wneuper/isa: src/HOL/Tools/Nitpick/nitpick

     1 (*  Title:      HOL/Nitpick/Tools/nitpick_mono.ML

     2     Author:     Jasmin Blanchette, TU Muenchen

     3     Copyright   2009

     5 Monotonicity predicate for higher-order logic.

     6 *)

     8 signature NITPICK_MONO =

     9 sig

    10   type extended_context = Nitpick_HOL.extended_context

    12   val formulas_monotonic :

    13     extended_context -> typ -> term list -> term list -> term -> bool

    14 end;

    16 structure Nitpick_Mono : NITPICK_MONO =

    17 struct

    19 open Nitpick_Util

    20 open Nitpick_HOL

    22 type var = int

    24 datatype sign = Pos | Neg

    25 datatype sign_atom = S of sign | V of var

    27 type literal = var * sign

    29 datatype ctype =

    30   CAlpha |

    31   CFun of ctype * sign_atom * ctype |

    32   CPair of ctype * ctype |

    33   CType of string * ctype list |

    34   CRec of string * typ list

    36 type cdata =

    37   {ext_ctxt: extended_context,

    38    alpha_T: typ,

    39    max_fresh: int Unsynchronized.ref,

    40    datatype_cache: ((string * typ list) * ctype) list Unsynchronized.ref,

    41    constr_cache: (styp * ctype) list Unsynchronized.ref}

    43 exception CTYPE of string * ctype list

    45 (* string -> unit *)

    46 fun print_g (s : string) = ()

    48 (* var -> string *)

    49 val string_for_var = signed_string_of_int

    50 (* string -> var list -> string *)

    51 fun string_for_vars sep [] = "0\<^bsub>" ^ sep ^ "\<^esub>"

    52   | string_for_vars sep xs = space_implode sep (map string_for_var xs)

    53 fun subscript_string_for_vars sep xs =

    54   if null xs then "" else "\<^bsub>" ^ string_for_vars sep xs ^ "\<^esub>"

    56 (* sign -> string *)

    57 fun string_for_sign Pos = "+"

    58   | string_for_sign Neg = "-"

    60 (* sign -> sign -> sign *)

    61 fun xor sn1 sn2 = if sn1 = sn2 then Pos else Neg

    62 (* sign -> sign *)

    63 val negate = xor Neg

    65 (* sign_atom -> string *)

    66 fun string_for_sign_atom (S sn) = string_for_sign sn

    67   | string_for_sign_atom (V j) = string_for_var j

    69 (* literal -> string *)

    70 fun string_for_literal (x, sn) = string_for_var x ^ " = " ^ string_for_sign sn

    72 val bool_C = CType (@{type_name bool}, [])

    74 (* ctype -> bool *)

    75 fun is_CRec (CRec _) = true

    76   | is_CRec _ = false

    78 val no_prec = 100

    79 val prec_CFun = 1

    80 val prec_CPair = 2

    82 (* tuple_set -> int *)

    83 fun precedence_of_ctype (CFun _) = prec_CFun

    84   | precedence_of_ctype (CPair _) = prec_CPair

    85   | precedence_of_ctype _ = no_prec

    87 (* ctype -> string *)

    88 val string_for_ctype =

    89   let

    90     (* int -> ctype -> string *)

    91     fun aux outer_prec C =

    92       let

    93         val prec = precedence_of_ctype C

    94         val need_parens = (prec < outer_prec)

    95       in

    96         (if need_parens then "(" else "") ^

    97         (case C of

    98            CAlpha => "\<alpha>"

    99          | CFun (C1, a, C2) =>

   100            aux (prec + 1) C1 ^ " \<Rightarrow>\<^bsup>" ^

   101            string_for_sign_atom a ^ "\<^esup> " ^ aux prec C2

   102          | CPair (C1, C2) => aux (prec + 1) C1 ^ " \<times> " ^ aux prec C2

   103          | CType (s, []) =>

   104            if s mem [@{type_name prop}, @{type_name bool}] then "o" else s

   105          | CType (s, Cs) => "(" ^ commas (map (aux 0) Cs) ^ ") " ^ s

   106          | CRec (s, _) => "[" ^ s ^ "]") ^

   107         (if need_parens then ")" else "")

   108       end

   109   in aux 0 end

   111 (* ctype -> ctype list *)

   112 fun flatten_ctype (CPair (C1, C2)) = maps flatten_ctype [C1, C2]

   113   | flatten_ctype (CType (_, Cs)) = maps flatten_ctype Cs

   114   | flatten_ctype C = [C]

   116 (* extended_context -> typ -> cdata *)

   117 fun initial_cdata ext_ctxt alpha_T =

   118   ({ext_ctxt = ext_ctxt, alpha_T = alpha_T, max_fresh = Unsynchronized.ref 0,

   119     datatype_cache = Unsynchronized.ref [],

   120     constr_cache = Unsynchronized.ref []} : cdata)

   122 (* typ -> typ -> bool *)

   123 fun could_exist_alpha_subtype alpha_T (T as Type (_, Ts)) =

   124     T = alpha_T orelse (not (is_fp_iterator_type T)

   125                         andalso exists (could_exist_alpha_subtype alpha_T) Ts)

   126   | could_exist_alpha_subtype alpha_T T = (T = alpha_T)

   127 (* theory -> typ -> typ -> bool *)

   128 fun could_exist_alpha_sub_ctype _ (alpha_T as TFree _) =

   129     could_exist_alpha_subtype alpha_T

   130   | could_exist_alpha_sub_ctype thy alpha_T = equal alpha_T orf is_datatype thy

   132 (* ctype -> bool *)

   133 fun exists_alpha_sub_ctype CAlpha = true

   134   | exists_alpha_sub_ctype (CFun (C1, _, C2)) =

   135     exists exists_alpha_sub_ctype [C1, C2]

   136   | exists_alpha_sub_ctype (CPair (C1, C2)) =

   137     exists exists_alpha_sub_ctype [C1, C2]

   138   | exists_alpha_sub_ctype (CType (_, Cs)) = exists exists_alpha_sub_ctype Cs

   139   | exists_alpha_sub_ctype (CRec _) = true

   141 (* ctype -> bool *)

