wneuper/isa: src/HOL/Tools/refute.ML@600de0485859

     1 (*  Title:      HOL/Tools/refute.ML

     2     Author:     Tjark Weber, TU Muenchen

     4 Finite model generation for HOL formulas, using a SAT solver.

     5 *)

     7 (* ------------------------------------------------------------------------- *)

     8 (* Declares the 'REFUTE' signature as well as a structure 'Refute'.          *)

     9 (* Documentation is available in the Isabelle/Isar theory 'HOL/Refute.thy'.  *)

    10 (* ------------------------------------------------------------------------- *)

    12 signature REFUTE =

    13 sig

    15   exception REFUTE of string * string

    17 (* ------------------------------------------------------------------------- *)

    18 (* Model/interpretation related code (translation HOL -> propositional logic *)

    19 (* ------------------------------------------------------------------------- *)

    21   type params

    22   type interpretation

    23   type model

    24   type arguments

    26   exception MAXVARS_EXCEEDED

    28   val add_interpreter : string -> (Proof.context -> model -> arguments -> term ->

    29     (interpretation * model * arguments) option) -> theory -> theory

    30   val add_printer : string -> (Proof.context -> model -> typ ->

    31     interpretation -> (int -> bool) -> term option) -> theory -> theory

    33   val interpret : Proof.context -> model -> arguments -> term ->

    34     (interpretation * model * arguments)

    36   val print : Proof.context -> model -> typ -> interpretation -> (int -> bool) -> term

    37   val print_model : Proof.context -> model -> (int -> bool) -> string

    39 (* ------------------------------------------------------------------------- *)

    40 (* Interface                                                                 *)

    41 (* ------------------------------------------------------------------------- *)

    43   val set_default_param  : (string * string) -> theory -> theory

    44   val get_default_param  : Proof.context -> string -> string option

    45   val get_default_params : Proof.context -> (string * string) list

    46   val actual_params      : Proof.context -> (string * string) list -> params

    48   val find_model : Proof.context -> params -> term list -> term -> bool -> unit

    50   (* tries to find a model for a formula: *)

    51   val satisfy_term :

    52     Proof.context -> (string * string) list -> term list -> term -> unit

    53   (* tries to find a model that refutes a formula: *)

    54   val refute_term :

    55     Proof.context -> (string * string) list -> term list -> term -> unit

    56   val refute_goal :

    57     Proof.context -> (string * string) list -> thm -> int -> unit

    59   val setup : theory -> theory

    61 (* ------------------------------------------------------------------------- *)

    62 (* Additional functions used by Nitpick (to be factored out)                 *)

    63 (* ------------------------------------------------------------------------- *)

    65   val close_form : term -> term

    66   val get_classdef : theory -> string -> (string * term) option

    67   val norm_rhs : term -> term

    68   val get_def : theory -> string * typ -> (string * term) option

    69   val get_typedef : theory -> typ -> (string * term) option

    70   val is_IDT_constructor : theory -> string * typ -> bool

    71   val is_IDT_recursor : theory -> string * typ -> bool

    72   val is_const_of_class: theory -> string * typ -> bool

    73   val string_of_typ : typ -> string

    74   val typ_of_dtyp : Datatype.descr -> (Datatype.dtyp * typ) list -> Datatype.dtyp -> typ

    75 end;

    77 structure Refute : REFUTE =

    78 struct

    80 open PropLogic;

    82 (* We use 'REFUTE' only for internal error conditions that should    *)

    83 (* never occur in the first place (i.e. errors caused by bugs in our *)

    84 (* code).  Otherwise (e.g. to indicate invalid input data) we use    *)

    85 (* 'error'.                                                          *)

    86 exception REFUTE of string * string;  (* ("in function", "cause") *)

    88 (* should be raised by an interpreter when more variables would be *)

    89 (* required than allowed by 'maxvars'                              *)

    90 exception MAXVARS_EXCEEDED;

    93 (* ------------------------------------------------------------------------- *)

    94 (* TREES                                                                     *)

    95 (* ------------------------------------------------------------------------- *)

    97 (* ------------------------------------------------------------------------- *)

    98 (* tree: implements an arbitrarily (but finitely) branching tree as a list   *)

    99 (*       of (lists of ...) elements                                          *)

   100 (* ------------------------------------------------------------------------- *)

   102 datatype 'a tree =

   103     Leaf of 'a

   104   | Node of ('a tree) list;

   106 (* ('a -> 'b) -> 'a tree -> 'b tree *)

   108 fun tree_map f tr =

   109   case tr of

   110     Leaf x  => Leaf (f x)

   111   | Node xs => Node (map (tree_map f) xs);

   113 (* ('a * 'b -> 'a) -> 'a * ('b tree) -> 'a *)

   115 fun tree_foldl f =

   116   let

   117     fun itl (e, Leaf x)  = f(e,x)

   118       | itl (e, Node xs) = Library.foldl (tree_foldl f) (e,xs)

   119   in

   120     itl

   121   end;

   123 (* 'a tree * 'b tree -> ('a * 'b) tree *)

   125 fun tree_pair (t1, t2) =

   126   case t1 of

   127     Leaf x =>

   128       (case t2 of

   129           Leaf y => Leaf (x,y)

   130         | Node _ => raise REFUTE ("tree_pair",

   131             "trees are of different height (second tree is higher)"))

   132   | Node xs =>

   133       (case t2 of

   134           (* '~~' will raise an exception if the number of branches in   *)

   135           (* both trees is different at the current node                 *)

   136           Node ys => Node (map tree_pair (xs ~~ ys))

   137         | Leaf _  => raise REFUTE ("tree_pair",

   138             "trees are of different height (first tree is higher)"));

   140 (* ------------------------------------------------------------------------- *)

   141 (* params: parameters that control the translation into a propositional      *)

   142 (*         formula/model generation                                          *)

   143 (*                                                                           *)

   144 (* The following parameters are supported (and required (!), except for      *)

   145 (* "sizes" and "expect"):                                                    *)

   146 (*                                                                           *)

   147 (* Name          Type    Description                                         *)

   148 (*                                                                           *)

   149 (* "sizes"       (string * int) list                                         *)

   150 (*                       Size of ground types (e.g. 'a=2), or depth of IDTs. *)

   151 (* "minsize"     int     If >0, minimal size of each ground type/IDT depth.  *)

   152 (* "maxsize"     int     If >0, maximal size of each ground type/IDT depth.  *)

   153 (* "maxvars"     int     If >0, use at most 'maxvars' Boolean variables      *)

   154 (*                       when transforming the term into a propositional     *)

   155 (*                       formula.                                            *)

   156 (* "maxtime"     int     If >0, terminate after at most 'maxtime' seconds.   *)

   157 (* "satsolver"   string  SAT solver to be used.                              *)

   158 (* "no_assms"    bool    If "true", assumptions in structured proofs are     *)

   159 (*                       not considered.                                     *)

   160 (* "expect"      string  Expected result ("genuine", "potential", "none", or *)

   161 (*                       "unknown").                                         *)

   162 (* ------------------------------------------------------------------------- *)

   164 type params =

   165   {

   166     sizes    : (string * int) list,

   167     minsize  : int,

   168     maxsize  : int,

   169     maxvars  : int,

   170     maxtime  : int,

   171     satsolver: string,

   172     no_assms : bool,

   173     expect   : string

   174   };

   176 (* ------------------------------------------------------------------------- *)

   177 (* interpretation: a term's interpretation is given by a variable of type    *)

   178 (*                 'interpretation'                                          *)

   179 (* ------------------------------------------------------------------------- *)

   181 type interpretation =

   182   prop_formula list tree;

   184 (* ------------------------------------------------------------------------- *)

   185 (* model: a model specifies the size of types and the interpretation of      *)

   186 (*        terms                                                              *)

   187 (* ------------------------------------------------------------------------- *)

   189 type model =

   190   (typ * int) list * (term * interpretation) list;

   192 (* ------------------------------------------------------------------------- *)

   193 (* arguments: additional arguments required during interpretation of terms   *)

   194 (* ------------------------------------------------------------------------- *)

   196 type arguments =

   197   {

   198     (* just passed unchanged from 'params': *)

   199     maxvars   : int,

   200     (* whether to use 'make_equality' or 'make_def_equality': *)

   201     def_eq    : bool,

   202     (* the following may change during the translation: *)

   203     next_idx  : int,

   204     bounds    : interpretation list,

   205     wellformed: prop_formula

   206   };

   209 structure Data = Theory_Data

   210 (

   211   type T =

   212     {interpreters: (string * (Proof.context -> model -> arguments -> term ->

   213       (interpretation * model * arguments) option)) list,

   214      printers: (string * (Proof.context -> model -> typ -> interpretation ->

   215       (int -> bool) -> term option)) list,

   216      parameters: string Symtab.table};

   217   val empty = {interpreters = [], printers = [], parameters = Symtab.empty};

   218   val extend = I;

   219   fun merge

   220     ({interpreters = in1, printers = pr1, parameters = pa1},

   221      {interpreters = in2, printers = pr2, parameters = pa2}) : T =

   222     {interpreters = AList.merge (op =) (K true) (in1, in2),

   223      printers = AList.merge (op =) (K true) (pr1, pr2),

   224      parameters = Symtab.merge (op=) (pa1, pa2)};

   225 );

   227 val get_data = Data.get o ProofContext.theory_of;

   230 (* ------------------------------------------------------------------------- *)

   231 (* interpret: interprets the term 't' using a suitable interpreter; returns  *)

   232 (*            the interpretation and a (possibly extended) model that keeps  *)

   233 (*            track of the interpretation of subterms                        *)

   234 (* ------------------------------------------------------------------------- *)

   236 fun interpret ctxt model args t =

   237   case get_first (fn (_, f) => f ctxt model args t)

   238       (#interpreters (get_data ctxt)) of

   239     NONE => raise REFUTE ("interpret",

   240       "no interpreter for term " ^ quote (Syntax.string_of_term ctxt t))

   241   | SOME x => x;

   243 (* ------------------------------------------------------------------------- *)

   244 (* print: converts the interpretation 'intr', which must denote a term of    *)

   245 (*        type 'T', into a term using a suitable printer                     *)

   246 (* ------------------------------------------------------------------------- *)

   248 fun print ctxt model T intr assignment =

   249   case get_first (fn (_, f) => f ctxt model T intr assignment)

   250       (#printers (get_data ctxt)) of

   251     NONE => raise REFUTE ("print",

   252       "no printer for type " ^ quote (Syntax.string_of_typ ctxt T))

   253   | SOME x => x;

   255 (* ------------------------------------------------------------------------- *)

   256 (* print_model: turns the model into a string, using a fixed interpretation  *)

   257 (*              (given by an assignment for Boolean variables) and suitable  *)

   258 (*              printers                                                     *)

   259 (* ------------------------------------------------------------------------- *)

   261 fun print_model ctxt model assignment =

   262   let

   263     val (typs, terms) = model

   264     val typs_msg =

   265       if null typs then

   266         "empty universe (no type variables in term)\n"

   267       else

   268         "Size of types: " ^ commas (map (fn (T, i) =>

   269           Syntax.string_of_typ ctxt T ^ ": " ^ string_of_int i) typs) ^ "\n"

   270     val show_consts_msg =

   271       if not (Config.get ctxt show_consts) andalso Library.exists (is_Const o fst) terms then

   272         "enable \"show_consts\" to show the interpretation of constants\n"

   273       else

   274         ""

   275     val terms_msg =

   276       if null terms then

   277         "empty interpretation (no free variables in term)\n"

   278       else

   279         cat_lines (map_filter (fn (t, intr) =>

   280           (* print constants only if 'show_consts' is true *)

   281           if Config.get ctxt show_consts orelse not (is_Const t) then

   282             SOME (Syntax.string_of_term ctxt t ^ ": " ^

   283               Syntax.string_of_term ctxt

   284                 (print ctxt model (Term.type_of t) intr assignment))

   285           else

   286             NONE) terms) ^ "\n"

   287   in

   288     typs_msg ^ show_consts_msg ^ terms_msg

   289   end;

   292 (* ------------------------------------------------------------------------- *)

   293 (* PARAMETER MANAGEMENT                                                      *)

   294 (* ------------------------------------------------------------------------- *)

   296 fun add_interpreter name f = Data.map (fn {interpreters, printers, parameters} =>

   297   case AList.lookup (op =) interpreters name of

   298     NONE => {interpreters = (name, f) :: interpreters,

   299       printers = printers, parameters = parameters}

   300   | SOME _ => error ("Interpreter " ^ name ^ " already declared"));

   302 fun add_printer name f = Data.map (fn {interpreters, printers, parameters} =>

   303   case AList.lookup (op =) printers name of

   304     NONE => {interpreters = interpreters,

   305       printers = (name, f) :: printers, parameters = parameters}

   306   | SOME _ => error ("Printer " ^ name ^ " already declared"));

   308 (* ------------------------------------------------------------------------- *)

   309 (* set_default_param: stores the '(name, value)' pair in Data's              *)

   310 (*                    parameter table                                        *)

   311 (* ------------------------------------------------------------------------- *)

   313 fun set_default_param (name, value) = Data.map

   314   (fn {interpreters, printers, parameters} =>

   315     {interpreters = interpreters, printers = printers,

   316       parameters = Symtab.update (name, value) parameters});

   318 (* ------------------------------------------------------------------------- *)

   319 (* get_default_param: retrieves the value associated with 'name' from        *)

   320 (*                    Data's parameter table                                 *)

   321 (* ------------------------------------------------------------------------- *)

   323 val get_default_param = Symtab.lookup o #parameters o get_data;

   325 (* ------------------------------------------------------------------------- *)

   326 (* get_default_params: returns a list of all '(name, value)' pairs that are  *)

   327 (*                     stored in Data's parameter table                      *)

   328 (* ------------------------------------------------------------------------- *)

   330 val get_default_params = Symtab.dest o #parameters o get_data;

   332 (* ------------------------------------------------------------------------- *)

   333 (* actual_params: takes a (possibly empty) list 'params' of parameters that  *)

   334 (*      override the default parameters currently specified, and             *)

   335 (*      returns a record that can be passed to 'find_model'.                 *)

   336 (* ------------------------------------------------------------------------- *)

   338 fun actual_params ctxt override =

   339   let

   340     (* (string * string) list * string -> bool *)

   341     fun read_bool (parms, name) =

   342       case AList.lookup (op =) parms name of

   343         SOME "true" => true

   344       | SOME "false" => false

   345       | SOME s => error ("parameter " ^ quote name ^

   346           " (value is " ^ quote s ^ ") must be \"true\" or \"false\"")

   347       | NONE   => error ("parameter " ^ quote name ^

   348           " must be assigned a value")

   349     (* (string * string) list * string -> int *)

   350     fun read_int (parms, name) =

   351       case AList.lookup (op =) parms name of

   352         SOME s =>

   353           (case Int.fromString s of

   354             SOME i => i

   355           | NONE   => error ("parameter " ^ quote name ^

   356             " (value is " ^ quote s ^ ") must be an integer value"))

   357       | NONE => error ("parameter " ^ quote name ^

   358           " must be assigned a value")

   359     (* (string * string) list * string -> string *)

   360     fun read_string (parms, name) =

   361       case AList.lookup (op =) parms name of

   362         SOME s => s

   363       | NONE => error ("parameter " ^ quote name ^

   364         " must be assigned a value")

   365     (* 'override' first, defaults last: *)

   366     (* (string * string) list *)

   367     val allparams = override @ get_default_params ctxt

   368     (* int *)

   369     val minsize = read_int (allparams, "minsize")

   370     val maxsize = read_int (allparams, "maxsize")

   371     val maxvars = read_int (allparams, "maxvars")

   372     val maxtime = read_int (allparams, "maxtime")

   373     (* string *)

   374     val satsolver = read_string (allparams, "satsolver")

   375     val no_assms = read_bool (allparams, "no_assms")

   376     val expect = the_default "" (AList.lookup (op =) allparams "expect")

   377     (* all remaining parameters of the form "string=int" are collected in *)

   378     (* 'sizes'                                                            *)

   379     (* TODO: it is currently not possible to specify a size for a type    *)

   380     (*       whose name is one of the other parameters (e.g. 'maxvars')   *)

   381     (* (string * int) list *)

   382     val sizes = map_filter

   383       (fn (name, value) => Option.map (pair name) (Int.fromString value))

   384       (filter (fn (name, _) => name<>"minsize" andalso name<>"maxsize"

   385         andalso name<>"maxvars" andalso name<>"maxtime"

   386         andalso name<>"satsolver" andalso name<>"no_assms") allparams)

   387   in

   388     {sizes=sizes, minsize=minsize, maxsize=maxsize, maxvars=maxvars,

   389       maxtime=maxtime, satsolver=satsolver, no_assms=no_assms, expect=expect}

   390   end;

   393 (* ------------------------------------------------------------------------- *)

   394 (* TRANSLATION HOL -> PROPOSITIONAL LOGIC, BOOLEAN ASSIGNMENT -> MODEL       *)

   395 (* ------------------------------------------------------------------------- *)

   397 fun typ_of_dtyp descr typ_assoc (Datatype_Aux.DtTFree a) =

   398       (* replace a 'DtTFree' variable by the associated type *)

   399       the (AList.lookup (op =) typ_assoc (Datatype_Aux.DtTFree a))

   400   | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtType (s, ds)) =

   401       Type (s, map (typ_of_dtyp descr typ_assoc) ds)

   402   | typ_of_dtyp descr typ_assoc (Datatype_Aux.DtRec i) =

   403       let

   404         val (s, ds, _) = the (AList.lookup (op =) descr i)

   405       in

   406         Type (s, map (typ_of_dtyp descr typ_assoc) ds)

   407       end;

   409 (* ------------------------------------------------------------------------- *)

   410 (* close_form: universal closure over schematic variables in 't'             *)

   411 (* ------------------------------------------------------------------------- *)

   413 (* Term.term -> Term.term *)

   415 fun close_form t =

   416   let

   417     (* (Term.indexname * Term.typ) list *)

   418     val vars = sort_wrt (fst o fst) (map dest_Var (OldTerm.term_vars t))

   419   in

   420     fold (fn ((x, i), T) => fn t' =>

   421       Term.all T $ Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t

   422   end;

   424 val monomorphic_term = Sledgehammer_Util.monomorphic_term

   425 val specialize_type = Sledgehammer_Util.specialize_type

   427 (* ------------------------------------------------------------------------- *)

   428 (* is_const_of_class: returns 'true' iff 'Const (s, T)' is a constant that   *)

   429 (*                    denotes membership to an axiomatic type class          *)

   430 (* ------------------------------------------------------------------------- *)

   432 fun is_const_of_class thy (s, T) =

   433   let

   434     val class_const_names = map Logic.const_of_class (Sign.all_classes thy)

   435   in

   436     (* I'm not quite sure if checking the name 's' is sufficient, *)

   437     (* or if we should also check the type 'T'.                   *)

   438     member (op =) class_const_names s

   439   end;

   441 (* ------------------------------------------------------------------------- *)

