TFL/tfl.sml
author wenzelm
Mon Sep 11 18:00:47 2000 +0200 (2000-09-11)
changeset 9924 3370f6aa3200
parent 9878 b145613939c1
child 10015 8c16ec5ba62b
permissions -rw-r--r--
updated;
     1 (*  Title:      TFL/tfl.sml
     2     ID:         $Id$
     3     Author:     Konrad Slind, Cambridge University Computer Laboratory
     4     Copyright   1997  University of Cambridge
     5 
     6 First part of main module.
     7 *)
     8 
     9 signature TFL_sig =
    10 sig
    11   val trace: bool ref
    12   type pattern
    13   val mk_functional: theory -> term list -> {functional: term, pats: pattern list}
    14   val wfrec_definition0: theory -> string -> term -> term -> theory * thm
    15   val post_definition: thm list -> theory * (thm * pattern list) ->
    16    {theory: theory,
    17     rules: thm,
    18     rows: int list,
    19     TCs: term list list,
    20     full_pats_TCs: (term * term list) list}
    21   val wfrec_eqns: theory -> xstring -> thm list -> term list ->
    22    {WFR: term,
    23     SV: term list,
    24     proto_def: term,
    25     extracta: (thm * term list) list,
    26     pats: pattern list}
    27   val lazyR_def: theory -> xstring -> thm list -> term list ->
    28    {theory: theory,
    29     rules: thm,
    30     R: term,
    31     SV: term list,
    32     full_pats_TCs: (term * term list) list,
    33     patterns : pattern list}
    34   val mk_induction: theory ->
    35     {fconst: term, R: term, SV: term list, pat_TCs_list: (term * term list) list} -> thm
    36   val postprocess: {wf_tac: tactic, terminator: tactic, simplifier: cterm -> thm} -> theory ->
    37     {rules: thm, induction: thm, TCs: term list list} ->
    38     {rules: thm, induction: thm, nested_tcs: thm list}
    39 end;
    40 
    41 
    42 structure Prim: TFL_sig =
    43 struct
    44 
    45 val trace = ref false;
    46 
    47 open BasisLibrary;
    48 
    49 structure R = Rules;
    50 structure S = USyntax;
    51 structure U = Utils;
    52 
    53 
    54 fun TFL_ERR {func, mesg} = U.ERR {module = "Tfl", func = func, mesg = mesg};
    55 
    56 val concl = #2 o R.dest_thm;
    57 val hyp = #1 o R.dest_thm;
    58 
    59 val list_mk_type = U.end_itlist (curry (op -->));
    60 
    61 fun enumerate xs = ListPair.zip(xs, 0 upto (length xs - 1));
    62 
    63 fun front_last [] = raise TFL_ERR {func="front_last", mesg="empty list"}
    64   | front_last [x] = ([],x)
    65   | front_last (h::t) =
    66      let val (pref,x) = front_last t
    67      in
    68         (h::pref,x)
    69      end;
    70 
    71 
    72 (*---------------------------------------------------------------------------
    73  * The next function is common to pattern-match translation and
    74  * proof of completeness of cases for the induction theorem.
    75  *
    76  * The curried function "gvvariant" returns a function to generate distinct
    77  * variables that are guaranteed not to be in names.  The names of
    78  * the variables go u, v, ..., z, aa, ..., az, ...  The returned
    79  * function contains embedded refs!
    80  *---------------------------------------------------------------------------*)
    81 fun gvvariant names =
    82   let val slist = ref names
    83       val vname = ref "u"
    84       fun new() =
    85          if !vname mem_string (!slist)
    86          then (vname := bump_string (!vname);  new())
    87          else (slist := !vname :: !slist;  !vname)
    88   in
    89   fn ty => Free(new(), ty)
    90   end;
    91 
    92 
    93 (*---------------------------------------------------------------------------
    94  * Used in induction theorem production. This is the simple case of
    95  * partitioning up pattern rows by the leading constructor.
    96  *---------------------------------------------------------------------------*)
    97 fun ipartition gv (constructors,rows) =
    98   let fun pfail s = raise TFL_ERR{func = "partition.part", mesg = s}
    99       fun part {constrs = [],   rows = [],   A} = rev A
   100         | part {constrs = [],   rows = _::_, A} = pfail"extra cases in defn"
   101         | part {constrs = _::_, rows = [],   A} = pfail"cases missing in defn"
   102         | part {constrs = c::crst, rows,     A} =
   103           let val (Name,Ty) = dest_Const c
   104               val L = binder_types Ty
   105               val (in_group, not_in_group) =
   106                U.itlist (fn (row as (p::rst, rhs)) =>
   107                          fn (in_group,not_in_group) =>
   108                   let val (pc,args) = S.strip_comb p
   109                   in if (#1(dest_Const pc) = Name)
   110                      then ((args@rst, rhs)::in_group, not_in_group)
   111                      else (in_group, row::not_in_group)
   112                   end)      rows ([],[])
   113               val col_types = U.take type_of (length L, #1(hd in_group))
   114           in
   115           part{constrs = crst, rows = not_in_group,
   116                A = {constructor = c,
   117                     new_formals = map gv col_types,
   118                     group = in_group}::A}
   119           end
   120   in part{constrs = constructors, rows = rows, A = []}
   121   end;
   122 
   123 
   124 
   125 (*---------------------------------------------------------------------------
   126  * Each pattern carries with it a tag (i,b) where
   127  * i is the clause it came from and
   128  * b=true indicates that clause was given by the user
   129  * (or is an instantiation of a user supplied pattern)
   130  * b=false --> i = ~1
   131  *---------------------------------------------------------------------------*)
   132 
   133 type pattern = term * (int * bool)
   134 
   135 fun pattern_map f (tm,x) = (f tm, x);
   136 
   137 fun pattern_subst theta = pattern_map (subst_free theta);
   138 
   139 val pat_of = fst;
   140 fun row_of_pat x = fst (snd x);
   141 fun given x = snd (snd x);
   142 
   143 (*---------------------------------------------------------------------------
   144  * Produce an instance of a constructor, plus genvars for its arguments.
