TFL/tfl.sml
author paulson
Wed Jan 27 10:31:31 1999 +0100 (1999-01-27)
changeset 6153 bff90585cce5
parent 6092 d9db67970c73
child 6397 e70ae9b575cc
permissions -rw-r--r--
new typechecking solver for the simplifier
     1 (*  Title:      TFL/tfl
     2     ID:         $Id$
     3     Author:     Konrad Slind, Cambridge University Computer Laboratory
     4     Copyright   1997  University of Cambridge
     5 
     6 Main module
     7 *)
     8 
     9 structure Prim : TFL_sig =
    10 struct
    11 
    12 (* Abbreviations *)
    13 structure R = Rules;
    14 structure S = USyntax;
    15 structure U = S.Utils;
    16 
    17 val concl = #2 o R.dest_thm;
    18 val hyp = #1 o R.dest_thm;
    19 
    20 val list_mk_type = U.end_itlist (curry(op -->));
    21 
    22 fun enumerate l = 
    23      rev(#1(foldl (fn ((alist,i), x) => ((x,i)::alist, i+1)) (([],0), l)));
    24 
    25 fun stringize [] = ""
    26   | stringize [i] = Int.toString i
    27   | stringize (h::t) = (Int.toString h^", "^stringize t);
    28 
    29 
    30 fun TFL_ERR{func,mesg} = U.ERR{module = "Tfl", func = func, mesg = mesg};
    31 
    32 
    33 (*---------------------------------------------------------------------------
    34  * The next function is common to pattern-match translation and 
    35  * proof of completeness of cases for the induction theorem.
    36  *
    37  * The curried function "gvvariant" returns a function to generate distinct
    38  * variables that are guaranteed not to be in names.  The names of
    39  * the variables go u, v, ..., z, aa, ..., az, ...  The returned 
    40  * function contains embedded refs!
    41  *---------------------------------------------------------------------------*)
    42 fun gvvariant names =
    43   let val slist = ref names
    44       val vname = ref "u"
    45       fun new() = 
    46          if !vname mem_string (!slist)
    47          then (vname := bump_string (!vname);  new())
    48          else (slist := !vname :: !slist;  !vname)
    49   in 
    50   fn ty => Free(new(), ty)
    51   end;
    52 
    53 
    54 (*---------------------------------------------------------------------------
    55  * Used in induction theorem production. This is the simple case of
    56  * partitioning up pattern rows by the leading constructor.
    57  *---------------------------------------------------------------------------*)
    58 fun ipartition gv (constructors,rows) =
    59   let fun pfail s = raise TFL_ERR{func = "partition.part", mesg = s}
    60       fun part {constrs = [],   rows = [],   A} = rev A
    61         | part {constrs = [],   rows = _::_, A} = pfail"extra cases in defn"
    62         | part {constrs = _::_, rows = [],   A} = pfail"cases missing in defn"
    63         | part {constrs = c::crst, rows,     A} =
    64           let val (Name,Ty) = dest_Const c
    65               val L = binder_types Ty
    66               val (in_group, not_in_group) =
    67                U.itlist (fn (row as (p::rst, rhs)) =>
    68                          fn (in_group,not_in_group) =>
    69                   let val (pc,args) = S.strip_comb p
    70                   in if (#1(dest_Const pc) = Name)
    71                      then ((args@rst, rhs)::in_group, not_in_group)
    72                      else (in_group, row::not_in_group)
    73                   end)      rows ([],[])
    74               val col_types = U.take type_of (length L, #1(hd in_group))
    75           in 
    76           part{constrs = crst, rows = not_in_group, 
    77                A = {constructor = c, 
    78                     new_formals = map gv col_types,
    79                     group = in_group}::A}
    80           end
    81   in part{constrs = constructors, rows = rows, A = []}
    82   end;
    83 
    84 
    85 
    86 (*---------------------------------------------------------------------------
    87  * This datatype carries some information about the origin of a
    88  * clause in a function definition.
    89  *---------------------------------------------------------------------------*)
    90 datatype pattern = GIVEN   of term * int
    91                  | OMITTED of term * int
    92 
    93 fun pattern_map f (GIVEN (tm,i)) = GIVEN(f tm, i)
    94   | pattern_map f (OMITTED (tm,i)) = OMITTED(f tm, i);
    95 
    96 fun pattern_subst theta = pattern_map (subst_free theta);
    97 
    98 fun dest_pattern (GIVEN (tm,i)) = ((GIVEN,i),tm)
    99   | dest_pattern (OMITTED (tm,i)) = ((OMITTED,i),tm);
   100 
   101 val pat_of = #2 o dest_pattern;
   102 val row_of_pat = #2 o #1 o dest_pattern;
   103 
   104 (*---------------------------------------------------------------------------
   105  * Produce an instance of a constructor, plus genvars for its arguments.