   142 fun exists_alpha_sub_ctype_fresh CAlpha = true

   143   | exists_alpha_sub_ctype_fresh (CFun (_, V _, _)) = true

   144   | exists_alpha_sub_ctype_fresh (CFun (_, _, C2)) =

   145     exists_alpha_sub_ctype_fresh C2

   146   | exists_alpha_sub_ctype_fresh (CPair (C1, C2)) =

   147     exists exists_alpha_sub_ctype_fresh [C1, C2]

   148   | exists_alpha_sub_ctype_fresh (CType (_, Cs)) =

   149     exists exists_alpha_sub_ctype_fresh Cs

   150   | exists_alpha_sub_ctype_fresh (CRec _) = true

   152 (* string * typ list -> ctype list -> ctype *)

   153 fun constr_ctype_for_binders z Cs =

   154   fold_rev (fn C => curry3 CFun C (S Neg)) Cs (CRec z)

   156 (* ((string * typ list) * ctype) list -> ctype list -> ctype -> ctype *)

   157 fun repair_ctype _ _ CAlpha = CAlpha

   158   | repair_ctype cache seen (CFun (C1, a, C2)) =

   159     CFun (repair_ctype cache seen C1, a, repair_ctype cache seen C2)

   160   | repair_ctype cache seen (CPair Cp) =

   161     CPair (pairself (repair_ctype cache seen) Cp)

   162   | repair_ctype cache seen (CType (s, Cs)) =

   163     CType (s, maps (flatten_ctype o repair_ctype cache seen) Cs)

   164   | repair_ctype cache seen (CRec (z as (s, _))) =

   165     case AList.lookup (op =) cache z |> the of

   166       CRec _ => CType (s, [])

   167     | C => if C mem seen then CType (s, [])

   168            else repair_ctype cache (C :: seen) C

   170 (* ((string * typ list) * ctype) list Unsynchronized.ref -> unit *)

   171 fun repair_datatype_cache cache =

   172   let

   173     (* (string * typ list) * ctype -> unit *)

   174     fun repair_one (z, C) =

   175       Unsynchronized.change cache

   176           (AList.update (op =) (z, repair_ctype (!cache) [] C))

   177   in List.app repair_one (rev (!cache)) end

   179 (* (typ * ctype) list -> (styp * ctype) list Unsynchronized.ref -> unit *)

   180 fun repair_constr_cache dtype_cache constr_cache =

   181   let

   182     (* styp * ctype -> unit *)

   183     fun repair_one (x, C) =

   184       Unsynchronized.change constr_cache

   185           (AList.update (op =) (x, repair_ctype dtype_cache [] C))

   186   in List.app repair_one (!constr_cache) end

   188 (* cdata -> typ -> ctype *)

   189 fun fresh_ctype_for_type ({ext_ctxt as {thy, ...}, alpha_T, max_fresh,

   190                            datatype_cache, constr_cache, ...} : cdata) =

   191   let

   192     (* typ -> typ -> ctype *)

   193     fun do_fun T1 T2 =

   194       let

   195         val C1 = do_type T1

   196         val C2 = do_type T2

   197         val a = if is_boolean_type (body_type T2)

   198                    andalso exists_alpha_sub_ctype_fresh C1 then

   199                   V (Unsynchronized.inc max_fresh)

   200                 else

   201                   S Neg

   202       in CFun (C1, a, C2) end

   203     (* typ -> ctype *)

   204     and do_type T =

   205       if T = alpha_T then

   206         CAlpha

   207       else case T of

   208         Type ("fun", [T1, T2]) => do_fun T1 T2

   209       | Type (@{type_name fun_box}, [T1, T2]) => do_fun T1 T2

   210       | Type ("*", [T1, T2]) => CPair (pairself do_type (T1, T2))

   211       | Type (z as (s, _)) =>

   212         if could_exist_alpha_sub_ctype thy alpha_T T then

   213           case AList.lookup (op =) (!datatype_cache) z of

   214             SOME C => C

   215           | NONE =>

   216             let

   217               val _ = Unsynchronized.change datatype_cache (cons (z, CRec z))

   218               val xs = datatype_constrs thy T

   219               val (all_Cs, constr_Cs) =

   220                 fold_rev (fn (_, T') => fn (all_Cs, constr_Cs) =>

   221                              let

   222                                val binder_Cs = map do_type (binder_types T')

   223                                val new_Cs = filter exists_alpha_sub_ctype_fresh

   224                                                    binder_Cs

   225                                val constr_C = constr_ctype_for_binders z

   226                                                                        binder_Cs

   227                              in

   228                                (union (op =) new_Cs all_Cs,

   229                                 constr_C :: constr_Cs)

   230                              end)

   231                          xs ([], [])

   232               val C = CType (s, all_Cs)

   233               val _ = Unsynchronized.change datatype_cache

   234                           (AList.update (op =) (z, C))

   235               val _ = Unsynchronized.change constr_cache

   236                           (append (xs ~~ constr_Cs))

   237             in

   238               if forall (not o is_CRec o snd) (!datatype_cache) then

   239                 (repair_datatype_cache datatype_cache;

   240                  repair_constr_cache (!datatype_cache) constr_cache;

   241                  AList.lookup (op =) (!datatype_cache) z |> the)

   242               else

   243                 C

   244             end

   245         else

   246           CType (s, [])

   247       | _ => CType (Refute.string_of_typ T, [])

   248   in do_type end

   250 (* ctype -> ctype list *)

   251 fun prodC_factors (CPair (C1, C2)) = maps prodC_factors [C1, C2]

   252   | prodC_factors C = [C]

   253 (* ctype -> ctype list * ctype *)

   254 fun curried_strip_ctype (CFun (C1, S Neg, C2)) =

   255     curried_strip_ctype C2 |>> append (prodC_factors C1)

   256   | curried_strip_ctype C = ([], C)

   257 (* string -> ctype -> ctype *)

   258 fun sel_ctype_from_constr_ctype s C =

   259   let val (arg_Cs, dataC) = curried_strip_ctype C in

   260     CFun (dataC, S Neg,

   261           case sel_no_from_name s of ~1 => bool_C | n => nth arg_Cs n)

   262   end

   264 (* cdata -> styp -> ctype *)

