, unused_frame) = List.partition is_in frame in
   681     accum |>> set_frame used_frame
   682           ||> add_comp_frame (A Gen) Eq unused_frame
   683   end
   684 
   685 fun split_frame is_in_fun (gamma as {frame, ...} : mcontext, cset) =
   686   let
   687     fun bubble fun_frame arg_frame [] cset =
   688         ((rev fun_frame, rev arg_frame), cset)
   689       | bubble fun_frame arg_frame ((bound as (_, aa)) :: rest) cset =
   690         if is_in_fun bound then
   691           bubble (bound :: fun_frame) arg_frame rest
   692                  (cset |> add_comp_frame aa Leq arg_frame)
   693         else
   694           bubble fun_frame (bound :: arg_frame) rest cset
   695   in cset |> bubble [] [] frame ||> pair gamma end
   696 
   697 fun add_annotation_atom_comp_alt _ (A Gen) _ _ = I
   698   | add_annotation_atom_comp_alt _ (A _) _ _ =
   699     (trace_msg (K "*** Expected G"); raise UNSOLVABLE ())
   700   | add_annotation_atom_comp_alt cmp (V x) aa1 aa2 =
   701     add_annotation_atom_comp cmp [x] aa1 aa2
   702 
   703 fun add_arg_order1 ((_, aa), (_, prev_aa)) = I
   704   add_annotation_atom_comp_alt Neq prev_aa (A Gen) aa
   705 fun add_app1 fun_aa ((_, res_aa), (_, arg_aa)) = I
   706   let
   707     val clause = [(arg_aa, (Eq, New)), (res_aa, (Eq, Gen))]
   708                  |> assign_clause_from_quasi_clause
   709   in
   710     trace_msg (fn () => "*** Add " ^ string_for_assign_clause clause);
   711     apsnd (add_assign_clause clause)
   712     #> add_annotation_atom_comp_alt Leq arg_aa fun_aa res_aa
   713   end
   714 fun add_app _ [] [] = I
   715   | add_app fun_aa res_frame arg_frame =
   716     add_comp_frame (A New) Leq arg_frame
   717     #> fold add_arg_order1 (tl arg_frame ~~ (fst (split_last arg_frame)))
   718     #> fold (add_app1 fun_aa) (res_frame ~~ arg_frame)
   719 
   720 fun consider_connective mdata (conn, mk_quasi_clauses) do_t1 do_t2
   721                         (accum as ({frame, ...}, _)) =
   722   let
   723     val frame1 = fresh_frame mdata (SOME Tru) NONE frame
   724     val frame2 = fresh_frame mdata (SOME Fls) NONE frame
   725   in
   726     accum |>> set_frame frame1 |> do_t1
   727           |>> set_frame frame2 |> do_t2
   728           |>> set_frame frame
   729           ||> apsnd (add_connective_frames conn mk_quasi_clauses frame frame1
   730                                            frame2)
   731   end
   732 
   733 fun consider_term (mdata as {hol_ctxt = {thy, ctxt, stds, ...}, alpha_T,
   734                              max_fresh, ...}) =
   735   let
   736     fun is_enough_eta_expanded t =
   737       case strip_comb t of
   738         (Const x, ts) => the_default 0 (arity_of_built_in_const thy stds x)
   739         <= length ts
   740       | _ => true
   741     val mtype_for = fresh_mtype_for_type mdata false
   742     fun do_all T (gamma, cset) =
   743       let
   744         val abs_M = mtype_for (domain_type (domain_type T))
   745         val x = Unsynchronized.inc max_fresh
   746         val body_M = mtype_for (body_type T)
   747       in
   748         (MFun (MFun (abs_M, V x, body_M), A Gen, body_M),
   749          (gamma, cset |> add_mtype_is_complete [(x, (Plus, Tru))] abs_M))
   750       end
   751     fun do_equals T (gamma, cset) =
   752       let
   753         val M = mtype_for (domain_type T)
   754         val x = Unsynchronized.inc max_fresh
   755       in
   756         (MFun (M, A Gen, MFun (M, V x, mtype_for (nth_range_type 2 T))),
   757          (gamma, cset |> add_mtype_is_concrete [] M
   758                       |> add_annotation_atom_comp Leq [] (A Fls) (V x)))
   759       end
   760     fun do_robust_set_operation T (gamma, cset) =
   761       let
   762         val set_T = domain_type T
   763         val M1 = mtype_for set_T
   764         val M2 = mtype_for set_T
   765         val M3 = mtype_for set_T
   766       in
   767         (MFun (M1, A Gen, MFun (M2, A Gen, M3)),
   768          (gamma, cset |> add_is_sub_mtype M1 M3 |> add_is_sub_mtype M2 M3))