   145  *---------------------------------------------------------------------------*)
   146 fun fresh_constr ty_match colty gv c =
   147   let val (_,Ty) = dest_Const c
   148       val L = binder_types Ty
   149       and ty = body_type Ty
   150       val ty_theta = ty_match ty colty
   151       val c' = S.inst ty_theta c
   152       val gvars = map (S.inst ty_theta o gv) L
   153   in (c', gvars)
   154   end;
   155 
   156 
   157 (*---------------------------------------------------------------------------
   158  * Goes through a list of rows and picks out the ones beginning with a
   159  * pattern with constructor = Name.
   160  *---------------------------------------------------------------------------*)
   161 fun mk_group Name rows =
   162   U.itlist (fn (row as ((prfx, p::rst), rhs)) =>
   163             fn (in_group,not_in_group) =>
   164                let val (pc,args) = S.strip_comb p
   165                in if ((#1(dest_Const pc) = Name) handle _ => false)
   166                   then (((prfx,args@rst), rhs)::in_group, not_in_group)
   167                   else (in_group, row::not_in_group) end)
   168       rows ([],[]);
   169 
   170 (*---------------------------------------------------------------------------
   171  * Partition the rows. Not efficient: we should use hashing.
   172  *---------------------------------------------------------------------------*)
   173 fun partition _ _ (_,_,_,[]) = raise TFL_ERR{func="partition", mesg="no rows"}
   174   | partition gv ty_match
   175               (constructors, colty, res_ty, rows as (((prfx,_),_)::_)) =
   176 let val fresh = fresh_constr ty_match colty gv
   177      fun part {constrs = [],      rows, A} = rev A
   178        | part {constrs = c::crst, rows, A} =
   179          let val (c',gvars) = fresh c
   180              val (Name,Ty) = dest_Const c'
   181              val (in_group, not_in_group) = mk_group Name rows
   182              val in_group' =
   183                  if (null in_group)  (* Constructor not given *)
   184                  then [((prfx, #2(fresh c)), (S.ARB res_ty, (~1,false)))]
   185                  else in_group
   186          in
   187          part{constrs = crst,
   188               rows = not_in_group,
   189               A = {constructor = c',
   190                    new_formals = gvars,
   191                    group = in_group'}::A}
   192          end
   193 in part{constrs=constructors, rows=rows, A=[]}
   194 end;
   195 
   196 (*---------------------------------------------------------------------------
   197  * Misc. routines used in mk_case
   198  *---------------------------------------------------------------------------*)
   199 
   200 fun mk_pat (c,l) =
   201   let val L = length (binder_types (type_of c))
   202       fun build (prfx,tag,plist) =
   203           let val args   = take (L,plist)
   204               and plist' = drop(L,plist)
   205           in (prfx,tag,list_comb(c,args)::plist') end
   206   in map build l end;
   207 
   208 fun v_to_prfx (prfx, v::pats) = (v::prfx,pats)
   209   | v_to_prfx _ = raise TFL_ERR{func="mk_case", mesg="v_to_prfx"};
   210 
   211 fun v_to_pats (v::prfx,tag, pats) = (prfx, tag, v::pats)
   212   | v_to_pats _ = raise TFL_ERR{func="mk_case", mesg="v_to_pats"};
   213 
   214 
   215 (*----------------------------------------------------------------------------
   216  * Translation of pattern terms into nested case expressions.
   217  *
   218  * This performs the translation and also builds the full set of patterns.
   219  * Thus it supports the construction of induction theorems even when an
   220  * incomplete set of patterns is given.