   265 fun ctype_for_constr (cdata as {ext_ctxt as {thy, ...}, alpha_T, constr_cache,

   266                                 ...}) (x as (_, T)) =

   267   if could_exist_alpha_sub_ctype thy alpha_T T then

   268     case AList.lookup (op =) (!constr_cache) x of

   269       SOME C => C

   270     | NONE => (fresh_ctype_for_type cdata (body_type T);

   271                AList.lookup (op =) (!constr_cache) x |> the)

   272   else

   273     fresh_ctype_for_type cdata T

   274 fun ctype_for_sel (cdata as {ext_ctxt, ...}) (x as (s, _)) =

   275   x |> boxed_constr_for_sel ext_ctxt |> ctype_for_constr cdata

   276     |> sel_ctype_from_constr_ctype s

   278 (* literal list -> ctype -> ctype *)

   279 fun instantiate_ctype lits =

   280   let

   281     (* ctype -> ctype *)

   282     fun aux CAlpha = CAlpha

   283       | aux (CFun (C1, V x, C2)) =

   284         let

   285           val a = case AList.lookup (op =) lits x of

   286                     SOME sn => S sn

   287                   | NONE => V x

   288         in CFun (aux C1, a, aux C2) end

   289       | aux (CFun (C1, a, C2)) = CFun (aux C1, a, aux C2)

   290       | aux (CPair Cp) = CPair (pairself aux Cp)

   291       | aux (CType (s, Cs)) = CType (s, map aux Cs)

   292       | aux (CRec z) = CRec z

   293   in aux end

   295 datatype comp_op = Eq | Leq

   297 type comp = sign_atom * sign_atom * comp_op * var list

   298 type sign_expr = literal list

   300 datatype constraint_set =

   301   UnsolvableCSet |

   302   CSet of literal list * comp list * sign_expr list

   304 (* comp_op -> string *)

   305 fun string_for_comp_op Eq = "="

   306   | string_for_comp_op Leq = "\<le>"

   308 (* sign_expr -> string *)

   309 fun string_for_sign_expr [] = "\<bot>"

   310   | string_for_sign_expr lits =

   311     space_implode " \<or> " (map string_for_literal lits)

   313 (* constraint_set *)

   314 val slack = CSet ([], [], [])

   316 (* literal -> literal list option -> literal list option *)

   317 fun do_literal _ NONE = NONE

   318   | do_literal (x, sn) (SOME lits) =

   319     case AList.lookup (op =) lits x of

   320       SOME sn' => if sn = sn' then SOME lits else NONE

   321     | NONE => SOME ((x, sn) :: lits)

   323 (* comp_op -> var list -> sign_atom -> sign_atom -> literal list * comp list

   324    -> (literal list * comp list) option *)

   325 fun do_sign_atom_comp Eq [] a1 a2 (accum as (lits, comps)) =

   326     (case (a1, a2) of

   327        (S sn1, S sn2) => if sn1 = sn2 then SOME accum else NONE

   328      | (V x1, S sn2) =>

   329        Option.map (rpair comps) (do_literal (x1, sn2) (SOME lits))

   330      | (V _, V _) => SOME (lits, insert (op =) (a1, a2, Eq, []) comps)

   331      | _ => do_sign_atom_comp Eq [] a2 a1 accum)

   332   | do_sign_atom_comp Leq [] a1 a2 (accum as (lits, comps)) =

   333     (case (a1, a2) of

   334        (_, S Neg) => SOME accum

   335      | (S Pos, _) => SOME accum

   336      | (S Neg, S Pos) => NONE

   337      | (V _, V _) => SOME (lits, insert (op =) (a1, a2, Leq, []) comps)

   338      | _ => do_sign_atom_comp Eq [] a1 a2 accum)

   339   | do_sign_atom_comp cmp xs a1 a2 (accum as (lits, comps)) =

   340     SOME (lits, insert (op =) (a1, a2, cmp, xs) comps)

   342 (* comp -> var list -> ctype -> ctype -> (literal list * comp list) option

   343    -> (literal list * comp list) option *)

   344 fun do_ctype_comp _ _ _ _ NONE = NONE

   345   | do_ctype_comp _ _ CAlpha CAlpha accum = accum

   346   | do_ctype_comp Eq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))

   347                   (SOME accum) =

   348      accum |> do_sign_atom_comp Eq xs a1 a2 |> do_ctype_comp Eq xs C11 C21

   349            |> do_ctype_comp Eq xs C12 C22

   350   | do_ctype_comp Leq xs (CFun (C11, a1, C12)) (CFun (C21, a2, C22))

   351                   (SOME accum) =

   352     (if exists_alpha_sub_ctype C11 then

   353        accum |> do_sign_atom_comp Leq xs a1 a2

   354              |> do_ctype_comp Leq xs C21 C11

   355              |> (case a2 of

   356                    S Neg => I

   357                  | S Pos => do_ctype_comp Leq xs C11 C21

   358                  | V x => do_ctype_comp Leq (x :: xs) C11 C21)

   359      else

   360        SOME accum)

   361     |> do_ctype_comp Leq xs C12 C22

   362   | do_ctype_comp cmp xs (C1 as CPair (C11, C12)) (C2 as CPair (C21, C22))

   363                   accum =

   364     (accum |> fold (uncurry (do_ctype_comp cmp xs)) [(C11, C21), (C12, C22)]

   365      handle Library.UnequalLengths =>

   366             raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2]))

   367   | do_ctype_comp cmp xs (CType _) (CType _) accum =

   368     accum (* no need to compare them thanks to the cache *)

   369   | do_ctype_comp _ _ C1 C2 _ =

   370     raise CTYPE ("Nitpick_Mono.do_ctype_comp", [C1, C2])

   372 (* comp_op -> ctype -> ctype -> constraint_set -> constraint_set *)

   373 fun add_ctype_comp _ _ _ UnsolvableCSet = UnsolvableCSet

   374   | add_ctype_comp cmp C1 C2 (CSet (lits, comps, sexps)) =

   375     (print_g ("*** Add " ^ string_for_ctype C1 ^ " " ^ string_for_comp_op cmp ^

   376               " " ^ string_for_ctype C2);