   769       end
   770     fun do_fragile_set_operation T (gamma, cset) =
   771       let
   772         val set_T = domain_type T
   773         val set_M = mtype_for set_T
   774         fun custom_mtype_for (T as Type (@{type_name fun}, [T1, T2])) =
   775             if T = set_T then set_M
   776             else MFun (custom_mtype_for T1, A Gen, custom_mtype_for T2)
   777           | custom_mtype_for T = mtype_for T
   778       in
   779         (custom_mtype_for T, (gamma, cset |> add_mtype_is_concrete [] set_M))
   780       end
   781     fun do_pair_constr T accum =
   782       case mtype_for (nth_range_type 2 T) of
   783         M as MPair (a_M, b_M) =>
   784         (MFun (a_M, A Gen, MFun (b_M, A Gen, M)), accum)
   785       | M => raise MTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [M], [])
   786     fun do_nth_pair_sel n T =
   787       case mtype_for (domain_type T) of
   788         M as MPair (a_M, b_M) =>
   789         pair (MFun (M, A Gen, if n = 0 then a_M else b_M))
   790       | M => raise MTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [M], [])
   791     and do_connect spec t1 t2 accum =
   792       (bool_M, consider_connective mdata spec (snd o do_term t1)
   793                                    (snd o do_term t2) accum)
   794     and do_term t
   795             (accum as (gamma as {bound_Ts, bound_Ms, frame, frees, consts},
   796                        cset)) =
   797       (trace_msg (fn () => "  " ^ string_for_mcontext ctxt t gamma ^
   798                            " \<turnstile> " ^ Syntax.string_of_term ctxt t ^
   799                            " : _?");
   800        case t of
   801          @{const False} => (bool_M, accum ||> add_comp_frame (A Fls) Leq frame)
   802        | Const (@{const_name None}, T) =>
   803          (mtype_for T, accum ||> add_comp_frame (A Fls) Leq frame)
   804        | @{const True} => (bool_M, accum ||> add_comp_frame (A Tru) Leq frame)
   805        | (t0 as Const (@{const_name HOL.eq}, _)) $ Bound 0 $ t2 =>
   806          (* hack to exploit symmetry of equality when typing "insert" *)
   807          (if t2 = Bound 0 then do_term @{term True}
   808           else do_term (t0 $ t2 $ Bound 0)) accum
   809        | Const (x as (s, T)) =>
   810          (case AList.lookup (op =) consts x of
   811             SOME M => (M, accum)
   812           | NONE =>
   813             if not (could_exist_alpha_subtype alpha_T T) then
   814               (mtype_for T, accum)
   815             else case s of
   816               @{const_name all} => do_all T accum
   817             | @{const_name "=="} => do_equals T accum
   818             | @{const_name All} => do_all T accum
   819             | @{const_name Ex} =>
   820               let val set_T = domain_type T in
   821                 do_term (Abs (Name.uu, set_T,
   822                               @{const Not} $ (HOLogic.mk_eq
   823                                   (Abs (Name.uu, domain_type set_T,
   824                                         @{const False}),
   825                                    Bound 0)))) accum
   826               end
   827             | @{const_name HOL.eq} => do_equals T accum
   828             | @{const_name The} =>
   829               (trace_msg (K "*** The"); raise UNSOLVABLE ())
   830             | @{const_name Eps} =>
   831               (trace_msg (K "*** Eps"); raise UNSOLVABLE ())
   832             | @{const_name If} =>
   833               do_robust_set_operation (range_type T) accum
   834               |>> curry3 MFun bool_M (A Gen)
   835             | @{const_name Pair} => do_pair_constr T accum
   836             | @{const_name fst} => do_nth_pair_sel 0 T accum
   837             | @{const_name snd} => do_nth_pair_sel 1 T accum
   838             | @{const_name Id} =>
   839               (MFun (mtype_for (domain_type T), A Gen, bool_M), accum)
   840             | @{const_name converse} =>
   841               let