   221  *---------------------------------------------------------------------------*)
   222 
   223 fun mk_case ty_info ty_match usednames range_ty =
   224  let
   225  fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = s}
   226  val fresh_var = gvvariant usednames
   227  val divide = partition fresh_var ty_match
   228  fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
   229    | expand constructors ty (row as ((prfx, p::rst), rhs)) =
   230        if (is_Free p)
   231        then let val fresh = fresh_constr ty_match ty fresh_var
   232                 fun expnd (c,gvs) =
   233                   let val capp = list_comb(c,gvs)
   234                   in ((prfx, capp::rst), pattern_subst[(p,capp)] rhs)
   235                   end
   236             in map expnd (map fresh constructors)  end
   237        else [row]
   238  fun mk{rows=[],...} = mk_case_fail"no rows"
   239    | mk{path=[], rows = ((prfx, []), (tm,tag))::_} =  (* Done *)
   240         ([(prfx,tag,[])], tm)
   241    | mk{path=[], rows = _::_} = mk_case_fail"blunder"
   242    | mk{path as u::rstp, rows as ((prfx, []), rhs)::rst} =
   243         mk{path = path,
   244            rows = ((prfx, [fresh_var(type_of u)]), rhs)::rst}
   245    | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
   246      let val (pat_rectangle,rights) = ListPair.unzip rows
   247          val col0 = map(hd o #2) pat_rectangle
   248      in
   249      if (forall is_Free col0)
   250      then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
   251                                 (ListPair.zip (col0, rights))
   252               val pat_rectangle' = map v_to_prfx pat_rectangle
   253               val (pref_patl,tm) = mk{path = rstp,
   254                                       rows = ListPair.zip (pat_rectangle',
   255                                                            rights')}
   256           in (map v_to_pats pref_patl, tm)
   257           end
   258      else
   259      let val pty as Type (ty_name,_) = type_of p
   260      in
   261      case (ty_info ty_name)
   262      of None => mk_case_fail("Not a known datatype: "^ty_name)
   263       | Some{case_const,constructors} =>
   264         let
   265             val case_const_name = #1(dest_Const case_const)
   266             val nrows = List.concat (map (expand constructors pty) rows)
   267             val subproblems = divide(constructors, pty, range_ty, nrows)
   268             val groups      = map #group subproblems
   269             and new_formals = map #new_formals subproblems
   270             and constructors' = map #constructor subproblems
   271             val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
   272                            (ListPair.zip (new_formals, groups))
   273             val rec_calls = map mk news
   274             val (pat_rect,dtrees) = ListPair.unzip rec_calls
   275             val case_functions = map S.list_mk_abs
   276                                   (ListPair.zip (new_formals, dtrees))
   277             val types = map type_of (case_functions@[u]) @ [range_ty]
   278             val case_const' = Const(case_const_name, list_mk_type types)
   279             val tree = list_comb(case_const', case_functions@[u])
   280             val pat_rect1 = List.concat
   281                               (ListPair.map mk_pat (constructors', pat_rect))
   282         in (pat_rect1,tree)
   283         end
   284      end end
   285  in mk
   286  end;
   287 
   288 
   289 (* Repeated variable occurrences in a pattern are not allowed. *)
   290 fun FV_multiset tm =
   291    case (S.dest_term tm)
   292      of S.VAR{Name,Ty} => [Free(Name,Ty)]
   293       | S.CONST _ => []
   294       | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
   295       | S.LAMB _ => raise TFL_ERR{func = "FV_multiset", mesg = "lambda"};
   296 
   297 fun no_repeat_vars thy pat =
   298  let fun check [] = true
   299        | check (v::rst) =
   300          if mem_term (v,rst) then
   301             raise TFL_ERR{func = "no_repeat_vars",
   302                           mesg = quote(#1(dest_Free v)) ^
   303                           " occurs repeatedly in the pattern " ^
   304                           quote (string_of_cterm (Thry.typecheck thy pat))}
   305          else check rst
   306  in check (FV_multiset pat)
   307  end;
   308 
   309 fun dest_atom (Free p) = p
   310   | dest_atom (Const p) = p
   311   | dest_atom  _ = raise TFL_ERR {func="dest_atom",
   312                                   mesg="function name not an identifier"};
   313 
   314 fun same_name (p,q) = #1(dest_atom p) = #1(dest_atom q);
   315 
   316 local fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s}
   317       fun single [_$_] =
   318               mk_functional_err "recdef does not allow currying"
   319         | single [f] = f
   320         | single fs  =
   321               (*multiple function names?*)
   322               if length (gen_distinct same_name fs) < length fs
   323               then mk_functional_err
   324                    "The function being declared appears with multiple types"
   325               else mk_functional_err
   326                    (Int.toString (length fs) ^
   327                     " distinct function names being declared")
   328 in
   329 fun mk_functional thy clauses =
   330  let val (L,R) = ListPair.unzip (map HOLogic.dest_eq clauses)
   331                    handle _ => raise TFL_ERR
   332                        {func = "mk_functional",
   333                         mesg = "recursion equations must use the = relation"}
   334      val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
   335      val atom = single (gen_distinct (op aconv) funcs)
   336      val (fname,ftype) = dest_atom atom
   337      val dummy = map (no_repeat_vars thy) pats
   338      val rows = ListPair.zip (map (fn x => ([]:term list,[x])) pats,
   339                               map (fn (t,i) => (t,(i,true))) (enumerate R))
   340      val names = foldr add_term_names (R,[])
   341      val atype = type_of(hd pats)
   342      and aname = variant names "a"
   343      val a = Free(aname,atype)
   344      val ty_info = Thry.match_info thy
   345      val ty_match = Thry.match_type thy
   346      val range_ty = type_of (hd R)
   347      val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty
