src/HOL/Tools/Nitpick/nitpick_mono.ML
author blanchet
Mon Dec 06 13:29:23 2010 +0100 (2010-12-06)
changeset 40996 63112be4a469
parent 40995 3cae30b60577
child 40997 67e11a73532a
permissions -rw-r--r--
added "Neq" operator to monotonicity inference module
     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 -> int -> typ -> term list * term list -> bool
    15   val finitize_funs :
    16     hol_context -> bool -> (typ option * bool option) list -> int -> typ
    17     -> term list * term list -> term list * term list
    18 end;
    19 
    20 structure Nitpick_Mono : NITPICK_MONO =
    21 struct
    22 
    23 open Nitpick_Util
    24 open Nitpick_HOL
    25 
    26 structure PL = PropLogic
    27 
    28 datatype sign = Plus | Minus
    29 
    30 type var = int
    31 
    32 datatype annotation = Gen | New | Fls | Tru
    33 datatype annotation_atom = A of annotation | V of var
    34 
    35 type assign_literal = var * (sign * annotation)
    36 
    37 datatype mtyp =
    38   MAlpha |
    39   MFun of mtyp * annotation_atom * mtyp |
    40   MPair of mtyp * mtyp |
    41   MType of string * mtyp list |
    42   MRec of string * typ list
    43 
    44 datatype mterm =
    45   MRaw of term * mtyp |
    46   MAbs of string * typ * mtyp * annotation_atom * mterm |
    47   MApp of mterm * mterm
    48 
    49 type mdata =
    50   {hol_ctxt: hol_context,
    51    binarize: bool,
    52    alpha_T: typ,
    53    no_harmless: bool,
    54    max_fresh: int Unsynchronized.ref,
    55    datatype_mcache: ((string * typ list) * mtyp) list Unsynchronized.ref,
    56    constr_mcache: (styp * mtyp) list Unsynchronized.ref}
    57 
    58 exception UNSOLVABLE of unit
    59 exception MTYPE of string * mtyp list * typ list
    60 exception MTERM of string * mterm list
    61 
    62 val trace = Unsynchronized.ref false
    63 fun trace_msg msg = if !trace then tracing (msg ()) else ()
    64 
    65 fun string_for_sign Plus = "+"
    66   | string_for_sign Minus = "-"
    67 
    68 fun negate_sign Plus = Minus
    69   | negate_sign Minus = Plus
    70 
    71 val string_for_var = signed_string_of_int
    72 fun string_for_vars sep [] = "0\<^bsub>" ^ sep ^ "\<^esub>"
    73   | string_for_vars sep xs = space_implode sep (map string_for_var xs)
    74 fun subscript_string_for_vars sep xs =
    75   if null xs then "" else "\<^bsub>" ^ string_for_vars sep xs ^ "\<^esub>"
    76 
    77 fun string_for_annotation Gen = "G"
    78   | string_for_annotation New = "N"
    79   | string_for_annotation Fls = "F"
    80   | string_for_annotation Tru = "T"
    81 
    82 fun string_for_annotation_atom (A a) = string_for_annotation a
    83   | string_for_annotation_atom (V x) = string_for_var x
    84 
    85 fun string_for_assign_literal (x, (sn, a)) =
    86   string_for_var x ^ (case sn of Plus => " = " | Minus => " \<noteq> ") ^
    87   string_for_annotation a
    88 
    89 val bool_M = MType (@{type_name bool}, [])
    90 val dummy_M = MType (nitpick_prefix ^ "dummy", [])
    91 
    92 fun is_MRec (MRec _) = true
    93   | is_MRec _ = false
    94 fun dest_MFun (MFun z) = z
    95   | dest_MFun M = raise MTYPE ("Nitpick_Mono.dest_MFun", [M], [])
    96 
    97 val no_prec = 100
    98 
    99 fun precedence_of_mtype (MFun _) = 1
   100   | precedence_of_mtype (MPair _) = 2
   101   | precedence_of_mtype _ = no_prec
   102 
   103 val string_for_mtype =
   104   let
   105     fun aux outer_prec M =
   106       let
   107         val prec = precedence_of_mtype M
   108         val need_parens = (prec < outer_prec)
   109       in
   110         (if need_parens then "(" else "") ^
   111         (if M = dummy_M then
   112            "_"
   113          else case M of
   114              MAlpha => "\<alpha>"
   115            | MFun (M1, aa, M2) =>
   116              aux (prec + 1) M1 ^ " \<Rightarrow>\<^bsup>" ^
   117              string_for_annotation_atom aa ^ "\<^esup> " ^ aux prec M2
   118            | MPair (M1, M2) => aux (prec + 1) M1 ^ " \<times> " ^ aux prec M2
   119            | MType (s, []) =>
   120              if s = @{type_name prop} orelse s = @{type_name bool} then "o"
   121              else s
   122            | MType (s, Ms) => "(" ^ commas (map (aux 0) Ms) ^ ") " ^ s
   123            | MRec (s, _) => "[" ^ s ^ "]") ^
   124         (if need_parens then ")" else "")
   125       end
   126   in aux 0 end
   127 
   128 fun flatten_mtype (MPair (M1, M2)) = maps flatten_mtype [M1, M2]
   129   | flatten_mtype (MType (_, Ms)) = maps flatten_mtype Ms
   130   | flatten_mtype M = [M]
   131 
   132 fun precedence_of_mterm (MRaw _) = no_prec
   133   | precedence_of_mterm (MAbs _) = 1
   134   | precedence_of_mterm (MApp _) = 2
   135 
   136 fun string_for_mterm ctxt =
   137   let
   138     fun mtype_annotation M = "\<^bsup>" ^ string_for_mtype M ^ "\<^esup>"
   139     fun aux outer_prec m =
   140       let
   141         val prec = precedence_of_mterm m
   142         val need_parens = (prec < outer_prec)
   143       in
   144         (if need_parens then "(" else "") ^
   145         (case m of
   146            MRaw (t, M) => Syntax.string_of_term ctxt t ^ mtype_annotation M
   147          | MAbs (s, _, M, aa, m) =>
   148            "\<lambda>" ^ s ^ mtype_annotation M ^ ".\<^bsup>" ^
   149            string_for_annotation_atom aa ^ "\<^esup> " ^ aux prec m
   150          | MApp (m1, m2) => aux prec m1 ^ " " ^ aux (prec + 1) m2) ^
   151         (if need_parens then ")" else "")
   152       end
   153   in aux 0 end
   154 
   155 fun mtype_of_mterm (MRaw (_, M)) = M
   156   | mtype_of_mterm (MAbs (_, _, M, aa, m)) = MFun (M, aa, mtype_of_mterm m)
   157   | mtype_of_mterm (MApp (m1, _)) =
   158     case mtype_of_mterm m1 of
   159       MFun (_, _, M12) => M12
   160     | M1 => raise MTYPE ("Nitpick_Mono.mtype_of_mterm", [M1], [])
   161 
   162 fun strip_mcomb (MApp (m1, m2)) = strip_mcomb m1 ||> (fn ms => append ms [m2])
   163   | strip_mcomb m = (m, [])
   164 
   165 fun initial_mdata hol_ctxt binarize no_harmless alpha_T =
   166   ({hol_ctxt = hol_ctxt, binarize = binarize, alpha_T = alpha_T,
   167     no_harmless = no_harmless, max_fresh = Unsynchronized.ref 0,
   168     datatype_mcache = Unsynchronized.ref [],
   169     constr_mcache = Unsynchronized.ref []} : mdata)
   170 
   171 fun could_exist_alpha_subtype alpha_T (T as Type (_, Ts)) =
   172     T = alpha_T orelse (not (is_fp_iterator_type T) andalso
   173                         exists (could_exist_alpha_subtype alpha_T) Ts)
   174   | could_exist_alpha_subtype alpha_T T = (T = alpha_T)
   175 fun could_exist_alpha_sub_mtype _ (alpha_T as TFree _) T =
   176     could_exist_alpha_subtype alpha_T T
   177   | could_exist_alpha_sub_mtype ctxt alpha_T T =
   178     (T = alpha_T orelse is_datatype ctxt [(NONE, true)] T)
   179 
   180 fun exists_alpha_sub_mtype MAlpha = true
   181   | exists_alpha_sub_mtype (MFun (M1, _, M2)) =
   182     exists exists_alpha_sub_mtype [M1, M2]
   183   | exists_alpha_sub_mtype (MPair (M1, M2)) =
   184     exists exists_alpha_sub_mtype [M1, M2]
   185   | exists_alpha_sub_mtype (MType (_, Ms)) = exists exists_alpha_sub_mtype Ms
   186   | exists_alpha_sub_mtype (MRec _) = true
   187 
   188 fun exists_alpha_sub_mtype_fresh MAlpha = true
   189   | exists_alpha_sub_mtype_fresh (MFun (_, V _, _)) = true
   190   | exists_alpha_sub_mtype_fresh (MFun (_, _, M2)) =
   191     exists_alpha_sub_mtype_fresh M2
   192   | exists_alpha_sub_mtype_fresh (MPair (M1, M2)) =
   193     exists exists_alpha_sub_mtype_fresh [M1, M2]
   194   | exists_alpha_sub_mtype_fresh (MType (_, Ms)) =
   195     exists exists_alpha_sub_mtype_fresh Ms
   196   | exists_alpha_sub_mtype_fresh (MRec _) = true
   197 
   198 fun constr_mtype_for_binders z Ms =
   199   fold_rev (fn M => curry3 MFun M (A Gen)) Ms (MRec z)