   842                 val x = Unsynchronized.inc max_fresh
   843                 fun mtype_for_set T =
   844                   MFun (mtype_for (domain_type T), V x, bool_M)
   845                 val ab_set_M = domain_type T |> mtype_for_set
   846                 val ba_set_M = range_type T |> mtype_for_set
   847               in
   848                 (MFun (ab_set_M, A Gen, ba_set_M),
   849                  accum ||> add_annotation_atom_comp Neq [] (V x) (A New))
   850               end
   851             | @{const_name trancl} => do_fragile_set_operation T accum
   852             | @{const_name rel_comp} =>
   853               let
   854                 val x = Unsynchronized.inc max_fresh
   855                 fun mtype_for_set T =
   856                   MFun (mtype_for (domain_type T), V x, bool_M)
   857                 val bc_set_M = domain_type T |> mtype_for_set
   858                 val ab_set_M = domain_type (range_type T) |> mtype_for_set
   859                 val ac_set_M = nth_range_type 2 T |> mtype_for_set
   860               in
   861                 (MFun (bc_set_M, A Gen, MFun (ab_set_M, A Gen, ac_set_M)),
   862                  accum ||> add_annotation_atom_comp Neq [] (V x) (A New))
   863               end
   864             | @{const_name finite} =>
   865               let
   866                 val M1 = mtype_for (domain_type (domain_type T))
   867                 val a = if exists_alpha_sub_mtype M1 then Fls else Gen
   868               in (MFun (MFun (M1, A a, bool_M), A Gen, bool_M), accum) end
   869             | @{const_name prod} =>
   870               let
   871                 val x = Unsynchronized.inc max_fresh
   872                 fun mtype_for_set T =
   873                   MFun (mtype_for (domain_type T), V x, bool_M)
   874                 val a_set_M = mtype_for_set (domain_type T)
   875                 val b_set_M =
   876                   mtype_for_set (range_type (domain_type (range_type T)))
   877                 val ab_set_M = mtype_for_set (nth_range_type 2 T)
   878               in
   879                 (MFun (a_set_M, A Gen, MFun (b_set_M, A Gen, ab_set_M)),
   880                  accum ||> add_annotation_atom_comp Neq [] (V x) (A New))
   881               end
   882             | _ =>
   883               if s = @{const_name safe_The} then
   884                 let
   885                   val a_set_M = mtype_for (domain_type T)
   886                   val a_M = dest_MFun a_set_M |> #1
   887                 in (MFun (a_set_M, A Gen, a_M), accum) end
   888               else if s = @{const_name ord_class.less_eq} andalso
   889                       is_set_type (domain_type T) then
   890                 do_fragile_set_operation T accum
   891               else if is_sel s then
   892                 (mtype_for_sel mdata x, accum)
   893               else if is_constr ctxt stds x then
   894                 (mtype_for_constr mdata x, accum)
   895               else if is_built_in_const thy stds x then
   896                 (fresh_mtype_for_type mdata true T, accum)
   897               else
   898                 let val M = mtype_for T in
   899                   (M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms, frame = frame,
   900                         frees = frees, consts = (x, M) :: consts}, cset))
   901                 end)
   902            ||> apsnd (add_comp_frame (A Gen) Eq frame)
   903          | Free (x as (_, T)) =>
   904            (case AList.lookup (op =) frees x of
   905               SOME M => (M, accum)
   906             | NONE =>
   907               let val M = mtype_for T in
   908                 (M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms, frame = frame,
   909                       frees = (x, M) :: frees, consts = consts}, cset))
   910               end)
   911            ||> apsnd (add_comp_frame (A Gen) Eq frame)
   912          | Var _ => (trace_msg (K "*** Var"); raise UNSOLVABLE ())
   913          | Bound j =>
   914            (nth bound_Ms j,