   348                                     {path=[a], rows=rows}
   349      val patts1 = map (fn (_,tag,[pat]) => (pat,tag)) patts
   350           handle _ => mk_functional_err "error in pattern-match translation"
   351      val patts2 = U.sort(fn p1=>fn p2=> row_of_pat p1 < row_of_pat p2) patts1
   352      val finals = map row_of_pat patts2
   353      val originals = map (row_of_pat o #2) rows
   354      val dummy = case (originals\\finals)
   355              of [] => ()
   356           | L => mk_functional_err
   357  ("The following clauses are redundant (covered by preceding clauses): " ^
   358                    commas (map (fn i => Int.toString (i + 1)) L))
   359  in {functional = Abs(Sign.base_name fname, ftype,
   360                       abstract_over (atom,
   361                                      absfree(aname,atype, case_tm))),
   362      pats = patts2}
   363 end end;
   364 
   365 
   366 (*----------------------------------------------------------------------------
   367  *
   368  *                    PRINCIPLES OF DEFINITION
   369  *
   370  *---------------------------------------------------------------------------*)
   371 
   372 
   373 (*For Isabelle, the lhs of a definition must be a constant.*)
   374 fun mk_const_def sign (Name, Ty, rhs) =
   375     Sign.infer_types sign (K None) (K None) [] false
   376                ([Const("==",dummyT) $ Const(Name,Ty) $ rhs], propT)
   377     |> #1;
   378 
   379 (*Make all TVars available for instantiation by adding a ? to the front*)
   380 fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
   381   | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
   382   | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);
   383 
   384 local val f_eq_wfrec_R_M =
   385            #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
   386       val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
   387       val (fname,_) = dest_Free f
   388       val (wfrec,_) = S.strip_comb rhs
   389 in
   390 fun wfrec_definition0 thy fid R (functional as Abs(Name, Ty, _)) =
   391  let val def_name = if Name<>fid then
   392                         raise TFL_ERR{func = "wfrec_definition0",
   393                                       mesg = "Expected a definition of " ^
   394                                              quote fid ^ " but found one of " ^
   395                                       quote Name}
   396                     else Name ^ "_def"
   397      val wfrec_R_M =  map_term_types poly_tvars
   398                           (wfrec $ map_term_types poly_tvars R)
   399                       $ functional
   400      val def_term = mk_const_def (Theory.sign_of thy) (Name, Ty, wfrec_R_M)
   401      val (thy', [def]) = PureThy.add_defs_i false [Thm.no_attributes (def_name, def_term)] thy
   402  in (thy', def) end;
   403 end;
   404 
   405 
   406 
   407 (*---------------------------------------------------------------------------
   408  * This structure keeps track of congruence rules that aren't derived
   409  * from a datatype definition.
   410  *---------------------------------------------------------------------------*)
   411 fun extraction_thms thy =
   412  let val {case_rewrites,case_congs} = Thry.extract_info thy
   413  in (case_rewrites, case_congs)
   414  end;
   415 
   416 
   417 (*---------------------------------------------------------------------------
   418  * Pair patterns with termination conditions. The full list of patterns for
   419  * a definition is merged with the TCs arising from the user-given clauses.
   420  * There can be fewer clauses than the full list, if the user omitted some
   421  * cases. This routine is used to prepare input for mk_induction.
   422  *---------------------------------------------------------------------------*)
   423 fun merge full_pats TCs =
   424 let fun insert (p,TCs) =
   425       let fun insrt ((x as (h,[]))::rst) =
   426                  if (p aconv h) then (p,TCs)::rst else x::insrt rst
   427             | insrt (x::rst) = x::insrt rst
   428             | insrt[] = raise TFL_ERR{func="merge.insert",
   429                                       mesg="pattern not found"}
   430       in insrt end
   431     fun pass ([],ptcl_final) = ptcl_final
   432       | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
   433 in
   434   pass (TCs, map (fn p => (p,[])) full_pats)
   435 end;
   436 
   437 
   438 fun givens pats = map pat_of (filter given pats);
   439 
   440 fun post_definition meta_tflCongs (theory, (def, pats)) =
   441  let val tych = Thry.typecheck theory
   442      val f = #lhs(S.dest_eq(concl def))
   443      val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
   444      val pats' = filter given pats
   445      val given_pats = map pat_of pats'
   446      val rows = map row_of_pat pats'
   447      val WFR = #ant(S.dest_imp(concl corollary))
   448      val R = #Rand(S.dest_comb WFR)
   449      val corollary' = R.UNDISCH corollary  (* put WF R on assums *)
   450      val corollaries = map (fn pat => R.SPEC (tych pat) corollary')
   451                            given_pats
   452      val (case_rewrites,context_congs) = extraction_thms theory
   453      val corollaries' = map(rewrite_rule case_rewrites) corollaries
   454      val extract = R.CONTEXT_REWRITE_RULE
   455                      (f, [R], cut_apply, meta_tflCongs@context_congs)
   456      val (rules, TCs) = ListPair.unzip (map extract corollaries')
   457      val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
   458      val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
   459      val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
   460  in
   461  {theory = theory,   (* holds def, if it's needed *)
   462   rules = rules1,
   463   rows = rows,
   464   full_pats_TCs = merge (map pat_of pats) (ListPair.zip (given_pats, TCs)),
   465   TCs = TCs}
   466  end;
   467 
   468 
   469 (*---------------------------------------------------------------------------
   470  * Perform the extraction without making the definition. Definition and
   471  * extraction commute for the non-nested case.  (Deferred recdefs)
   472  *
   473  * The purpose of wfrec_eqns is merely to instantiate the recursion theorem
   474  * and extract termination conditions: no definition is made.