   200 
   201 fun repair_mtype _ _ MAlpha = MAlpha
   202   | repair_mtype cache seen (MFun (M1, aa, M2)) =
   203     MFun (repair_mtype cache seen M1, aa, repair_mtype cache seen M2)
   204   | repair_mtype cache seen (MPair Mp) =
   205     MPair (pairself (repair_mtype cache seen) Mp)
   206   | repair_mtype cache seen (MType (s, Ms)) =
   207     MType (s, maps (flatten_mtype o repair_mtype cache seen) Ms)
   208   | repair_mtype cache seen (MRec (z as (s, _))) =
   209     case AList.lookup (op =) cache z |> the of
   210       MRec _ => MType (s, [])
   211     | M => if member (op =) seen M then MType (s, [])
   212            else repair_mtype cache (M :: seen) M
   213 
   214 fun repair_datatype_mcache cache =
   215   let
   216     fun repair_one (z, M) =
   217       Unsynchronized.change cache
   218           (AList.update (op =) (z, repair_mtype (!cache) [] M))
   219   in List.app repair_one (rev (!cache)) end
   220 
   221 fun repair_constr_mcache dtype_cache constr_mcache =
   222   let
   223     fun repair_one (x, M) =
   224       Unsynchronized.change constr_mcache
   225           (AList.update (op =) (x, repair_mtype dtype_cache [] M))
   226   in List.app repair_one (!constr_mcache) end
   227 
   228 fun is_fin_fun_supported_type @{typ prop} = true
   229   | is_fin_fun_supported_type @{typ bool} = true
   230   | is_fin_fun_supported_type (Type (@{type_name option}, _)) = true
   231   | is_fin_fun_supported_type _ = false
   232 fun fin_fun_body _ _ (t as @{term False}) = SOME t
   233   | fin_fun_body _ _ (t as Const (@{const_name None}, _)) = SOME t
   234   | fin_fun_body dom_T ran_T
   235                  ((t0 as Const (@{const_name If}, _))
   236                   $ (t1 as Const (@{const_name HOL.eq}, _) $ Bound 0 $ t1')
   237                   $ t2 $ t3) =
   238     (if loose_bvar1 (t1', 0) then
   239        NONE
   240      else case fin_fun_body dom_T ran_T t3 of
   241        NONE => NONE
   242      | SOME t3 =>
   243        SOME (t0 $ (Const (@{const_name is_unknown}, dom_T --> bool_T) $ t1')
   244                 $ (Const (@{const_name unknown}, ran_T)) $ (t0 $ t1 $ t2 $ t3)))
   245   | fin_fun_body _ _ _ = NONE
   246 
   247 fun fresh_mfun_for_fun_type (mdata as {max_fresh, ...} : mdata) all_minus
   248                             T1 T2 =
   249   let
   250     val M1 = fresh_mtype_for_type mdata all_minus T1
   251     val M2 = fresh_mtype_for_type mdata all_minus T2
   252     val aa = if not all_minus andalso exists_alpha_sub_mtype_fresh M1 andalso
   253                 is_fin_fun_supported_type (body_type T2) then
   254                V (Unsynchronized.inc max_fresh)
   255              else
   256                A Gen
   257   in (M1, aa, M2) end
   258 and fresh_mtype_for_type (mdata as {hol_ctxt as {ctxt, ...}, binarize, alpha_T,
   259                                     datatype_mcache, constr_mcache, ...})
   260                          all_minus =
   261   let
   262     fun do_type T =
   263       if T = alpha_T then
   264         MAlpha
   265       else case T of
   266         Type (@{type_name fun}, [T1, T2]) =>
   267         MFun (fresh_mfun_for_fun_type mdata all_minus T1 T2)
   268       | Type (@{type_name prod}, [T1, T2]) => MPair (pairself do_type (T1, T2))
   269       | Type (z as (s, _)) =>
   270         if could_exist_alpha_sub_mtype ctxt alpha_T T then
   271           case AList.lookup (op =) (!datatype_mcache) z of
   272             SOME M => M
   273           | NONE =>
   274             let
   275               val _ = Unsynchronized.change datatype_mcache (cons (z, MRec z))
   276               val xs = binarized_and_boxed_datatype_constrs hol_ctxt binarize T
   277               val (all_Ms, constr_Ms) =
   278                 fold_rev (fn (_, T') => fn (all_Ms, constr_Ms) =>
   279                              let
   280                                val binder_Ms = map do_type (binder_types T')
   281                                val new_Ms = filter exists_alpha_sub_mtype_fresh
   282                                                    binder_Ms
   283                                val constr_M = constr_mtype_for_binders z
   284                                                                        binder_Ms
   285                              in
   286                                (union (op =) new_Ms all_Ms,
   287                                 constr_M :: constr_Ms)
   288                              end)
   289                          xs ([], [])
   290               val M = MType (s, all_Ms)
   291               val _ = Unsynchronized.change datatype_mcache
   292                           (AList.update (op =) (z, M))
   293               val _ = Unsynchronized.change constr_mcache
   294                           (append (xs ~~ constr_Ms))
   295             in
   296               if forall (not o is_MRec o snd) (!datatype_mcache) then
   297                 (repair_datatype_mcache datatype_mcache;
   298                  repair_constr_mcache (!datatype_mcache) constr_mcache;
   299                  AList.lookup (op =) (!datatype_mcache) z |> the)
   300               else
   301                 M
   302             end
   303         else
   304           MType (s, [])
   305       | _ => MType (simple_string_of_typ T, [])
   306   in do_type end
   307 
   308 fun prodM_factors (MPair (M1, M2)) = maps prodM_factors [M1, M2]
   309   | prodM_factors M = [M]
   310 fun curried_strip_mtype (MFun (M1, _, M2)) =
   311     curried_strip_mtype M2 |>> append (prodM_factors M1)
   312   | curried_strip_mtype M = ([], M)
   313 fun sel_mtype_from_constr_mtype s M =
   314   let val (arg_Ms, dataM) = curried_strip_mtype M in
   315     MFun (dataM, A Gen,
   316           case sel_no_from_name s of ~1 => bool_M | n => nth arg_Ms n)
   317   end
   318 
   319 fun mtype_for_constr (mdata as {hol_ctxt = {ctxt, ...}, alpha_T, constr_mcache,
   320                                 ...}) (x as (_, T)) =
   321   if could_exist_alpha_sub_mtype ctxt alpha_T T then
   322     case AList.lookup (op =) (!constr_mcache) x of
   323       SOME M => M
   324     | NONE => if T = alpha_T then
   325                 let val M = fresh_mtype_for_type mdata false T in
   326                   (Unsynchronized.change constr_mcache (cons (x, M)); M)
   327                 end
   328               else
   329                 (fresh_mtype_for_type mdata false (body_type T);
   330                  AList.lookup (op =) (!constr_mcache) x |> the)
   331   else
   332     fresh_mtype_for_type mdata false T
   333 fun mtype_for_sel (mdata as {hol_ctxt, binarize, ...}) (x as (s, _)) =
   334   x |> binarized_and_boxed_constr_for_sel hol_ctxt binarize
   335     |> mtype_for_constr mdata |> sel_mtype_from_constr_mtype s
   336 
   337 fun resolve_annotation_atom asgs (V x) =
   338     x |> AList.lookup (op =) asgs |> Option.map A |> the_default (V x)
   339   | resolve_annotation_atom _ aa = aa
   340 fun resolve_mtype asgs =
   341   let
   342     fun aux MAlpha = MAlpha
   343       | aux (MFun (M1, aa, M2)) =
   344         MFun (aux M1, resolve_annotation_atom asgs aa, aux M2)
   345       | aux (MPair Mp) = MPair (pairself aux Mp)
   346       | aux (MType (s, Ms)) = MType (s, map aux Ms)
   347       | aux (MRec z) = MRec z
   348   in aux end
   349 
   350 datatype comp_op = Eq | Neq | Leq
   351 
   352 type comp = annotation_atom * annotation_atom * comp_op * var list
   353 type assign_clause = assign_literal list
   354 
   355 type constraint_set = comp list * assign_clause list
   356 
   357 fun string_for_comp_op Eq = "="
   358   | string_for_comp_op Neq = "\<noteq>"
   359   | string_for_comp_op Leq = "\<le>"
   360 
   361 fun string_for_comp (aa1, aa2, cmp, xs) =