   377      case do_ctype_comp cmp [] C1 C2 (SOME (lits, comps)) of

   378        NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

   379      | SOME (lits, comps) => CSet (lits, comps, sexps))

   381 (* ctype -> ctype -> constraint_set -> constraint_set *)

   382 val add_ctypes_equal = add_ctype_comp Eq

   383 val add_is_sub_ctype = add_ctype_comp Leq

   385 (* sign -> sign_expr -> ctype -> (literal list * sign_expr list) option

   386    -> (literal list * sign_expr list) option *)

   387 fun do_notin_ctype_fv _ _ _ NONE = NONE

   388   | do_notin_ctype_fv Neg _ CAlpha accum = accum

   389   | do_notin_ctype_fv Pos [] CAlpha _ = NONE

   390   | do_notin_ctype_fv Pos [(x, sn)] CAlpha (SOME (lits, sexps)) =

   391     SOME lits |> do_literal (x, sn) |> Option.map (rpair sexps)

   392   | do_notin_ctype_fv Pos sexp CAlpha (SOME (lits, sexps)) =

   393     SOME (lits, insert (op =) sexp sexps)

   394   | do_notin_ctype_fv sn sexp (CFun (C1, S sn', C2)) accum =

   395     accum |> (if sn' = Pos andalso sn = Pos then do_notin_ctype_fv Pos sexp C1

   396               else I)

   397           |> (if sn' = Neg orelse sn = Pos then do_notin_ctype_fv Neg sexp C1

   398               else I)

   399           |> do_notin_ctype_fv sn sexp C2

   400   | do_notin_ctype_fv Pos sexp (CFun (C1, V x, C2)) accum =

   401     accum |> (case do_literal (x, Neg) (SOME sexp) of

   402                 NONE => I

   403               | SOME sexp' => do_notin_ctype_fv Pos sexp' C1)

   404           |> do_notin_ctype_fv Neg sexp C1

   405           |> do_notin_ctype_fv Pos sexp C2

   406   | do_notin_ctype_fv Neg sexp (CFun (C1, V x, C2)) accum =

   407     accum |> (case do_literal (x, Pos) (SOME sexp) of

   408                 NONE => I

   409               | SOME sexp' => do_notin_ctype_fv Pos sexp' C1)

   410           |> do_notin_ctype_fv Neg sexp C2

   411   | do_notin_ctype_fv sn sexp (CPair (C1, C2)) accum =

   412     accum |> fold (do_notin_ctype_fv sn sexp) [C1, C2]

   413   | do_notin_ctype_fv sn sexp (CType (_, Cs)) accum =

   414     accum |> fold (do_notin_ctype_fv sn sexp) Cs

   415   | do_notin_ctype_fv _ _ C _ =

   416     raise CTYPE ("Nitpick_Mono.do_notin_ctype_fv", [C])

   418 (* sign -> ctype -> constraint_set -> constraint_set *)

   419 fun add_notin_ctype_fv _ _ UnsolvableCSet = UnsolvableCSet

   420   | add_notin_ctype_fv sn C (CSet (lits, comps, sexps)) =

   421     (print_g ("*** Add " ^ string_for_ctype C ^ " is right-" ^

   422               (case sn of Neg => "unique" | Pos => "total") ^ ".");

   423      case do_notin_ctype_fv sn [] C (SOME (lits, sexps)) of

   424        NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

   425      | SOME (lits, sexps) => CSet (lits, comps, sexps))

   427 (* ctype -> constraint_set -> constraint_set *)

   428 val add_ctype_is_right_unique = add_notin_ctype_fv Neg

   429 val add_ctype_is_right_total = add_notin_ctype_fv Pos

   431 (* constraint_set -> constraint_set -> constraint_set *)

   432 fun unite (CSet (lits1, comps1, sexps1)) (CSet (lits2, comps2, sexps2)) =

   433     (case SOME lits1 |> fold do_literal lits2 of

   434        NONE => (print_g "**** Unsolvable"; UnsolvableCSet)

   435      | SOME lits => CSet (lits, comps1 @ comps2, sexps1 @ sexps2))

   436   | unite _ _ = UnsolvableCSet

   438 (* sign -> bool *)

   439 fun bool_from_sign Pos = false

   440   | bool_from_sign Neg = true

   441 (* bool -> sign *)

   442 fun sign_from_bool false = Pos

   443   | sign_from_bool true = Neg

   445 (* literal -> PropLogic.prop_formula *)

   446 fun prop_for_literal (x, sn) =

   447   (not (bool_from_sign sn) ? PropLogic.Not) (PropLogic.BoolVar x)

   448 (* sign_atom -> PropLogic.prop_formula *)

   449 fun prop_for_sign_atom_eq (S sn', sn) =

   450     if sn = sn' then PropLogic.True else PropLogic.False

   451   | prop_for_sign_atom_eq (V x, sn) = prop_for_literal (x, sn)

   452 (* sign_expr -> PropLogic.prop_formula *)

   453 fun prop_for_sign_expr xs = PropLogic.exists (map prop_for_literal xs)

   454 (* var list -> sign -> PropLogic.prop_formula *)

   455 fun prop_for_exists_eq xs sn =

   456   PropLogic.exists (map (fn x => prop_for_literal (x, sn)) xs)

   457 (* comp -> PropLogic.prop_formula *)

   458 fun prop_for_comp (a1, a2, Eq, []) =

   459     PropLogic.SAnd (prop_for_comp (a1, a2, Leq, []),

   460                     prop_for_comp (a2, a1, Leq, []))

   461   | prop_for_comp (a1, a2, Leq, []) =

   462     PropLogic.SOr (prop_for_sign_atom_eq (a1, Pos),

   463                    prop_for_sign_atom_eq (a2, Neg))

   464   | prop_for_comp (a1, a2, cmp, xs) =

   465     PropLogic.SOr (prop_for_exists_eq xs Neg, prop_for_comp (a1, a2, cmp, []))

   467 (* var -> (int -> bool option) -> literal list -> literal list *)

   468 fun literals_from_assignments max_var asgns lits =

   469   fold (fn x => fn accum =>

   470            if AList.defined (op =) lits x then

   471              accum

   472            else case asgns x of

   473              SOME b => (x, sign_from_bool b) :: accum

   474            | NONE => accum) (max_var downto 1) lits

   476 (* literal list -> sign_atom -> sign option *)