   475  *---------------------------------------------------------------------------*)
   476 
   477 fun wfrec_eqns thy fid tflCongs eqns =
   478  let val {lhs,rhs} = S.dest_eq (hd eqns)
   479      val (f,args) = S.strip_comb lhs
   480      val (fname,fty) = dest_atom f
   481      val (SV,a) = front_last args    (* SV = schematic variables *)
   482      val g = list_comb(f,SV)
   483      val h = Free(fname,type_of g)
   484      val eqns1 = map (subst_free[(g,h)]) eqns
   485      val {functional as Abs(Name, Ty, _),  pats} = mk_functional thy eqns1
   486      val given_pats = givens pats
   487      (* val f = Free(Name,Ty) *)
   488      val Type("fun", [f_dty, f_rty]) = Ty
   489      val dummy = if Name<>fid then
   490                         raise TFL_ERR{func = "wfrec_eqns",
   491                                       mesg = "Expected a definition of " ^
   492                                       quote fid ^ " but found one of " ^
   493                                       quote Name}
   494                  else ()
   495      val (case_rewrites,context_congs) = extraction_thms thy
   496      val tych = Thry.typecheck thy
   497      val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
   498      val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0
   499      val R = Free (variant (foldr add_term_names (eqns,[])) Rname,
   500                    Rtype)
   501      val WFREC_THM = R.ISPECL [tych R, tych g] WFREC_THM0
   502      val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
   503      val dummy =
   504            if !trace then
   505                writeln ("ORIGINAL PROTO_DEF: " ^
   506                           Sign.string_of_term (Theory.sign_of thy) proto_def)
   507            else ()
   508      val R1 = S.rand WFR
   509      val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
   510      val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats
   511      val corollaries' = map (rewrite_rule case_rewrites) corollaries
   512      fun extract X = R.CONTEXT_REWRITE_RULE
   513                        (f, R1::SV, cut_apply, tflCongs@context_congs) X
   514  in {proto_def = proto_def,
   515      SV=SV,
   516      WFR=WFR,
   517      pats=pats,
   518      extracta = map extract corollaries'}
   519  end;
   520 
   521 
   522 (*---------------------------------------------------------------------------
   523  * Define the constant after extracting the termination conditions. The
   524  * wellfounded relation used in the definition is computed by using the
   525  * choice operator on the extracted conditions (plus the condition that
   526  * such a relation must be wellfounded).
   527  *---------------------------------------------------------------------------*)
   528 
   529 fun lazyR_def thy fid tflCongs eqns =
   530  let val {proto_def,WFR,pats,extracta,SV} =
   531            wfrec_eqns thy fid tflCongs eqns
   532      val R1 = S.rand WFR
   533      val f = #lhs(S.dest_eq proto_def)
   534      val (extractants,TCl) = ListPair.unzip extracta
   535      val dummy = if !trace
   536                  then (writeln "Extractants = ";
   537                        prths extractants;
   538                        ())
   539                  else ()
   540      val TCs = foldr (gen_union (op aconv)) (TCl, [])
   541      val full_rqt = WFR::TCs
   542      val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt}
   543      val R'abs = S.rand R'
   544      val proto_def' = subst_free[(R1,R')] proto_def
   545      val dummy = if !trace then writeln ("proto_def' = " ^
   546                                          Sign.string_of_term
   547                                          (Theory.sign_of thy) proto_def')
   548                            else ()
   549      val {lhs,rhs} = S.dest_eq proto_def'
   550      val (c,args) = S.strip_comb lhs
   551      val (Name,Ty) = dest_atom c
   552      val defn = mk_const_def (Theory.sign_of thy)
   553                  (Name, Ty, S.list_mk_abs (args,rhs))
   554      val (theory, [def0]) =
   555        thy
   556        |> PureThy.add_defs_i false
   557             [Thm.no_attributes (fid ^ "_def", defn)]
   558      val def = freezeT def0;
   559      val dummy = if !trace then writeln ("DEF = " ^ string_of_thm def)
   560                            else ()
   561      (* val fconst = #lhs(S.dest_eq(concl def))  *)
   562      val tych = Thry.typecheck theory
   563      val full_rqt_prop = map (Dcterm.mk_prop o tych) full_rqt
   564          (*lcp: a lot of object-logic inference to remove*)
   565      val baz = R.DISCH_ALL
   566                  (U.itlist R.DISCH full_rqt_prop
   567                   (R.LIST_CONJ extractants))
   568      val dum = if !trace then writeln ("baz = " ^ string_of_thm baz)
   569                            else ()
   570      val f_free = Free (fid, fastype_of f)  (*'cos f is a Const*)
   571      val SV' = map tych SV;
   572      val SVrefls = map reflexive SV'
   573      val def0 = (U.rev_itlist (fn x => fn th => R.rbeta(combination th x))
   574                    SVrefls def)
   575                 RS meta_eq_to_obj_eq
   576      val def' = R.MP (R.SPEC (tych R') (R.GEN (tych R1) baz)) def0
   577      val body_th = R.LIST_CONJ (map R.ASSUME full_rqt_prop)
   578      val bar = R.MP (R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX)
   579                     body_th
   580  in {theory = theory, R=R1, SV=SV,
   581      rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def',
   582      full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
   583      patterns = pats}
   584  end;
   585 
   586 
   587 
   588 (*----------------------------------------------------------------------------
   589  *
   590  *                           INDUCTION THEOREM
   591  *
   592  *---------------------------------------------------------------------------*)
   593 
   594 
   595 (*------------------------  Miscellaneous function  --------------------------
   596  *
   597  *           [x_1,...,x_n]     ?v_1...v_n. M[v_1,...,v_n]
   598  *     -----------------------------------------------------------
   599  *     ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
   600  *                        ...
   601  *                        (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
   602  *
   603  * This function is totally ad hoc. Used in the production of the induction
   604  * theorem. The nchotomy theorem can have clauses that look like
   605  *
   606  *     ?v1..vn. z = C vn..v1
   607  *
   608  * in which the order of quantification is not the order of occurrence of the
   609  * quantified variables as arguments to C. Since we have no control over this
   610  * aspect of the nchotomy theorem, we make the correspondence explicit by
   611  * pairing the incoming new variable with the term it gets beta-reduced into.