   362   string_for_annotation_atom aa1 ^ " " ^ string_for_comp_op cmp ^
   363   subscript_string_for_vars " \<and> " xs ^ " " ^ string_for_annotation_atom aa2
   364 
   365 fun string_for_assign_clause [] = "\<bot>"
   366   | string_for_assign_clause asgs =
   367     space_implode " \<or> " (map string_for_assign_literal asgs)
   368 
   369 fun add_assign_literal (x, (sn, a)) clauses =
   370   if exists (fn [(x', (sn', a'))] =>
   371                 x = x' andalso ((sn = sn' andalso a <> a') orelse
   372                                 (sn <> sn' andalso a = a'))
   373               | _ => false) clauses then
   374     NONE
   375   else
   376     SOME ([(x, a)] :: clauses)
   377 
   378 fun add_assign_disjunct _ NONE = NONE
   379   | add_assign_disjunct asg (SOME asgs) = SOME (insert (op =) asg asgs)
   380 
   381 fun annotation_comp Eq a1 a2 = (a1 = a2)
   382   | annotation_comp Neq a1 a2 = (a1 <> a2)
   383   | annotation_comp Leq a1 a2 = (a1 = a2 orelse a2 = Gen)
   384 
   385 fun comp_op_sign Eq = Plus
   386   | comp_op_sign Neq = Minus
   387   | comp_op_sign Leq =
   388     raise BAD ("Nitpick_Mono.comp_op_sign", "unexpected \"Leq\"")
   389 
   390 fun do_annotation_atom_comp Leq [] aa1 aa2 (cset as (comps, clauses)) =
   391     (case (aa1, aa2) of
   392        (A a1, A a2) => if annotation_comp Leq a1 a2 then SOME cset else NONE
   393      | _ => SOME (insert (op =) (aa1, aa2, Leq, []) comps, clauses))
   394   | do_annotation_atom_comp cmp [] aa1 aa2 (cset as (comps, clauses)) =
   395     (case (aa1, aa2) of
   396        (A a1, A a2) => if annotation_comp cmp a1 a2 then SOME cset else NONE
   397      | (V x1, A a2) =>
   398        clauses |> add_assign_literal (x1, (comp_op_sign cmp, a2))
   399                |> Option.map (pair comps)
   400      | (A _, V _) => do_annotation_atom_comp cmp [] aa2 aa1 cset
   401      | (V _, V _) => SOME (insert (op =) (aa1, aa2, cmp, []) comps, clauses))
   402   | do_annotation_atom_comp cmp xs aa1 aa2 (comps, clauses) =
   403     SOME (insert (op =) (aa1, aa2, cmp, xs) comps, clauses)
   404 
   405 fun add_annotation_atom_comp cmp xs aa1 aa2
   406                              ((comps, clauses) : constraint_set) =
   407   (trace_msg (fn () => "*** Add " ^ string_for_annotation_atom aa1 ^ " " ^
   408                        string_for_comp_op cmp ^ " " ^
   409                        string_for_annotation_atom aa2);
   410    case do_annotation_atom_comp cmp xs aa1 aa2 (comps, clauses) of
   411      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   412    | SOME cset => cset)
   413 
   414 fun do_mtype_comp _ _ _ _ NONE = NONE
   415   | do_mtype_comp _ _ MAlpha MAlpha cset = cset
   416   | do_mtype_comp Eq xs (MFun (M11, aa1, M12)) (MFun (M21, aa2, M22))
   417                   (SOME cset) =
   418     cset |> do_annotation_atom_comp Eq xs aa1 aa2
   419          |> do_mtype_comp Eq xs M11 M21 |> do_mtype_comp Eq xs M12 M22
   420   | do_mtype_comp Leq xs (MFun (M11, aa1, M12)) (MFun (M21, aa2, M22))
   421                   (SOME cset) =
   422     (if exists_alpha_sub_mtype M11 then
   423        cset |> do_annotation_atom_comp Leq xs aa1 aa2
   424             |> do_mtype_comp Leq xs M21 M11
   425             |> (case aa2 of
   426                   A Gen => I
   427                 | A _ => do_mtype_comp Leq xs M11 M21
   428                 | V x => do_mtype_comp Leq (x :: xs) M11 M21)
   429      else
   430        SOME cset)
   431     |> do_mtype_comp Leq xs M12 M22
   432   | do_mtype_comp cmp xs (M1 as MPair (M11, M12)) (M2 as MPair (M21, M22))
   433                   cset =
   434     (cset |> fold (uncurry (do_mtype_comp cmp xs)) [(M11, M21), (M12, M22)]
   435      handle Library.UnequalLengths =>
   436             raise MTYPE ("Nitpick_Mono.do_mtype_comp", [M1, M2], []))
   437   | do_mtype_comp _ _ (MType _) (MType _) cset =
   438     cset (* no need to compare them thanks to the cache *)
   439   | do_mtype_comp cmp _ M1 M2 _ =
   440     raise MTYPE ("Nitpick_Mono.do_mtype_comp (" ^ string_for_comp_op cmp ^ ")",
   441                  [M1, M2], [])
   442 
   443 fun add_mtype_comp cmp M1 M2 cset =
   444   (trace_msg (fn () => "*** Add " ^ string_for_mtype M1 ^ " " ^
   445                        string_for_comp_op cmp ^ " " ^ string_for_mtype M2);
   446    case SOME cset |> do_mtype_comp cmp [] M1 M2 of
   447      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   448    | SOME cset => cset)
   449 
   450 val add_mtypes_equal = add_mtype_comp Eq
   451 val add_is_sub_mtype = add_mtype_comp Leq
   452 
   453 fun do_notin_mtype_fv _ _ _ NONE = NONE
   454   | do_notin_mtype_fv Minus _ MAlpha cset = cset
   455   | do_notin_mtype_fv Plus [] MAlpha _ = NONE
   456   | do_notin_mtype_fv Plus [asg] MAlpha (SOME clauses) =
   457     clauses |> add_assign_literal asg
   458   | do_notin_mtype_fv Plus clause MAlpha (SOME clauses) =
   459     SOME (insert (op =) clause clauses)
   460   | do_notin_mtype_fv sn clause (MFun (M1, A a, M2)) cset =
   461     cset |> (if a <> Gen andalso sn = Plus then do_notin_mtype_fv Plus clause M1
   462              else I)
   463          |> (if a = Gen orelse sn = Plus then do_notin_mtype_fv Minus clause M1
   464              else I)
   465          |> do_notin_mtype_fv sn clause M2
   466   | do_notin_mtype_fv Plus clause (MFun (M1, V x, M2)) cset =
   467     cset |> (case add_assign_disjunct (x, (Plus, Gen)) (SOME clause) of
   468                NONE => I
   469              | SOME clause' => do_notin_mtype_fv Plus clause' M1)
   470          |> do_notin_mtype_fv Minus clause M1
   471          |> do_notin_mtype_fv Plus clause M2
   472   | do_notin_mtype_fv Minus clause (MFun (M1, V x, M2)) cset =
   473     cset |> (case fold (fn a => add_assign_disjunct (x, (Plus, a))) [Fls, Tru]
   474                        (SOME clause) of
   475                NONE => I
   476              | SOME clause' => do_notin_mtype_fv Plus clause' M1)
   477          |> do_notin_mtype_fv Minus clause M2
   478   | do_notin_mtype_fv sn clause (MPair (M1, M2)) cset =
   479     cset |> fold (do_notin_mtype_fv sn clause) [M1, M2]
   480   | do_notin_mtype_fv sn clause (MType (_, Ms)) cset =
   481     cset |> fold (do_notin_mtype_fv sn clause) Ms
   482  | do_notin_mtype_fv _ _ M _ =
   483    raise MTYPE ("Nitpick_Mono.do_notin_mtype_fv", [M], [])
   484 
   485 fun add_notin_mtype_fv sn M ((comps, clauses) : constraint_set) =
   486   (trace_msg (fn () => "*** Add " ^ string_for_mtype M ^ " is " ^
   487                        (case sn of Minus => "concrete" | Plus => "complete"));
   488    case SOME clauses |> do_notin_mtype_fv sn [] M of
   489      NONE => (trace_msg (K "**** Unsolvable"); raise UNSOLVABLE ())
   490    | SOME clauses => (comps, clauses))
   491 
   492 val add_mtype_is_concrete = add_notin_mtype_fv Minus
   493 val add_mtype_is_complete = add_notin_mtype_fv Plus
   494 
   495 val bool_table =
   496   [(Gen, (false, false)),
   497    (New, (false, true)),
   498    (Fls, (true, false)),
   499    (Tru, (true, true))]
   500 
   501 fun fst_var n = 2 * n
   502 fun snd_var n = 2 * n + 1
   503 
   504 val bools_from_annotation = AList.lookup (op =) bool_table #> the
   505 val annotation_from_bools = AList.find (op =) bool_table #> the_single
   506 
   507 fun prop_for_bool b = if b then PL.True else PL.False
   508 fun prop_for_bool_var_equality (v1, v2) =
   509   PL.And (PL.Or (PL.BoolVar v1, PL.Not (PL.BoolVar v2)),
   510           PL.Or (PL.Not (PL.BoolVar v1), PL.BoolVar v2))
   511 fun prop_for_assign (x, a) =
   512   let val (b1, b2) = bools_from_annotation a in
   513     PL.And (PL.BoolVar (fst_var x) |> not b1 ? PL.Not,
   514             PL.BoolVar (snd_var x) |> not b2 ? PL.Not)
   515   end
   516 fun prop_for_assign_literal (x, (Plus, a)) = prop_for_assign (x, a)