   442 (* is_IDT_constructor: returns 'true' iff 'Const (s, T)' is the constructor  *)

   443 (*                     of an inductive datatype in 'thy'                     *)

   444 (* ------------------------------------------------------------------------- *)

   446 fun is_IDT_constructor thy (s, T) =

   447   (case body_type T of

   448     Type (s', _) =>

   449       (case Datatype.get_constrs thy s' of

   450         SOME constrs =>

   451           List.exists (fn (cname, cty) =>

   452             cname = s andalso Sign.typ_instance thy (T, cty)) constrs

   453       | NONE => false)

   454   | _  => false);

   456 (* ------------------------------------------------------------------------- *)

   457 (* is_IDT_recursor: returns 'true' iff 'Const (s, T)' is the recursion       *)

   458 (*                  operator of an inductive datatype in 'thy'               *)

   459 (* ------------------------------------------------------------------------- *)

   461 fun is_IDT_recursor thy (s, T) =

   462   let

   463     val rec_names = Symtab.fold (append o #rec_names o snd)

   464       (Datatype.get_all thy) []

   465   in

   466     (* I'm not quite sure if checking the name 's' is sufficient, *)

   467     (* or if we should also check the type 'T'.                   *)

   468     member (op =) rec_names s

   469   end;

   471 (* ------------------------------------------------------------------------- *)

   472 (* norm_rhs: maps  f ?t1 ... ?tn == rhs  to  %t1...tn. rhs                   *)

   473 (* ------------------------------------------------------------------------- *)

   475 fun norm_rhs eqn =

   476   let

   477     fun lambda (v as Var ((x, _), T)) t = Abs (x, T, abstract_over (v, t))

   478       | lambda v t = raise TERM ("lambda", [v, t])

   479     val (lhs, rhs) = Logic.dest_equals eqn

   480     val (_, args) = Term.strip_comb lhs

   481   in

   482     fold lambda (rev args) rhs

   483   end

   485 (* ------------------------------------------------------------------------- *)

   486 (* get_def: looks up the definition of a constant                            *)

   487 (* ------------------------------------------------------------------------- *)

   489 fun get_def thy (s, T) =

   490   let

   491     (* (string * Term.term) list -> (string * Term.term) option *)

   492     fun get_def_ax [] = NONE

   493       | get_def_ax ((axname, ax) :: axioms) =

   494           (let

   495             val (lhs, _) = Logic.dest_equals ax  (* equations only *)

   496             val c        = Term.head_of lhs

   497             val (s', T') = Term.dest_Const c

   498           in

   499             if s=s' then

   500               let

   501                 val typeSubs = Sign.typ_match thy (T', T) Vartab.empty

   502                 val ax'      = monomorphic_term typeSubs ax

   503                 val rhs      = norm_rhs ax'

   504               in

   505                 SOME (axname, rhs)

   506               end

   507             else

   508               get_def_ax axioms

   509           end handle ERROR _         => get_def_ax axioms

   510                    | TERM _          => get_def_ax axioms

   511                    | Type.TYPE_MATCH => get_def_ax axioms)

   512   in

   513     get_def_ax (Theory.all_axioms_of thy)

   514   end;

   516 (* ------------------------------------------------------------------------- *)

   517 (* get_typedef: looks up the definition of a type, as created by "typedef"   *)

   518 (* ------------------------------------------------------------------------- *)

   520 fun get_typedef thy T =

   521   let

   522     (* (string * Term.term) list -> (string * Term.term) option *)

   523     fun get_typedef_ax [] = NONE

   524       | get_typedef_ax ((axname, ax) :: axioms) =

   525           (let

   526             (* Term.term -> Term.typ option *)

   527             fun type_of_type_definition (Const (s', T')) =

   528                   if s'= @{const_name type_definition} then

   529                     SOME T'

   530                   else

   531                     NONE

   532               | type_of_type_definition (Free _) = NONE

   533               | type_of_type_definition (Var _) = NONE

   534               | type_of_type_definition (Bound _) = NONE

   535               | type_of_type_definition (Abs (_, _, body)) =

   536                   type_of_type_definition body

   537               | type_of_type_definition (t1 $ t2) =

   538                   (case type_of_type_definition t1 of

   539                     SOME x => SOME x

   540                   | NONE => type_of_type_definition t2)

   541           in

   542             case type_of_type_definition ax of

   543               SOME T' =>

   544                 let

   545                   val T'' = domain_type (domain_type T')

   546                   val typeSubs = Sign.typ_match thy (T'', T) Vartab.empty

   547                 in

   548                   SOME (axname, monomorphic_term typeSubs ax)

   549                 end

   550             | NONE => get_typedef_ax axioms

   551           end handle ERROR _         => get_typedef_ax axioms

   552                    | TERM _          => get_typedef_ax axioms

   553                    | Type.TYPE_MATCH => get_typedef_ax axioms)

   554   in

   555     get_typedef_ax (Theory.all_axioms_of thy)

   556   end;

   558 (* ------------------------------------------------------------------------- *)

   559 (* get_classdef: looks up the defining axiom for an axiomatic type class, as *)

   560 (*               created by the "axclass" command                            *)

   561 (* ------------------------------------------------------------------------- *)

   563 fun get_classdef thy class =

   564   let

   565     val axname = class ^ "_class_def"

   566   in

   567     Option.map (pair axname)

   568       (AList.lookup (op =) (Theory.all_axioms_of thy) axname)

   569   end;

   571 (* ------------------------------------------------------------------------- *)

   572 (* unfold_defs: unfolds all defined constants in a term 't', beta-eta        *)

   573 (*              normalizes the result term; certain constants are not        *)

   574 (*              unfolded (cf. 'collect_axioms' and the various interpreters  *)

   575 (*              below): if the interpretation respects a definition anyway,  *)

   576 (*              that definition does not need to be unfolded                 *)

   577 (* ------------------------------------------------------------------------- *)

   579 (* Note: we could intertwine unfolding of constants and beta-(eta-)       *)

   580 (*       normalization; this would save some unfolding for terms where    *)

   581 (*       constants are eliminated by beta-reduction (e.g. 'K c1 c2').  On *)

   582 (*       the other hand, this would cause additional work for terms where *)

   583 (*       constants are duplicated by beta-reduction (e.g. 'S c1 c2 c3').  *)

   585 fun unfold_defs thy t =

   586   let

   587     (* Term.term -> Term.term *)

   588     fun unfold_loop t =

   589       case t of

   590       (* Pure *)

   591         Const (@{const_name all}, _) => t

   592       | Const (@{const_name "=="}, _) => t

   593       | Const (@{const_name "==>"}, _) => t

   594       | Const (@{const_name TYPE}, _) => t  (* axiomatic type classes *)

   595       (* HOL *)

   596       | Const (@{const_name Trueprop}, _) => t

   597       | Const (@{const_name Not}, _) => t

   598       | (* redundant, since 'True' is also an IDT constructor *)

   599         Const (@{const_name True}, _) => t

   600       | (* redundant, since 'False' is also an IDT constructor *)

   601         Const (@{const_name False}, _) => t

   602       | Const (@{const_name undefined}, _) => t

   603       | Const (@{const_name The}, _) => t

   604       | Const (@{const_name Hilbert_Choice.Eps}, _) => t

   605       | Const (@{const_name All}, _) => t

   606       | Const (@{const_name Ex}, _) => t

   607       | Const (@{const_name HOL.eq}, _) => t

   608       | Const (@{const_name HOL.conj}, _) => t

   609       | Const (@{const_name HOL.disj}, _) => t

   610       | Const (@{const_name HOL.implies}, _) => t

   611       (* sets *)

   612       | Const (@{const_name Collect}, _) => t

   613       | Const (@{const_name Set.member}, _) => t

   614       (* other optimizations *)

   615       | Const (@{const_name Finite_Set.card}, _) => t

   616       | Const (@{const_name Finite_Set.finite}, _) => t

   617       | Const (@{const_name Orderings.less}, Type ("fun", [@{typ nat},

   618           Type ("fun", [@{typ nat}, @{typ bool}])])) => t

   619       | Const (@{const_name Groups.plus}, Type ("fun", [@{typ nat},

   620           Type ("fun", [@{typ nat}, @{typ nat}])])) => t

   621       | Const (@{const_name Groups.minus}, Type ("fun", [@{typ nat},

   622           Type ("fun", [@{typ nat}, @{typ nat}])])) => t

   623       | Const (@{const_name Groups.times}, Type ("fun", [@{typ nat},

   624           Type ("fun", [@{typ nat}, @{typ nat}])])) => t

   625       | Const (@{const_name List.append}, _) => t

   626 (* UNSOUND

   627       | Const (@{const_name lfp}, _) => t

   628       | Const (@{const_name gfp}, _) => t

   629 *)

   630       | Const (@{const_name fst}, _) => t

   631       | Const (@{const_name snd}, _) => t

   632       (* simply-typed lambda calculus *)

   633       | Const (s, T) =>

   634           (if is_IDT_constructor thy (s, T)

   635             orelse is_IDT_recursor thy (s, T) then

   636             t  (* do not unfold IDT constructors/recursors *)

   637           (* unfold the constant if there is a defining equation *)

   638           else

   639             case get_def thy (s, T) of

   640               SOME (axname, rhs) =>

   641               (* Note: if the term to be unfolded (i.e. 'Const (s, T)')  *)

   642               (* occurs on the right-hand side of the equation, i.e. in  *)

   643               (* 'rhs', we must not use this equation to unfold, because *)

   644               (* that would loop.  Here would be the right place to      *)

   645               (* check this.  However, getting this really right seems   *)

   646               (* difficult because the user may state arbitrary axioms,  *)

   647               (* which could interact with overloading to create loops.  *)

   648               ((*tracing (" unfolding: " ^ axname);*)

   649                unfold_loop rhs)

   650             | NONE => t)

   651       | Free _ => t

   652       | Var _ => t

   653       | Bound _ => t

   654       | Abs (s, T, body) => Abs (s, T, unfold_loop body)

   655       | t1 $ t2 => (unfold_loop t1) $ (unfold_loop t2)

   656     val result = Envir.beta_eta_contract (unfold_loop t)

   657   in

   658     result

   659   end;

   661 (* ------------------------------------------------------------------------- *)

   662 (* collect_axioms: collects (monomorphic, universally quantified, unfolded   *)

   663 (*                 versions of) all HOL axioms that are relevant w.r.t 't'   *)

   664 (* ------------------------------------------------------------------------- *)

   666 (* Note: to make the collection of axioms more easily extensible, this    *)

   667 (*       function could be based on user-supplied "axiom collectors",     *)

   668 (*       similar to 'interpret'/interpreters or 'print'/printers          *)

   670 (* Note: currently we use "inverse" functions to the definitional         *)

   671 (*       mechanisms provided by Isabelle/HOL, e.g. for "axclass",         *)

   672 (*       "typedef", "definition".  A more general approach could consider *)

   673 (*       *every* axiom of the theory and collect it if it has a constant/ *)

   674 (*       type/typeclass in common with the term 't'.                      *)

   676 (* Which axioms are "relevant" for a particular term/type goes hand in    *)

   677 (* hand with the interpretation of that term/type by its interpreter (see *)

   678 (* way below): if the interpretation respects an axiom anyway, the axiom  *)

   679 (* does not need to be added as a constraint here.                        *)

   681 (* To avoid collecting the same axiom multiple times, we use an           *)

   682 (* accumulator 'axs' which contains all axioms collected so far.          *)

   684 fun collect_axioms ctxt t =

   685   let

   686     val thy = ProofContext.theory_of ctxt

   687     val _ = tracing "Adding axioms..."

   688     val axioms = Theory.all_axioms_of thy

   689     fun collect_this_axiom (axname, ax) axs =

   690       let

   691         val ax' = unfold_defs thy ax

   692       in

   693         if member (op aconv) axs ax' then axs

   694         else (tracing axname; collect_term_axioms ax' (ax' :: axs))

   695       end

   696     and collect_sort_axioms T axs =

   697       let

   698         val sort =

   699           (case T of

   700             TFree (_, sort) => sort

   701           | TVar (_, sort)  => sort

   702           | _ => raise REFUTE ("collect_axioms",

   703               "type " ^ Syntax.string_of_typ ctxt T ^ " is not a variable"))

   704         (* obtain axioms for all superclasses *)

   705         val superclasses = sort @ maps (Sign.super_classes thy) sort

   706         (* merely an optimization, because 'collect_this_axiom' disallows *)

   707         (* duplicate axioms anyway:                                       *)

   708         val superclasses = distinct (op =) superclasses

   709         val class_axioms = maps (fn class => map (fn ax =>

   710           ("<" ^ class ^ ">", Thm.prop_of ax))

   711           (#axioms (AxClass.get_info thy class) handle ERROR _ => []))

   712           superclasses

   713         (* replace the (at most one) schematic type variable in each axiom *)

   714         (* by the actual type 'T'                                          *)

   715         val monomorphic_class_axioms = map (fn (axname, ax) =>

   716           (case Term.add_tvars ax [] of

   717             [] => (axname, ax)

   718           | [(idx, S)] => (axname, monomorphic_term (Vartab.make [(idx, (S, T))]) ax)

   719           | _ =>

   720             raise REFUTE ("collect_axioms", "class axiom " ^ axname ^ " (" ^

   721               Syntax.string_of_term ctxt ax ^

   722               ") contains more than one type variable")))

   723           class_axioms

   724       in

   725         fold collect_this_axiom monomorphic_class_axioms axs

   726       end

   727     and collect_type_axioms T axs =

   728       case T of

   729       (* simple types *)

   730         Type ("prop", []) => axs

   731       | Type ("fun", [T1, T2]) => collect_type_axioms T2 (collect_type_axioms T1 axs)

   732       (* axiomatic type classes *)

   733       | Type ("itself", [T1]) => collect_type_axioms T1 axs

   734       | Type (s, Ts) =>

   735         (case Datatype.get_info thy s of

   736           SOME info =>  (* inductive datatype *)

   737             (* only collect relevant type axioms for the argument types *)

   738             fold collect_type_axioms Ts axs

   739         | NONE =>

   740           (case get_typedef thy T of

   741             SOME (axname, ax) =>

   742               collect_this_axiom (axname, ax) axs

   743           | NONE =>

   744             (* unspecified type, perhaps introduced with "typedecl" *)

   745             (* at least collect relevant type axioms for the argument types *)

   746             fold collect_type_axioms Ts axs))

   747       (* axiomatic type classes *)

   748       | TFree _ => collect_sort_axioms T axs

   749       (* axiomatic type classes *)

   750       | TVar _ => collect_sort_axioms T axs

   751     and collect_term_axioms t axs =

   752       case t of

   753       (* Pure *)

   754         Const (@{const_name all}, _) => axs

   755       | Const (@{const_name "=="}, _) => axs

   756       | Const (@{const_name "==>"}, _) => axs

   757       (* axiomatic type classes *)

   758       | Const (@{const_name TYPE}, T) => collect_type_axioms T axs

   759       (* HOL *)

   760       | Const (@{const_name Trueprop}, _) => axs

   761       | Const (@{const_name Not}, _) => axs

   762       (* redundant, since 'True' is also an IDT constructor *)

   763       | Const (@{const_name True}, _) => axs

   764       (* redundant, since 'False' is also an IDT constructor *)

   765       | Const (@{const_name False}, _) => axs

   766       | Const (@{const_name undefined}, T) => collect_type_axioms T axs

   767       | Const (@{const_name The}, T) =>

   768           let

   769             val ax = specialize_type thy (@{const_name The}, T)

   770               (the (AList.lookup (op =) axioms "HOL.the_eq_trivial"))

   771           in

   772             collect_this_axiom ("HOL.the_eq_trivial", ax) axs

   773           end

   774       | Const (@{const_name Hilbert_Choice.Eps}, T) =>

   775           let

   776             val ax = specialize_type thy (@{const_name Hilbert_Choice.Eps}, T)

   777               (the (AList.lookup (op =) axioms "Hilbert_Choice.someI"))

   778           in

   779             collect_this_axiom ("Hilbert_Choice.someI", ax) axs

   780           end

   781       | Const (@{const_name All}, T) => collect_type_axioms T axs

   782       | Const (@{const_name Ex}, T) => collect_type_axioms T axs

   783       | Const (@{const_name HOL.eq}, T) => collect_type_axioms T axs

   784       | Const (@{const_name HOL.conj}, _) => axs

   785       | Const (@{const_name HOL.disj}, _) => axs

   786       | Const (@{const_name HOL.implies}, _) => axs

   787       (* sets *)

   788       | Const (@{const_name Collect}, T) => collect_type_axioms T axs

   789       | Const (@{const_name Set.member}, T) => collect_type_axioms T axs

   790       (* other optimizations *)

   791       | Const (@{const_name Finite_Set.card}, T) => collect_type_axioms T axs

   792       | Const (@{const_name Finite_Set.finite}, T) =>

   793         collect_type_axioms T axs

   794       | Const (@{const_name Orderings.less}, T as Type ("fun", [@{typ nat},

   795         Type ("fun", [@{typ nat}, @{typ bool}])])) =>

   796           collect_type_axioms T axs

   797       | Const (@{const_name Groups.plus}, T as Type ("fun", [@{typ nat},

   798         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

   799           collect_type_axioms T axs

   800       | Const (@{const_name Groups.minus}, T as Type ("fun", [@{typ nat},

   801         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

   802           collect_type_axioms T axs

   803       | Const (@{const_name Groups.times}, T as Type ("fun", [@{typ nat},

   804         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

   805           collect_type_axioms T axs

   806       | Const (@{const_name List.append}, T) => collect_type_axioms T axs

   807 (* UNSOUND

   808       | Const (@{const_name lfp}, T) => collect_type_axioms T axs

   809       | Const (@{const_name gfp}, T) => collect_type_axioms T axs

   810 *)

   811       | Const (@{const_name fst}, T) => collect_type_axioms T axs

   812       | Const (@{const_name snd}, T) => collect_type_axioms T axs

   813       (* simply-typed lambda calculus *)

   814       | Const (s, T) =>

   815           if is_const_of_class thy (s, T) then

   816             (* axiomatic type classes: add "OFCLASS(?'a::c, c_class)" *)

   817             (* and the class definition                               *)

   818             let

   819               val class = Logic.class_of_const s

   820               val of_class = Logic.mk_of_class (TVar (("'a", 0), [class]), class)

   821               val ax_in = SOME (specialize_type thy (s, T) of_class)

   822                 (* type match may fail due to sort constraints *)

   823                 handle Type.TYPE_MATCH => NONE

   824               val ax_1 = Option.map (fn ax => (Syntax.string_of_term ctxt ax, ax)) ax_in

   825               val ax_2 = Option.map (apsnd (specialize_type thy (s, T))) (get_classdef thy class)

   826             in

   827               collect_type_axioms T (fold collect_this_axiom (map_filter I [ax_1, ax_2]) axs)

   828             end

   829           else if is_IDT_constructor thy (s, T)

   830             orelse is_IDT_recursor thy (s, T)

   831           then

   832             (* only collect relevant type axioms *)

   833             collect_type_axioms T axs

   834           else

   835             (* other constants should have been unfolded, with some *)

   836             (* exceptions: e.g. Abs_xxx/Rep_xxx functions for       *)

   837             (* typedefs, or type-class related constants            *)

   838             (* only collect relevant type axioms *)

   839             collect_type_axioms T axs

   840       | Free (_, T) => collect_type_axioms T axs

   841       | Var (_, T) => collect_type_axioms T axs

   842       | Bound _ => axs

   843       | Abs (_, T, body) => collect_term_axioms body (collect_type_axioms T axs)

   844       | t1 $ t2 => collect_term_axioms t2 (collect_term_axioms t1 axs)

   845     val result = map close_form (collect_term_axioms t [])

   846     val _ = tracing " ...done."