   612  *---------------------------------------------------------------------------*)
   613 
   614 fun alpha_ex_unroll (xlist, tm) =
   615   let val (qvars,body) = S.strip_exists tm
   616       val vlist = #2(S.strip_comb (S.rhs body))
   617       val plist = ListPair.zip (vlist, xlist)
   618       val args = map (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars
   619                    handle OPTION => error
   620                        "TFL fault [alpha_ex_unroll]: no correspondence"
   621       fun build ex      []   = []
   622         | build (_$rex) (v::rst) =
   623            let val ex1 = betapply(rex, v)
   624            in  ex1 :: build ex1 rst
   625            end
   626      val (nex::exl) = rev (tm::build tm args)
   627   in
   628   (nex, ListPair.zip (args, rev exl))
   629   end;
   630 
   631 
   632 
   633 (*----------------------------------------------------------------------------
   634  *
   635  *             PROVING COMPLETENESS OF PATTERNS
   636  *
   637  *---------------------------------------------------------------------------*)
   638 
   639 fun mk_case ty_info usednames thy =
   640  let
   641  val divide = ipartition (gvvariant usednames)
   642  val tych = Thry.typecheck thy
   643  fun tych_binding(x,y) = (tych x, tych y)
   644  fun fail s = raise TFL_ERR{func = "mk_case", mesg = s}
   645  fun mk{rows=[],...} = fail"no rows"
   646    | mk{path=[], rows = [([], (thm, bindings))]} =
   647                          R.IT_EXISTS (map tych_binding bindings) thm
   648    | mk{path = u::rstp, rows as (p::_, _)::_} =
   649      let val (pat_rectangle,rights) = ListPair.unzip rows
   650          val col0 = map hd pat_rectangle
   651          val pat_rectangle' = map tl pat_rectangle
   652      in
   653      if (forall is_Free col0) (* column 0 is all variables *)
   654      then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
   655                                 (ListPair.zip (rights, col0))
   656           in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
   657           end
   658      else                     (* column 0 is all constructors *)
   659      let val Type (ty_name,_) = type_of p
   660      in
   661      case (ty_info ty_name)
   662      of None => fail("Not a known datatype: "^ty_name)
   663       | Some{constructors,nchotomy} =>
   664         let val thm' = R.ISPEC (tych u) nchotomy
   665             val disjuncts = S.strip_disj (concl thm')
   666             val subproblems = divide(constructors, rows)
   667             val groups      = map #group subproblems
   668             and new_formals = map #new_formals subproblems
   669             val existentials = ListPair.map alpha_ex_unroll
   670                                    (new_formals, disjuncts)
   671             val constraints = map #1 existentials
   672             val vexl = map #2 existentials
   673             fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
   674             val news = map (fn (nf,rows,c) => {path = nf@rstp,
   675                                                rows = map (expnd c) rows})
   676                            (U.zip3 new_formals groups constraints)
   677             val recursive_thms = map mk news
   678             val build_exists = foldr
   679                                 (fn((x,t), th) =>
   680                                  R.CHOOSE (tych x, R.ASSUME (tych t)) th)
   681             val thms' = ListPair.map build_exists (vexl, recursive_thms)
   682             val same_concls = R.EVEN_ORS thms'
   683         in R.DISJ_CASESL thm' same_concls
   684         end
   685      end end
   686  in mk
   687  end;
   688 
   689 
   690 fun complete_cases thy =
   691  let val tych = Thry.typecheck thy
   692      val ty_info = Thry.induct_info thy
   693  in fn pats =>
   694  let val names = foldr add_term_names (pats,[])
   695      val T = type_of (hd pats)
   696      val aname = Term.variant names "a"
   697      val vname = Term.variant (aname::names) "v"
   698      val a = Free (aname, T)
   699      val v = Free (vname, T)
   700      val a_eq_v = HOLogic.mk_eq(a,v)
   701      val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
   702                            (R.REFL (tych a))
   703      val th0 = R.ASSUME (tych a_eq_v)
   704      val rows = map (fn x => ([x], (th0,[]))) pats
   705  in
   706  R.GEN (tych a)
   707        (R.RIGHT_ASSOC
   708           (R.CHOOSE(tych v, ex_th0)
   709                 (mk_case ty_info (vname::aname::names)
   710                  thy {path=[v], rows=rows})))
   711  end end;
   712 
   713 
   714 (*---------------------------------------------------------------------------
   715  * Constructing induction hypotheses: one for each recursive call.
   716  *
   717  * Note. R will never occur as a variable in the ind_clause, because
   718  * to do so, it would have to be from a nested definition, and we don't
   719  * allow nested defns to have R variable.
   720  *
   721  * Note. When the context is empty, there can be no local variables.