   106  *---------------------------------------------------------------------------*)
   107 fun fresh_constr ty_match colty gv c =
   108   let val (_,Ty) = dest_Const c
   109       val L = binder_types Ty
   110       and ty = body_type Ty
   111       val ty_theta = ty_match ty colty
   112       val c' = S.inst ty_theta c
   113       val gvars = map (S.inst ty_theta o gv) L
   114   in (c', gvars)
   115   end;
   116 
   117 
   118 (*---------------------------------------------------------------------------
   119  * Goes through a list of rows and picks out the ones beginning with a
   120  * pattern with constructor = Name.
   121  *---------------------------------------------------------------------------*)
   122 fun mk_group Name rows =
   123   U.itlist (fn (row as ((prefix, p::rst), rhs)) =>
   124             fn (in_group,not_in_group) =>
   125                let val (pc,args) = S.strip_comb p
   126                in if ((#1(dest_Const pc) = Name) handle _ => false)
   127                   then (((prefix,args@rst), rhs)::in_group, not_in_group)
   128                   else (in_group, row::not_in_group) end)
   129       rows ([],[]);
   130 
   131 (*---------------------------------------------------------------------------
   132  * Partition the rows. Not efficient: we should use hashing.
   133  *---------------------------------------------------------------------------*)
   134 fun partition _ _ (_,_,_,[]) = raise TFL_ERR{func="partition", mesg="no rows"}
   135   | partition gv ty_match
   136               (constructors, colty, res_ty, rows as (((prefix,_),_)::_)) =
   137 let val fresh = fresh_constr ty_match colty gv
   138      fun part {constrs = [],      rows, A} = rev A
   139        | part {constrs = c::crst, rows, A} =
   140          let val (c',gvars) = fresh c
   141              val (Name,Ty) = dest_Const c'
   142              val (in_group, not_in_group) = mk_group Name rows
   143              val in_group' =
   144                  if (null in_group)  (* Constructor not given *)
   145                  then [((prefix, #2(fresh c)), OMITTED (S.ARB res_ty, ~1))]
   146                  else in_group
   147          in 
   148          part{constrs = crst, 
   149               rows = not_in_group, 
   150               A = {constructor = c', 
   151                    new_formals = gvars,
   152                    group = in_group'}::A}
   153          end
   154 in part{constrs=constructors, rows=rows, A=[]}
   155 end;
   156 
   157 (*---------------------------------------------------------------------------
   158  * Misc. routines used in mk_case
   159  *---------------------------------------------------------------------------*)
   160 
   161 fun mk_pat (c,l) =
   162   let val L = length (binder_types (type_of c))
   163       fun build (prefix,tag,plist) =
   164           let val args   = take (L,plist)
   165               and plist' = drop(L,plist)
   166           in (prefix,tag,list_comb(c,args)::plist') end
   167   in map build l end;
   168 
   169 fun v_to_prefix (prefix, v::pats) = (v::prefix,pats)
   170   | v_to_prefix _ = raise TFL_ERR{func="mk_case", mesg="v_to_prefix"};
   171 
   172 fun v_to_pats (v::prefix,tag, pats) = (prefix, tag, v::pats)
   173   | v_to_pats _ = raise TFL_ERR{func="mk_case", mesg="v_to_pats"};
   174  
   175 
   176 (*----------------------------------------------------------------------------
   177  * Translation of pattern terms into nested case expressions.
   178  *
   179  * This performs the translation and also builds the full set of patterns. 
   180  * Thus it supports the construction of induction theorems even when an 
   181  * incomplete set of patterns is given.