   477 fun lookup_sign_atom _ (S sn) = SOME sn

   478   | lookup_sign_atom lit (V x) = AList.lookup (op =) lit x

   480 (* comp -> string *)

   481 fun string_for_comp (a1, a2, cmp, xs) =

   482   string_for_sign_atom a1 ^ " " ^ string_for_comp_op cmp ^

   483   subscript_string_for_vars " \<and> " xs ^ " " ^ string_for_sign_atom a2

   485 (* literal list -> comp list -> sign_expr list -> unit *)

   486 fun print_problem lits comps sexps =

   487   print_g ("*** Problem:\n" ^ cat_lines (map string_for_literal lits @

   488                                          map string_for_comp comps @

   489                                          map string_for_sign_expr sexps))

   491 (* literal list -> unit *)

   492 fun print_solution lits =

   493   let val (pos, neg) = List.partition (equal Pos o snd) lits in

   494     print_g ("*** Solution:\n" ^

   495              "+: " ^ commas (map (string_for_var o fst) pos) ^ "\n" ^

   496              "-: " ^ commas (map (string_for_var o fst) neg))

   497   end

   499 (* var -> constraint_set -> literal list list option *)

   500 fun solve _ UnsolvableCSet = (print_g "*** Problem: Unsolvable"; NONE)

   501   | solve max_var (CSet (lits, comps, sexps)) =

   502     let

   503       val _ = print_problem lits comps sexps

   504       val prop = PropLogic.all (map prop_for_literal lits @

   505                                 map prop_for_comp comps @

   506                                 map prop_for_sign_expr sexps)

   507     in

   508       case SatSolver.invoke_solver "dpll" prop of

   509         SatSolver.SATISFIABLE asgns =>

   510         SOME (literals_from_assignments max_var asgns lits

   511               |> tap print_solution)

   512       | _ => NONE

   513     end

   515 (* var -> constraint_set -> bool *)

   516 val is_solvable = is_some oo solve

   518 type ctype_schema = ctype * constraint_set

   519 type ctype_context =

   520   {bounds: ctype list,

   521    frees: (styp * ctype) list,

   522    consts: (styp * ctype_schema) list}

   524 type accumulator = ctype_context * constraint_set

   526 val initial_gamma = {bounds = [], frees = [], consts = []}

   527 val unsolvable_accum = (initial_gamma, UnsolvableCSet)

   529 (* ctype -> ctype_context -> ctype_context *)

   530 fun push_bound C {bounds, frees, consts} =

   531   {bounds = C :: bounds, frees = frees, consts = consts}

   532 (* ctype_context -> ctype_context *)

   533 fun pop_bound {bounds, frees, consts} =

   534   {bounds = tl bounds, frees = frees, consts = consts}

   535   handle List.Empty => initial_gamma

   537 (* cdata -> term -> accumulator -> ctype * accumulator *)

   538 fun consider_term (cdata as {ext_ctxt as {ctxt, thy, def_table, ...}, alpha_T,

   539                              max_fresh, ...}) =

   540   let

   541     (* typ -> ctype *)

   542     val ctype_for = fresh_ctype_for_type cdata

   543     (* ctype -> ctype *)

   544     fun pos_set_ctype_for_dom C =

   545       CFun (C, S (if exists_alpha_sub_ctype C then Pos else Neg), bool_C)

   546     (* typ -> accumulator -> ctype * accumulator *)

   547     fun do_quantifier T (gamma, cset) =

   548       let

   549         val abs_C = ctype_for (domain_type (domain_type T))

   550         val body_C = ctype_for (range_type T)

   551       in

   552         (CFun (CFun (abs_C, S Neg, body_C), S Neg, body_C),

   553          (gamma, cset |> add_ctype_is_right_total abs_C))

   554       end

   555     fun do_equals T (gamma, cset) =

   556       let val C = ctype_for (domain_type T) in

   557         (CFun (C, S Neg, CFun (C, S Neg, ctype_for (nth_range_type 2 T))),

   558          (gamma, cset |> add_ctype_is_right_unique C))

   559       end

   560     fun do_robust_set_operation T (gamma, cset) =

   561       let

   562         val set_T = domain_type T

   563         val C1 = ctype_for set_T

   564         val C2 = ctype_for set_T

   565         val C3 = ctype_for set_T

   566       in

   567         (CFun (C1, S Neg, CFun (C2, S Neg, C3)),

   568          (gamma, cset |> add_is_sub_ctype C1 C3 |> add_is_sub_ctype C2 C3))

   569       end

   570     fun do_fragile_set_operation T (gamma, cset) =

   571       let

   572         val set_T = domain_type T

   573         val set_C = ctype_for set_T

   574         (* typ -> ctype *)

   575         fun custom_ctype_for (T as Type ("fun", [T1, T2])) =

   576             if T = set_T then set_C

   577             else CFun (custom_ctype_for T1, S Neg, custom_ctype_for T2)

   578           | custom_ctype_for T = ctype_for T

   579       in

   580         (custom_ctype_for T, (gamma, cset |> add_ctype_is_right_unique set_C))

   581       end

   582     (* typ -> accumulator -> ctype * accumulator *)

   583     fun do_pair_constr T accum =

   584       case ctype_for (nth_range_type 2 T) of

   585         C as CPair (a_C, b_C) =>

   586         (CFun (a_C, S Neg, CFun (b_C, S Neg, C)), accum)

   587       | C => raise CTYPE ("Nitpick_Mono.consider_term.do_pair_constr", [C])

   588     (* int -> typ -> accumulator -> ctype * accumulator *)

   589     fun do_nth_pair_sel n T =

   590       case ctype_for (domain_type T) of

   591         C as CPair (a_C, b_C) =>

   592         pair (CFun (C, S Neg, if n = 0 then a_C else b_C))

   593       | C => raise CTYPE ("Nitpick_Mono.consider_term.do_nth_pair_sel", [C])

   594     val unsolvable = (CType ("unsolvable", []), unsolvable_accum)

   595     (* typ -> term -> accumulator -> ctype * accumulator *)