   517   | prop_for_assign_literal (x, (Minus, a)) = PL.Not (prop_for_assign (x, a))
   518 fun prop_for_atom_assign (A a', a) = prop_for_bool (a = a')
   519   | prop_for_atom_assign (V x, a) = prop_for_assign_literal (x, (Plus, a))
   520 fun prop_for_atom_equality (aa1, A a2) = prop_for_atom_assign (aa1, a2)
   521   | prop_for_atom_equality (A a1, aa2) = prop_for_atom_assign (aa2, a1)
   522   | prop_for_atom_equality (V x1, V x2) =
   523     PL.And (prop_for_bool_var_equality (pairself fst_var (x1, x2)),
   524             prop_for_bool_var_equality (pairself snd_var (x1, x2)))
   525 val prop_for_assign_clause = PL.exists o map prop_for_assign_literal
   526 fun prop_for_exists_var_assign_literal xs a =
   527   PL.exists (map (fn x => prop_for_assign_literal (x, (Plus, a))) xs)
   528 fun prop_for_comp (aa1, aa2, Eq, []) =
   529     PL.SAnd (prop_for_comp (aa1, aa2, Leq, []),
   530              prop_for_comp (aa2, aa1, Leq, []))
   531   | prop_for_comp (aa1, aa2, Neq, []) =
   532     PL.Not (prop_for_comp (aa1, aa2, Eq, []))
   533   | prop_for_comp (aa1, aa2, Leq, []) =
   534     PL.SOr (prop_for_atom_equality (aa1, aa2), prop_for_atom_assign (aa2, Gen))
   535   | prop_for_comp (aa1, aa2, cmp, xs) =
   536     PL.SOr (prop_for_exists_var_assign_literal xs Gen,
   537             prop_for_comp (aa1, aa2, cmp, []))
   538 
   539 (* The "calculus" parameter may be 1, 2, or 3, corresponding approximately to
   540    the M1, M2, and M3 calculi from Blanchette & Krauss 2011. *)
   541 fun variable_domain calculus =
   542   [Gen] @ (if calculus > 1 then [Fls] else [])
   543   @ (if calculus > 2 then [New, Tru] else [])
   544 
   545 fun prop_for_variable_domain calculus x =
   546   PL.exists (map (fn a => prop_for_assign_literal (x, (Plus, a)))
   547                  (variable_domain calculus))
   548 
   549 fun extract_assigns max_var assigns asgs =
   550   fold (fn x => fn accum =>
   551            if AList.defined (op =) asgs x then
   552              accum
   553            else case (fst_var x, snd_var x) |> pairself assigns of
   554              (NONE, NONE) => accum
   555            | bp => (x, annotation_from_bools (pairself (the_default false) bp))
   556                    :: accum)
   557        (max_var downto 1) asgs
   558 
   559 fun print_problem calculus comps clauses =
   560   trace_msg (fn () =>
   561                 "*** Problem (calculus M" ^ string_of_int calculus ^ "):\n" ^
   562                 cat_lines (map string_for_comp comps @
   563                            map string_for_assign_clause clauses))
   564 
   565 fun print_solution asgs =
   566   trace_msg (fn () => "*** Solution:\n" ^
   567       (asgs
   568        |> map swap
   569        |> AList.group (op =)
   570        |> map (fn (a, xs) => string_for_annotation a ^ ": " ^
   571                              string_for_vars ", " xs)
   572        |> space_implode "\n"))
   573 
   574 fun solve calculus max_var (comps, clauses) =
   575   let
   576     val asgs = map_filter (fn [(x, (_, a))] => SOME (x, a) | _ => NONE) clauses
   577     fun do_assigns assigns =
   578       SOME (extract_assigns max_var assigns asgs |> tap print_solution)
   579     val _ = print_problem calculus comps clauses
   580     val prop =
   581       map prop_for_comp comps @
   582       map prop_for_assign_clause clauses @
   583       (if calculus < 3 then
   584          map (prop_for_variable_domain calculus) (1 upto max_var)
   585        else
   586          [])
   587       |> PL.all
   588   in
   589     if PL.eval (K false) prop then
   590       do_assigns (K (SOME false))
   591     else if PL.eval (K true) prop then
   592       do_assigns (K (SOME true))
   593     else
   594       let
   595         (* use the first ML solver (to avoid startup overhead) *)
   596         val solvers = !SatSolver.solvers
   597                       |> filter (member (op =) ["dptsat", "dpll"] o fst)
   598       in
   599         case snd (hd solvers) prop of
   600           SatSolver.SATISFIABLE assigns => do_assigns assigns
   601         | _ => (trace_msg (K "*** Unsolvable"); NONE)
   602       end
   603   end
   604 
   605 type mtype_schema = mtyp * constraint_set
   606 type mtype_context =
   607   {bound_Ts: typ list,
   608    bound_Ms: mtyp list,
   609    bound_frame: (int * annotation_atom) list,
   610    frees: (styp * mtyp) list,
   611    consts: (styp * mtyp) list}
   612 
   613 val string_for_frame =
   614   map (fn (j, aa) => "#" ^ string_of_int j ^ " |-> " ^
   615                      string_for_annotation_atom aa)
   616   #> commas #> enclose "[" "]"
   617 
   618 type accumulator = mtype_context * constraint_set
   619 
   620 val initial_gamma =
   621   {bound_Ts = [], bound_Ms = [], bound_frame = [], frees = [], consts = []}
   622 
   623 fun push_bound aa T M {bound_Ts, bound_Ms, bound_frame, frees, consts} =
   624   {bound_Ts = T :: bound_Ts, bound_Ms = M :: bound_Ms,
   625    bound_frame = (length bound_Ts, aa) :: bound_frame, frees = frees,
   626    consts = consts}
   627 fun pop_bound {bound_Ts, bound_Ms, bound_frame, frees, consts} =
   628   {bound_Ts = tl bound_Ts, bound_Ms = tl bound_Ms,
   629    bound_frame = bound_frame
   630                  |> filter_out (fn (j, _) => j = length bound_Ts - 1),
   631    frees = frees, consts = consts}
   632   handle List.Empty => initial_gamma (* FIXME: needed? *)
   633 
   634 fun set_frame bound_frame ({bound_Ts, bound_Ms, frees, consts, ...}
   635                            : mtype_context) =
   636   {bound_Ts = bound_Ts, bound_Ms = bound_Ms, bound_frame = bound_frame,
   637    frees = frees, consts = consts}
   638 
   639 (* FIXME: make sure tracing messages are complete *)
   640 
   641 fun add_comp_frame a cmp frame =
   642   (trace_msg (fn () => "*** Add " ^ string_for_annotation a ^ " " ^
   643                        string_for_comp_op cmp ^ " " ^
   644                        string_for_frame frame);
   645    fold (add_annotation_atom_comp cmp [] (A a) o snd) frame)
   646 
   647 fun add_bound_frame j frame =
   648   let
   649     val (new_frame, gen_frame) = List.partition (curry (op =) j o fst) frame
   650   in
   651     add_comp_frame New Leq new_frame
   652     #> add_comp_frame Gen Eq gen_frame
   653   end
   654 
   655 fun fresh_imp_frame ({max_fresh, ...} : mdata) sn =
   656   let
   657     fun do_var (j, A Fls) = (j, A (case sn of Plus => Fls | Minus => Tru))
   658       | do_var (j, A Gen) = (j, A Gen)
   659       | do_var (j, _) = (j, V (Unsynchronized.inc max_fresh))
   660   in map do_var end
   661 
   662 fun do_not_var j aa0 aa1 _ = I
   663 (*
   664 x1 ~= T | x0 = F
   665 x1 ~= F | x0 = T
   666 x1 ~= G | x0 = G
   667 x1 ~= N | x0 = G
   668 *)
   669 
   670 fun do_conj_var j aa0 aa1 aa2 = I
   671 (*
   672   (x1 ≠ T | x2 ≠ T | x0 = T) &
   673   (x1 ≠ F | x0 = F) &
   674   (x2 ≠ F | x0 = F) &
   675   (x1 ≠ G | x2 = F | x0 = G) &
   676   (x1 ≠ N | x2 = F | x0 = G) &
   677   (x1 = F | x2 ≠ G | x0 = G) &
   678   (x1 = F | x2 ≠ N | x0 = G)"
   679 *)
   680 
   681 fun do_disj_var j aa0 aa1 aa2 = I
   682 fun do_imp_var j aa0 aa1 aa2 = I
   683 
   684 fun add_connective_frames do_var res_frame frame1 frame2 =
   685   fold I (map3 (fn (j, aa0) => fn (_, aa1) => fn (_, aa2) =>
   686                    do_var j aa0 aa1 aa2) res_frame frame1 frame2)
   687 
   688 fun consider_term (mdata as {hol_ctxt = {thy, ctxt, stds, ...}, alpha_T,
   689                              max_fresh, ...}) =
   690   let
   691     fun is_enough_eta_expanded t =
   692       case strip_comb t of
   693         (Const x, ts) => the_default 0 (arity_of_built_in_const thy stds x)
   694         <= length ts
   695       | _ => true
   696     val mtype_for = fresh_mtype_for_type mdata false
   697     fun plus_set_mtype_for_dom M =