   182  *---------------------------------------------------------------------------*)
   183 
   184 fun mk_case ty_info ty_match usednames range_ty =
   185  let 
   186  fun mk_case_fail s = raise TFL_ERR{func = "mk_case", mesg = s}
   187  val fresh_var = gvvariant usednames 
   188  val divide = partition fresh_var ty_match
   189  fun expand constructors ty ((_,[]), _) = mk_case_fail"expand_var_row"
   190    | expand constructors ty (row as ((prefix, p::rst), rhs)) = 
   191        if (is_Free p) 
   192        then let val fresh = fresh_constr ty_match ty fresh_var
   193                 fun expnd (c,gvs) = 
   194                   let val capp = list_comb(c,gvs)
   195                   in ((prefix, capp::rst), pattern_subst[(p,capp)] rhs)
   196                   end
   197             in map expnd (map fresh constructors)  end
   198        else [row]
   199  fun mk{rows=[],...} = mk_case_fail"no rows"
   200    | mk{path=[], rows = ((prefix, []), rhs)::_} =  (* Done *)
   201         let val (tag,tm) = dest_pattern rhs
   202         in ([(prefix,tag,[])], tm)
   203         end
   204    | mk{path=[], rows = _::_} = mk_case_fail"blunder"
   205    | mk{path as u::rstp, rows as ((prefix, []), rhs)::rst} = 
   206         mk{path = path, 
   207            rows = ((prefix, [fresh_var(type_of u)]), rhs)::rst}
   208    | mk{path = u::rstp, rows as ((_, p::_), _)::_} =
   209      let val (pat_rectangle,rights) = ListPair.unzip rows
   210          val col0 = map(hd o #2) pat_rectangle
   211      in 
   212      if (forall is_Free col0) 
   213      then let val rights' = map (fn(v,e) => pattern_subst[(v,u)] e)
   214                                 (ListPair.zip (col0, rights))
   215               val pat_rectangle' = map v_to_prefix pat_rectangle
   216               val (pref_patl,tm) = mk{path = rstp,
   217                                       rows = ListPair.zip (pat_rectangle',
   218                                                            rights')}
   219           in (map v_to_pats pref_patl, tm)
   220           end
   221      else
   222      let val pty as Type (ty_name,_) = type_of p
   223      in
   224      case (ty_info ty_name)
   225      of None => mk_case_fail("Not a known datatype: "^ty_name)
   226       | Some{case_const,constructors} =>
   227         let open BasisLibrary (*restore original List*)
   228 	    val case_const_name = #1(dest_Const case_const)
   229             val nrows = List.concat (map (expand constructors pty) rows)
   230             val subproblems = divide(constructors, pty, range_ty, nrows)
   231             val groups      = map #group subproblems
   232             and new_formals = map #new_formals subproblems
   233             and constructors' = map #constructor subproblems
   234             val news = map (fn (nf,rows) => {path = nf@rstp, rows=rows})
   235                            (ListPair.zip (new_formals, groups))
   236             val rec_calls = map mk news
   237             val (pat_rect,dtrees) = ListPair.unzip rec_calls
   238             val case_functions = map S.list_mk_abs
   239                                   (ListPair.zip (new_formals, dtrees))
   240             val types = map type_of (case_functions@[u]) @ [range_ty]
   241             val case_const' = Const(case_const_name, list_mk_type types)
   242             val tree = list_comb(case_const', case_functions@[u])
   243             val pat_rect1 = List.concat
   244                               (ListPair.map mk_pat (constructors', pat_rect))
   245         in (pat_rect1,tree)
   246         end 
   247      end end
   248  in mk
   249  end;
   250 
   251 
   252 (* Repeated variable occurrences in a pattern are not allowed. *)
   253 fun FV_multiset tm = 
   254    case (S.dest_term tm)
   255      of S.VAR{Name,Ty} => [Free(Name,Ty)]
   256       | S.CONST _ => []
   257       | S.COMB{Rator, Rand} => FV_multiset Rator @ FV_multiset Rand
   258       | S.LAMB _ => raise TFL_ERR{func = "FV_multiset", mesg = "lambda"};
   259 
   260 fun no_repeat_vars thy pat =
   261  let fun check [] = true
   262        | check (v::rst) =
   263          if mem_term (v,rst) then
   264 	    raise TFL_ERR{func = "no_repeat_vars",
   265 			  mesg = quote(#1(dest_Free v)) ^
   266 			  " occurs repeatedly in the pattern " ^
   267 			  quote (string_of_cterm (Thry.typecheck thy pat))}
   268          else check rst
   269  in check (FV_multiset pat)
   270  end;
   271 
   272 local fun mk_functional_err s = raise TFL_ERR{func = "mk_functional", mesg=s}
   273       fun single [f] = f
   274         | single fs  = mk_functional_err (Int.toString (length fs) ^ 
   275                                           " distinct function names!")