   722  *---------------------------------------------------------------------------*)
   723 (*
   724 local infix 5 ==>
   725       fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
   726 in
   727 fun build_ih f P (pat,TCs) =
   728  let val globals = S.free_vars_lr pat
   729      fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
   730      fun dest_TC tm =
   731          let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
   732              val (R,y,_) = S.dest_relation R_y_pat
   733              val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
   734          in case cntxt
   735               of [] => (P_y, (tm,[]))
   736                | _  => let
   737                     val imp = S.list_mk_conj cntxt ==> P_y
   738                     val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
   739                     val locals = #2(U.pluck (curry (op aconv) P) lvs) handle _ => lvs
   740                     in (S.list_mk_forall(locals,imp), (tm,locals)) end
   741          end
   742  in case TCs
   743     of [] => (S.list_mk_forall(globals, P$pat), [])
   744      |  _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
   745                  val ind_clause = S.list_mk_conj ihs ==> P$pat
   746              in (S.list_mk_forall(globals,ind_clause), TCs_locals)
   747              end
   748  end
   749 end;
   750 *)
   751 
   752 local infix 5 ==>
   753       fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
   754 in
   755 fun build_ih f (P,SV) (pat,TCs) =
   756  let val pat_vars = S.free_vars_lr pat
   757      val globals = pat_vars@SV
   758      fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
   759      fun dest_TC tm =
   760          let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
   761              val (R,y,_) = S.dest_relation R_y_pat
   762              val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
   763          in case cntxt
   764               of [] => (P_y, (tm,[]))
   765                | _  => let
   766                     val imp = S.list_mk_conj cntxt ==> P_y
   767                     val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
   768                     val locals = #2(U.pluck (curry (op aconv) P) lvs) handle _ => lvs
   769                     in (S.list_mk_forall(locals,imp), (tm,locals)) end
   770          end
   771  in case TCs
   772     of [] => (S.list_mk_forall(pat_vars, P$pat), [])
   773      |  _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
   774                  val ind_clause = S.list_mk_conj ihs ==> P$pat
   775              in (S.list_mk_forall(pat_vars,ind_clause), TCs_locals)
   776              end
   777  end
   778 end;
   779 
   780 (*---------------------------------------------------------------------------
   781  * This function makes good on the promise made in "build_ih".
   782  *
   783  * Input  is tm = "(!y. R y pat ==> P y) ==> P pat",
   784  *           TCs = TC_1[pat] ... TC_n[pat]
   785  *           thm = ih1 /\ ... /\ ih_n |- ih[pat]
   786  *---------------------------------------------------------------------------*)
   787 fun prove_case f thy (tm,TCs_locals,thm) =
   788  let val tych = Thry.typecheck thy
   789      val antc = tych(#ant(S.dest_imp tm))
   790      val thm' = R.SPEC_ALL thm
   791      fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
   792      fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
   793      fun mk_ih ((TC,locals),th2,nested) =
   794          R.GENL (map tych locals)
   795             (if nested
   796               then R.DISCH (get_cntxt TC) th2 handle _ => th2
   797                else if S.is_imp(concl TC)
   798                      then R.IMP_TRANS TC th2
   799                       else R.MP th2 TC)
   800  in
   801  R.DISCH antc
   802  (if S.is_imp(concl thm') (* recursive calls in this clause *)
   803   then let val th1 = R.ASSUME antc
   804            val TCs = map #1 TCs_locals
   805            val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o
   806                             #2 o S.strip_forall) TCs
   807            val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
   808                             TCs_locals
   809            val th2list = map (U.C R.SPEC th1 o tych) ylist
   810            val nlist = map nested TCs
   811            val triples = U.zip3 TClist th2list nlist
   812            val Pylist = map mk_ih triples
   813        in R.MP thm' (R.LIST_CONJ Pylist) end
   814   else thm')
   815  end;
   816 
   817 
   818 (*---------------------------------------------------------------------------
   819  *
   820  *         x = (v1,...,vn)  |- M[x]
   821  *    ---------------------------------------------
   822  *      ?v1 ... vn. x = (v1,...,vn) |- M[x]
   823  *
   824  *---------------------------------------------------------------------------*)
   825 fun LEFT_ABS_VSTRUCT tych thm =
   826   let fun CHOOSER v (tm,thm) =
   827         let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
   828         in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
   829         end
   830       val [veq] = filter (U.can S.dest_eq) (#1 (R.dest_thm thm))
   831       val {lhs,rhs} = S.dest_eq veq
   832       val L = S.free_vars_lr rhs
   833   in  #2 (U.itlist CHOOSER L (veq,thm))  end;
   834 
   835 
   836 (*----------------------------------------------------------------------------
   837  * Input : f, R,  and  [(pat1,TCs1),..., (patn,TCsn)]
   838  *
   839  * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
   840  * recursion induction (Rinduct) by proving the antecedent of Sinduct from
   841  * the antecedent of Rinduct.