   698       MFun (M, A (if exists_alpha_sub_mtype M then Fls else Gen), bool_M)
   699     fun do_all T (gamma, cset) =
   700       let
   701         val abs_M = mtype_for (domain_type (domain_type T))
   702         val body_M = mtype_for (body_type T)
   703       in
   704         (MFun (MFun (abs_M, A Gen, body_M), A Gen, body_M),
   705          (gamma, cset |> add_mtype_is_complete abs_M))
   706       end
   707     fun do_equals T (gamma, cset) =
   708       let val M = mtype_for (domain_type T) in
   709         (MFun (M, A Gen, MFun (M, V (Unsynchronized.inc max_fresh),
   710                                mtype_for (nth_range_type 2 T))),
   711          (gamma, cset |> add_mtype_is_concrete M))
   712       end
   713     fun do_robust_set_operation T (gamma, cset) =
   714       let
   715         val set_T = domain_type T
   716         val M1 = mtype_for set_T
   717         val M2 = mtype_for set_T
   718         val M3 = mtype_for set_T
   719       in
   720         (MFun (M1, A Gen, MFun (M2, A Gen, M3)),
   721          (gamma, cset |> add_is_sub_mtype M1 M3 |> add_is_sub_mtype M2 M3))
   722       end
   723     fun do_fragile_set_operation T (gamma, cset) =
   724       let
   725         val set_T = domain_type T
   726         val set_M = mtype_for set_T
   727         fun custom_mtype_for (T as Type (@{type_name fun}, [T1, T2])) =
   728             if T = set_T then set_M
   729             else MFun (custom_mtype_for T1, A Gen, custom_mtype_for T2)
   730           | custom_mtype_for T = mtype_for T
   731       in
   732         (custom_mtype_for T, (gamma, cset |> add_mtype_is_concrete set_M))
   733       end
   734     fun do_pair_constr T accum =
   735       case mtype_for (nth_range_type 2 T) of
   736         M as MPair (a_M, b_M) =>
   737         (MFun (a_M, A Gen, MFun (b_M, A Gen, M)), accum)
   738       | M => raise MTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [M], [])
   739     fun do_nth_pair_sel n T =
   740       case mtype_for (domain_type T) of
   741         M as MPair (a_M, b_M) =>
   742         pair (MFun (M, A Gen, if n = 0 then a_M else b_M))
   743       | M => raise MTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [M], [])
   744     fun do_bounded_quantifier t0 abs_s abs_T connective_t bound_t body_t accum =
   745       let
   746         val abs_M = mtype_for abs_T
   747         val aa = V (Unsynchronized.inc max_fresh)
   748         val (bound_m, accum) =
   749           accum |>> push_bound aa abs_T abs_M |> do_term bound_t
   750         val expected_bound_M = plus_set_mtype_for_dom abs_M
   751         val (body_m, accum) =
   752           accum ||> add_mtypes_equal expected_bound_M (mtype_of_mterm bound_m)
   753                 |> do_term body_t ||> apfst pop_bound
   754         val bound_M = mtype_of_mterm bound_m
   755         val (M1, aa', _) = dest_MFun bound_M
   756       in
   757         (MApp (MRaw (t0, MFun (bound_M, aa, bool_M)),
   758                MAbs (abs_s, abs_T, M1, aa',
   759                      MApp (MApp (MRaw (connective_t,
   760                                        mtype_for (fastype_of connective_t)),
   761                                  MApp (bound_m, MRaw (Bound 0, M1))),
   762                            body_m))), accum)
   763       end
   764     and do_connect do_var t0 t1 t2 (accum as ({bound_frame, ...}, _)) =
   765       let
   766         val frame1 = fresh_imp_frame mdata Minus bound_frame
   767         val frame2 = fresh_imp_frame mdata Plus bound_frame
   768         val (m1, accum) = accum |>> set_frame frame1 |> do_term t1
   769         val (m2, accum) = accum |>> set_frame frame2 |> do_term t2
   770       in
   771         (MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2),
   772          accum |>> set_frame bound_frame
   773                ||> add_connective_frames do_var bound_frame frame1 frame2)
   774       end
   775     and do_term t (accum as ({bound_Ts, bound_Ms, bound_frame, frees, consts},
   776                              cset)) =
   777       (trace_msg (fn () => "  \<Gamma> \<turnstile> " ^
   778                            Syntax.string_of_term ctxt t ^ " : _?");
   779        case t of
   780          @{const False} =>
   781          (MRaw (t, bool_M), accum ||> add_comp_frame Fls Leq bound_frame)
   782        | @{const True} =>
   783          (MRaw (t, bool_M), accum ||> add_comp_frame Tru Leq bound_frame)
   784        | Const (x as (s, T)) =>
   785          (case AList.lookup (op =) consts x of
   786             SOME M => (M, accum)
   787           | NONE =>
   788             if not (could_exist_alpha_subtype alpha_T T) then
   789               (mtype_for T, accum)
   790             else case s of
   791               @{const_name all} => do_all T accum
   792             | @{const_name "=="} => do_equals T accum
   793             | @{const_name All} => do_all T accum
   794             | @{const_name Ex} =>
   795               let val set_T = domain_type T in
   796                 do_term (Abs (Name.uu, set_T,
   797                               @{const Not} $ (HOLogic.mk_eq
   798                                   (Abs (Name.uu, domain_type set_T,
   799                                         @{const False}),
   800                                    Bound 0)))) accum
   801                 |>> mtype_of_mterm
   802               end
   803             | @{const_name HOL.eq} => do_equals T accum
   804             | @{const_name The} =>
   805               (trace_msg (K "*** The"); raise UNSOLVABLE ())
   806             | @{const_name Eps} =>
   807               (trace_msg (K "*** Eps"); raise UNSOLVABLE ())
   808             | @{const_name If} =>
   809               do_robust_set_operation (range_type T) accum
   810               |>> curry3 MFun bool_M (A Gen)
   811             | @{const_name Pair} => do_pair_constr T accum
   812             | @{const_name fst} => do_nth_pair_sel 0 T accum
   813             | @{const_name snd} => do_nth_pair_sel 1 T accum
   814             | @{const_name Id} =>
   815               (MFun (mtype_for (domain_type T), A Gen, bool_M), accum)
   816             | @{const_name converse} =>
   817               let
   818                 val x = Unsynchronized.inc max_fresh
   819                 fun mtype_for_set T =
   820                   MFun (mtype_for (domain_type T), V x, bool_M)
   821                 val ab_set_M = domain_type T |> mtype_for_set
   822                 val ba_set_M = range_type T |> mtype_for_set
   823               in (MFun (ab_set_M, A Gen, ba_set_M), accum) end
   824             | @{const_name trancl} => do_fragile_set_operation T accum
   825             | @{const_name rel_comp} =>
   826               let
   827                 val x = Unsynchronized.inc max_fresh
   828                 fun mtype_for_set T =
   829                   MFun (mtype_for (domain_type T), V x, bool_M)
   830                 val bc_set_M = domain_type T |> mtype_for_set
   831                 val ab_set_M = domain_type (range_type T) |> mtype_for_set
   832                 val ac_set_M = nth_range_type 2 T |> mtype_for_set
   833               in
   834                 (MFun (bc_set_M, A Gen, MFun (ab_set_M, A Gen, ac_set_M)),
   835                  accum)
   836               end
   837             | @{const_name image} =>
   838               let
   839                 val a_M = mtype_for (domain_type (domain_type T))
   840                 val b_M = mtype_for (range_type (domain_type T))
   841               in
   842                 (MFun (MFun (a_M, A Gen, b_M), A Gen,
   843                        MFun (plus_set_mtype_for_dom a_M, A Gen,
   844                              plus_set_mtype_for_dom b_M)), accum)
   845               end
   846             | @{const_name finite} =>