   276 in
   277 fun mk_functional thy clauses =
   278  let val (L,R) = ListPair.unzip 
   279                     (map (fn (Const("op =",_) $ t $ u) => (t,u)) clauses)
   280      val (funcs,pats) = ListPair.unzip (map (fn (t$u) =>(t,u)) L)
   281      val fcon as Const (fname, ftype) = single (gen_distinct (op aconv) funcs)
   282      val dummy = map (no_repeat_vars thy) pats
   283      val rows = ListPair.zip (map (fn x => ([],[x])) pats,
   284                               map GIVEN (enumerate R))
   285      val names = foldr add_term_names (R,[])
   286      val atype = type_of(hd pats)
   287      and aname = variant names "a"
   288      val a = Free(aname,atype)
   289      val ty_info = Thry.match_info thy
   290      val ty_match = Thry.match_type thy
   291      val range_ty = type_of (hd R)
   292      val (patts, case_tm) = mk_case ty_info ty_match (aname::names) range_ty 
   293                                     {path=[a], rows=rows}
   294      val patts1 = map (fn (_,(tag,i),[pat]) => tag (pat,i)) patts 
   295 	  handle _ => mk_functional_err "error in pattern-match translation"
   296      val patts2 = U.sort(fn p1=>fn p2=> row_of_pat p1 < row_of_pat p2) patts1
   297      val finals = map row_of_pat patts2
   298      val originals = map (row_of_pat o #2) rows
   299      val dummy = case (originals\\finals)
   300              of [] => ()
   301           | L => mk_functional_err("The following rows (counting from zero)\
   302                                    \ are inaccessible: "^stringize L)
   303  in {functional = Abs(Sign.base_name fname, ftype,
   304 		      abstract_over (fcon, 
   305 				     absfree(aname,atype, case_tm))),
   306      pats = patts2}
   307 end end;
   308 
   309 
   310 (*----------------------------------------------------------------------------
   311  *
   312  *                    PRINCIPLES OF DEFINITION
   313  *
   314  *---------------------------------------------------------------------------*)
   315 
   316 
   317 (*Make all TVars available for instantiation by adding a ? to the front*)
   318 fun poly_tvars (Type(a,Ts)) = Type(a, map (poly_tvars) Ts)
   319   | poly_tvars (TFree (a,sort)) = TVar (("?" ^ a, 0), sort)
   320   | poly_tvars (TVar ((a,i),sort)) = TVar (("?" ^ a, i+1), sort);
   321 
   322 local val f_eq_wfrec_R_M = 
   323            #ant(S.dest_imp(#2(S.strip_forall (concl Thms.WFREC_COROLLARY))))
   324       val {lhs=f, rhs} = S.dest_eq f_eq_wfrec_R_M
   325       val (fname,_) = dest_Free f
   326       val (wfrec,_) = S.strip_comb rhs
   327 in
   328 fun wfrec_definition0 thy fid R (functional as Abs(Name, Ty, _)) =
   329  let val def_name = if Name<>fid then 
   330 			raise TFL_ERR{func = "wfrec_definition0",
   331 				      mesg = "Expected a definition of " ^ 
   332 					     quote fid ^ " but found one of " ^
   333 				      quote Name}
   334 		    else Name ^ "_def"
   335      val wfrec_R_M =  map_term_types poly_tvars 
   336 	                  (wfrec $ map_term_types poly_tvars R) 
   337 	              $ functional
   338      val (def_term, _) = 
   339 	 Sign.infer_types (sign_of thy) (K None) (K None) [] false
   340 	       ([Const("==",dummyT) $ Const(Name,Ty) $ wfrec_R_M], 
   341 		propT)
   342   in  PureThy.add_defs_i [Thm.no_attributes (def_name, def_term)] thy  end
   343 end;
   344 
   345 
   346 
   347 (*---------------------------------------------------------------------------
   348  * This structure keeps track of congruence rules that aren't derived
   349  * from a datatype definition.
   350  *---------------------------------------------------------------------------*)
   351 fun extraction_thms thy = 
   352  let val {case_rewrites,case_congs} = Thry.extract_info thy
   353  in (case_rewrites, case_congs)
   354  end;
   355 
   356 
   357 (*---------------------------------------------------------------------------
   358  * Pair patterns with termination conditions. The full list of patterns for
   359  * a definition is merged with the TCs arising from the user-given clauses.
   360  * There can be fewer clauses than the full list, if the user omitted some 
   361  * cases. This routine is used to prepare input for mk_induction.