   842  *---------------------------------------------------------------------------*)
   843 fun mk_induction thy {fconst, R, SV, pat_TCs_list} =
   844 let val tych = Thry.typecheck thy
   845     val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
   846     val (pats,TCsl) = ListPair.unzip pat_TCs_list
   847     val case_thm = complete_cases thy pats
   848     val domain = (type_of o hd) pats
   849     val Pname = Term.variant (foldr (foldr add_term_names)
   850                               (pats::TCsl, [])) "P"
   851     val P = Free(Pname, domain --> HOLogic.boolT)
   852     val Sinduct = R.SPEC (tych P) Sinduction
   853     val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
   854     val Rassums_TCl' = map (build_ih fconst (P,SV)) pat_TCs_list
   855     val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
   856     val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
   857     val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats
   858     val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
   859     val proved_cases = map (prove_case fconst thy) tasks
   860     val v = Free (variant (foldr add_term_names (map concl proved_cases, []))
   861                     "v",
   862                   domain)
   863     val vtyped = tych v
   864     val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
   865     val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th')
   866                           (substs, proved_cases)
   867     val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
   868     val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
   869     val dc = R.MP Sinduct dant
   870     val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
   871     val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
   872     val dc' = U.itlist (R.GEN o tych) vars
   873                        (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
   874 in
   875    R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
   876 end
   877 handle _ => raise TFL_ERR{func = "mk_induction", mesg = "failed derivation"};
   878 
   879 
   880 
   881 
   882 (*---------------------------------------------------------------------------
   883  *
   884  *                        POST PROCESSING
   885  *
   886  *---------------------------------------------------------------------------*)
   887 
   888 
   889 fun simplify_induction thy hth ind =
   890   let val tych = Thry.typecheck thy
   891       val (asl,_) = R.dest_thm ind
   892       val (_,tc_eq_tc') = R.dest_thm hth
   893       val tc = S.lhs tc_eq_tc'
   894       fun loop [] = ind
   895         | loop (asm::rst) =
   896           if (U.can (Thry.match_term thy asm) tc)
   897           then R.UNDISCH
   898                  (R.MATCH_MP
   899                      (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind))
   900                      hth)
   901          else loop rst
   902   in loop asl
   903 end;
   904 
   905 
   906 (*---------------------------------------------------------------------------
   907  * The termination condition is an antecedent to the rule, and an
   908  * assumption to the theorem.
   909  *---------------------------------------------------------------------------*)
   910 fun elim_tc tcthm (rule,induction) =
   911    (R.MP rule tcthm, R.PROVE_HYP tcthm induction)
   912 
   913 
   914 fun postprocess{wf_tac, terminator, simplifier} theory {rules,induction,TCs} =
   915 let val tych = Thry.typecheck theory
   916 
   917    (*---------------------------------------------------------------------
   918     * Attempt to eliminate WF condition. It's the only assumption of rules
   919     *---------------------------------------------------------------------*)
   920    val (rules1,induction1)  =
   921        let val thm = R.prove(tych(HOLogic.mk_Trueprop
   922                                   (hd(#1(R.dest_thm rules)))),
   923                              wf_tac)
   924        in (R.PROVE_HYP thm rules,  R.PROVE_HYP thm induction)
   925        end handle _ => (rules,induction)
   926 
   927    (*----------------------------------------------------------------------
   928     * The termination condition (tc) is simplified to |- tc = tc' (there
   929     * might not be a change!) and then 3 attempts are made:
   930     *
   931     *   1. if |- tc = T, then eliminate it with eqT; otherwise,
   932     *   2. apply the terminator to tc'. If |- tc' = T then eliminate; else
   933     *   3. replace tc by tc' in both the rules and the induction theorem.
   934     *---------------------------------------------------------------------*)
   935 
   936    fun print_thms s L =
   937      if !trace then writeln (cat_lines (s :: map string_of_thm L))
   938      else ();
   939 
   940    fun print_cterms s L =
   941      if !trace then writeln (cat_lines (s :: map string_of_cterm L))
   942      else ();;
   943 
   944    fun simplify_tc tc (r,ind) =
   945        let val tc1 = tych tc
   946            val _ = print_cterms "TC before simplification: " [tc1]
   947            val tc_eq = simplifier tc1
   948            val _ = print_thms "result: " [tc_eq]
   949        in
   950        elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
   951        handle _ =>
   952         (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
   953                   (R.prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))),
   954                            terminator)))
   955                  (r,ind)
   956          handle _ =>
   957           (R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq),
   958            simplify_induction theory tc_eq ind))
   959        end
   960 
   961    (*----------------------------------------------------------------------
   962     * Nested termination conditions are harder to get at, since they are
   963     * left embedded in the body of the function (and in induction
   964     * theorem hypotheses). Our "solution" is to simplify them, and try to
   965     * prove termination, but leave the application of the resulting theorem
   966     * to a higher level. So things go much as in "simplify_tc": the
   967     * termination condition (tc) is simplified to |- tc = tc' (there might
   968     * not be a change) and then 2 attempts are made:
   969     *
   970     *   1. if |- tc = T, then return |- tc; otherwise,
   971     *   2. apply the terminator to tc'. If |- tc' = T then return |- tc; else
   972     *   3. return |- tc = tc'
   973     *---------------------------------------------------------------------*)
   974    fun simplify_nested_tc tc =
   975       let val tc_eq = simplifier (tych (#2 (S.strip_forall tc)))
   976       in
   977       R.GEN_ALL
   978        (R.MATCH_MP Thms.eqT tc_eq
   979         handle _
   980         => (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
   981                       (R.prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))),
   982                                terminator))
   983             handle _ => tc_eq))
   984       end
   985 
   986    (*-------------------------------------------------------------------
   987     * Attempt to simplify the termination conditions in each rule and
   988     * in the induction theorem.
   989     *-------------------------------------------------------------------*)
   990    fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm
   991    fun loop ([],extras,R,ind) = (rev R, ind, extras)
   992      | loop ((r,ftcs)::rst, nthms, R, ind) =
   993         let val tcs = #1(strip_imp (concl r))
   994             val extra_tcs = gen_rems (op aconv) (ftcs, tcs)
   995             val extra_tc_thms = map simplify_nested_tc extra_tcs
   996             val (r1,ind1) = U.rev_itlist simplify_tc tcs (r,ind)
   997             val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1
   998         in loop(rst, nthms@extra_tc_thms, r2::R, ind1)
   999         end
  1000    val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs)
  1001    val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1)
  1002 in
  1003   {induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras}
  1004 end;
  1005 
  1006 end;