   596     fun do_bounded_quantifier abs_T bound_t body_t accum =

   597       let

   598         val abs_C = ctype_for abs_T

   599         val (bound_C, accum) = accum |>> push_bound abs_C |> do_term bound_t

   600         val expected_bound_C = pos_set_ctype_for_dom abs_C

   601       in

   602         accum ||> add_ctypes_equal expected_bound_C bound_C |> do_term body_t

   603               ||> apfst pop_bound

   604       end

   605     (* term -> accumulator -> ctype * accumulator *)

   606     and do_term _ (_, UnsolvableCSet) = unsolvable

   607       | do_term t (accum as (gamma as {bounds, frees, consts}, cset)) =

   608         (case t of

   609            Const (x as (s, T)) =>

   610            (case AList.lookup (op =) consts x of

   611               SOME (C, cset') => (C, (gamma, cset |> unite cset'))

   612             | NONE =>

   613               if not (could_exist_alpha_subtype alpha_T T) then

   614                 (ctype_for T, accum)

   615               else case s of

   616                 @{const_name all} => do_quantifier T accum

   617               | @{const_name "=="} => do_equals T accum

   618               | @{const_name All} => do_quantifier T accum

   619               | @{const_name Ex} => do_quantifier T accum

   620               | @{const_name "op ="} => do_equals T accum

   621               | @{const_name The} => (print_g "*** The"; unsolvable)

   622               | @{const_name Eps} => (print_g "*** Eps"; unsolvable)

   623               | @{const_name If} =>

   624                 do_robust_set_operation (range_type T) accum

   625                 |>> curry3 CFun bool_C (S Neg)

   626               | @{const_name Pair} => do_pair_constr T accum

   627               | @{const_name fst} => do_nth_pair_sel 0 T accum

   628               | @{const_name snd} => do_nth_pair_sel 1 T accum

   629               | @{const_name Id} =>

   630                 (CFun (ctype_for (domain_type T), S Neg, bool_C), accum)

   631               | @{const_name insert} =>

   632                 let

   633                   val set_T = domain_type (range_type T)

   634                   val C1 = ctype_for (domain_type set_T)

   635                   val C1' = pos_set_ctype_for_dom C1

   636                   val C2 = ctype_for set_T

   637                   val C3 = ctype_for set_T

   638                 in

   639                   (CFun (C1, S Neg, CFun (C2, S Neg, C3)),

   640                    (gamma, cset |> add_ctype_is_right_unique C1

   641                                 |> add_is_sub_ctype C1' C3

   642                                 |> add_is_sub_ctype C2 C3))

   643                 end

   644               | @{const_name converse} =>

   645                 let

   646                   val x = Unsynchronized.inc max_fresh

   647                   (* typ -> ctype *)

   648                   fun ctype_for_set T =

   649                     CFun (ctype_for (domain_type T), V x, bool_C)

   650                   val ab_set_C = domain_type T |> ctype_for_set

   651                   val ba_set_C = range_type T |> ctype_for_set

   652                 in (CFun (ab_set_C, S Neg, ba_set_C), accum) end

   653               | @{const_name trancl} => do_fragile_set_operation T accum

   654               | @{const_name rtrancl} => (print_g "*** rtrancl"; unsolvable)

   655               | @{const_name lower_semilattice_fun_inst.inf_fun} =>

   656                 do_robust_set_operation T accum

   657               | @{const_name upper_semilattice_fun_inst.sup_fun} =>

   658                 do_robust_set_operation T accum

   659               | @{const_name finite} =>

   660                 let val C1 = ctype_for (domain_type (domain_type T)) in

   661                   (CFun (pos_set_ctype_for_dom C1, S Neg, bool_C), accum)

   662                 end

   663               | @{const_name rel_comp} =>

   664                 let

   665                   val x = Unsynchronized.inc max_fresh

   666                   (* typ -> ctype *)

   667                   fun ctype_for_set T =

   668                     CFun (ctype_for (domain_type T), V x, bool_C)

   669                   val bc_set_C = domain_type T |> ctype_for_set

   670                   val ab_set_C = domain_type (range_type T) |> ctype_for_set

   671                   val ac_set_C = nth_range_type 2 T |> ctype_for_set

   672                 in

   673                   (CFun (bc_set_C, S Neg, CFun (ab_set_C, S Neg, ac_set_C)),

   674                    accum)

   675                 end

   676               | @{const_name image} =>

   677                 let

   678                   val a_C = ctype_for (domain_type (domain_type T))

   679                   val b_C = ctype_for (range_type (domain_type T))

   680                 in

   681                   (CFun (CFun (a_C, S Neg, b_C), S Neg,

   682                          CFun (pos_set_ctype_for_dom a_C, S Neg,

   683                                pos_set_ctype_for_dom b_C)), accum)

   684                 end

   685               | @{const_name Sigma} =>

   686                 let

   687                   val x = Unsynchronized.inc max_fresh

   688                   (* typ -> ctype *)

   689                   fun ctype_for_set T =

   690                     CFun (ctype_for (domain_type T), V x, bool_C)

   691                   val a_set_T = domain_type T

   692                   val a_C = ctype_for (domain_type a_set_T)

   693                   val b_set_C = ctype_for_set (range_type (domain_type

   694                                                                (range_type T)))

   695                   val a_set_C = ctype_for_set a_set_T

   696                   val a_to_b_set_C = CFun (a_C, S Neg, b_set_C)

   697                   val ab_set_C = ctype_for_set (nth_range_type 2 T)

   698                 in

   699                   (CFun (a_set_C, S Neg, CFun (a_to_b_set_C, S Neg, ab_set_C)),

   700                    accum)

   701                 end

   702               | @{const_name minus_fun_inst.minus_fun} =>

   703                 let

   704                   val set_T = domain_type T

   705                   val left_set_C = ctype_for set_T

   706                   val right_set_C = ctype_for set_T

   707                 in

   708                   (CFun (left_set_C, S Neg,

   709                          CFun (right_set_C, S Neg, left_set_C)),

   710                    (gamma, cset |> add_ctype_is_right_unique right_set_C

   711                                 |> add_is_sub_ctype right_set_C left_set_C))