   362  *---------------------------------------------------------------------------*)
   363 fun merge full_pats TCs =
   364 let fun insert (p,TCs) =
   365       let fun insrt ((x as (h,[]))::rst) = 
   366                  if (p aconv h) then (p,TCs)::rst else x::insrt rst
   367             | insrt (x::rst) = x::insrt rst
   368             | insrt[] = raise TFL_ERR{func="merge.insert",
   369 				      mesg="pattern not found"}
   370       in insrt end
   371     fun pass ([],ptcl_final) = ptcl_final
   372       | pass (ptcs::tcl, ptcl) = pass(tcl, insert ptcs ptcl)
   373 in 
   374   pass (TCs, map (fn p => (p,[])) full_pats)
   375 end;
   376 
   377 
   378 fun givens [] = []
   379   | givens (GIVEN(tm,_)::pats) = tm :: givens pats
   380   | givens (OMITTED _::pats)   = givens pats;
   381 
   382 fun post_definition (ss, tflCongs) (theory, (def, pats)) =
   383  let val tych = Thry.typecheck theory 
   384      val f = #lhs(S.dest_eq(concl def))
   385      val corollary = R.MATCH_MP Thms.WFREC_COROLLARY def
   386      val given_pats = givens pats
   387      val WFR = #ant(S.dest_imp(concl corollary))
   388      val R = #Rand(S.dest_comb WFR)
   389      val corollary' = R.UNDISCH corollary  (* put WF R on assums *)
   390      val corollaries = map (fn pat => R.SPEC (tych pat) corollary') 
   391 	                   given_pats
   392      val (case_rewrites,context_congs) = extraction_thms theory
   393      val corollaries' = map(rewrite_rule case_rewrites) corollaries
   394      val extract = R.CONTEXT_REWRITE_RULE 
   395 	             (ss, f, R,
   396 		      R.ISPECL (map tych [f,R]) Thms.CUT_LEMMA,
   397 		      tflCongs@context_congs)
   398      val (rules, TCs) = ListPair.unzip (map extract corollaries')
   399      val rules0 = map (rewrite_rule [Thms.CUT_DEF]) rules
   400      val mk_cond_rule = R.FILTER_DISCH_ALL(not o curry (op aconv) WFR)
   401      val rules1 = R.LIST_CONJ(map mk_cond_rule rules0)
   402  in
   403  {theory = theory,   (* holds def, if it's needed *)
   404   rules = rules1,
   405   full_pats_TCs = merge (map pat_of pats) 
   406                         (ListPair.zip (given_pats, TCs)), 
   407   TCs = TCs, 
   408   patterns = pats}
   409  end;
   410 
   411 (*---------------------------------------------------------------------------
   412  * Perform the extraction without making the definition. Definition and
   413  * extraction commute for the non-nested case. For hol90 users, this 
   414  * function can be invoked without being in draft mode.
   415  * CURRENTLY UNUSED
   416 fun wfrec_eqns (ss, tflCongs) thy eqns =
   417  let val {functional,pats} = mk_functional thy eqns
   418      val given_pats = givens pats
   419      val {Bvar = f, Body} = S.dest_abs functional
   420      val {Bvar = x, ...} = S.dest_abs Body
   421      val (Name, Type("fun", [f_dty, f_rty])) = dest_Free f
   422      val (case_rewrites,context_congs) = extraction_thms thy
   423      val tych = Thry.typecheck thy
   424      val WFREC_THM0 = R.ISPEC (tych functional) Thms.WFREC_COROLLARY
   425      val Const("All",_) $ Abs(Rname,Rtype,_) = concl WFREC_THM0
   426      val R = Free (variant (foldr add_term_names (eqns,[])) Rname,
   427 		   Rtype)
   428      val WFREC_THM = R.ISPECL [tych R, tych f] WFREC_THM0
   429      val ([proto_def, WFR],_) = S.strip_imp(concl WFREC_THM)
   430      val R1 = S.rand WFR
   431      val corollary' = R.UNDISCH(R.UNDISCH WFREC_THM)
   432      val corollaries = map (fn pat => R.SPEC (tych pat) corollary') given_pats
   433      val corollaries' = map (rewrite_rule case_rewrites) corollaries
   434      val extract = R.CONTEXT_REWRITE_RULE 
   435 	               (ss, f, R1, 
   436 		        R.ISPECL (map tych [f,R1]) Thms.CUT_LEMMA, 
   437 			tflCongs@context_congs)
   438  in {proto_def=proto_def, 
   439      WFR=WFR, 
   440      pats=pats,
   441      extracta = map extract corollaries'}
   442  end;
   443  *---------------------------------------------------------------------------*)
   444 
   445 
   446 (*---------------------------------------------------------------------------
   447  * Define the constant after extracting the termination conditions. The 
   448  * wellfounded relation used in the definition is computed by using the
   449  * choice operator on the extracted conditions (plus the condition that
   450  * such a relation must be wellfounded).