   847   in

   848     result

   849   end;

   851 (* ------------------------------------------------------------------------- *)

   852 (* ground_types: collects all ground types in a term (including argument     *)

   853 (*               types of other types), suppressing duplicates.  Does not    *)

   854 (*               return function types, set types, non-recursive IDTs, or    *)

   855 (*               'propT'.  For IDTs, also the argument types of constructors *)

   856 (*               and all mutually recursive IDTs are considered.             *)

   857 (* ------------------------------------------------------------------------- *)

   859 fun ground_types ctxt t =

   860   let

   861     val thy = ProofContext.theory_of ctxt

   862     fun collect_types T acc =

   863       (case T of

   864         Type ("fun", [T1, T2]) => collect_types T1 (collect_types T2 acc)

   865       | Type ("prop", []) => acc

   866       | Type (s, Ts) =>

   867           (case Datatype.get_info thy s of

   868             SOME info =>  (* inductive datatype *)

   869               let

   870                 val index = #index info

   871                 val descr = #descr info

   872                 val (_, typs, _) = the (AList.lookup (op =) descr index)

   873                 val typ_assoc = typs ~~ Ts

   874                 (* sanity check: every element in 'dtyps' must be a *)

   875                 (* 'DtTFree'                                        *)

   876                 val _ = if Library.exists (fn d =>

   877                   case d of Datatype_Aux.DtTFree _ => false | _ => true) typs then

   878                   raise REFUTE ("ground_types", "datatype argument (for type "

   879                     ^ Syntax.string_of_typ ctxt T ^ ") is not a variable")

   880                 else ()

   881                 (* required for mutually recursive datatypes; those need to   *)

   882                 (* be added even if they are an instance of an otherwise non- *)

   883                 (* recursive datatype                                         *)

   884                 fun collect_dtyp d acc =

   885                   let

   886                     val dT = typ_of_dtyp descr typ_assoc d

   887                   in

   888                     case d of

   889                       Datatype_Aux.DtTFree _ =>

   890                       collect_types dT acc

   891                     | Datatype_Aux.DtType (_, ds) =>

   892                       collect_types dT (fold_rev collect_dtyp ds acc)

   893                     | Datatype_Aux.DtRec i =>

   894                       if member (op =) acc dT then

   895                         acc  (* prevent infinite recursion *)

   896                       else

   897                         let

   898                           val (_, dtyps, dconstrs) = the (AList.lookup (op =) descr i)

   899                           (* if the current type is a recursive IDT (i.e. a depth *)

   900                           (* is required), add it to 'acc'                        *)

   901                           val acc_dT = if Library.exists (fn (_, ds) =>

   902                             Library.exists Datatype_Aux.is_rec_type ds) dconstrs then

   903                               insert (op =) dT acc

   904                             else acc

   905                           (* collect argument types *)

   906                           val acc_dtyps = fold_rev collect_dtyp dtyps acc_dT

   907                           (* collect constructor types *)

   908                           val acc_dconstrs = fold_rev collect_dtyp (maps snd dconstrs) acc_dtyps

   909                         in

   910                           acc_dconstrs

   911                         end

   912                   end

   913               in

   914                 (* argument types 'Ts' could be added here, but they are also *)

   915                 (* added by 'collect_dtyp' automatically                      *)

   916                 collect_dtyp (Datatype_Aux.DtRec index) acc

   917               end

   918           | NONE =>

   919             (* not an inductive datatype, e.g. defined via "typedef" or *)

   920             (* "typedecl"                                               *)

   921             insert (op =) T (fold collect_types Ts acc))

   922       | TFree _ => insert (op =) T acc

   923       | TVar _ => insert (op =) T acc)

   924   in

   925     fold_types collect_types t []

   926   end;

   928 (* ------------------------------------------------------------------------- *)

   929 (* string_of_typ: (rather naive) conversion from types to strings, used to   *)

   930 (*                look up the size of a type in 'sizes'.  Parameterized      *)

   931 (*                types with different parameters (e.g. "'a list" vs. "bool  *)

   932 (*                list") are identified.                                     *)

   933 (* ------------------------------------------------------------------------- *)

   935 (* Term.typ -> string *)

   937 fun string_of_typ (Type (s, _))     = s

   938   | string_of_typ (TFree (s, _))    = s

   939   | string_of_typ (TVar ((s,_), _)) = s;

   941 (* ------------------------------------------------------------------------- *)

   942 (* first_universe: returns the "first" (i.e. smallest) universe by assigning *)

   943 (*                 'minsize' to every type for which no size is specified in *)

   944 (*                 'sizes'                                                   *)

   945 (* ------------------------------------------------------------------------- *)

   947 (* Term.typ list -> (string * int) list -> int -> (Term.typ * int) list *)

   949 fun first_universe xs sizes minsize =

   950   let

   951     fun size_of_typ T =

   952       case AList.lookup (op =) sizes (string_of_typ T) of

   953         SOME n => n

   954       | NONE => minsize

   955   in

   956     map (fn T => (T, size_of_typ T)) xs

   957   end;

   959 (* ------------------------------------------------------------------------- *)

   960 (* next_universe: enumerates all universes (i.e. assignments of sizes to     *)

   961 (*                types), where the minimal size of a type is given by       *)

   962 (*                'minsize', the maximal size is given by 'maxsize', and a   *)

   963 (*                type may have a fixed size given in 'sizes'                *)

   964 (* ------------------------------------------------------------------------- *)

   966 (* (Term.typ * int) list -> (string * int) list -> int -> int ->

   967   (Term.typ * int) list option *)

   969 fun next_universe xs sizes minsize maxsize =

   970   let

   971     (* creates the "first" list of length 'len', where the sum of all list *)

   972     (* elements is 'sum', and the length of the list is 'len'              *)

   973     (* int -> int -> int -> int list option *)

   974     fun make_first _ 0 sum =

   975           if sum = 0 then

   976             SOME []

   977           else

   978             NONE

   979       | make_first max len sum =

   980           if sum <= max orelse max < 0 then

   981             Option.map (fn xs' => sum :: xs') (make_first max (len-1) 0)

   982           else

   983             Option.map (fn xs' => max :: xs') (make_first max (len-1) (sum-max))

   984     (* enumerates all int lists with a fixed length, where 0<=x<='max' for *)

   985     (* all list elements x (unless 'max'<0)                                *)

   986     (* int -> int -> int -> int list -> int list option *)

   987     fun next max len sum [] =

   988           NONE

   989       | next max len sum [x] =

   990           (* we've reached the last list element, so there's no shift possible *)

   991           make_first max (len+1) (sum+x+1)  (* increment 'sum' by 1 *)

   992       | next max len sum (x1::x2::xs) =

   993           if x1>0 andalso (x2<max orelse max<0) then

   994             (* we can shift *)

   995             SOME (the (make_first max (len+1) (sum+x1-1)) @ (x2+1) :: xs)

   996           else

   997             (* continue search *)

   998             next max (len+1) (sum+x1) (x2::xs)

   999     (* only consider those types for which the size is not fixed *)

  1000     val mutables = filter_out (AList.defined (op =) sizes o string_of_typ o fst) xs

  1001     (* subtract 'minsize' from every size (will be added again at the end) *)

  1002     val diffs = map (fn (_, n) => n-minsize) mutables

  1003   in

  1004     case next (maxsize-minsize) 0 0 diffs of

  1005       SOME diffs' =>

  1006         (* merge with those types for which the size is fixed *)

  1007         SOME (fst (fold_map (fn (T, _) => fn ds =>

  1008           case AList.lookup (op =) sizes (string_of_typ T) of

  1009           (* return the fixed size *)

  1010             SOME n => ((T, n), ds)

  1011           (* consume the head of 'ds', add 'minsize' *)

  1012           | NONE   => ((T, minsize + hd ds), tl ds))

  1013           xs diffs'))

  1014     | NONE => NONE

  1015   end;

  1017 (* ------------------------------------------------------------------------- *)

  1018 (* toTrue: converts the interpretation of a Boolean value to a propositional *)

  1019 (*         formula that is true iff the interpretation denotes "true"        *)

  1020 (* ------------------------------------------------------------------------- *)

  1022 (* interpretation -> prop_formula *)

  1024 fun toTrue (Leaf [fm, _]) = fm

  1025   | toTrue _ = raise REFUTE ("toTrue", "interpretation does not denote a Boolean value");

  1027 (* ------------------------------------------------------------------------- *)

  1028 (* toFalse: converts the interpretation of a Boolean value to a              *)

  1029 (*          propositional formula that is true iff the interpretation        *)

  1030 (*          denotes "false"                                                  *)

  1031 (* ------------------------------------------------------------------------- *)

  1033 (* interpretation -> prop_formula *)

  1035 fun toFalse (Leaf [_, fm]) = fm

  1036   | toFalse _ = raise REFUTE ("toFalse", "interpretation does not denote a Boolean value");

  1038 (* ------------------------------------------------------------------------- *)

  1039 (* find_model: repeatedly calls 'interpret' with appropriate parameters,     *)

  1040 (*             applies a SAT solver, and (in case a model is found) displays *)

  1041 (*             the model to the user by calling 'print_model'                *)

  1042 (* {...}     : parameters that control the translation/model generation      *)

  1043 (* assm_ts   : assumptions to be considered unless "no_assms" is specified   *)

  1044 (* t         : term to be translated into a propositional formula            *)

  1045 (* negate    : if true, find a model that makes 't' false (rather than true) *)

  1046 (* ------------------------------------------------------------------------- *)

  1048 fun find_model ctxt

  1049     {sizes, minsize, maxsize, maxvars, maxtime, satsolver, no_assms, expect}

  1050     assm_ts t negate =

  1051   let

  1052     val thy = ProofContext.theory_of ctxt

  1053     (* string -> unit *)

  1054     fun check_expect outcome_code =

  1055       if expect = "" orelse outcome_code = expect then ()

  1056       else error ("Unexpected outcome: " ^ quote outcome_code ^ ".")

  1057     (* unit -> unit *)

  1058     fun wrapper () =

  1059       let

  1060         val timer = Timer.startRealTimer ()

  1061         val t =

  1062           if no_assms then t

  1063           else if negate then Logic.list_implies (assm_ts, t)

  1064           else Logic.mk_conjunction_list (t :: assm_ts)

  1065         val u = unfold_defs thy t

  1066         val _ = tracing ("Unfolded term: " ^ Syntax.string_of_term ctxt u)

  1067         val axioms = collect_axioms ctxt u

  1068         (* Term.typ list *)

  1069         val types = fold (union (op =) o ground_types ctxt) (u :: axioms) []

  1070         val _ = tracing ("Ground types: "

  1071           ^ (if null types then "none."

  1072              else commas (map (Syntax.string_of_typ ctxt) types)))

  1073         (* we can only consider fragments of recursive IDTs, so we issue a  *)

  1074         (* warning if the formula contains a recursive IDT                  *)

  1075         (* TODO: no warning needed for /positive/ occurrences of IDTs       *)

  1076         val maybe_spurious = Library.exists (fn

  1077             Type (s, _) =>

  1078               (case Datatype.get_info thy s of

  1079                 SOME info =>  (* inductive datatype *)

  1080                   let

  1081                     val index           = #index info

  1082                     val descr           = #descr info

  1083                     val (_, _, constrs) = the (AList.lookup (op =) descr index)

  1084                   in

  1085                     (* recursive datatype? *)

  1086                     Library.exists (fn (_, ds) =>

  1087                       Library.exists Datatype_Aux.is_rec_type ds) constrs

  1088                   end

  1089               | NONE => false)

  1090           | _ => false) types

  1091         val _ =

  1092           if maybe_spurious then

  1093             warning ("Term contains a recursive datatype; "

  1094               ^ "countermodel(s) may be spurious!")

  1095           else

  1096             ()

  1097         (* (Term.typ * int) list -> string *)

  1098         fun find_model_loop universe =

  1099           let

  1100             val msecs_spent = Time.toMilliseconds (Timer.checkRealTimer timer)

  1101             val _ = maxtime = 0 orelse msecs_spent < 1000 * maxtime

  1102                     orelse raise TimeLimit.TimeOut

  1103             val init_model = (universe, [])

  1104             val init_args  = {maxvars = maxvars, def_eq = false, next_idx = 1,

  1105               bounds = [], wellformed = True}

  1106             val _ = tracing ("Translating term (sizes: "

  1107               ^ commas (map (fn (_, n) => string_of_int n) universe) ^ ") ...")

  1108             (* translate 'u' and all axioms *)

  1109             val (intrs, (model, args)) = fold_map (fn t' => fn (m, a) =>

  1110               let

  1111                 val (i, m', a') = interpret ctxt m a t'

  1112               in

  1113                 (* set 'def_eq' to 'true' *)

  1114                 (i, (m', {maxvars = #maxvars a', def_eq = true,

  1115                   next_idx = #next_idx a', bounds = #bounds a',

  1116                   wellformed = #wellformed a'}))

  1117               end) (u :: axioms) (init_model, init_args)

  1118             (* make 'u' either true or false, and make all axioms true, and *)

  1119             (* add the well-formedness side condition                       *)

  1120             val fm_u = (if negate then toFalse else toTrue) (hd intrs)

  1121             val fm_ax = PropLogic.all (map toTrue (tl intrs))

  1122             val fm = PropLogic.all [#wellformed args, fm_ax, fm_u]

  1123             val _ =

  1124               (if satsolver = "dpll" orelse satsolver = "enumerate" then

  1125                 warning ("Using SAT solver " ^ quote satsolver ^

  1126                          "; for better performance, consider installing an \

  1127                          \external solver.")

  1128                else ());

  1129             val solver =

  1130               SatSolver.invoke_solver satsolver

  1131               handle Option.Option =>

  1132                      error ("Unknown SAT solver: " ^ quote satsolver ^

  1133                             ". Available solvers: " ^

  1134                             commas (map (quote o fst) (!SatSolver.solvers)) ^ ".")

  1135           in

  1136             priority "Invoking SAT solver...";

  1137             (case solver fm of

  1138               SatSolver.SATISFIABLE assignment =>

  1139                 (priority ("*** Model found: ***\n" ^ print_model ctxt model

  1140                   (fn i => case assignment i of SOME b => b | NONE => true));

  1141                  if maybe_spurious then "potential" else "genuine")

  1142             | SatSolver.UNSATISFIABLE _ =>

  1143                 (priority "No model exists.";

  1144                 case next_universe universe sizes minsize maxsize of

  1145                   SOME universe' => find_model_loop universe'

  1146                 | NONE => (priority

  1147                   "Search terminated, no larger universe within the given limits.";

  1148                   "none"))

  1149             | SatSolver.UNKNOWN =>

  1150                 (priority "No model found.";

  1151                 case next_universe universe sizes minsize maxsize of

  1152                   SOME universe' => find_model_loop universe'

  1153                 | NONE           => (priority

  1154                   "Search terminated, no larger universe within the given limits.";

  1155                   "unknown"))) handle SatSolver.NOT_CONFIGURED =>

  1156               (error ("SAT solver " ^ quote satsolver ^ " is not configured.");

  1157                "unknown")

  1158           end

  1159           handle MAXVARS_EXCEEDED =>

  1160             (priority ("Search terminated, number of Boolean variables ("

  1161               ^ string_of_int maxvars ^ " allowed) exceeded.");

  1162               "unknown")

  1164         val outcome_code = find_model_loop (first_universe types sizes minsize)

  1165       in

  1166         check_expect outcome_code

  1167       end

  1168   in

  1169     (* some parameter sanity checks *)

  1170     minsize>=1 orelse

  1171       error ("\"minsize\" is " ^ string_of_int minsize ^ ", must be at least 1");

  1172     maxsize>=1 orelse

  1173       error ("\"maxsize\" is " ^ string_of_int maxsize ^ ", must be at least 1");

  1174     maxsize>=minsize orelse

  1175       error ("\"maxsize\" (=" ^ string_of_int maxsize ^

  1176       ") is less than \"minsize\" (=" ^ string_of_int minsize ^ ").");

  1177     maxvars>=0 orelse

  1178       error ("\"maxvars\" is " ^ string_of_int maxvars ^ ", must be at least 0");

  1179     maxtime>=0 orelse

  1180       error ("\"maxtime\" is " ^ string_of_int maxtime ^ ", must be at least 0");

  1181     (* enter loop with or without time limit *)

  1182     priority ("Trying to find a model that "

  1183       ^ (if negate then "refutes" else "satisfies") ^ ": "

  1184       ^ Syntax.string_of_term ctxt t);

  1185     if maxtime > 0 then (

  1186       TimeLimit.timeLimit (Time.fromSeconds maxtime)

  1187         wrapper ()

  1188       handle TimeLimit.TimeOut =>

  1189         (priority ("Search terminated, time limit (" ^

  1190             string_of_int maxtime

  1191             ^ (if maxtime=1 then " second" else " seconds") ^ ") exceeded.");

  1192          check_expect "unknown")

  1193     ) else wrapper ()

  1194   end;

  1197 (* ------------------------------------------------------------------------- *)

  1198 (* INTERFACE, PART 2: FINDING A MODEL                                        *)

  1199 (* ------------------------------------------------------------------------- *)

  1201 (* ------------------------------------------------------------------------- *)

  1202 (* satisfy_term: calls 'find_model' to find a model that satisfies 't'       *)

  1203 (* params      : list of '(name, value)' pairs used to override default      *)

  1204 (*               parameters                                                  *)

  1205 (* ------------------------------------------------------------------------- *)

  1207 fun satisfy_term ctxt params assm_ts t =

  1208   find_model ctxt (actual_params ctxt params) assm_ts t false;

  1210 (* ------------------------------------------------------------------------- *)

  1211 (* refute_term: calls 'find_model' to find a model that refutes 't'          *)

  1212 (* params     : list of '(name, value)' pairs used to override default       *)

  1213 (*              parameters                                                   *)

  1214 (* ------------------------------------------------------------------------- *)

  1216 fun refute_term ctxt params assm_ts t =

  1217   let

  1218     (* disallow schematic type variables, since we cannot properly negate  *)

  1219     (* terms containing them (their logical meaning is that there EXISTS a *)

  1220     (* type s.t. ...; to refute such a formula, we would have to show that *)

  1221     (* for ALL types, not ...)                                             *)

  1222     val _ = null (Term.add_tvars t []) orelse

  1223       error "Term to be refuted contains schematic type variables"