   915             accum ||> add_bound_frame (length bound_Ts - j - 1) frame)
   916          | Abs (_, T, t') =>
   917            (case fin_fun_body T (fastype_of1 (T :: bound_Ts, t')) t' of
   918               SOME t' =>
   919               let
   920                 val M = mtype_for T
   921                 val x = Unsynchronized.inc max_fresh
   922                 val (M', accum) = do_term t' (accum |>> push_bound (V x) T M)
   923               in
   924                 (MFun (M, V x, M'),
   925                  accum |>> pop_bound
   926                        ||> add_annotation_atom_comp Leq [] (A Fls) (V x))
   927               end
   928             | NONE =>
   929               ((case t' of
   930                   t1' $ Bound 0 =>
   931                   if not (loose_bvar1 (t1', 0)) andalso
   932                      is_enough_eta_expanded t1' then
   933                     do_term (incr_boundvars ~1 t1') accum
   934                   else
   935                     raise SAME ()
   936                 | (t11 as Const (@{const_name HOL.eq}, _)) $ Bound 0 $ t13 =>
   937                   if not (loose_bvar1 (t13, 0)) then
   938                     do_term (incr_boundvars ~1 (t11 $ t13)) accum
   939                   else
   940                     raise SAME ()
   941                 | _ => raise SAME ())
   942                handle SAME () =>
   943                       let
   944                         val M = mtype_for T
   945                         val x = Unsynchronized.inc max_fresh
   946                         val (M', accum) =
   947                           do_term t' (accum |>> push_bound (V x) T M)
   948                       in (MFun (M, V x, M'), accum |>> pop_bound) end))
   949          | @{const Not} $ t1 => do_connect imp_spec t1 @{const False} accum
   950          | @{const conj} $ t1 $ t2 => do_connect conj_spec t1 t2 accum
   951          | @{const disj} $ t1 $ t2 => do_connect disj_spec t1 t2 accum
   952          | @{const implies} $ t1 $ t2 => do_connect imp_spec t1 t2 accum
   953          | Const (@{const_name Let}, _) $ t1 $ t2 =>
   954            do_term (betapply (t2, t1)) accum
   955          | t1 $ t2 =>
   956            let
   957              fun is_in t (j, _) = loose_bvar1 (t, length bound_Ts - j - 1)
   958              val accum as ({frame, ...}, _) =
   959                accum |> kill_unused_in_frame (is_in t)
   960              val ((frame1a, frame1b), accum) = accum |> split_frame (is_in t1)
   961              val frame2a = frame1a |> map (apsnd (K (A Gen)))
   962              val frame2b =
   963                frame1b |> map (apsnd (fn _ => V (Unsynchronized.inc max_fresh)))
   964              val frame2 = frame2a @ frame2b
   965              val (M1, accum) = accum |>> set_frame frame1a |> do_term t1
   966              val (M2, accum) = accum |>> set_frame frame2 |> do_term t2
   967            in
   968              let
   969                val (M11, aa, M12) = M1 |> dest_MFun
   970              in
   971                (M12, accum |>> set_frame frame
   972                            ||> add_is_sub_mtype M2 M11
   973                            ||> add_app aa frame1b frame2b)
   974              end
   975            end)
   976         |> tap (fn (M, (gamma, _)) =>
   977                    trace_msg (fn () => "  " ^ string_for_mcontext ctxt t gamma ^
   978                                        " \<turnstile> " ^
   979                                        Syntax.string_of_term ctxt t ^ " : " ^
   980                                        string_for_mtype M))
   981   in do_term end
   982 
   983 fun force_gen_funs 0 _ = I
   984   | force_gen_funs n (M as MFun (M1, _, M2)) =
   985     add_mtypes_equal M (MFun (M1, A Gen, M2)) #> force_gen_funs (n - 1) M2
   986   | force_gen_funs _ M = raise MTYPE ("Nitpick_Mono.force_gen_funs", [M], [])
   987 fun consider_general_equals mdata def t1 t2 accum =
   988   let
   989     val (M1, accum) = consider_term mdata t1 accum