   847               let val M1 = mtype_for (domain_type (domain_type T)) in
   848                 (MFun (plus_set_mtype_for_dom M1, A Gen, bool_M), accum)
   849               end
   850             | @{const_name Sigma} =>
   851               let
   852                 val x = Unsynchronized.inc max_fresh
   853                 fun mtype_for_set T =
   854                   MFun (mtype_for (domain_type T), V x, bool_M)
   855                 val a_set_T = domain_type T
   856                 val a_M = mtype_for (domain_type a_set_T)
   857                 val b_set_M =
   858                   mtype_for_set (range_type (domain_type (range_type T)))
   859                 val a_set_M = mtype_for_set a_set_T
   860                 val a_to_b_set_M = MFun (a_M, A Gen, b_set_M)
   861                 val ab_set_M = mtype_for_set (nth_range_type 2 T)
   862               in
   863                 (MFun (a_set_M, A Gen, MFun (a_to_b_set_M, A Gen, ab_set_M)),
   864                  accum)
   865               end
   866             | _ =>
   867               if s = @{const_name safe_The} then
   868                 let
   869                   val a_set_M = mtype_for (domain_type T)
   870                   val a_M = dest_MFun a_set_M |> #1
   871                 in (MFun (a_set_M, A Gen, a_M), accum) end
   872               else if s = @{const_name ord_class.less_eq} andalso
   873                       is_set_type (domain_type T) then
   874                 do_fragile_set_operation T accum
   875               else if is_sel s then
   876                 (mtype_for_sel mdata x, accum)
   877               else if is_constr ctxt stds x then
   878                 (mtype_for_constr mdata x, accum)
   879               else if is_built_in_const thy stds x then
   880                 (fresh_mtype_for_type mdata true T, accum)
   881               else
   882                 let val M = mtype_for T in
   883                   (M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms,
   884                         bound_frame = bound_frame, frees = frees,
   885                         consts = (x, M) :: consts}, cset))
   886                 end)
   887            |>> curry MRaw t
   888            ||> apsnd (add_comp_frame Gen Eq bound_frame)
   889          | Free (x as (_, T)) =>
   890            (case AList.lookup (op =) frees x of
   891               SOME M => (M, accum)
   892             | NONE =>
   893               let val M = mtype_for T in
   894                 (M, ({bound_Ts = bound_Ts, bound_Ms = bound_Ms,
   895                       bound_frame = bound_frame, frees = (x, M) :: frees,
   896                       consts = consts}, cset))
   897               end)
   898            |>> curry MRaw t ||> apsnd (add_comp_frame Gen Eq bound_frame)
   899          | Var _ => (trace_msg (K "*** Var"); raise UNSOLVABLE ())
   900          | Bound j =>
   901            (MRaw (t, nth bound_Ms j),
   902             accum ||> add_bound_frame (length bound_Ts - j - 1) bound_frame)
   903          | Abs (s, T, t') =>
   904            (case fin_fun_body T (fastype_of1 (T :: bound_Ts, t')) t' of
   905               SOME t' =>
   906               let
   907                 val M = mtype_for T
   908                 val (m', accum) = do_term t' (accum |>> push_bound (A Fls) T M)
   909               in (MAbs (s, T, M, A Fls, m'), accum |>> pop_bound) end
   910             | NONE =>
   911               ((case t' of
   912                   t1' $ Bound 0 =>
   913                   if not (loose_bvar1 (t1', 0)) andalso
   914                      is_enough_eta_expanded t1' then
   915                     do_term (incr_boundvars ~1 t1') accum
   916                   else
   917                     raise SAME ()
   918                 | (t11 as Const (@{const_name HOL.eq}, _)) $ Bound 0 $ t13 =>
   919                   if not (loose_bvar1 (t13, 0)) then
   920                     do_term (incr_boundvars ~1 (t11 $ t13)) accum
   921                   else
   922                     raise SAME ()
   923                 | _ => raise SAME ())
   924                handle SAME () =>
   925                       let
   926                         val M = mtype_for T
   927                         val aa = V (Unsynchronized.inc max_fresh)
   928                         val (m', accum) =
   929                           do_term t' (accum |>> push_bound aa T M)
   930                       in (MAbs (s, T, M, aa, m'), accum |>> pop_bound) end))
   931          | (t0 as Const (@{const_name All}, _))
   932            $ Abs (s', T', (t10 as @{const HOL.implies})
   933                           $ (t11 $ Bound 0) $ t12) =>
   934            do_bounded_quantifier t0 s' T' t10 t11 t12 accum
   935          | (t0 as Const (@{const_name Ex}, _))
   936            $ Abs (s', T', (t10 as @{const HOL.conj})
   937                           $ (t11 $ Bound 0) $ t12) =>
   938            do_bounded_quantifier t0 s' T' t10 t11 t12 accum
   939          | (t0 as @{const Not}) $ t1 =>
   940            let
   941              val frame1 = fresh_imp_frame mdata Minus bound_frame
   942              val (m1, accum) = accum |>> set_frame frame1 |> do_term t1
   943            in
   944              (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1),
   945               accum |>> set_frame bound_frame
   946                     ||> add_connective_frames do_not_var bound_frame frame1
   947                                                          frame1)
   948            end
   949          | (t0 as @{const conj}) $ t1 $ t2 =>
   950            do_connect do_conj_var t0 t1 t2 accum
   951          | (t0 as @{const disj}) $ t1 $ t2 =>
   952            do_connect do_disj_var t0 t1 t2 accum
   953          | (t0 as @{const implies}) $ t1 $ t2 =>
   954            do_connect do_imp_var t0 t1 t2 accum
   955          | Const (@{const_name Let}, _) $ t1 $ t2 =>
   956            do_term (betapply (t2, t1)) accum
   957          | t1 $ t2 =>
   958            let
   959              val (m1, accum) = do_term t1 accum
   960              val (m2, accum) = do_term t2 accum
   961            in
   962              let
   963                val M11 = mtype_of_mterm m1 |> dest_MFun |> #1
   964                val M2 = mtype_of_mterm m2
   965              in (MApp (m1, m2), accum ||> add_is_sub_mtype M2 M11) end
   966            end)
   967         |> tap (fn (m, _) => trace_msg (fn () => "  \<Gamma> \<turnstile> " ^
   968                                                  string_for_mterm ctxt m))
   969   in do_term end
   970 
   971 fun force_minus_funs 0 _ = I
   972   | force_minus_funs n (M as MFun (M1, _, M2)) =
   973     add_mtypes_equal M (MFun (M1, A Gen, M2)) #> force_minus_funs (n - 1) M2
   974   | force_minus_funs _ M =
   975     raise MTYPE ("Nitpick_Mono.force_minus_funs", [M], [])
   976 fun consider_general_equals mdata def (x as (_, T)) t1 t2 accum =
   977   let
   978     val (m1, accum) = consider_term mdata t1 accum
   979     val (m2, accum) = consider_term mdata t2 accum
   980     val M1 = mtype_of_mterm m1
   981     val M2 = mtype_of_mterm m2
   982     val accum = accum ||> add_mtypes_equal M1 M2
   983     val body_M = fresh_mtype_for_type mdata false (nth_range_type 2 T)
   984     val m = MApp (MApp (MRaw (Const x,
   985                            MFun (M1, A Gen, MFun (M2, A Gen, body_M))), m1), m2)
   986   in
   987     (m, if def then
   988           let val (head_m, arg_ms) = strip_mcomb m1 in
   989             accum ||> force_minus_funs (length arg_ms) (mtype_of_mterm head_m)
   990           end
   991         else
   992           accum)
   993   end
   994 
   995 fun consider_general_formula (mdata as {hol_ctxt = {ctxt, ...}, ...}) =
   996   let
   997     val mtype_for = fresh_mtype_for_type mdata false