  1225     (* existential closure over schematic variables *)

  1226     (* (Term.indexname * Term.typ) list *)

  1227     val vars = sort_wrt (fst o fst) (map dest_Var (OldTerm.term_vars t))

  1228     (* Term.term *)

  1229     val ex_closure = fold (fn ((x, i), T) => fn t' =>

  1230       HOLogic.exists_const T $

  1231         Abs (x, T, abstract_over (Var ((x, i), T), t'))) vars t

  1232     (* Note: If 't' is of type 'propT' (rather than 'boolT'), applying   *)

  1233     (* 'HOLogic.exists_const' is not type-correct.  However, this is not *)

  1234     (* really a problem as long as 'find_model' still interprets the     *)

  1235     (* resulting term correctly, without checking its type.              *)

  1237     (* replace outermost universally quantified variables by Free's:     *)

  1238     (* refuting a term with Free's is generally faster than refuting a   *)

  1239     (* term with (nested) quantifiers, because quantifiers are expanded, *)

  1240     (* while the SAT solver searches for an interpretation for Free's.   *)

  1241     (* Also we get more information back that way, namely an             *)

  1242     (* interpretation which includes values for the (formerly)           *)

  1243     (* quantified variables.                                             *)

  1244     (* maps  !!x1...xn. !xk...xm. t   to   t  *)

  1245     fun strip_all_body (Const (@{const_name all}, _) $ Abs (_, _, t)) =

  1246           strip_all_body t

  1247       | strip_all_body (Const (@{const_name Trueprop}, _) $ t) =

  1248           strip_all_body t

  1249       | strip_all_body (Const (@{const_name All}, _) $ Abs (_, _, t)) =

  1250           strip_all_body t

  1251       | strip_all_body t = t

  1252     (* maps  !!x1...xn. !xk...xm. t   to   [x1, ..., xn, xk, ..., xm]  *)

  1253     fun strip_all_vars (Const (@{const_name all}, _) $ Abs (a, T, t)) =

  1254           (a, T) :: strip_all_vars t

  1255       | strip_all_vars (Const (@{const_name Trueprop}, _) $ t) =

  1256           strip_all_vars t

  1257       | strip_all_vars (Const (@{const_name All}, _) $ Abs (a, T, t)) =

  1258           (a, T) :: strip_all_vars t

  1259       | strip_all_vars t = [] : (string * typ) list

  1260     val strip_t = strip_all_body ex_closure

  1261     val frees = Term.rename_wrt_term strip_t (strip_all_vars ex_closure)

  1262     val subst_t = Term.subst_bounds (map Free frees, strip_t)

  1263   in

  1264     find_model ctxt (actual_params ctxt params) assm_ts subst_t true

  1265   end;

  1267 (* ------------------------------------------------------------------------- *)

  1268 (* refute_goal                                                               *)

  1269 (* ------------------------------------------------------------------------- *)

  1271 fun refute_goal ctxt params th i =

  1272   let

  1273     val t = th |> prop_of

  1274   in

  1275     if Logic.count_prems t = 0 then

  1276       priority "No subgoal!"

  1277     else

  1278       let

  1279         val assms = map term_of (Assumption.all_assms_of ctxt)

  1280         val (t, frees) = Logic.goal_params t i

  1281       in

  1282         refute_term ctxt params assms (subst_bounds (frees, t))

  1283       end

  1284   end

  1287 (* ------------------------------------------------------------------------- *)

  1288 (* INTERPRETERS: Auxiliary Functions                                         *)

  1289 (* ------------------------------------------------------------------------- *)

  1291 (* ------------------------------------------------------------------------- *)

  1292 (* make_constants: returns all interpretations for type 'T' that consist of  *)

  1293 (*                 unit vectors with 'True'/'False' only (no Boolean         *)

  1294 (*                 variables)                                                *)

  1295 (* ------------------------------------------------------------------------- *)

  1297 fun make_constants ctxt model T =

  1298   let

  1299     (* returns a list with all unit vectors of length n *)

  1300     (* int -> interpretation list *)

  1301     fun unit_vectors n =

  1302       let

  1303         (* returns the k-th unit vector of length n *)

  1304         (* int * int -> interpretation *)

  1305         fun unit_vector (k, n) =

  1306           Leaf ((replicate (k-1) False) @ (True :: (replicate (n-k) False)))

  1307         (* int -> interpretation list *)

  1308         fun unit_vectors_loop k =

  1309           if k>n then [] else unit_vector (k,n) :: unit_vectors_loop (k+1)

  1310       in

  1311         unit_vectors_loop 1

  1312       end

  1313     (* returns a list of lists, each one consisting of n (possibly *)

  1314     (* identical) elements from 'xs'                               *)

  1315     (* int -> 'a list -> 'a list list *)

  1316     fun pick_all 1 xs = map single xs

  1317       | pick_all n xs =

  1318           let val rec_pick = pick_all (n - 1) xs in

  1319             maps (fn x => map (cons x) rec_pick) xs

  1320           end

  1321     (* returns all constant interpretations that have the same tree *)

  1322     (* structure as the interpretation argument                     *)

  1323     (* interpretation -> interpretation list *)

  1324     fun make_constants_intr (Leaf xs) = unit_vectors (length xs)

  1325       | make_constants_intr (Node xs) = map Node (pick_all (length xs)

  1326           (make_constants_intr (hd xs)))

  1327     (* obtain the interpretation for a variable of type 'T' *)

  1328     val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,

  1329       bounds=[], wellformed=True} (Free ("dummy", T))

  1330   in

  1331     make_constants_intr i

  1332   end;

  1334 (* ------------------------------------------------------------------------- *)

  1335 (* size_of_type: returns the number of elements in a type 'T' (i.e. 'length  *)

  1336 (*               (make_constants T)', but implemented more efficiently)      *)

  1337 (* ------------------------------------------------------------------------- *)

  1339 (* returns 0 for an empty ground type or a function type with empty      *)

  1340 (* codomain, but fails for a function type with empty domain --          *)

  1341 (* admissibility of datatype constructor argument types (see "Inductive  *)

  1342 (* datatypes in HOL - lessons learned ...", S. Berghofer, M. Wenzel,     *)

  1343 (* TPHOLs 99) ensures that recursive, possibly empty, datatype fragments *)

  1344 (* never occur as the domain of a function type that is the type of a    *)

  1345 (* constructor argument                                                  *)

  1347 fun size_of_type ctxt model T =

  1348   let

  1349     (* returns the number of elements that have the same tree structure as a *)

  1350     (* given interpretation                                                  *)

  1351     fun size_of_intr (Leaf xs) = length xs

  1352       | size_of_intr (Node xs) = Integer.pow (length xs) (size_of_intr (hd xs))

  1353     (* obtain the interpretation for a variable of type 'T' *)

  1354     val (i, _, _) = interpret ctxt model {maxvars=0, def_eq=false, next_idx=1,

  1355       bounds=[], wellformed=True} (Free ("dummy", T))

  1356   in

  1357     size_of_intr i

  1358   end;

  1360 (* ------------------------------------------------------------------------- *)

  1361 (* TT/FF: interpretations that denote "true" or "false", respectively        *)

  1362 (* ------------------------------------------------------------------------- *)

  1364 (* interpretation *)

  1366 val TT = Leaf [True, False];

  1368 val FF = Leaf [False, True];

  1370 (* ------------------------------------------------------------------------- *)

  1371 (* make_equality: returns an interpretation that denotes (extensional)       *)

  1372 (*                equality of two interpretations                            *)

  1373 (* - two interpretations are 'equal' iff they are both defined and denote    *)

  1374 (*   the same value                                                          *)

  1375 (* - two interpretations are 'not_equal' iff they are both defined at least  *)

  1376 (*   partially, and a defined part denotes different values                  *)

  1377 (* - a completely undefined interpretation is neither 'equal' nor            *)

  1378 (*   'not_equal' to another interpretation                                   *)

  1379 (* ------------------------------------------------------------------------- *)

  1381 (* We could in principle represent '=' on a type T by a particular        *)

  1382 (* interpretation.  However, the size of that interpretation is quadratic *)

  1383 (* in the size of T.  Therefore comparing the interpretations 'i1' and    *)

  1384 (* 'i2' directly is more efficient than constructing the interpretation   *)

  1385 (* for equality on T first, and "applying" this interpretation to 'i1'    *)

  1386 (* and 'i2' in the usual way (cf. 'interpretation_apply') then.           *)

  1388 (* interpretation * interpretation -> interpretation *)

  1390 fun make_equality (i1, i2) =

  1391   let

  1392     (* interpretation * interpretation -> prop_formula *)

  1393     fun equal (i1, i2) =

  1394       (case i1 of

  1395         Leaf xs =>

  1396           (case i2 of

  1397             Leaf ys => PropLogic.dot_product (xs, ys)  (* defined and equal *)

  1398           | Node _  => raise REFUTE ("make_equality",

  1399             "second interpretation is higher"))

  1400       | Node xs =>

  1401           (case i2 of

  1402             Leaf _  => raise REFUTE ("make_equality",

  1403             "first interpretation is higher")

  1404           | Node ys => PropLogic.all (map equal (xs ~~ ys))))

  1405     (* interpretation * interpretation -> prop_formula *)

  1406     fun not_equal (i1, i2) =

  1407       (case i1 of

  1408         Leaf xs =>

  1409           (case i2 of

  1410             (* defined and not equal *)

  1411             Leaf ys => PropLogic.all ((PropLogic.exists xs)

  1412             :: (PropLogic.exists ys)

  1413             :: (map (fn (x,y) => SOr (SNot x, SNot y)) (xs ~~ ys)))

  1414           | Node _  => raise REFUTE ("make_equality",

  1415             "second interpretation is higher"))

  1416       | Node xs =>

  1417           (case i2 of

  1418             Leaf _  => raise REFUTE ("make_equality",

  1419             "first interpretation is higher")

  1420           | Node ys => PropLogic.exists (map not_equal (xs ~~ ys))))

  1421   in

  1422     (* a value may be undefined; therefore 'not_equal' is not just the *)

  1423     (* negation of 'equal'                                             *)

  1424     Leaf [equal (i1, i2), not_equal (i1, i2)]

  1425   end;

  1427 (* ------------------------------------------------------------------------- *)

  1428 (* make_def_equality: returns an interpretation that denotes (extensional)   *)

  1429 (*                    equality of two interpretations                        *)

  1430 (* This function treats undefined/partially defined interpretations          *)

  1431 (* different from 'make_equality': two undefined interpretations are         *)

  1432 (* considered equal, while a defined interpretation is considered not equal  *)

  1433 (* to an undefined interpretation.                                           *)

  1434 (* ------------------------------------------------------------------------- *)

  1436 (* interpretation * interpretation -> interpretation *)

  1438 fun make_def_equality (i1, i2) =

  1439   let

  1440     (* interpretation * interpretation -> prop_formula *)

  1441     fun equal (i1, i2) =

  1442       (case i1 of

  1443         Leaf xs =>

  1444           (case i2 of

  1445             (* defined and equal, or both undefined *)

  1446             Leaf ys => SOr (PropLogic.dot_product (xs, ys),

  1447             SAnd (PropLogic.all (map SNot xs), PropLogic.all (map SNot ys)))

  1448           | Node _  => raise REFUTE ("make_def_equality",

  1449             "second interpretation is higher"))

  1450       | Node xs =>

  1451           (case i2 of

  1452             Leaf _  => raise REFUTE ("make_def_equality",

  1453             "first interpretation is higher")

  1454           | Node ys => PropLogic.all (map equal (xs ~~ ys))))

  1455     (* interpretation *)

  1456     val eq = equal (i1, i2)

  1457   in

  1458     Leaf [eq, SNot eq]

  1459   end;

  1461 (* ------------------------------------------------------------------------- *)

  1462 (* interpretation_apply: returns an interpretation that denotes the result   *)

  1463 (*                       of applying the function denoted by 'i1' to the     *)

  1464 (*                       argument denoted by 'i2'                            *)

  1465 (* ------------------------------------------------------------------------- *)

  1467 (* interpretation * interpretation -> interpretation *)

  1469 fun interpretation_apply (i1, i2) =

  1470   let

  1471     (* interpretation * interpretation -> interpretation *)

  1472     fun interpretation_disjunction (tr1,tr2) =

  1473       tree_map (fn (xs,ys) => map (fn (x,y) => SOr(x,y)) (xs ~~ ys))

  1474         (tree_pair (tr1,tr2))

  1475     (* prop_formula * interpretation -> interpretation *)

  1476     fun prop_formula_times_interpretation (fm,tr) =

  1477       tree_map (map (fn x => SAnd (fm,x))) tr

  1478     (* prop_formula list * interpretation list -> interpretation *)

  1479     fun prop_formula_list_dot_product_interpretation_list ([fm],[tr]) =

  1480           prop_formula_times_interpretation (fm,tr)

  1481       | prop_formula_list_dot_product_interpretation_list (fm::fms,tr::trees) =

  1482           interpretation_disjunction (prop_formula_times_interpretation (fm,tr),

  1483             prop_formula_list_dot_product_interpretation_list (fms,trees))

  1484       | prop_formula_list_dot_product_interpretation_list (_,_) =

  1485           raise REFUTE ("interpretation_apply", "empty list (in dot product)")

  1486     (* concatenates 'x' with every list in 'xss', returning a new list of *)

  1487     (* lists                                                              *)

  1488     (* 'a -> 'a list list -> 'a list list *)

  1489     fun cons_list x xss = map (cons x) xss

  1490     (* returns a list of lists, each one consisting of one element from each *)

  1491     (* element of 'xss'                                                      *)

  1492     (* 'a list list -> 'a list list *)

  1493     fun pick_all [xs] = map single xs

  1494       | pick_all (xs::xss) =

  1495           let val rec_pick = pick_all xss in

  1496             maps (fn x => map (cons x) rec_pick) xs

  1497           end

  1498       | pick_all _ = raise REFUTE ("interpretation_apply", "empty list (in pick_all)")

  1499     (* interpretation -> prop_formula list *)

  1500     fun interpretation_to_prop_formula_list (Leaf xs) = xs

  1501       | interpretation_to_prop_formula_list (Node trees) =

  1502           map PropLogic.all (pick_all

  1503             (map interpretation_to_prop_formula_list trees))

  1504   in

  1505     case i1 of

  1506       Leaf _ =>

  1507         raise REFUTE ("interpretation_apply", "first interpretation is a leaf")

  1508     | Node xs =>

  1509         prop_formula_list_dot_product_interpretation_list

  1510           (interpretation_to_prop_formula_list i2, xs)

  1511   end;

  1513 (* ------------------------------------------------------------------------- *)

  1514 (* eta_expand: eta-expands a term 't' by adding 'i' lambda abstractions      *)

  1515 (* ------------------------------------------------------------------------- *)

  1517 (* Term.term -> int -> Term.term *)

  1519 fun eta_expand t i =

  1520   let

  1521     val Ts = Term.binder_types (Term.fastype_of t)

  1522     val t' = Term.incr_boundvars i t

  1523   in

  1524     fold_rev (fn T => fn term => Abs ("<eta_expand>", T, term))

  1525       (List.take (Ts, i))

  1526       (Term.list_comb (t', map Bound (i-1 downto 0)))

  1527   end;

  1529 (* ------------------------------------------------------------------------- *)

  1530 (* size_of_dtyp: the size of (an initial fragment of) an inductive data type *)

  1531 (*               is the sum (over its constructors) of the product (over     *)

  1532 (*               their arguments) of the size of the argument types          *)

  1533 (* ------------------------------------------------------------------------- *)

  1535 fun size_of_dtyp ctxt typ_sizes descr typ_assoc constructors =

  1536   Integer.sum (map (fn (_, dtyps) =>

  1537     Integer.prod (map (size_of_type ctxt (typ_sizes, []) o

  1538       (typ_of_dtyp descr typ_assoc)) dtyps))

  1539         constructors);

  1542 (* ------------------------------------------------------------------------- *)

  1543 (* INTERPRETERS: Actual Interpreters                                         *)

  1544 (* ------------------------------------------------------------------------- *)

  1546 (* simply typed lambda calculus: Isabelle's basic term syntax, with type *)

  1547 (* variables, function types, and propT                                  *)

  1549 fun stlc_interpreter ctxt model args t =

  1550   let

  1551     val thy = ProofContext.theory_of ctxt

  1552     val (typs, terms) = model

  1553     val {maxvars, def_eq, next_idx, bounds, wellformed} = args

  1554     (* Term.typ -> (interpretation * model * arguments) option *)

  1555     fun interpret_groundterm T =

  1556       let

  1557         (* unit -> (interpretation * model * arguments) option *)

  1558         fun interpret_groundtype () =

  1559           let

  1560             (* the model must specify a size for ground types *)

  1561             val size =

  1562               if T = Term.propT then 2

  1563               else the (AList.lookup (op =) typs T)

  1564             val next = next_idx + size

  1565             (* check if 'maxvars' is large enough *)

  1566             val _ = (if next - 1 > maxvars andalso maxvars > 0 then

  1567               raise MAXVARS_EXCEEDED else ())

  1568             (* prop_formula list *)

  1569             val fms  = map BoolVar (next_idx upto (next_idx + size - 1))

  1570             (* interpretation *)

  1571             val intr = Leaf fms

  1572             (* prop_formula list -> prop_formula *)

  1573             fun one_of_two_false [] = True

  1574               | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>

  1575                   SOr (SNot x, SNot x')) xs), one_of_two_false xs)

  1576             (* prop_formula *)

  1577             val wf = one_of_two_false fms

  1578           in

  1579             (* extend the model, increase 'next_idx', add well-formedness *)

  1580             (* condition                                                  *)

  1581             SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,

  1582               def_eq = def_eq, next_idx = next, bounds = bounds,

  1583               wellformed = SAnd (wellformed, wf)})

  1584           end

  1585       in

  1586         case T of

  1587           Type ("fun", [T1, T2]) =>

  1588             let

  1589               (* we create 'size_of_type ... T1' different copies of the        *)

  1590               (* interpretation for 'T2', which are then combined into a single *)

  1591               (* new interpretation                                             *)

  1592               (* make fresh copies, with different variable indices *)

  1593               (* 'idx': next variable index                         *)

  1594               (* 'n'  : number of copies                            *)

  1595               (* int -> int -> (int * interpretation list * prop_formula *)

  1596               fun make_copies idx 0 = (idx, [], True)

  1597                 | make_copies idx n =

  1598                     let

  1599                       val (copy, _, new_args) = interpret ctxt (typs, [])

  1600                         {maxvars = maxvars, def_eq = false, next_idx = idx,

  1601                         bounds = [], wellformed = True} (Free ("dummy", T2))

  1602                       val (idx', copies, wf') = make_copies (#next_idx new_args) (n-1)

  1603                     in

  1604                       (idx', copy :: copies, SAnd (#wellformed new_args, wf'))

  1605                     end

  1606               val (next, copies, wf) = make_copies next_idx

  1607                 (size_of_type ctxt model T1)

  1608               (* combine copies into a single interpretation *)