   990     val (M2, accum) = consider_term mdata t2 accum
   991     val accum = accum ||> add_mtypes_equal M1 M2
   992   in
   993     if def then
   994       let
   995         val (head1, args1) = strip_comb t1
   996         val (head_M1, accum) = consider_term mdata head1 accum
   997       in accum ||> force_gen_funs (length args1) head_M1 end
   998     else
   999       accum
  1000   end
  1001 
  1002 fun consider_general_formula (mdata as {hol_ctxt = {ctxt, ...}, max_fresh,
  1003                                         ...}) =
  1004   let
  1005     val mtype_for = fresh_mtype_for_type mdata false
  1006     val do_term = snd oo consider_term mdata
  1007     fun do_formula sn t (accum as (gamma, _)) =
  1008       let
  1009         fun do_quantifier quant_s abs_T body_t =
  1010           let
  1011             val abs_M = mtype_for abs_T
  1012             val x = Unsynchronized.inc max_fresh
  1013             val side_cond = ((sn = Minus) = (quant_s = @{const_name Ex}))
  1014             fun ann () = if quant_s = @{const_name Ex} then Fls else Tru
  1015           in
  1016             accum ||> side_cond
  1017                       ? add_mtype_is_complete [(x, (Plus, ann ()))] abs_M
  1018                   |>> push_bound (V x) abs_T abs_M
  1019                   |> do_formula sn body_t
  1020                   |>> pop_bound
  1021           end
  1022         fun do_connect spec neg1 t1 t2 =
  1023           consider_connective mdata spec
  1024               (do_formula (sn |> neg1 ? negate_sign) t1) (do_formula sn t2)
  1025         fun do_equals t1 t2 =
  1026           case sn of
  1027             Plus => do_term t accum
  1028           | Minus => consider_general_equals mdata false t1 t2 accum
  1029       in
  1030         trace_msg (fn () => "  " ^ string_for_mcontext ctxt t gamma ^
  1031                             " \<turnstile> " ^ Syntax.string_of_term ctxt t ^
  1032                             " : o\<^sup>" ^ string_for_sign sn ^ "?");
  1033         case t of
  1034           Const (s as @{const_name all}, _) $ Abs (_, T1, t1) =>
  1035           do_quantifier s T1 t1
  1036         | Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2
  1037         | @{const Trueprop} $ t1 => do_formula sn t1 accum
  1038         | Const (s as @{const_name All}, _) $ Abs (_, T1, t1) =>
  1039           do_quantifier s T1 t1
  1040         | Const (s as @{const_name Ex}, T0) $ (t1 as Abs (_, T1, t1')) =>
  1041           (case sn of
  1042              Plus => do_quantifier s T1 t1'
  1043            | Minus =>
  1044              (* FIXME: Needed? *)
  1045              do_term (@{const Not}
  1046                       $ (HOLogic.eq_const (domain_type T0) $ t1
  1047                          $ Abs (Name.uu, T1, @{const False}))) accum)
  1048         | Const (@{const_name HOL.eq}, _) $ t1 $ t2 => do_equals t1 t2
  1049         | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1050           do_formula sn (betapply (t2, t1)) accum
  1051         | @{const Pure.conjunction} $ t1 $ t2 =>
  1052           do_connect meta_conj_spec false t1 t2 accum
  1053         | @{const "==>"} $ t1 $ t2 => do_connect meta_imp_spec true t1 t2 accum
  1054         | @{const Not} $ t1 => do_connect imp_spec true t1 @{const False} accum
  1055         | @{const conj} $ t1 $ t2 => do_connect conj_spec false t1 t2 accum
  1056         | @{const disj} $ t1 $ t2 => do_connect disj_spec false t1 t2 accum
  1057         | @{const implies} $ t1 $ t2 => do_connect imp_spec true t1 t2 accum
  1058         | _ => do_term t accum
  1059       end
  1060       |> tap (fn (gamma, _) =>
  1061                  trace_msg (fn () => string_for_mcontext ctxt t gamma ^
  1062                                      " \<turnstile> " ^
  1063                                      Syntax.string_of_term ctxt t ^
  1064                                      " : o\<^sup>" ^ string_for_sign sn))
  1065   in do_formula end
  1066 
  1067 (* The harmless axiom optimization below is somewhat too aggressive in the face