   998     val do_term = consider_term mdata
   999     fun do_formula sn t accum =
  1000         let
  1001           fun do_quantifier (quant_x as (quant_s, _)) abs_s abs_T body_t =
  1002             let
  1003               val abs_M = mtype_for abs_T
  1004               val a = Gen (* FIXME: strengthen *)
  1005               val side_cond = ((sn = Minus) = (quant_s = @{const_name Ex}))
  1006               val (body_m, accum) =
  1007                 accum ||> side_cond ? add_mtype_is_complete abs_M
  1008                       |>> push_bound (A a) abs_T abs_M |> do_formula sn body_t
  1009               val body_M = mtype_of_mterm body_m
  1010             in
  1011               (MApp (MRaw (Const quant_x,
  1012                            MFun (MFun (abs_M, A Gen, body_M), A Gen, body_M)),
  1013                      MAbs (abs_s, abs_T, abs_M, A Gen, body_m)),
  1014                accum |>> pop_bound)
  1015             end
  1016           fun do_equals x t1 t2 =
  1017             case sn of
  1018               Plus => do_term t accum
  1019             | Minus => consider_general_equals mdata false x t1 t2 accum
  1020         in
  1021           (trace_msg (fn () => "  \<Gamma> \<turnstile> " ^
  1022                                Syntax.string_of_term ctxt t ^ " : o\<^sup>" ^
  1023                                string_for_sign sn ^ "?");
  1024            case t of
  1025              Const (x as (@{const_name all}, _)) $ Abs (s1, T1, t1) =>
  1026              do_quantifier x s1 T1 t1
  1027            | Const (x as (@{const_name "=="}, _)) $ t1 $ t2 => do_equals x t1 t2
  1028            | @{const Trueprop} $ t1 =>
  1029              let val (m1, accum) = do_formula sn t1 accum in
  1030                (MApp (MRaw (@{const Trueprop}, mtype_for (bool_T --> prop_T)),
  1031                       m1), accum)
  1032              end
  1033            | @{const Not} $ t1 =>
  1034              let val (m1, accum) = do_formula (negate_sign sn) t1 accum in
  1035                (MApp (MRaw (@{const Not}, mtype_for (bool_T --> bool_T)), m1),
  1036                 accum)
  1037              end
  1038            | Const (x as (@{const_name All}, _)) $ Abs (s1, T1, t1) =>
  1039              do_quantifier x s1 T1 t1
  1040            | Const (x0 as (@{const_name Ex}, T0))
  1041              $ (t1 as Abs (s1, T1, t1')) =>
  1042              (case sn of
  1043                 Plus => do_quantifier x0 s1 T1 t1'
  1044               | Minus =>
  1045                 (* FIXME: Move elsewhere *)
  1046                 do_term (@{const Not}
  1047                          $ (HOLogic.eq_const (domain_type T0) $ t1
  1048                             $ Abs (Name.uu, T1, @{const False}))) accum)
  1049            | Const (x as (@{const_name HOL.eq}, _)) $ t1 $ t2 =>
  1050              do_equals x t1 t2
  1051            | Const (@{const_name Let}, _) $ t1 $ t2 =>
  1052              do_formula sn (betapply (t2, t1)) accum
  1053            | (t0 as Const (s0, _)) $ t1 $ t2 =>
  1054              if s0 = @{const_name "==>"} orelse
  1055                 s0 = @{const_name Pure.conjunction} orelse
  1056                 s0 = @{const_name HOL.conj} orelse
  1057                 s0 = @{const_name HOL.disj} orelse
  1058                 s0 = @{const_name HOL.implies} then
  1059                let
  1060                  val impl = (s0 = @{const_name "==>"} orelse
  1061                              s0 = @{const_name HOL.implies})
  1062                  val (m1, accum) =
  1063                    do_formula (sn |> impl ? negate_sign) t1 accum
  1064                  val (m2, accum) = do_formula sn t2 accum
  1065                in
  1066                  (MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2),
  1067                   accum)
  1068                end
  1069              else
  1070                do_term t accum
  1071            | _ => do_term t accum)
  1072         end
  1073         |> tap (fn (m, _) =>
  1074                    trace_msg (fn () => "\<Gamma> \<turnstile> " ^
  1075                                        string_for_mterm ctxt m ^ " : o\<^sup>" ^
  1076                                        string_for_sign sn))
  1077   in do_formula end
  1078 
  1079 (* The harmless axiom optimization below is somewhat too aggressive in the face
  1080    of (rather peculiar) user-defined axioms. *)
  1081 val harmless_consts =
  1082   [@{const_name ord_class.less}, @{const_name ord_class.less_eq}]
  1083 val bounteous_consts = [@{const_name bisim}]
  1084 
  1085 fun is_harmless_axiom ({no_harmless = true, ...} : mdata) _ = false
  1086   | is_harmless_axiom {hol_ctxt = {thy, stds, ...}, ...} t =
  1087     Term.add_consts t []
  1088     |> filter_out (is_built_in_const thy stds)
  1089     |> (forall (member (op =) harmless_consts o original_name o fst) orf
  1090         exists (member (op =) bounteous_consts o fst))
  1091 
  1092 fun consider_nondefinitional_axiom mdata t =
  1093   if is_harmless_axiom mdata t then pair (MRaw (t, dummy_M))
  1094   else consider_general_formula mdata Plus t
  1095 
  1096 fun consider_definitional_axiom (mdata as {hol_ctxt = {ctxt, ...}, ...}) t =
  1097   if not (is_constr_pattern_formula ctxt t) then
  1098     consider_nondefinitional_axiom mdata t
  1099   else if is_harmless_axiom mdata t then
  1100     pair (MRaw (t, dummy_M))
  1101   else
  1102     let
  1103       val mtype_for = fresh_mtype_for_type mdata false
  1104       val do_term = consider_term mdata
  1105       fun do_all quant_t abs_s abs_T body_t accum =
  1106         let
  1107           val abs_M = mtype_for abs_T
  1108           val (body_m, accum) =
  1109             accum |>> push_bound (A Gen) abs_T abs_M |> do_formula body_t
  1110           val body_M = mtype_of_mterm body_m
  1111         in
  1112           (MApp (MRaw (quant_t, MFun (MFun (abs_M, A Gen, body_M), A Gen,
  1113                        body_M)),
  1114                  MAbs (abs_s, abs_T, abs_M, A Gen, body_m)),
  1115            accum |>> pop_bound)
  1116         end
  1117       and do_conjunction t0 t1 t2 accum =
  1118         let
  1119           val (m1, accum) = do_formula t1 accum
  1120           val (m2, accum) = do_formula t2 accum
  1121         in
  1122           (MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2), accum)
  1123         end
  1124       and do_implies t0 t1 t2 accum =
  1125         let
  1126           val (m1, accum) = do_term t1 accum
  1127           val (m2, accum) = do_formula t2 accum
  1128         in
  1129           (MApp (MApp (MRaw (t0, mtype_for (fastype_of t0)), m1), m2), accum)
  1130         end
  1131       and do_formula t accum =
  1132           case t of
  1133             (t0 as Const (@{const_name all}, _)) $ Abs (s1, T1, t1) =>
  1134             do_all t0 s1 T1 t1 accum
  1135           | @{const Trueprop} $ t1 =>
  1136             let val (m1, accum) = do_formula t1 accum in
  1137               (MApp (MRaw (@{const Trueprop}, mtype_for (bool_T --> prop_T)),
  1138                      m1), accum)
  1139             end
  1140           | Const (x as (@{const_name "=="}, _)) $ t1 $ t2 =>
  1141             consider_general_equals mdata true x t1 t2 accum
  1142           | (t0 as @{const "==>"}) $ t1 $ t2 => do_implies t0 t1 t2 accum
  1143           | (t0 as @{const Pure.conjunction}) $ t1 $ t2 =>
  1144             do_conjunction t0 t1 t2 accum
  1145           | (t0 as Const (@{const_name All}, _)) $ Abs (s0, T1, t1) =>
  1146             do_all t0 s0 T1 t1 accum
  1147           | Const (x as (@{const_name HOL.eq}, _)) $ t1 $ t2 =>
  1148             consider_general_equals mdata true x t1 t2 accum
  1149           | (t0 as @{const HOL.conj}) $ t1 $ t2 => do_conjunction t0 t1 t2 accum