  1609               val intr = Node copies

  1610             in

  1611               (* extend the model, increase 'next_idx', add well-formedness *)

  1612               (* condition                                                  *)

  1613               SOME (intr, (typs, (t, intr)::terms), {maxvars = maxvars,

  1614                 def_eq = def_eq, next_idx = next, bounds = bounds,

  1615                 wellformed = SAnd (wellformed, wf)})

  1616             end

  1617         | Type _  => interpret_groundtype ()

  1618         | TFree _ => interpret_groundtype ()

  1619         | TVar  _ => interpret_groundtype ()

  1620       end

  1621   in

  1622     case AList.lookup (op =) terms t of

  1623       SOME intr =>

  1624         (* return an existing interpretation *)

  1625         SOME (intr, model, args)

  1626     | NONE =>

  1627         (case t of

  1628           Const (_, T) => interpret_groundterm T

  1629         | Free (_, T) => interpret_groundterm T

  1630         | Var (_, T) => interpret_groundterm T

  1631         | Bound i => SOME (List.nth (#bounds args, i), model, args)

  1632         | Abs (x, T, body) =>

  1633             let

  1634               (* create all constants of type 'T' *)

  1635               val constants = make_constants ctxt model T

  1636               (* interpret the 'body' separately for each constant *)

  1637               val (bodies, (model', args')) = fold_map

  1638                 (fn c => fn (m, a) =>

  1639                   let

  1640                     (* add 'c' to 'bounds' *)

  1641                     val (i', m', a') = interpret ctxt m {maxvars = #maxvars a,

  1642                       def_eq = #def_eq a, next_idx = #next_idx a,

  1643                       bounds = (c :: #bounds a), wellformed = #wellformed a} body

  1644                   in

  1645                     (* keep the new model m' and 'next_idx' and 'wellformed', *)

  1646                     (* but use old 'bounds'                                   *)

  1647                     (i', (m', {maxvars = maxvars, def_eq = def_eq,

  1648                       next_idx = #next_idx a', bounds = bounds,

  1649                       wellformed = #wellformed a'}))

  1650                   end)

  1651                 constants (model, args)

  1652             in

  1653               SOME (Node bodies, model', args')

  1654             end

  1655         | t1 $ t2 =>

  1656             let

  1657               (* interpret 't1' and 't2' separately *)

  1658               val (intr1, model1, args1) = interpret ctxt model args t1

  1659               val (intr2, model2, args2) = interpret ctxt model1 args1 t2

  1660             in

  1661               SOME (interpretation_apply (intr1, intr2), model2, args2)

  1662             end)

  1663   end;

  1665 fun Pure_interpreter ctxt model args t =

  1666   case t of

  1667     Const (@{const_name all}, _) $ t1 =>

  1668       let

  1669         val (i, m, a) = interpret ctxt model args t1

  1670       in

  1671         case i of

  1672           Node xs =>

  1673             (* 3-valued logic *)

  1674             let

  1675               val fmTrue  = PropLogic.all (map toTrue xs)

  1676               val fmFalse = PropLogic.exists (map toFalse xs)

  1677             in

  1678               SOME (Leaf [fmTrue, fmFalse], m, a)

  1679             end

  1680         | _ =>

  1681           raise REFUTE ("Pure_interpreter",

  1682             "\"all\" is followed by a non-function")

  1683       end

  1684   | Const (@{const_name all}, _) =>

  1685       SOME (interpret ctxt model args (eta_expand t 1))

  1686   | Const (@{const_name "=="}, _) $ t1 $ t2 =>

  1687       let

  1688         val (i1, m1, a1) = interpret ctxt model args t1

  1689         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1690       in

  1691         (* we use either 'make_def_equality' or 'make_equality' *)

  1692         SOME ((if #def_eq args then make_def_equality else make_equality)

  1693           (i1, i2), m2, a2)

  1694       end

  1695   | Const (@{const_name "=="}, _) $ t1 =>

  1696       SOME (interpret ctxt model args (eta_expand t 1))

  1697   | Const (@{const_name "=="}, _) =>

  1698       SOME (interpret ctxt model args (eta_expand t 2))

  1699   | Const (@{const_name "==>"}, _) $ t1 $ t2 =>

  1700       (* 3-valued logic *)

  1701       let

  1702         val (i1, m1, a1) = interpret ctxt model args t1

  1703         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1704         val fmTrue       = PropLogic.SOr (toFalse i1, toTrue i2)

  1705         val fmFalse      = PropLogic.SAnd (toTrue i1, toFalse i2)

  1706       in

  1707         SOME (Leaf [fmTrue, fmFalse], m2, a2)

  1708       end

  1709   | Const (@{const_name "==>"}, _) $ t1 =>

  1710       SOME (interpret ctxt model args (eta_expand t 1))

  1711   | Const (@{const_name "==>"}, _) =>

  1712       SOME (interpret ctxt model args (eta_expand t 2))

  1713   | _ => NONE;

  1715 fun HOLogic_interpreter ctxt model args t =

  1716 (* Providing interpretations directly is more efficient than unfolding the *)

  1717 (* logical constants.  In HOL however, logical constants can themselves be *)

  1718 (* arguments.  They are then translated using eta-expansion.               *)

  1719   case t of

  1720     Const (@{const_name Trueprop}, _) =>

  1721       SOME (Node [TT, FF], model, args)

  1722   | Const (@{const_name Not}, _) =>

  1723       SOME (Node [FF, TT], model, args)

  1724   (* redundant, since 'True' is also an IDT constructor *)

  1725   | Const (@{const_name True}, _) =>

  1726       SOME (TT, model, args)

  1727   (* redundant, since 'False' is also an IDT constructor *)

  1728   | Const (@{const_name False}, _) =>

  1729       SOME (FF, model, args)

  1730   | Const (@{const_name All}, _) $ t1 =>  (* similar to "all" (Pure) *)

  1731       let

  1732         val (i, m, a) = interpret ctxt model args t1

  1733       in

  1734         case i of

  1735           Node xs =>

  1736             (* 3-valued logic *)

  1737             let

  1738               val fmTrue  = PropLogic.all (map toTrue xs)

  1739               val fmFalse = PropLogic.exists (map toFalse xs)

  1740             in

  1741               SOME (Leaf [fmTrue, fmFalse], m, a)

  1742             end

  1743         | _ =>

  1744           raise REFUTE ("HOLogic_interpreter",

  1745             "\"All\" is followed by a non-function")

  1746       end

  1747   | Const (@{const_name All}, _) =>

  1748       SOME (interpret ctxt model args (eta_expand t 1))

  1749   | Const (@{const_name Ex}, _) $ t1 =>

  1750       let

  1751         val (i, m, a) = interpret ctxt model args t1

  1752       in

  1753         case i of

  1754           Node xs =>

  1755             (* 3-valued logic *)

  1756             let

  1757               val fmTrue  = PropLogic.exists (map toTrue xs)

  1758               val fmFalse = PropLogic.all (map toFalse xs)

  1759             in

  1760               SOME (Leaf [fmTrue, fmFalse], m, a)

  1761             end

  1762         | _ =>

  1763           raise REFUTE ("HOLogic_interpreter",

  1764             "\"Ex\" is followed by a non-function")

  1765       end

  1766   | Const (@{const_name Ex}, _) =>

  1767       SOME (interpret ctxt model args (eta_expand t 1))

  1768   | Const (@{const_name HOL.eq}, _) $ t1 $ t2 =>  (* similar to "==" (Pure) *)

  1769       let

  1770         val (i1, m1, a1) = interpret ctxt model args t1

  1771         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1772       in

  1773         SOME (make_equality (i1, i2), m2, a2)

  1774       end

  1775   | Const (@{const_name HOL.eq}, _) $ t1 =>

  1776       SOME (interpret ctxt model args (eta_expand t 1))

  1777   | Const (@{const_name HOL.eq}, _) =>

  1778       SOME (interpret ctxt model args (eta_expand t 2))

  1779   | Const (@{const_name HOL.conj}, _) $ t1 $ t2 =>

  1780       (* 3-valued logic *)

  1781       let

  1782         val (i1, m1, a1) = interpret ctxt model args t1

  1783         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1784         val fmTrue       = PropLogic.SAnd (toTrue i1, toTrue i2)

  1785         val fmFalse      = PropLogic.SOr (toFalse i1, toFalse i2)

  1786       in

  1787         SOME (Leaf [fmTrue, fmFalse], m2, a2)

  1788       end

  1789   | Const (@{const_name HOL.conj}, _) $ t1 =>

  1790       SOME (interpret ctxt model args (eta_expand t 1))

  1791   | Const (@{const_name HOL.conj}, _) =>

  1792       SOME (interpret ctxt model args (eta_expand t 2))

  1793       (* this would make "undef" propagate, even for formulae like *)

  1794       (* "False & undef":                                          *)

  1795       (* SOME (Node [Node [TT, FF], Node [FF, FF]], model, args) *)

  1796   | Const (@{const_name HOL.disj}, _) $ t1 $ t2 =>

  1797       (* 3-valued logic *)

  1798       let

  1799         val (i1, m1, a1) = interpret ctxt model args t1

  1800         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1801         val fmTrue       = PropLogic.SOr (toTrue i1, toTrue i2)

  1802         val fmFalse      = PropLogic.SAnd (toFalse i1, toFalse i2)

  1803       in

  1804         SOME (Leaf [fmTrue, fmFalse], m2, a2)

  1805       end

  1806   | Const (@{const_name HOL.disj}, _) $ t1 =>

  1807       SOME (interpret ctxt model args (eta_expand t 1))

  1808   | Const (@{const_name HOL.disj}, _) =>

  1809       SOME (interpret ctxt model args (eta_expand t 2))

  1810       (* this would make "undef" propagate, even for formulae like *)

  1811       (* "True | undef":                                           *)

  1812       (* SOME (Node [Node [TT, TT], Node [TT, FF]], model, args) *)

  1813   | Const (@{const_name HOL.implies}, _) $ t1 $ t2 =>  (* similar to "==>" (Pure) *)

  1814       (* 3-valued logic *)

  1815       let

  1816         val (i1, m1, a1) = interpret ctxt model args t1

  1817         val (i2, m2, a2) = interpret ctxt m1 a1 t2

  1818         val fmTrue       = PropLogic.SOr (toFalse i1, toTrue i2)

  1819         val fmFalse      = PropLogic.SAnd (toTrue i1, toFalse i2)

  1820       in

  1821         SOME (Leaf [fmTrue, fmFalse], m2, a2)

  1822       end

  1823   | Const (@{const_name HOL.implies}, _) $ t1 =>

  1824       SOME (interpret ctxt model args (eta_expand t 1))

  1825   | Const (@{const_name HOL.implies}, _) =>

  1826       SOME (interpret ctxt model args (eta_expand t 2))

  1827       (* this would make "undef" propagate, even for formulae like *)

  1828       (* "False --> undef":                                        *)

  1829       (* SOME (Node [Node [TT, FF], Node [TT, TT]], model, args) *)

  1830   | _ => NONE;

  1832 (* interprets variables and constants whose type is an IDT (this is        *)

  1833 (* relatively easy and merely requires us to compute the size of the IDT); *)

  1834 (* constructors of IDTs however are properly interpreted by                *)

  1835 (* 'IDT_constructor_interpreter'                                           *)

  1837 fun IDT_interpreter ctxt model args t =

  1838   let

  1839     val thy = ProofContext.theory_of ctxt

  1840     val (typs, terms) = model

  1841     (* Term.typ -> (interpretation * model * arguments) option *)

  1842     fun interpret_term (Type (s, Ts)) =

  1843           (case Datatype.get_info thy s of

  1844             SOME info =>  (* inductive datatype *)

  1845               let

  1846                 (* int option -- only recursive IDTs have an associated depth *)

  1847                 val depth = AList.lookup (op =) typs (Type (s, Ts))

  1848                 (* sanity check: depth must be at least 0 *)

  1849                 val _ =

  1850                   (case depth of SOME n =>

  1851                     if n < 0 then

  1852                       raise REFUTE ("IDT_interpreter", "negative depth")

  1853                     else ()

  1854                   | _ => ())

  1855               in

  1856                 (* termination condition to avoid infinite recursion *)

  1857                 if depth = (SOME 0) then

  1858                   (* return a leaf of size 0 *)

  1859                   SOME (Leaf [], model, args)

  1860                 else

  1861                   let

  1862                     val index               = #index info

  1863                     val descr               = #descr info

  1864                     val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)

  1865                     val typ_assoc           = dtyps ~~ Ts

  1866                     (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)

  1867                     val _ =

  1868                       if Library.exists (fn d =>

  1869                         case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps

  1870                       then

  1871                         raise REFUTE ("IDT_interpreter",

  1872                           "datatype argument (for type "

  1873                           ^ Syntax.string_of_typ ctxt (Type (s, Ts))

  1874                           ^ ") is not a variable")

  1875                       else ()

  1876                     (* if the model specifies a depth for the current type, *)

  1877                     (* decrement it to avoid infinite recursion             *)

  1878                     val typs' = case depth of NONE => typs | SOME n =>

  1879                       AList.update (op =) (Type (s, Ts), n-1) typs

  1880                     (* recursively compute the size of the datatype *)

  1881                     val size     = size_of_dtyp ctxt typs' descr typ_assoc constrs

  1882                     val next_idx = #next_idx args

  1883                     val next     = next_idx+size

  1884                     (* check if 'maxvars' is large enough *)

  1885                     val _        = (if next-1 > #maxvars args andalso

  1886                       #maxvars args > 0 then raise MAXVARS_EXCEEDED else ())

  1887                     (* prop_formula list *)

  1888                     val fms      = map BoolVar (next_idx upto (next_idx+size-1))

  1889                     (* interpretation *)

  1890                     val intr     = Leaf fms

  1891                     (* prop_formula list -> prop_formula *)

  1892                     fun one_of_two_false [] = True

  1893                       | one_of_two_false (x::xs) = SAnd (PropLogic.all (map (fn x' =>

  1894                           SOr (SNot x, SNot x')) xs), one_of_two_false xs)

  1895                     (* prop_formula *)

  1896                     val wf = one_of_two_false fms

  1897                   in

  1898                     (* extend the model, increase 'next_idx', add well-formedness *)

  1899                     (* condition                                                  *)

  1900                     SOME (intr, (typs, (t, intr)::terms), {maxvars = #maxvars args,

  1901                       def_eq = #def_eq args, next_idx = next, bounds = #bounds args,

  1902                       wellformed = SAnd (#wellformed args, wf)})

  1903                   end

  1904               end

  1905           | NONE =>  (* not an inductive datatype *)

  1906               NONE)

  1907       | interpret_term _ =  (* a (free or schematic) type variable *)

  1908           NONE

  1909   in

  1910     case AList.lookup (op =) terms t of

  1911       SOME intr =>

  1912         (* return an existing interpretation *)

  1913         SOME (intr, model, args)

  1914     | NONE =>

  1915         (case t of

  1916           Free (_, T) => interpret_term T

  1917         | Var (_, T) => interpret_term T

  1918         | Const (_, T) => interpret_term T

  1919         | _ => NONE)

  1920   end;

  1922 (* This function imposes an order on the elements of a datatype fragment  *)

  1923 (* as follows: C_i x_1 ... x_n < C_j y_1 ... y_m iff i < j or             *)

  1924 (* (x_1, ..., x_n) < (y_1, ..., y_m).  With this order, a constructor is  *)

  1925 (* a function C_i that maps some argument indices x_1, ..., x_n to the    *)

  1926 (* datatype element given by index C_i x_1 ... x_n.  The idea remains the *)

  1927 (* same for recursive datatypes, although the computation of indices gets *)

  1928 (* a little tricky.                                                       *)

  1930 fun IDT_constructor_interpreter ctxt model args t =

  1931   let

  1932     val thy = ProofContext.theory_of ctxt

  1933     (* returns a list of canonical representations for terms of the type 'T' *)

  1934     (* It would be nice if we could just use 'print' for this, but 'print'   *)

  1935     (* for IDTs calls 'IDT_constructor_interpreter' again, and this could    *)

  1936     (* lead to infinite recursion when we have (mutually) recursive IDTs.    *)

  1937     (* (Term.typ * int) list -> Term.typ -> Term.term list *)

  1938     fun canonical_terms typs T =

  1939           (case T of

  1940             Type ("fun", [T1, T2]) =>

  1941             (* 'T2' might contain a recursive IDT, so we cannot use 'print' (at *)

  1942             (* least not for 'T2'                                               *)

  1943             let

  1944               (* returns a list of lists, each one consisting of n (possibly *)

  1945               (* identical) elements from 'xs'                               *)

  1946               (* int -> 'a list -> 'a list list *)

  1947               fun pick_all 1 xs = map single xs

  1948                 | pick_all n xs =

  1949                     let val rec_pick = pick_all (n-1) xs in

  1950                       maps (fn x => map (cons x) rec_pick) xs

  1951                     end

  1952               (* ["x1", ..., "xn"] *)

  1953               val terms1 = canonical_terms typs T1

  1954               (* ["y1", ..., "ym"] *)

  1955               val terms2 = canonical_terms typs T2

  1956               (* [[("x1", "y1"), ..., ("xn", "y1")], ..., *)

  1957               (*   [("x1", "ym"), ..., ("xn", "ym")]]     *)

  1958               val functions = map (curry (op ~~) terms1)

  1959                 (pick_all (length terms1) terms2)

  1960               (* [["(x1, y1)", ..., "(xn, y1)"], ..., *)

  1961               (*   ["(x1, ym)", ..., "(xn, ym)"]]     *)

  1962               val pairss = map (map HOLogic.mk_prod) functions

  1963               (* Term.typ *)

  1964               val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)

  1965               val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT

  1966               (* Term.term *)

  1967               val HOLogic_empty_set = Const (@{const_abbrev Set.empty}, HOLogic_setT)

  1968               val HOLogic_insert    =

  1969                 Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)

  1970             in

  1971               (* functions as graphs, i.e. as a (HOL) set of pairs "(x, y)" *)

  1972               map (fn ps => fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) ps

  1973                 HOLogic_empty_set) pairss

  1974             end

  1975       | Type (s, Ts) =>

  1976           (case Datatype.get_info thy s of

  1977             SOME info =>

  1978               (case AList.lookup (op =) typs T of

  1979                 SOME 0 =>

  1980                   (* termination condition to avoid infinite recursion *)

  1981                   []  (* at depth 0, every IDT is empty *)

  1982               | _ =>

  1983                 let

  1984                   val index = #index info

  1985                   val descr = #descr info

  1986                   val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)

  1987                   val typ_assoc = dtyps ~~ Ts

  1988                   (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)

  1989                   val _ =

  1990                     if Library.exists (fn d =>

  1991                       case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps

  1992                     then

  1993                       raise REFUTE ("IDT_constructor_interpreter",

  1994                         "datatype argument (for type "

  1995                         ^ Syntax.string_of_typ ctxt T

  1996                         ^ ") is not a variable")

  1997                     else ()

  1998                   (* decrement depth for the IDT 'T' *)