   712                 end

   713               | @{const_name ord_fun_inst.less_eq_fun} =>

   714                 do_fragile_set_operation T accum

   715               | @{const_name Tha} =>

   716                 let

   717                   val a_C = ctype_for (domain_type (domain_type T))

   718                   val a_set_C = pos_set_ctype_for_dom a_C

   719                 in (CFun (a_set_C, S Neg, a_C), accum) end

   720               | @{const_name FunBox} =>

   721                 let val dom_C = ctype_for (domain_type T) in

   722                   (CFun (dom_C, S Neg, dom_C), accum)

   723                 end

   724               | _ => if is_sel s then

   725                        if constr_name_for_sel_like s = @{const_name FunBox} then

   726                          let val dom_C = ctype_for (domain_type T) in

   727                            (CFun (dom_C, S Neg, dom_C), accum)

   728                          end

   729                        else

   730                          (ctype_for_sel cdata x, accum)

   731                      else if is_constr thy x then

   732                        (ctype_for_constr cdata x, accum)

   733                      else if is_built_in_const true x then

   734                        case def_of_const thy def_table x of

   735                          SOME t' => do_term t' accum

   736                        | NONE => (print_g ("*** built-in " ^ s); unsolvable)

   737                      else

   738                        (ctype_for T, accum))

   739          | Free (x as (_, T)) =>

   740            (case AList.lookup (op =) frees x of

   741               SOME C => (C, accum)

   742             | NONE =>

   743               let val C = ctype_for T in

   744                 (C, ({bounds = bounds, frees = (x, C) :: frees,

   745                       consts = consts}, cset))

   746               end)

   747          | Var _ => (print_g "*** Var"; unsolvable)

   748          | Bound j => (nth bounds j, accum)

   749          | Abs (_, T, @{const False}) => (ctype_for (T --> bool_T), accum)

   750          | Abs (s, T, t') =>

   751            let

   752              val C = ctype_for T

   753              val (C', accum) = do_term t' (accum |>> push_bound C)

   754            in (CFun (C, S Neg, C'), accum |>> pop_bound) end

   755          | Const (@{const_name All}, _)

   756            $ Abs (_, T', @{const "op -->"} $ (t1 $ Bound 0) $ t2) =>

   757            do_bounded_quantifier T' t1 t2 accum

   758          | Const (@{const_name Ex}, _)

   759            $ Abs (_, T', @{const "op &"} $ (t1 $ Bound 0) $ t2) =>

   760            do_bounded_quantifier T' t1 t2 accum

   761          | Const (@{const_name Let}, _) $ t1 $ t2 =>

   762            do_term (betapply (t2, t1)) accum

   763          | t1 $ t2 =>

   764            let

   765              val (C1, accum) = do_term t1 accum

   766              val (C2, accum) = do_term t2 accum

   767            in

   768              case accum of

   769                (_, UnsolvableCSet) => unsolvable

   770              | _ => case C1 of

   771                       CFun (C11, _, C12) =>

   772                       (C12, accum ||> add_is_sub_ctype C2 C11)

   773                     | _ => raise CTYPE ("Nitpick_Mono.consider_term.do_term \

   774                                         \(op $)", [C1])

   775            end)

   776         |> tap (fn (C, _) =>

   777                    print_g ("  \<Gamma> \<turnstile> " ^

   778                             Syntax.string_of_term ctxt t ^ " : " ^

   779                             string_for_ctype C))

   780   in do_term end

   782 (* cdata -> sign -> term -> accumulator -> accumulator *)

   783 fun consider_general_formula (cdata as {ext_ctxt as {ctxt, ...}, ...}) =

   784   let

   785     (* typ -> ctype *)

   786     val ctype_for = fresh_ctype_for_type cdata

   787     (* term -> accumulator -> accumulator *)

   788     val do_term = snd oo consider_term cdata

   789     (* sign -> term -> accumulator -> accumulator *)

   790     fun do_formula _ _ (_, UnsolvableCSet) = unsolvable_accum

   791       | do_formula sn t (accum as (gamma as {bounds, frees, consts}, cset)) =

   792         let

   793           (* term -> accumulator -> accumulator *)

   794           val do_co_formula = do_formula sn

   795           val do_contra_formula = do_formula (negate sn)

   796           (* string -> typ -> term -> accumulator *)

   797           fun do_quantifier quant_s abs_T body_t =

   798             let

   799               val abs_C = ctype_for abs_T

   800               val side_cond = ((sn = Neg) = (quant_s = @{const_name Ex}))

   801               val cset = cset |> side_cond ? add_ctype_is_right_total abs_C

   802             in

   803               (gamma |> push_bound abs_C, cset) |> do_co_formula body_t

   804                                                 |>> pop_bound

   805             end

   806           (* typ -> term -> accumulator *)

   807           fun do_bounded_quantifier abs_T body_t =

   808             accum |>> push_bound (ctype_for abs_T) |> do_co_formula body_t

   809                   |>> pop_bound

   810           (* term -> term -> accumulator *)

   811           fun do_equals t1 t2 =

   812             case sn of

   813               Pos => do_term t accum

   814             | Neg => fold do_term [t1, t2] accum

   815         in

   816           case t of

   817             Const (s0 as @{const_name all}, _) $ Abs (_, T1, t1) =>

   818             do_quantifier s0 T1 t1

   819           | Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2

   820           | @{const "==>"} $ t1 $ t2 =>

   821             accum |> do_contra_formula t1 |> do_co_formula t2

   822           | @{const Trueprop} $ t1 => do_co_formula t1 accum

   823           | @{const Not} $ t1 => do_contra_formula t1 accum

   824           | Const (@{const_name All}, _)

   825             $ Abs (_, T1, t1 as @{const "op -->"} $ (_ $ Bound 0) $ _) =>

   826             do_bounded_quantifier T1 t1

   827           | Const (s0 as @{const_name All}, _) $ Abs (_, T1, t1) =>

   828             do_quantifier s0 T1 t1

   829           | Const (@{const_name Ex}, _)