  1068    of (rather peculiar) user-defined axioms. *)
  1069 val harmless_consts =
  1070   [@{const_name ord_class.less}, @{const_name ord_class.less_eq}]
  1071 val bounteous_consts = [@{const_name bisim}]
  1072 
  1073 fun is_harmless_axiom ({hol_ctxt = {thy, stds, ...}, ...} : mdata) t =
  1074     Term.add_consts t []
  1075     |> filter_out (is_built_in_const thy stds)
  1076     |> (forall (member (op =) harmless_consts o original_name o fst) orf
  1077         exists (member (op =) bounteous_consts o fst))
  1078 
  1079 fun consider_nondefinitional_axiom mdata t =
  1080   if is_harmless_axiom mdata t then I
  1081   else consider_general_formula mdata Plus t
  1082 
  1083 fun consider_definitional_axiom (mdata as {hol_ctxt = {ctxt, ...}, ...} : mdata) t =
  1084   if not (is_constr_pattern_formula ctxt t) then
  1085     consider_nondefinitional_axiom mdata t
  1086   else if is_harmless_axiom mdata t then
  1087     I
  1088   else
  1089     let
  1090       val mtype_for = fresh_mtype_for_type mdata false
  1091       val do_term = snd oo consider_term mdata
  1092       fun do_all abs_T body_t accum =
  1093         accum |>> push_bound (A Gen) abs_T (mtype_for abs_T)
  1094               |> do_formula body_t
  1095               |>> pop_bound
  1096       and do_implies t1 t2 = do_term t1 #> do_formula t2
  1097       and do_formula t accum =
  1098         case t of
  1099           Const (@{const_name all}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum
  1100         | @{const Trueprop} $ t1 => do_formula t1 accum
  1101         | Const (@{const_name "=="}, _) $ t1 $ t2 =>
  1102           consider_general_equals mdata true t1 t2 accum
  1103         | @{const "==>"} $ t1 $ t2 => do_implies t1 t2 accum
  1104         | @{const Pure.conjunction} $ t1 $ t2 =>
  1105           fold (do_formula) [t1, t2] accum
  1106         | Const (@{const_name All}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum
  1107         | Const (@{const_name HOL.eq}, _) $ t1 $ t2 =>
  1108           consider_general_equals mdata true t1 t2 accum
  1109         | @{const conj} $ t1 $ t2 => fold (do_formula) [t1, t2] accum
  1110         | @{const implies} $ t1 $ t2 => do_implies t1 t2 accum
  1111         | _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\
  1112                            \do_formula", [t])
  1113     in do_formula t end
  1114 
  1115 fun string_for_mtype_of_term ctxt asgs t M =
  1116   Syntax.string_of_term ctxt t ^ " : " ^ string_for_mtype (resolve_mtype asgs M)
  1117 
  1118 fun print_mcontext ctxt asgs ({frees, consts, ...} : mcontext) =
  1119   trace_msg (fn () =>
  1120       map (fn (x, M) => string_for_mtype_of_term ctxt asgs (Free x) M) frees @
  1121       map (fn (x, M) => string_for_mtype_of_term ctxt asgs (Const x) M) consts
  1122       |> cat_lines)
  1123 
  1124 fun formulas_monotonic (hol_ctxt as {ctxt, tac_timeout, ...}) binarize alpha_T
  1125                        (nondef_ts, def_ts) =
  1126   let
  1127     val _ = trace_msg (fn () => "****** Monotonicity analysis: " ^
  1128                                 string_for_mtype MAlpha ^ " is " ^
  1129                                 Syntax.string_of_typ ctxt alpha_T)
  1130     val mdata as {max_fresh, ...} = initial_mdata hol_ctxt binarize alpha_T
  1131     val (gamma, cset) =
  1132       (initial_gamma, ([], []))
  1133       |> consider_general_formula mdata Plus (hd nondef_ts)
  1134       |> fold (consider_nondefinitional_axiom mdata) (tl nondef_ts)
  1135       |> fold (consider_definitional_axiom mdata) def_ts
  1136   in
  1137     case solve tac_timeout (!max_fresh) cset of
  1138       SOME asgs => (print_mcontext ctxt asgs gamma; true)
  1139     | _ => false
  1140   end
  1141   handle UNSOLVABLE () => false
  1142        | MTYPE (loc, Ms, Ts) =>
  1143          raise BAD (loc, commas (map string_for_mtype Ms @
  1144                                  map (Syntax.string_of_typ ctxt) Ts))
  1145 
  1146 end;