  1999                   val typs' =

  2000                     (case AList.lookup (op =) typs T of NONE => typs

  2001                     | SOME n => AList.update (op =) (T, n-1) typs)

  2002                   fun constructor_terms terms [] = terms

  2003                     | constructor_terms terms (d::ds) =

  2004                         let

  2005                           val dT = typ_of_dtyp descr typ_assoc d

  2006                           val d_terms = canonical_terms typs' dT

  2007                         in

  2008                           (* C_i x_1 ... x_n < C_i y_1 ... y_n if *)

  2009                           (* (x_1, ..., x_n) < (y_1, ..., y_n)    *)

  2010                           constructor_terms

  2011                             (map_product (curry op $) terms d_terms) ds

  2012                         end

  2013                 in

  2014                   (* C_i ... < C_j ... if i < j *)

  2015                   maps (fn (cname, ctyps) =>

  2016                     let

  2017                       val cTerm = Const (cname,

  2018                         map (typ_of_dtyp descr typ_assoc) ctyps ---> T)

  2019                     in

  2020                       constructor_terms [cTerm] ctyps

  2021                     end) constrs

  2022                 end)

  2023           | NONE =>

  2024               (* not an inductive datatype; in this case the argument types in *)

  2025               (* 'Ts' may not be IDTs either, so 'print' should be safe        *)

  2026               map (fn intr => print ctxt (typs, []) T intr (K false))

  2027                 (make_constants ctxt (typs, []) T))

  2028       | _ =>  (* TFree ..., TVar ... *)

  2029           map (fn intr => print ctxt (typs, []) T intr (K false))

  2030             (make_constants ctxt (typs, []) T))

  2031     val (typs, terms) = model

  2032   in

  2033     case AList.lookup (op =) terms t of

  2034       SOME intr =>

  2035         (* return an existing interpretation *)

  2036         SOME (intr, model, args)

  2037     | NONE =>

  2038         (case t of

  2039           Const (s, T) =>

  2040             (case body_type T of

  2041               Type (s', Ts') =>

  2042                 (case Datatype.get_info thy s' of

  2043                   SOME info =>  (* body type is an inductive datatype *)

  2044                     let

  2045                       val index               = #index info

  2046                       val descr               = #descr info

  2047                       val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)

  2048                       val typ_assoc           = dtyps ~~ Ts'

  2049                       (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)

  2050                       val _ = if Library.exists (fn d =>

  2051                           case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps

  2052                         then

  2053                           raise REFUTE ("IDT_constructor_interpreter",

  2054                             "datatype argument (for type "

  2055                             ^ Syntax.string_of_typ ctxt (Type (s', Ts'))

  2056                             ^ ") is not a variable")

  2057                         else ()

  2058                       (* split the constructors into those occuring before/after *)

  2059                       (* 'Const (s, T)'                                          *)

  2060                       val (constrs1, constrs2) = take_prefix (fn (cname, ctypes) =>

  2061                         not (cname = s andalso Sign.typ_instance thy (T,

  2062                           map (typ_of_dtyp descr typ_assoc) ctypes

  2063                             ---> Type (s', Ts')))) constrs

  2064                     in

  2065                       case constrs2 of

  2066                         [] =>

  2067                           (* 'Const (s, T)' is not a constructor of this datatype *)

  2068                           NONE

  2069                       | (_, ctypes)::cs =>

  2070                           let

  2071                             (* int option -- only /recursive/ IDTs have an associated *)

  2072                             (*               depth                                    *)

  2073                             val depth = AList.lookup (op =) typs (Type (s', Ts'))

  2074                             (* this should never happen: at depth 0, this IDT fragment *)

  2075                             (* is definitely empty, and in this case we don't need to  *)

  2076                             (* interpret its constructors                              *)

  2077                             val _ = (case depth of SOME 0 =>

  2078                                 raise REFUTE ("IDT_constructor_interpreter",

  2079                                   "depth is 0")

  2080                               | _ => ())

  2081                             val typs' = (case depth of NONE => typs | SOME n =>

  2082                               AList.update (op =) (Type (s', Ts'), n-1) typs)

  2083                             (* elements of the datatype come before elements generated *)

  2084                             (* by 'Const (s, T)' iff they are generated by a           *)

  2085                             (* constructor in constrs1                                 *)

  2086                             val offset = size_of_dtyp ctxt typs' descr typ_assoc constrs1

  2087                             (* compute the total (current) size of the datatype *)

  2088                             val total = offset +

  2089                               size_of_dtyp ctxt typs' descr typ_assoc constrs2

  2090                             (* sanity check *)

  2091                             val _ = if total <> size_of_type ctxt (typs, [])

  2092                               (Type (s', Ts')) then

  2093                                 raise REFUTE ("IDT_constructor_interpreter",

  2094                                   "total is not equal to current size")

  2095                               else ()

  2096                             (* returns an interpretation where everything is mapped to *)

  2097                             (* an "undefined" element of the datatype                  *)

  2098                             fun make_undef [] = Leaf (replicate total False)

  2099                               | make_undef (d::ds) =

  2100                                   let

  2101                                     (* compute the current size of the type 'd' *)

  2102                                     val dT   = typ_of_dtyp descr typ_assoc d

  2103                                     val size = size_of_type ctxt (typs, []) dT

  2104                                   in

  2105                                     Node (replicate size (make_undef ds))

  2106                                   end

  2107                             (* returns the interpretation for a constructor *)

  2108                             fun make_constr [] offset =

  2109                                   if offset < total then

  2110                                     (Leaf (replicate offset False @ True ::

  2111                                       (replicate (total - offset - 1) False)), offset + 1)

  2112                                   else

  2113                                     raise REFUTE ("IDT_constructor_interpreter",

  2114                                       "offset >= total")

  2115                               | make_constr (d::ds) offset =

  2116                                   let

  2117                                     (* Term.typ *)

  2118                                     val dT = typ_of_dtyp descr typ_assoc d

  2119                                     (* compute canonical term representations for all   *)

  2120                                     (* elements of the type 'd' (with the reduced depth *)

  2121                                     (* for the IDT)                                     *)

  2122                                     val terms' = canonical_terms typs' dT

  2123                                     (* sanity check *)

  2124                                     val _ =

  2125                                       if length terms' <> size_of_type ctxt (typs', []) dT

  2126                                       then

  2127                                         raise REFUTE ("IDT_constructor_interpreter",

  2128                                           "length of terms' is not equal to old size")

  2129                                       else ()

  2130                                     (* compute canonical term representations for all   *)

  2131                                     (* elements of the type 'd' (with the current depth *)

  2132                                     (* for the IDT)                                     *)

  2133                                     val terms = canonical_terms typs dT

  2134                                     (* sanity check *)

  2135                                     val _ =

  2136                                       if length terms <> size_of_type ctxt (typs, []) dT

  2137                                       then

  2138                                         raise REFUTE ("IDT_constructor_interpreter",

  2139                                           "length of terms is not equal to current size")

  2140                                       else ()

  2141                                     (* sanity check *)

  2142                                     val _ =

  2143                                       if length terms < length terms' then

  2144                                         raise REFUTE ("IDT_constructor_interpreter",

  2145                                           "current size is less than old size")

  2146                                       else ()

  2147                                     (* sanity check: every element of terms' must also be *)

  2148                                     (*               present in terms                     *)

  2149                                     val _ =

  2150                                       if forall (member (op =) terms) terms' then ()

  2151                                       else

  2152                                         raise REFUTE ("IDT_constructor_interpreter",

  2153                                           "element has disappeared")

  2154                                     (* sanity check: the order on elements of terms' is    *)

  2155                                     (*               the same in terms, for those elements *)

  2156                                     val _ =

  2157                                       let

  2158                                         fun search (x::xs) (y::ys) =

  2159                                               if x = y then search xs ys else search (x::xs) ys

  2160                                           | search (x::xs) [] =

  2161                                               raise REFUTE ("IDT_constructor_interpreter",

  2162                                                 "element order not preserved")

  2163                                           | search [] _ = ()

  2164                                       in search terms' terms end

  2165                                     (* int * interpretation list *)

  2166                                     val (intrs, new_offset) =

  2167                                       fold_map (fn t_elem => fn off =>

  2168                                         (* if 't_elem' existed at the previous depth,    *)

  2169                                         (* proceed recursively, otherwise map the entire *)

  2170                                         (* subtree to "undefined"                        *)

  2171                                         if member (op =) terms' t_elem then

  2172                                           make_constr ds off

  2173                                         else

  2174                                           (make_undef ds, off))

  2175                                       terms offset

  2176                                   in

  2177                                     (Node intrs, new_offset)

  2178                                   end

  2179                           in

  2180                             SOME (fst (make_constr ctypes offset), model, args)

  2181                           end

  2182                     end

  2183                 | NONE =>  (* body type is not an inductive datatype *)

  2184                     NONE)

  2185             | _ =>  (* body type is a (free or schematic) type variable *)

  2186               NONE)

  2187         | _ =>  (* term is not a constant *)

  2188           NONE)

  2189   end;

  2191 (* Difficult code ahead.  Make sure you understand the                *)

  2192 (* 'IDT_constructor_interpreter' and the order in which it enumerates *)

  2193 (* elements of an IDT before you try to understand this function.     *)

  2195 fun IDT_recursion_interpreter ctxt model args t =

  2196   let

  2197     val thy = ProofContext.theory_of ctxt

  2198   in

  2199     (* careful: here we descend arbitrarily deep into 't', possibly before *)

  2200     (* any other interpreter for atomic terms has had a chance to look at  *)

  2201     (* 't'                                                                 *)

  2202     case strip_comb t of

  2203       (Const (s, T), params) =>

  2204         (* iterate over all datatypes in 'thy' *)

  2205         Symtab.fold (fn (_, info) => fn result =>

  2206           case result of

  2207             SOME _ =>

  2208               result  (* just keep 'result' *)

  2209           | NONE =>

  2210               if member (op =) (#rec_names info) s then

  2211                 (* we do have a recursion operator of one of the (mutually *)

  2212                 (* recursive) datatypes given by 'info'                    *)

  2213                 let

  2214                   (* number of all constructors, including those of different  *)

  2215                   (* (mutually recursive) datatypes within the same descriptor *)

  2216                   val mconstrs_count =

  2217                     Integer.sum (map (fn (_, (_, _, cs)) => length cs) (#descr info))

  2218                 in

  2219                   if mconstrs_count < length params then

  2220                     (* too many actual parameters; for now we'll use the *)

  2221                     (* 'stlc_interpreter' to strip off one application   *)

  2222                     NONE

  2223                   else if mconstrs_count > length params then

  2224                     (* too few actual parameters; we use eta expansion          *)

  2225                     (* Note that the resulting expansion of lambda abstractions *)

  2226                     (* by the 'stlc_interpreter' may be rather slow (depending  *)

  2227                     (* on the argument types and the size of the IDT, of        *)

  2228                     (* course).                                                 *)

  2229                     SOME (interpret ctxt model args (eta_expand t

  2230                       (mconstrs_count - length params)))

  2231                   else  (* mconstrs_count = length params *)

  2232                     let

  2233                       (* interpret each parameter separately *)

  2234                       val (p_intrs, (model', args')) = fold_map (fn p => fn (m, a) =>

  2235                         let

  2236                           val (i, m', a') = interpret ctxt m a p

  2237                         in

  2238                           (i, (m', a'))

  2239                         end) params (model, args)

  2240                       val (typs, _) = model'

  2241                       (* 'index' is /not/ necessarily the index of the IDT that *)

  2242                       (* the recursion operator is associated with, but merely  *)

  2243                       (* the index of some mutually recursive IDT               *)

  2244                       val index         = #index info

  2245                       val descr         = #descr info

  2246                       val (_, dtyps, _) = the (AList.lookup (op =) descr index)

  2247                       (* sanity check: we assume that the order of constructors *)

  2248                       (*               in 'descr' is the same as the order of   *)

  2249                       (*               corresponding parameters, otherwise the  *)

  2250                       (*               association code below won't match the   *)

  2251                       (*               right constructors/parameters; we also   *)

  2252                       (*               assume that the order of recursion       *)

  2253                       (*               operators in '#rec_names info' is the    *)

  2254                       (*               same as the order of corresponding       *)

  2255                       (*               datatypes in 'descr'                     *)

  2256                       val _ = if map fst descr <> (0 upto (length descr - 1)) then

  2257                           raise REFUTE ("IDT_recursion_interpreter",

  2258                             "order of constructors and corresponding parameters/" ^

  2259                               "recursion operators and corresponding datatypes " ^

  2260                               "different?")

  2261                         else ()

  2262                       (* sanity check: every element in 'dtyps' must be a *)

  2263                       (*               'DtTFree'                          *)

  2264                       val _ =

  2265                         if Library.exists (fn d =>

  2266                           case d of Datatype_Aux.DtTFree _ => false

  2267                                   | _ => true) dtyps

  2268                         then

  2269                           raise REFUTE ("IDT_recursion_interpreter",

  2270                             "datatype argument is not a variable")

  2271                         else ()

  2272                       (* the type of a recursion operator is *)

  2273                       (* [T1, ..., Tn, IDT] ---> Tresult     *)

  2274                       val IDT = List.nth (binder_types T, mconstrs_count)

  2275                       (* by our assumption on the order of recursion operators *)

  2276                       (* and datatypes, this is the index of the datatype      *)

  2277                       (* corresponding to the given recursion operator         *)

  2278                       val idt_index = find_index (fn s' => s' = s) (#rec_names info)

  2279                       (* mutually recursive types must have the same type   *)

  2280                       (* parameters, unless the mutual recursion comes from *)

  2281                       (* indirect recursion                                 *)

  2282                       fun rec_typ_assoc acc [] = acc

  2283                         | rec_typ_assoc acc ((d, T)::xs) =

  2284                             (case AList.lookup op= acc d of

  2285                               NONE =>

  2286                                 (case d of

  2287                                   Datatype_Aux.DtTFree _ =>

  2288                                   (* add the association, proceed *)

  2289                                   rec_typ_assoc ((d, T)::acc) xs

  2290                                 | Datatype_Aux.DtType (s, ds) =>

  2291                                     let

  2292                                       val (s', Ts) = dest_Type T

  2293                                     in

  2294                                       if s=s' then

  2295                                         rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)

  2296                                       else

  2297                                         raise REFUTE ("IDT_recursion_interpreter",

  2298                                           "DtType/Type mismatch")

  2299                                     end

  2300                                 | Datatype_Aux.DtRec i =>

  2301                                     let

  2302                                       val (_, ds, _) = the (AList.lookup (op =) descr i)

  2303                                       val (_, Ts)    = dest_Type T

  2304                                     in

  2305                                       rec_typ_assoc ((d, T)::acc) ((ds ~~ Ts) @ xs)

  2306                                     end)

  2307                             | SOME T' =>

  2308                                 if T=T' then

  2309                                   (* ignore the association since it's already *)

  2310                                   (* present, proceed                          *)

  2311                                   rec_typ_assoc acc xs

  2312                                 else

  2313                                   raise REFUTE ("IDT_recursion_interpreter",

  2314                                     "different type associations for the same dtyp"))

  2315                       val typ_assoc = filter

  2316                         (fn (Datatype_Aux.DtTFree _, _) => true | (_, _) => false)

  2317                         (rec_typ_assoc []

  2318                           (#2 (the (AList.lookup (op =) descr idt_index)) ~~ (snd o dest_Type) IDT))

  2319                       (* sanity check: typ_assoc must associate types to the   *)

  2320                       (*               elements of 'dtyps' (and only to those) *)

  2321                       val _ =

  2322                         if not (eq_set (op =) (dtyps, map fst typ_assoc))

  2323                         then

  2324                           raise REFUTE ("IDT_recursion_interpreter",

  2325                             "type association has extra/missing elements")

  2326                         else ()

  2327                       (* interpret each constructor in the descriptor (including *)

  2328                       (* those of mutually recursive datatypes)                  *)

  2329                       (* (int * interpretation list) list *)

  2330                       val mc_intrs = map (fn (idx, (_, _, cs)) =>

  2331                         let

  2332                           val c_return_typ = typ_of_dtyp descr typ_assoc

  2333                             (Datatype_Aux.DtRec idx)

  2334                         in

  2335                           (idx, map (fn (cname, cargs) =>

  2336                             (#1 o interpret ctxt (typs, []) {maxvars=0,

  2337                               def_eq=false, next_idx=1, bounds=[],

  2338                               wellformed=True}) (Const (cname, map (typ_of_dtyp

  2339                               descr typ_assoc) cargs ---> c_return_typ))) cs)

  2340                         end) descr

  2341                       (* associate constructors with corresponding parameters *)

  2342                       (* (int * (interpretation * interpretation) list) list *)

  2343                       val (mc_p_intrs, p_intrs') = fold_map

  2344                         (fn (idx, c_intrs) => fn p_intrs' =>

  2345                           let

  2346                             val len = length c_intrs

  2347                           in

  2348                             ((idx, c_intrs ~~ List.take (p_intrs', len)),

  2349                               List.drop (p_intrs', len))

  2350                           end) mc_intrs p_intrs

  2351                       (* sanity check: no 'p_intr' may be left afterwards *)

  2352                       val _ =

  2353                         if p_intrs' <> [] then

  2354                           raise REFUTE ("IDT_recursion_interpreter",

  2355                             "more parameter than constructor interpretations")

  2356                         else ()

  2357                       (* The recursion operator, applied to 'mconstrs_count'     *)

  2358                       (* arguments, is a function that maps every element of the *)

  2359                       (* inductive datatype to an element of some result type.   *)

  2360                       (* Recursion operators for mutually recursive IDTs are     *)

  2361                       (* translated simultaneously.                              *)

  2362                       (* Since the order on datatype elements is given by an     *)

  2363                       (* order on constructors (and then by the order on         *)

  2364                       (* argument tuples), we can simply copy corresponding      *)

  2365                       (* subtrees from 'p_intrs', in the order in which they are *)

  2366                       (* given.                                                  *)

  2367                       (* interpretation * interpretation -> interpretation list *)

  2368                       fun ci_pi (Leaf xs, pi) =

  2369                             (* if the constructor does not match the arguments to a *)

  2370                             (* defined element of the IDT, the corresponding value  *)

  2371                             (* of the parameter must be ignored                     *)

  2372                             if List.exists (equal True) xs then [pi] else []

  2373                         | ci_pi (Node xs, Node ys) = maps ci_pi (xs ~~ ys)

  2374                         | ci_pi (Node _, Leaf _) =

  2375                             raise REFUTE ("IDT_recursion_interpreter",

  2376                               "constructor takes more arguments than the " ^

  2377                                 "associated parameter")

  2378                       (* (int * interpretation list) list *)

  2379                       val rec_operators = map (fn (idx, c_p_intrs) =>

  2380                         (idx, maps ci_pi c_p_intrs)) mc_p_intrs

  2381                       (* sanity check: every recursion operator must provide as  *)

  2382                       (*               many values as the corresponding datatype *)

  2383                       (*               has elements                              *)

  2384                       val _ = map (fn (idx, intrs) =>

  2385                         let

  2386                           val T = typ_of_dtyp descr typ_assoc

  2387                             (Datatype_Aux.DtRec idx)