  1150           | (t0 as @{const HOL.implies}) $ t1 $ t2 => do_implies t0 t1 t2 accum
  1151           | _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\
  1152                              \do_formula", [t])
  1153     in do_formula t end
  1154 
  1155 fun string_for_mtype_of_term ctxt asgs t M =
  1156   Syntax.string_of_term ctxt t ^ " : " ^ string_for_mtype (resolve_mtype asgs M)
  1157 
  1158 fun print_mtype_context ctxt asgs ({frees, consts, ...} : mtype_context) =
  1159   trace_msg (fn () =>
  1160       map (fn (x, M) => string_for_mtype_of_term ctxt asgs (Free x) M) frees @
  1161       map (fn (x, M) => string_for_mtype_of_term ctxt asgs (Const x) M) consts
  1162       |> cat_lines)
  1163 
  1164 fun amass f t (ms, accum) =
  1165   let val (m, accum) = f t accum in (m :: ms, accum) end
  1166 
  1167 fun infer which no_harmless (hol_ctxt as {ctxt, ...}) binarize calculus alpha_T
  1168           (nondef_ts, def_ts) =
  1169   let
  1170     val _ = trace_msg (fn () => "****** " ^ which ^ " analysis: " ^
  1171                                 string_for_mtype MAlpha ^ " is " ^
  1172                                 Syntax.string_of_typ ctxt alpha_T)
  1173     val mdata as {max_fresh, constr_mcache, ...} =
  1174       initial_mdata hol_ctxt binarize no_harmless alpha_T
  1175     val accum = (initial_gamma, ([], []))
  1176     val (nondef_ms, accum) =
  1177       ([], accum) |> amass (consider_general_formula mdata Plus) (hd nondef_ts)
  1178                   |> fold (amass (consider_nondefinitional_axiom mdata))
  1179                           (tl nondef_ts)
  1180     val (def_ms, (gamma, cset)) =
  1181       ([], accum) |> fold (amass (consider_definitional_axiom mdata)) def_ts
  1182   in
  1183     case solve calculus (!max_fresh) cset of
  1184       SOME asgs => (print_mtype_context ctxt asgs gamma;
  1185                     SOME (asgs, (nondef_ms, def_ms), !constr_mcache))
  1186     | _ => NONE
  1187   end
  1188   handle UNSOLVABLE () => NONE
  1189        | MTYPE (loc, Ms, Ts) =>
  1190          raise BAD (loc, commas (map string_for_mtype Ms @
  1191                                  map (Syntax.string_of_typ ctxt) Ts))
  1192        | MTERM (loc, ms) =>
  1193          raise BAD (loc, commas (map (string_for_mterm ctxt) ms))
  1194 
  1195 fun formulas_monotonic hol_ctxt =
  1196   is_some oooo infer "Monotonicity" false hol_ctxt
  1197 
  1198 fun fin_fun_constr T1 T2 =
  1199   (@{const_name FinFun}, (T1 --> T2) --> Type (@{type_name fin_fun}, [T1, T2]))
  1200 
  1201 fun finitize_funs (hol_ctxt as {thy, ctxt, stds, constr_cache, ...}) binarize
  1202                   finitizes calculus alpha_T tsp =
  1203   case infer "Finiteness" true hol_ctxt binarize calculus alpha_T tsp of
  1204     SOME (asgs, msp, constr_mtypes) =>
  1205     if forall (curry (op =) Gen o snd) asgs then
  1206       tsp
  1207     else
  1208       let
  1209         fun should_finitize T aa =
  1210           case triple_lookup (type_match thy) finitizes T of
  1211             SOME (SOME false) => false
  1212           | _ => resolve_annotation_atom asgs aa = A Fls
  1213         fun type_from_mtype T M =
  1214           case (M, T) of
  1215             (MAlpha, _) => T
  1216           | (MFun (M1, aa, M2), Type (@{type_name fun}, Ts)) =>
  1217             Type (if should_finitize T aa then @{type_name fin_fun}
  1218                   else @{type_name fun}, map2 type_from_mtype Ts [M1, M2])
  1219           | (MPair (M1, M2), Type (@{type_name prod}, Ts)) =>
  1220             Type (@{type_name prod}, map2 type_from_mtype Ts [M1, M2])
  1221           | (MType _, _) => T
  1222           | _ => raise MTYPE ("Nitpick_Mono.finitize_funs.type_from_mtype",
  1223                               [M], [T])
  1224         fun finitize_constr (x as (s, T)) =
  1225           (s, case AList.lookup (op =) constr_mtypes x of
  1226                 SOME M => type_from_mtype T M
  1227               | NONE => T)
  1228         fun term_from_mterm new_Ts old_Ts m =
  1229           case m of
  1230             MRaw (t, M) =>
  1231             let
  1232               val T = fastype_of1 (old_Ts, t)
  1233               val T' = type_from_mtype T M
  1234             in
  1235               case t of
  1236                 Const (x as (s, _)) =>
  1237                 if s = @{const_name finite} then
  1238                   case domain_type T' of
  1239                     set_T' as Type (@{type_name fin_fun}, _) =>
  1240                     Abs (Name.uu, set_T', @{const True})
  1241                   | _ => Const (s, T')
  1242                 else if s = @{const_name "=="} orelse
  1243                         s = @{const_name HOL.eq} then
  1244                   let
  1245                     val T =
  1246                       case T' of
  1247                         Type (_, [T1, Type (_, [T2, T3])]) =>
  1248                         T1 --> T2 --> T3
  1249                       | _ => raise TYPE ("Nitpick_Mono.finitize_funs.\
  1250                                          \term_from_mterm", [T, T'], [])
  1251                   in coerce_term hol_ctxt new_Ts T' T (Const (s, T)) end
  1252                 else if is_built_in_const thy stds x then
  1253                   coerce_term hol_ctxt new_Ts T' T t
  1254                 else if is_constr ctxt stds x then
  1255                   Const (finitize_constr x)
  1256                 else if is_sel s then
  1257                   let
  1258                     val n = sel_no_from_name s
  1259                     val x' =
  1260                       x |> binarized_and_boxed_constr_for_sel hol_ctxt binarize
  1261                         |> finitize_constr
  1262                     val x'' =
  1263                       binarized_and_boxed_nth_sel_for_constr hol_ctxt binarize
  1264                                                              x' n
  1265                   in Const x'' end
  1266                 else
  1267                   Const (s, T')
  1268               | Free (s, T) => Free (s, type_from_mtype T M)
  1269               | Bound _ => t
  1270               | _ => raise MTERM ("Nitpick_Mono.finitize_funs.term_from_mterm",
  1271                                   [m])
  1272             end
  1273           | MApp (m1, m2) =>
  1274             let
  1275               val (t1, t2) = pairself (term_from_mterm new_Ts old_Ts) (m1, m2)
  1276               val (T1, T2) = pairself (curry fastype_of1 new_Ts) (t1, t2)
  1277               val (t1', T2') =
  1278                 case T1 of
  1279                   Type (s, [T11, T12]) =>
  1280                   (if s = @{type_name fin_fun} then
  1281                      select_nth_constr_arg ctxt stds (fin_fun_constr T11 T12) t1
  1282                                            0 (T11 --> T12)
  1283                    else
  1284                      t1, T11)
  1285                 | _ => raise TYPE ("Nitpick_Mono.finitize_funs.term_from_mterm",
  1286                                    [T1], [])
  1287             in betapply (t1', coerce_term hol_ctxt new_Ts T2' T2 t2) end
  1288           | MAbs (s, old_T, M, aa, m') =>
  1289             let
  1290               val new_T = type_from_mtype old_T M
  1291               val t' = term_from_mterm (new_T :: new_Ts) (old_T :: old_Ts) m'
  1292               val T' = fastype_of1 (new_T :: new_Ts, t')
  1293             in
  1294               Abs (s, new_T, t')
  1295               |> should_finitize (new_T --> T') aa
  1296                  ? construct_value ctxt stds (fin_fun_constr new_T T') o single
  1297             end
  1298       in
  1299         Unsynchronized.change constr_cache (map (apsnd (map finitize_constr)));
  1300         pairself (map (term_from_mterm [] [])) msp
  1301       end
  1302   | NONE => tsp
  1303 
  1304 end;