   830             $ Abs (_, T1, t1 as @{const "op &"} $ (_ $ Bound 0) $ _) =>

   831             do_bounded_quantifier T1 t1

   832           | Const (s0 as @{const_name Ex}, _) $ Abs (_, T1, t1) =>

   833             do_quantifier s0 T1 t1

   834           | Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2

   835           | @{const "op &"} $ t1 $ t2 =>

   836             accum |> do_co_formula t1 |> do_co_formula t2

   837           | @{const "op |"} $ t1 $ t2 =>

   838             accum |> do_co_formula t1 |> do_co_formula t2

   839           | @{const "op -->"} $ t1 $ t2 =>

   840             accum |> do_contra_formula t1 |> do_co_formula t2

   841           | Const (@{const_name If}, _) $ t1 $ t2 $ t3 =>

   842             accum |> do_term t1 |> fold do_co_formula [t2, t3]

   843           | Const (@{const_name Let}, _) $ t1 $ t2 =>

   844             do_co_formula (betapply (t2, t1)) accum

   845           | _ => do_term t accum

   846         end

   847         |> tap (fn _ => print_g ("\<Gamma> \<turnstile> " ^

   848                                  Syntax.string_of_term ctxt t ^

   849                                  " : o\<^sup>" ^ string_for_sign sn))

   850   in do_formula end

   852 (* The harmless axiom optimization below is somewhat too aggressive in the face

   853    of (rather peculiar) user-defined axioms. *)

   854 val harmless_consts =

   855   [@{const_name ord_class.less}, @{const_name ord_class.less_eq}]

   856 val bounteous_consts = [@{const_name bisim}]

   858 (* term -> bool *)

   859 fun is_harmless_axiom t =

   860   Term.add_consts t [] |> filter_out (is_built_in_const true)

   861   |> (forall (member (op =) harmless_consts o original_name o fst)

   862       orf exists (member (op =) bounteous_consts o fst))

   864 (* cdata -> sign -> term -> accumulator -> accumulator *)

   865 fun consider_nondefinitional_axiom cdata sn t =

   866   not (is_harmless_axiom t) ? consider_general_formula cdata sn t

   868 (* cdata -> term -> accumulator -> accumulator *)

   869 fun consider_definitional_axiom (cdata as {ext_ctxt as {thy, ...}, ...}) t =

   870   if not (is_constr_pattern_formula thy t) then

   871     consider_nondefinitional_axiom cdata Pos t

   872   else if is_harmless_axiom t then

   873     I

   874   else

   875     let

   876       (* term -> accumulator -> accumulator *)

   877       val do_term = consider_term cdata

   878       (* typ -> term -> accumulator -> accumulator *)

   879       fun do_all abs_T body_t accum =

   880         let val abs_C = fresh_ctype_for_type cdata abs_T in

   881           accum |>> push_bound abs_C |> do_formula body_t |>> pop_bound

   882         end

   883       (* term -> term -> accumulator -> accumulator *)

   884       and do_implies t1 t2 = do_term t1 #> snd #> do_formula t2

   885       and do_equals t1 t2 accum =

   886         let

   887           val (C1, accum) = do_term t1 accum

   888           val (C2, accum) = do_term t2 accum

   889         in accum ||> add_ctypes_equal C1 C2 end

   890       (* term -> accumulator -> accumulator *)

   891       and do_formula _ (_, UnsolvableCSet) = unsolvable_accum

   892         | do_formula t accum =

   893           case t of

   894             Const (@{const_name all}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum

   895           | @{const Trueprop} $ t1 => do_formula t1 accum

   896           | Const (@{const_name "=="}, _) $ t1 $ t2 => do_equals t1 t2 accum

   897           | @{const "==>"} $ t1 $ t2 => do_implies t1 t2 accum

   898           | @{const Pure.conjunction} $ t1 $ t2 =>

   899             accum |> do_formula t1 |> do_formula t2

   900           | Const (@{const_name All}, _) $ Abs (_, T1, t1) => do_all T1 t1 accum

   901           | Const (@{const_name "op ="}, _) $ t1 $ t2 => do_equals t1 t2 accum

   902           | @{const "op &"} $ t1 $ t2 => accum |> do_formula t1 |> do_formula t2

   903           | @{const "op -->"} $ t1 $ t2 => do_implies t1 t2 accum

   904           | _ => raise TERM ("Nitpick_Mono.consider_definitional_axiom.\

   905                              \do_formula", [t])

   906     in do_formula t end

   908 (* Proof.context -> literal list -> term -> ctype -> string *)

   909 fun string_for_ctype_of_term ctxt lits t C =

   910   Syntax.string_of_term ctxt t ^ " : " ^

   911   string_for_ctype (instantiate_ctype lits C)

   913 (* theory -> literal list -> ctype_context -> unit *)

   914 fun print_ctype_context ctxt lits ({frees, consts, ...} : ctype_context) =

   915   map (fn (x, C) => string_for_ctype_of_term ctxt lits (Free x) C) frees @

   916   map (fn (x, (C, _)) => string_for_ctype_of_term ctxt lits (Const x) C) consts

   917   |> cat_lines |> print_g

   919 (* extended_context -> typ -> term list -> term list -> term -> bool *)

   920 fun formulas_monotonic (ext_ctxt as {ctxt, ...}) alpha_T def_ts nondef_ts

   921                        core_t =

   922   let

   923     val _ = print_g ("****** " ^ string_for_ctype CAlpha ^ " is " ^

   924                      Syntax.string_of_typ ctxt alpha_T)

   925     val cdata as {max_fresh, ...} = initial_cdata ext_ctxt alpha_T

   926     val (gamma, cset) =

   927       (initial_gamma, slack)

   928       |> fold (consider_definitional_axiom cdata) def_ts

   929       |> fold (consider_nondefinitional_axiom cdata Pos) nondef_ts

   930       |> consider_general_formula cdata Pos core_t

   931   in

   932     case solve (!max_fresh) cset of

   933       SOME lits => (print_ctype_context ctxt lits gamma; true)

   934     | _ => false

   935   end

   936   handle CTYPE (loc, Cs) => raise BAD (loc, commas (map string_for_ctype Cs))

   938 end;

author	blanchet
	Thu, 29 Oct 2009 21:57:59 +0100
changeset 33565	113e235e84e3
parent 33562	3655e51f9958
child 33571	45c33e97cb86
permissions	-rw-r--r--