  2388                         in

  2389                           if length intrs <> size_of_type ctxt (typs, []) T then

  2390                             raise REFUTE ("IDT_recursion_interpreter",

  2391                               "wrong number of interpretations for rec. operator")

  2392                           else ()

  2393                         end) rec_operators

  2394                       (* For non-recursive datatypes, we are pretty much done at *)

  2395                       (* this point.  For recursive datatypes however, we still  *)

  2396                       (* need to apply the interpretations in 'rec_operators' to *)

  2397                       (* (recursively obtained) interpretations for recursive    *)

  2398                       (* constructor arguments.  To do so more efficiently, we   *)

  2399                       (* copy 'rec_operators' into arrays first.  Each Boolean   *)

  2400                       (* indicates whether the recursive arguments have been     *)

  2401                       (* considered already.                                     *)

  2402                       (* (int * (bool * interpretation) Array.array) list *)

  2403                       val REC_OPERATORS = map (fn (idx, intrs) =>

  2404                         (idx, Array.fromList (map (pair false) intrs)))

  2405                         rec_operators

  2406                       (* takes an interpretation, and if some leaf of this     *)

  2407                       (* interpretation is the 'elem'-th element of the type,  *)

  2408                       (* the indices of the arguments leading to this leaf are *)

  2409                       (* returned                                              *)

  2410                       (* interpretation -> int -> int list option *)

  2411                       fun get_args (Leaf xs) elem =

  2412                             if find_index (fn x => x = True) xs = elem then

  2413                               SOME []

  2414                             else

  2415                               NONE

  2416                         | get_args (Node xs) elem =

  2417                             let

  2418                               (* interpretation list * int -> int list option *)

  2419                               fun search ([], _) =

  2420                                 NONE

  2421                                 | search (x::xs, n) =

  2422                                 (case get_args x elem of

  2423                                   SOME result => SOME (n::result)

  2424                                 | NONE        => search (xs, n+1))

  2425                             in

  2426                               search (xs, 0)

  2427                             end

  2428                       (* returns the index of the constructor and indices for *)

  2429                       (* its arguments that generate the 'elem'-th element of *)

  2430                       (* the datatype given by 'idx'                          *)

  2431                       (* int -> int -> int * int list *)

  2432                       fun get_cargs idx elem =

  2433                         let

  2434                           (* int * interpretation list -> int * int list *)

  2435                           fun get_cargs_rec (_, []) =

  2436                                 raise REFUTE ("IDT_recursion_interpreter",

  2437                                   "no matching constructor found for datatype element")

  2438                             | get_cargs_rec (n, x::xs) =

  2439                                 (case get_args x elem of

  2440                                   SOME args => (n, args)

  2441                                 | NONE => get_cargs_rec (n+1, xs))

  2442                         in

  2443                           get_cargs_rec (0, the (AList.lookup (op =) mc_intrs idx))

  2444                         end

  2445                       (* computes one entry in 'REC_OPERATORS', and recursively *)

  2446                       (* all entries needed for it, where 'idx' gives the       *)

  2447                       (* datatype and 'elem' the element of it                  *)

  2448                       (* int -> int -> interpretation *)

  2449                       fun compute_array_entry idx elem =

  2450                         let

  2451                           val arr = the (AList.lookup (op =) REC_OPERATORS idx)

  2452                           val (flag, intr) = Array.sub (arr, elem)

  2453                         in

  2454                           if flag then

  2455                             (* simply return the previously computed result *)

  2456                             intr

  2457                           else

  2458                             (* we have to apply 'intr' to interpretations for all *)

  2459                             (* recursive arguments                                *)

  2460                             let

  2461                               (* int * int list *)

  2462                               val (c, args) = get_cargs idx elem

  2463                               (* find the indices of the constructor's /recursive/ *)

  2464                               (* arguments                                         *)

  2465                               val (_, _, constrs) = the (AList.lookup (op =) descr idx)

  2466                               val (_, dtyps)      = List.nth (constrs, c)

  2467                               val rec_dtyps_args  = filter

  2468                                 (Datatype_Aux.is_rec_type o fst) (dtyps ~~ args)

  2469                               (* map those indices to interpretations *)

  2470                               val rec_dtyps_intrs = map (fn (dtyp, arg) =>

  2471                                 let

  2472                                   val dT     = typ_of_dtyp descr typ_assoc dtyp

  2473                                   val consts = make_constants ctxt (typs, []) dT

  2474                                   val arg_i  = List.nth (consts, arg)

  2475                                 in

  2476                                   (dtyp, arg_i)

  2477                                 end) rec_dtyps_args

  2478                               (* takes the dtyp and interpretation of an element, *)

  2479                               (* and computes the interpretation for the          *)

  2480                               (* corresponding recursive argument                 *)

  2481                               fun rec_intr (Datatype_Aux.DtRec i) (Leaf xs) =

  2482                                     (* recursive argument is "rec_i params elem" *)

  2483                                     compute_array_entry i (find_index (fn x => x = True) xs)

  2484                                 | rec_intr (Datatype_Aux.DtRec _) (Node _) =

  2485                                     raise REFUTE ("IDT_recursion_interpreter",

  2486                                       "interpretation for IDT is a node")

  2487                                 | rec_intr (Datatype_Aux.DtType ("fun", [dt1, dt2])) (Node xs) =

  2488                                     (* recursive argument is something like     *)

  2489                                     (* "\<lambda>x::dt1. rec_? params (elem x)" *)

  2490                                     Node (map (rec_intr dt2) xs)

  2491                                 | rec_intr (Datatype_Aux.DtType ("fun", [_, _])) (Leaf _) =

  2492                                     raise REFUTE ("IDT_recursion_interpreter",

  2493                                       "interpretation for function dtyp is a leaf")

  2494                                 | rec_intr _ _ =

  2495                                     (* admissibility ensures that every recursive type *)

  2496                                     (* is of the form 'Dt_1 -> ... -> Dt_k ->          *)

  2497                                     (* (DtRec i)'                                      *)

  2498                                     raise REFUTE ("IDT_recursion_interpreter",

  2499                                       "non-recursive codomain in recursive dtyp")

  2500                               (* obtain interpretations for recursive arguments *)

  2501                               (* interpretation list *)

  2502                               val arg_intrs = map (uncurry rec_intr) rec_dtyps_intrs

  2503                               (* apply 'intr' to all recursive arguments *)

  2504                               val result = fold (fn arg_i => fn i =>

  2505                                 interpretation_apply (i, arg_i)) arg_intrs intr

  2506                               (* update 'REC_OPERATORS' *)

  2507                               val _ = Array.update (arr, elem, (true, result))

  2508                             in

  2509                               result

  2510                             end

  2511                         end

  2512                       val idt_size = Array.length (the (AList.lookup (op =) REC_OPERATORS idt_index))

  2513                       (* sanity check: the size of 'IDT' should be 'idt_size' *)

  2514                       val _ =

  2515                           if idt_size <> size_of_type ctxt (typs, []) IDT then

  2516                             raise REFUTE ("IDT_recursion_interpreter",

  2517                               "unexpected size of IDT (wrong type associated?)")

  2518                           else ()

  2519                       (* interpretation *)

  2520                       val rec_op = Node (map_range (compute_array_entry idt_index) idt_size)

  2521                     in

  2522                       SOME (rec_op, model', args')

  2523                     end

  2524                 end

  2525               else

  2526                 NONE  (* not a recursion operator of this datatype *)

  2527           ) (Datatype.get_all thy) NONE

  2528     | _ =>  (* head of term is not a constant *)

  2529       NONE

  2530   end;

  2532 fun set_interpreter ctxt model args t =

  2533   let

  2534     val (typs, terms) = model

  2535   in

  2536     case AList.lookup (op =) terms t of

  2537       SOME intr =>

  2538         (* return an existing interpretation *)

  2539         SOME (intr, model, args)

  2540     | NONE =>

  2541         (case t of

  2542         (* 'Collect' == identity *)

  2543           Const (@{const_name Collect}, _) $ t1 =>

  2544             SOME (interpret ctxt model args t1)

  2545         | Const (@{const_name Collect}, _) =>

  2546             SOME (interpret ctxt model args (eta_expand t 1))

  2547         (* 'op :' == application *)

  2548         | Const (@{const_name Set.member}, _) $ t1 $ t2 =>

  2549             SOME (interpret ctxt model args (t2 $ t1))

  2550         | Const (@{const_name Set.member}, _) $ t1 =>

  2551             SOME (interpret ctxt model args (eta_expand t 1))

  2552         | Const (@{const_name Set.member}, _) =>

  2553             SOME (interpret ctxt model args (eta_expand t 2))

  2554         | _ => NONE)

  2555   end;

  2557 (* only an optimization: 'card' could in principle be interpreted with *)

  2558 (* interpreters available already (using its definition), but the code *)

  2559 (* below is more efficient                                             *)

  2561 fun Finite_Set_card_interpreter ctxt model args t =

  2562   case t of

  2563     Const (@{const_name Finite_Set.card},

  2564         Type ("fun", [Type ("fun", [T, @{typ bool}]), @{typ nat}])) =>

  2565       let

  2566         (* interpretation -> int *)

  2567         fun number_of_elements (Node xs) =

  2568             fold (fn x => fn n =>

  2569               if x = TT then

  2570                 n + 1

  2571               else if x = FF then

  2572                 n

  2573               else

  2574                 raise REFUTE ("Finite_Set_card_interpreter",

  2575                   "interpretation for set type does not yield a Boolean"))

  2576               xs 0

  2577           | number_of_elements (Leaf _) =

  2578               raise REFUTE ("Finite_Set_card_interpreter",

  2579                 "interpretation for set type is a leaf")

  2580         val size_of_nat = size_of_type ctxt model (@{typ nat})

  2581         (* takes an interpretation for a set and returns an interpretation *)

  2582         (* for a 'nat' denoting the set's cardinality                      *)

  2583         (* interpretation -> interpretation *)

  2584         fun card i =

  2585           let

  2586             val n = number_of_elements i

  2587           in

  2588             if n < size_of_nat then

  2589               Leaf ((replicate n False) @ True ::

  2590                 (replicate (size_of_nat-n-1) False))

  2591             else

  2592               Leaf (replicate size_of_nat False)

  2593           end

  2594         val set_constants =

  2595           make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))

  2596       in

  2597         SOME (Node (map card set_constants), model, args)

  2598       end

  2599   | _ => NONE;

  2601 (* only an optimization: 'finite' could in principle be interpreted with  *)

  2602 (* interpreters available already (using its definition), but the code    *)

  2603 (* below is more efficient                                                *)

  2605 fun Finite_Set_finite_interpreter ctxt model args t =

  2606   case t of

  2607     Const (@{const_name Finite_Set.finite},

  2608       Type ("fun", [Type ("fun", [T, @{typ bool}]),

  2609                     @{typ bool}])) $ _ =>

  2610         (* we only consider finite models anyway, hence EVERY set is *)

  2611         (* "finite"                                                  *)

  2612         SOME (TT, model, args)

  2613   | Const (@{const_name Finite_Set.finite},

  2614       Type ("fun", [Type ("fun", [T, @{typ bool}]),

  2615                     @{typ bool}])) =>

  2616       let

  2617         val size_of_set =

  2618           size_of_type ctxt model (Type ("fun", [T, HOLogic.boolT]))

  2619       in

  2620         (* we only consider finite models anyway, hence EVERY set is *)

  2621         (* "finite"                                                  *)

  2622         SOME (Node (replicate size_of_set TT), model, args)

  2623       end

  2624   | _ => NONE;

  2626 (* only an optimization: 'less' could in principle be interpreted with *)

  2627 (* interpreters available already (using its definition), but the code     *)

  2628 (* below is more efficient                                                 *)

  2630 fun Nat_less_interpreter ctxt model args t =

  2631   case t of

  2632     Const (@{const_name Orderings.less}, Type ("fun", [@{typ nat},

  2633         Type ("fun", [@{typ nat}, @{typ bool}])])) =>

  2634       let

  2635         val size_of_nat = size_of_type ctxt model (@{typ nat})

  2636         (* the 'n'-th nat is not less than the first 'n' nats, while it *)

  2637         (* is less than the remaining 'size_of_nat - n' nats            *)

  2638         (* int -> interpretation *)

  2639         fun less n = Node ((replicate n FF) @ (replicate (size_of_nat - n) TT))

  2640       in

  2641         SOME (Node (map less (1 upto size_of_nat)), model, args)

  2642       end

  2643   | _ => NONE;

  2645 (* only an optimization: 'plus' could in principle be interpreted with *)

  2646 (* interpreters available already (using its definition), but the code     *)

  2647 (* below is more efficient                                                 *)

  2649 fun Nat_plus_interpreter ctxt model args t =

  2650   case t of

  2651     Const (@{const_name Groups.plus}, Type ("fun", [@{typ nat},

  2652         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

  2653       let

  2654         val size_of_nat = size_of_type ctxt model (@{typ nat})

  2655         (* int -> int -> interpretation *)

  2656         fun plus m n =

  2657           let

  2658             val element = m + n

  2659           in

  2660             if element > size_of_nat - 1 then

  2661               Leaf (replicate size_of_nat False)

  2662             else

  2663               Leaf ((replicate element False) @ True ::

  2664                 (replicate (size_of_nat - element - 1) False))

  2665           end

  2666       in

  2667         SOME (Node (map_range (fn m => Node (map_range (plus m) size_of_nat)) size_of_nat),

  2668           model, args)

  2669       end

  2670   | _ => NONE;

  2672 (* only an optimization: 'minus' could in principle be interpreted *)

  2673 (* with interpreters available already (using its definition), but the *)

  2674 (* code below is more efficient                                        *)

  2676 fun Nat_minus_interpreter ctxt model args t =

  2677   case t of

  2678     Const (@{const_name Groups.minus}, Type ("fun", [@{typ nat},

  2679         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

  2680       let

  2681         val size_of_nat = size_of_type ctxt model (@{typ nat})

  2682         (* int -> int -> interpretation *)

  2683         fun minus m n =

  2684           let

  2685             val element = Int.max (m-n, 0)

  2686           in

  2687             Leaf ((replicate element False) @ True ::

  2688               (replicate (size_of_nat - element - 1) False))

  2689           end

  2690       in

  2691         SOME (Node (map_range (fn m => Node (map_range (minus m) size_of_nat)) size_of_nat),

  2692           model, args)

  2693       end

  2694   | _ => NONE;

  2696 (* only an optimization: 'times' could in principle be interpreted *)

  2697 (* with interpreters available already (using its definition), but the *)

  2698 (* code below is more efficient                                        *)

  2700 fun Nat_times_interpreter ctxt model args t =

  2701   case t of

  2702     Const (@{const_name Groups.times}, Type ("fun", [@{typ nat},

  2703         Type ("fun", [@{typ nat}, @{typ nat}])])) =>

  2704       let

  2705         val size_of_nat = size_of_type ctxt model (@{typ nat})

  2706         (* nat -> nat -> interpretation *)

  2707         fun mult m n =

  2708           let

  2709             val element = m * n

  2710           in

  2711             if element > size_of_nat - 1 then

  2712               Leaf (replicate size_of_nat False)

  2713             else

  2714               Leaf ((replicate element False) @ True ::

  2715                 (replicate (size_of_nat - element - 1) False))

  2716           end

  2717       in

  2718         SOME (Node (map_range (fn m => Node (map_range (mult m) size_of_nat)) size_of_nat),

  2719           model, args)

  2720       end

  2721   | _ => NONE;

  2723 (* only an optimization: 'append' could in principle be interpreted with *)

  2724 (* interpreters available already (using its definition), but the code   *)

  2725 (* below is more efficient                                               *)

  2727 fun List_append_interpreter ctxt model args t =

  2728   case t of

  2729     Const (@{const_name List.append}, Type ("fun", [Type ("List.list", [T]), Type ("fun",

  2730         [Type ("List.list", [_]), Type ("List.list", [_])])])) =>

  2731       let

  2732         val size_elem = size_of_type ctxt model T

  2733         val size_list = size_of_type ctxt model (Type ("List.list", [T]))

  2734         (* maximal length of lists; 0 if we only consider the empty list *)

  2735         val list_length =

  2736           let

  2737             (* int -> int -> int -> int *)

  2738             fun list_length_acc len lists total =

  2739               if lists = total then

  2740                 len

  2741               else if lists < total then

  2742                 list_length_acc (len+1) (lists*size_elem) (total-lists)

  2743               else

  2744                 raise REFUTE ("List_append_interpreter",

  2745                   "size_list not equal to 1 + size_elem + ... + " ^

  2746                     "size_elem^len, for some len")

  2747           in

  2748             list_length_acc 0 1 size_list

  2749           end

  2750         val elements = 0 upto (size_list-1)

  2751         (* FIXME: there should be a nice formula, which computes the same as *)

  2752         (*        the following, but without all this intermediate tree      *)

  2753         (*        length/offset stuff                                        *)

  2754         (* associate each list with its length and offset in a complete tree *)

  2755         (* of width 'size_elem' and depth 'length_list' (with 'size_list'    *)

  2756         (* nodes total)                                                      *)

  2757         (* (int * (int * int)) list *)

  2758         val (lenoff_lists, _) = fold_map (fn elem => fn (offsets, off) =>

  2759           (* corresponds to a pre-order traversal of the tree *)

  2760           let

  2761             val len = length offsets

  2762             (* associate the given element with len/off *)

  2763             val assoc = (elem, (len, off))

  2764           in

  2765             if len < list_length then

  2766               (* go to first child node *)

  2767               (assoc, (off :: offsets, off * size_elem))

  2768             else if off mod size_elem < size_elem - 1 then

  2769               (* go to next sibling node *)

  2770               (assoc, (offsets, off + 1))

  2771             else

  2772               (* go back up the stack until we find a level where we can go *)

  2773               (* to the next sibling node                                   *)

  2774               let

  2775                 val offsets' = dropwhile

  2776                   (fn off' => off' mod size_elem = size_elem - 1) offsets

  2777               in

  2778                 case offsets' of

  2779                   [] =>

  2780                     (* we're at the last node in the tree; the next value *)

  2781                     (* won't be used anyway                               *)

  2782                     (assoc, ([], 0))

  2783                 | off'::offs' =>

  2784                     (* go to next sibling node *)

  2785                     (assoc, (offs', off' + 1))

  2786               end

  2787           end) elements ([], 0)

  2788         (* we also need the reverse association (from length/offset to *)

  2789         (* index)                                                      *)

  2790         val lenoff'_lists = map Library.swap lenoff_lists

  2791         (* returns the interpretation for "(list no. m) @ (list no. n)" *)

  2792         (* nat -> nat -> interpretation *)

  2793         fun append m n =

  2794           let

  2795             val (len_m, off_m) = the (AList.lookup (op =) lenoff_lists m)

  2796             val (len_n, off_n) = the (AList.lookup (op =) lenoff_lists n)

  2797             val len_elem = len_m + len_n

  2798             val off_elem = off_m * Integer.pow len_n size_elem + off_n

  2799           in

  2800             case AList.lookup op= lenoff'_lists (len_elem, off_elem) of

  2801               NONE =>

  2802                 (* undefined *)

  2803                 Leaf (replicate size_list False)

  2804             | SOME element =>

  2805                 Leaf ((replicate element False) @ True ::

  2806                   (replicate (size_list - element - 1) False))

  2807           end

  2808       in

  2809         SOME (Node (map (fn m => Node (map (append m) elements)) elements),

  2810           model, args)

  2811       end

  2812   | _ => NONE;

  2814 (* UNSOUND

  2816 (* only an optimization: 'lfp' could in principle be interpreted with  *)

  2817 (* interpreters available already (using its definition), but the code *)

  2818 (* below is more efficient                                             *)

  2820 fun lfp_interpreter ctxt model args t =

  2821   case t of

  2822     Const (@{const_name lfp}, Type ("fun", [Type ("fun",

  2823       [Type ("fun", [T, @{typ bool}]),

  2824        Type ("fun", [_, @{typ bool}])]),

  2825        Type ("fun", [_, @{typ bool}])])) =>

  2826       let

  2827         val size_elem = size_of_type ctxt model T

  2828         (* the universe (i.e. the set that contains every element) *)

  2829         val i_univ = Node (replicate size_elem TT)

  2830         (* all sets with elements from type 'T' *)

  2831         val i_sets =

  2832           make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))

  2833         (* all functions that map sets to sets *)

  2834         val i_funs = make_constants ctxt model (Type ("fun",

  2835           [Type ("fun", [T, @{typ bool}]),

  2836            Type ("fun", [T, @{typ bool}])]))

  2837         (* "lfp(f) == Inter({u. f(u) <= u})" *)

  2838         (* interpretation * interpretation -> bool *)

  2839         fun is_subset (Node subs, Node sups) =

  2840               forall (fn (sub, sup) => (sub = FF) orelse (sup = TT)) (subs ~~ sups)

  2841           | is_subset (_, _) =

  2842               raise REFUTE ("lfp_interpreter",

  2843                 "is_subset: interpretation for set is not a node")

  2844         (* interpretation * interpretation -> interpretation *)

  2845         fun intersection (Node xs, Node ys) =

  2846               Node (map (fn (x, y) => if x=TT andalso y=TT then TT else FF)

  2847                 (xs ~~ ys))

  2848           | intersection (_, _) =

  2849               raise REFUTE ("lfp_interpreter",

  2850                 "intersection: interpretation for set is not a node")

  2851         (* interpretation -> interpretaion *)

  2852         fun lfp (Node resultsets) =

  2853               fold (fn (set, resultset) => fn acc =>

  2854                 if is_subset (resultset, set) then

  2855                   intersection (acc, set)

  2856                 else

  2857                   acc) (i_sets ~~ resultsets) i_univ

  2858           | lfp _ =

  2859               raise REFUTE ("lfp_interpreter",

  2860                 "lfp: interpretation for function is not a node")

  2861       in

  2862         SOME (Node (map lfp i_funs), model, args)

  2863       end

  2864   | _ => NONE;

  2866 (* only an optimization: 'gfp' could in principle be interpreted with  *)

  2867 (* interpreters available already (using its definition), but the code *)

  2868 (* below is more efficient                                             *)

  2870 fun gfp_interpreter ctxt model args t =

  2871   case t of

  2872     Const (@{const_name gfp}, Type ("fun", [Type ("fun",

  2873       [Type ("fun", [T, @{typ bool}]),

  2874        Type ("fun", [_, @{typ bool}])]),

  2875        Type ("fun", [_, @{typ bool}])])) =>

  2876     let

  2877       val size_elem = size_of_type ctxt model T

  2878       (* the universe (i.e. the set that contains every element) *)

  2879       val i_univ = Node (replicate size_elem TT)

  2880       (* all sets with elements from type 'T' *)

  2881       val i_sets =

  2882         make_constants ctxt model (Type ("fun", [T, HOLogic.boolT]))

  2883       (* all functions that map sets to sets *)

  2884       val i_funs = make_constants ctxt model (Type ("fun",

  2885         [Type ("fun", [T, HOLogic.boolT]),

  2886          Type ("fun", [T, HOLogic.boolT])]))

  2887       (* "gfp(f) == Union({u. u <= f(u)})" *)

  2888       (* interpretation * interpretation -> bool *)

  2889       fun is_subset (Node subs, Node sups) =

  2890             forall (fn (sub, sup) => (sub = FF) orelse (sup = TT))

  2891               (subs ~~ sups)

  2892         | is_subset (_, _) =

  2893             raise REFUTE ("gfp_interpreter",

  2894               "is_subset: interpretation for set is not a node")

  2895       (* interpretation * interpretation -> interpretation *)

  2896       fun union (Node xs, Node ys) =

  2897             Node (map (fn (x,y) => if x=TT orelse y=TT then TT else FF)

  2898                  (xs ~~ ys))

  2899         | union (_, _) =

  2900             raise REFUTE ("gfp_interpreter",

  2901               "union: interpretation for set is not a node")

  2902       (* interpretation -> interpretaion *)

  2903       fun gfp (Node resultsets) =

  2904             fold (fn (set, resultset) => fn acc =>

  2905               if is_subset (set, resultset) then

  2906                 union (acc, set)

  2907               else

  2908                 acc) (i_sets ~~ resultsets) i_univ

  2909         | gfp _ =

  2910             raise REFUTE ("gfp_interpreter",

  2911               "gfp: interpretation for function is not a node")

  2912     in

  2913       SOME (Node (map gfp i_funs), model, args)

  2914     end

  2915   | _ => NONE;

  2916 *)

  2918 (* only an optimization: 'fst' could in principle be interpreted with  *)

  2919 (* interpreters available already (using its definition), but the code *)

  2920 (* below is more efficient                                             *)

  2922 fun Product_Type_fst_interpreter ctxt model args t =

  2923   case t of

  2924     Const (@{const_name fst}, Type ("fun", [Type (@{type_name Product_Type.prod}, [T, U]), _])) =>

  2925       let

  2926         val constants_T = make_constants ctxt model T

  2927         val size_U = size_of_type ctxt model U

  2928       in

  2929         SOME (Node (maps (replicate size_U) constants_T), model, args)

  2930       end

  2931   | _ => NONE;

  2933 (* only an optimization: 'snd' could in principle be interpreted with  *)

  2934 (* interpreters available already (using its definition), but the code *)

  2935 (* below is more efficient                                             *)

  2937 fun Product_Type_snd_interpreter ctxt model args t =

  2938   case t of

  2939     Const (@{const_name snd}, Type ("fun", [Type (@{type_name Product_Type.prod}, [T, U]), _])) =>

  2940       let

  2941         val size_T = size_of_type ctxt model T

  2942         val constants_U = make_constants ctxt model U

  2943       in

  2944         SOME (Node (flat (replicate size_T constants_U)), model, args)

  2945       end

  2946   | _ => NONE;

  2949 (* ------------------------------------------------------------------------- *)

  2950 (* PRINTERS                                                                  *)

  2951 (* ------------------------------------------------------------------------- *)

  2953 fun stlc_printer ctxt model T intr assignment =

  2954   let

  2955     (* string -> string *)

  2956     fun strip_leading_quote s =

  2957       (implode o (fn [] => [] | x::xs => if x="'" then xs else x::xs)

  2958         o explode) s

  2959     (* Term.typ -> string *)

  2960     fun string_of_typ (Type (s, _)) = s

  2961       | string_of_typ (TFree (x, _)) = strip_leading_quote x

  2962       | string_of_typ (TVar ((x,i), _)) =

  2963           strip_leading_quote x ^ string_of_int i

  2964     (* interpretation -> int *)

  2965     fun index_from_interpretation (Leaf xs) =

  2966           find_index (PropLogic.eval assignment) xs

  2967       | index_from_interpretation _ =

  2968           raise REFUTE ("stlc_printer",

  2969             "interpretation for ground type is not a leaf")

  2970   in

  2971     case T of

  2972       Type ("fun", [T1, T2]) =>

  2973         let

  2974           (* create all constants of type 'T1' *)

  2975           val constants = make_constants ctxt model T1

  2976           (* interpretation list *)

  2977           val results =

  2978             (case intr of

  2979               Node xs => xs

  2980             | _ => raise REFUTE ("stlc_printer",

  2981               "interpretation for function type is a leaf"))

  2982           (* Term.term list *)

  2983           val pairs = map (fn (arg, result) =>

  2984             HOLogic.mk_prod

  2985               (print ctxt model T1 arg assignment,

  2986                print ctxt model T2 result assignment))

  2987             (constants ~~ results)

  2988           (* Term.typ *)

  2989           val HOLogic_prodT = HOLogic.mk_prodT (T1, T2)

  2990           val HOLogic_setT  = HOLogic.mk_setT HOLogic_prodT

  2991           (* Term.term *)

  2992           val HOLogic_empty_set = Const (@{const_abbrev Set.empty}, HOLogic_setT)

  2993           val HOLogic_insert    =

  2994             Const (@{const_name insert}, HOLogic_prodT --> HOLogic_setT --> HOLogic_setT)

  2995         in

  2996           SOME (fold_rev (fn pair => fn acc => HOLogic_insert $ pair $ acc) pairs HOLogic_empty_set)

  2997         end

  2998     | Type ("prop", []) =>

  2999         (case index_from_interpretation intr of

  3000           ~1 => SOME (HOLogic.mk_Trueprop (Const (@{const_name undefined}, HOLogic.boolT)))

  3001         | 0  => SOME (HOLogic.mk_Trueprop HOLogic.true_const)

  3002         | 1  => SOME (HOLogic.mk_Trueprop HOLogic.false_const)

  3003         | _  => raise REFUTE ("stlc_interpreter",

  3004           "illegal interpretation for a propositional value"))

  3005     | Type _  =>

  3006         if index_from_interpretation intr = (~1) then

  3007           SOME (Const (@{const_name undefined}, T))

  3008         else

  3009           SOME (Const (string_of_typ T ^

  3010             string_of_int (index_from_interpretation intr), T))

  3011     | TFree _ =>

  3012         if index_from_interpretation intr = (~1) then

  3013           SOME (Const (@{const_name undefined}, T))

  3014         else

  3015           SOME (Const (string_of_typ T ^

  3016             string_of_int (index_from_interpretation intr), T))

  3017     | TVar _  =>

  3018         if index_from_interpretation intr = (~1) then

  3019           SOME (Const (@{const_name undefined}, T))

  3020         else

  3021           SOME (Const (string_of_typ T ^

  3022             string_of_int (index_from_interpretation intr), T))

  3023   end;

  3025 fun IDT_printer ctxt model T intr assignment =

  3026   let

  3027     val thy = ProofContext.theory_of ctxt

  3028   in

  3029     (case T of

  3030       Type (s, Ts) =>

  3031         (case Datatype.get_info thy s of

  3032           SOME info =>  (* inductive datatype *)

  3033             let

  3034               val (typs, _)           = model

  3035               val index               = #index info

  3036               val descr               = #descr info

  3037               val (_, dtyps, constrs) = the (AList.lookup (op =) descr index)

  3038               val typ_assoc           = dtyps ~~ Ts

  3039               (* sanity check: every element in 'dtyps' must be a 'DtTFree' *)

  3040               val _ =

  3041                 if Library.exists (fn d =>

  3042                   case d of Datatype_Aux.DtTFree _ => false | _ => true) dtyps

  3043                 then

  3044                   raise REFUTE ("IDT_printer", "datatype argument (for type " ^

  3045                     Syntax.string_of_typ ctxt (Type (s, Ts)) ^ ") is not a variable")

  3046                 else ()

  3047               (* the index of the element in the datatype *)

  3048               val element =

  3049                 (case intr of

  3050                   Leaf xs => find_index (PropLogic.eval assignment) xs

  3051                 | Node _  => raise REFUTE ("IDT_printer",

  3052                   "interpretation is not a leaf"))

  3053             in

  3054               if element < 0 then

  3055                 SOME (Const (@{const_name undefined}, Type (s, Ts)))

  3056               else

  3057                 let

  3058                   (* takes a datatype constructor, and if for some arguments this  *)

  3059                   (* constructor generates the datatype's element that is given by *)

  3060                   (* 'element', returns the constructor (as a term) as well as the *)

  3061                   (* indices of the arguments                                      *)

  3062                   fun get_constr_args (cname, cargs) =

  3063                     let

  3064                       val cTerm      = Const (cname,

  3065                         map (typ_of_dtyp descr typ_assoc) cargs ---> Type (s, Ts))

  3066                       val (iC, _, _) = interpret ctxt (typs, []) {maxvars=0,

  3067                         def_eq=false, next_idx=1, bounds=[], wellformed=True} cTerm

  3068                       (* interpretation -> int list option *)

  3069                       fun get_args (Leaf xs) =

  3070                             if find_index (fn x => x = True) xs = element then

  3071                               SOME []

  3072                             else

  3073                               NONE

  3074                         | get_args (Node xs) =

  3075                             let

  3076                               (* interpretation * int -> int list option *)

  3077                               fun search ([], _) =

  3078                                 NONE

  3079                                 | search (x::xs, n) =

  3080                                 (case get_args x of

  3081                                   SOME result => SOME (n::result)

  3082                                 | NONE        => search (xs, n+1))

  3083                             in

  3084                               search (xs, 0)

  3085                             end

  3086                     in

  3087                       Option.map (fn args => (cTerm, cargs, args)) (get_args iC)

  3088                     end

  3089                   val (cTerm, cargs, args) =

  3090                     (* we could speed things up by computing the correct          *)

  3091                     (* constructor directly (rather than testing all              *)

  3092                     (* constructors), based on the order in which constructors    *)

  3093                     (* generate elements of datatypes; the current implementation *)

  3094                     (* of 'IDT_printer' however is independent of the internals   *)

  3095                     (* of 'IDT_constructor_interpreter'                           *)

  3096                     (case get_first get_constr_args constrs of

  3097                       SOME x => x

  3098                     | NONE   => raise REFUTE ("IDT_printer",

  3099                       "no matching constructor found for element " ^

  3100                       string_of_int element))

  3101                   val argsTerms = map (fn (d, n) =>

  3102                     let

  3103                       val dT = typ_of_dtyp descr typ_assoc d

  3104                       (* we only need the n-th element of this list, so there   *)

  3105                       (* might be a more efficient implementation that does not *)

  3106                       (* generate all constants                                 *)

  3107                       val consts = make_constants ctxt (typs, []) dT

  3108                     in

  3109                       print ctxt (typs, []) dT (List.nth (consts, n)) assignment

  3110                     end) (cargs ~~ args)

  3111                 in

  3112                   SOME (list_comb (cTerm, argsTerms))

  3113                 end

  3114             end

  3115         | NONE =>  (* not an inductive datatype *)

  3116             NONE)

  3117     | _ =>  (* a (free or schematic) type variable *)

  3118         NONE)

  3119   end;

  3122 (* ------------------------------------------------------------------------- *)

  3123 (* use 'setup Refute.setup' in an Isabelle theory to initialize the 'Refute' *)

  3124 (* structure                                                                 *)

  3125 (* ------------------------------------------------------------------------- *)

  3127 (* ------------------------------------------------------------------------- *)

  3128 (* Note: the interpreters and printers are used in reverse order; however,   *)

  3129 (*       an interpreter that can handle non-atomic terms ends up being       *)

  3130 (*       applied before the 'stlc_interpreter' breaks the term apart into    *)

  3131 (*       subterms that are then passed to other interpreters!                *)

  3132 (* ------------------------------------------------------------------------- *)

  3134 val setup =

  3135    add_interpreter "stlc"    stlc_interpreter #>

  3136    add_interpreter "Pure"    Pure_interpreter #>

  3137    add_interpreter "HOLogic" HOLogic_interpreter #>

  3138    add_interpreter "set"     set_interpreter #>

  3139    add_interpreter "IDT"             IDT_interpreter #>

  3140    add_interpreter "IDT_constructor" IDT_constructor_interpreter #>

  3141    add_interpreter "IDT_recursion"   IDT_recursion_interpreter #>

  3142    add_interpreter "Finite_Set.card"    Finite_Set_card_interpreter #>

  3143    add_interpreter "Finite_Set.finite"  Finite_Set_finite_interpreter #>

  3144    add_interpreter "Nat_Orderings.less" Nat_less_interpreter #>

  3145    add_interpreter "Nat_HOL.plus"       Nat_plus_interpreter #>

  3146    add_interpreter "Nat_HOL.minus"      Nat_minus_interpreter #>

  3147    add_interpreter "Nat_HOL.times"      Nat_times_interpreter #>

  3148    add_interpreter "List.append" List_append_interpreter #>

  3149 (* UNSOUND

  3150    add_interpreter "lfp" lfp_interpreter #>

  3151    add_interpreter "gfp" gfp_interpreter #>

  3152 *)

  3153    add_interpreter "Product_Type.fst" Product_Type_fst_interpreter #>

  3154    add_interpreter "Product_Type.snd" Product_Type_snd_interpreter #>

  3155    add_printer "stlc" stlc_printer #>

  3156    add_printer "IDT"  IDT_printer;

  3160 (** outer syntax commands 'refute' and 'refute_params' **)

  3162 (* argument parsing *)

  3164 (*optional list of arguments of the form [name1=value1, name2=value2, ...]*)

  3166 val scan_parm = Parse.name -- (Scan.optional (Parse.$$$ "=" |-- Parse.name) "true")

  3167 val scan_parms = Scan.optional (Parse.$$$ "[" |-- Parse.list scan_parm --| Parse.$$$ "]") [];

  3170 (* 'refute' command *)

  3172 val _ =

  3173   Outer_Syntax.improper_command "refute"

  3174     "try to find a model that refutes a given subgoal" Keyword.diag

  3175     (scan_parms -- Scan.optional Parse.nat 1 >>

  3176       (fn (parms, i) =>

  3177         Toplevel.keep (fn state =>

  3178           let

  3179             val ctxt = Toplevel.context_of state;

  3180             val {goal = st, ...} = Proof.raw_goal (Toplevel.proof_of state);

  3181           in refute_goal ctxt parms st i end)));

  3184 (* 'refute_params' command *)

  3186 val _ =

  3187   Outer_Syntax.command "refute_params"

  3188     "show/store default parameters for the 'refute' command" Keyword.thy_decl

  3189     (scan_parms >> (fn parms =>

  3190       Toplevel.theory (fn thy =>

  3191         let

  3192           val thy' = fold set_default_param parms thy;

  3193           val output =

  3194             (case get_default_params (ProofContext.init_global thy') of

  3195               [] => "none"

  3196             | new_defaults => cat_lines (map (fn (x, y) => x ^ "=" ^ y) new_defaults));

  3197           val _ = writeln ("Default parameters for 'refute':\n" ^ output);

  3198         in thy' end)));

  3200 end;

author	wenzelm
	Fri, 03 Sep 2010 11:21:58 +0200
changeset 39348	600de0485859
parent 39347	423b72f2d242
child 40055	dac3c3106746
permissions	-rw-r--r--