   451  * CURRENTLY UNUSED
   452 fun lazyR_def ss thy eqns =
   453  let val {proto_def,WFR,pats,extracta} = wfrec_eqns ss thy eqns
   454      val R1 = S.rand WFR
   455      val f = S.lhs proto_def
   456      val (Name,_) = dest_Free f
   457      val (extractants,TCl) = ListPair.unzip extracta
   458      val TCs = foldr (gen_union (op aconv)) (TCl, [])
   459      val full_rqt = WFR::TCs
   460      val R' = S.mk_select{Bvar=R1, Body=S.list_mk_conj full_rqt}
   461      val R'abs = S.rand R'
   462      val theory =
   463        thy
   464        |> PureThy.add_defs_i [Thm.no_attributes (Name ^ "_def", subst_free[(R1,R')] proto_def)];
   465      val def = freezeT((get_axiom theory (Name ^ "_def")) RS meta_eq_to_obj_eq)
   466      val fconst = #lhs(S.dest_eq(concl def)) 
   467      val tych = Thry.typecheck theory
   468      val baz = R.DISCH (tych proto_def)
   469                  (U.itlist (R.DISCH o tych) full_rqt (R.LIST_CONJ extractants))
   470      val def' = R.MP (R.SPEC (tych fconst) 
   471                              (R.SPEC (tych R') (R.GENL[tych R1, tych f] baz)))
   472                      def
   473      val body_th = R.LIST_CONJ (map (R.ASSUME o tych) full_rqt)
   474      val bar = R.MP (R.ISPECL[tych R'abs, tych R1] Thms.SELECT_AX)
   475                     body_th
   476  in {theory = theory, R=R1,
   477      rules = U.rev_itlist (U.C R.MP) (R.CONJUNCTS bar) def',
   478      full_pats_TCs = merge (map pat_of pats) (ListPair.zip (givens pats, TCl)),
   479      patterns = pats}
   480  end;
   481  *---------------------------------------------------------------------------*)
   482 
   483 
   484 
   485 (*----------------------------------------------------------------------------
   486  *
   487  *                           INDUCTION THEOREM
   488  *
   489  *---------------------------------------------------------------------------*)
   490 
   491 
   492 (*------------------------  Miscellaneous function  --------------------------
   493  *
   494  *           [x_1,...,x_n]     ?v_1...v_n. M[v_1,...,v_n]
   495  *     -----------------------------------------------------------
   496  *     ( M[x_1,...,x_n], [(x_i,?v_1...v_n. M[v_1,...,v_n]),
   497  *                        ... 
   498  *                        (x_j,?v_n. M[x_1,...,x_(n-1),v_n])] )
   499  *
   500  * This function is totally ad hoc. Used in the production of the induction 
   501  * theorem. The nchotomy theorem can have clauses that look like
   502  *
   503  *     ?v1..vn. z = C vn..v1
   504  *
   505  * in which the order of quantification is not the order of occurrence of the
   506  * quantified variables as arguments to C. Since we have no control over this
   507  * aspect of the nchotomy theorem, we make the correspondence explicit by
   508  * pairing the incoming new variable with the term it gets beta-reduced into.
   509  *---------------------------------------------------------------------------*)
   510 
   511 fun alpha_ex_unroll (xlist, tm) =
   512   let val (qvars,body) = S.strip_exists tm
   513       val vlist = #2(S.strip_comb (S.rhs body))
   514       val plist = ListPair.zip (vlist, xlist)
   515       val args = map (fn qv => the (gen_assoc (op aconv) (plist, qv))) qvars
   516                    handle OPTION => error 
   517                        "TFL fault [alpha_ex_unroll]: no correspondence"
   518       fun build ex      []   = []
   519         | build (_$rex) (v::rst) =
   520            let val ex1 = betapply(rex, v)
   521            in  ex1 :: build ex1 rst
   522            end
   523      val (nex::exl) = rev (tm::build tm args)
   524   in 
   525   (nex, ListPair.zip (args, rev exl))
   526   end;
   527 
   528 
   529 
   530 (*----------------------------------------------------------------------------
   531  *
   532  *             PROVING COMPLETENESS OF PATTERNS
   533  *
   534  *---------------------------------------------------------------------------*)
   535 
   536 fun mk_case ty_info usednames thy =
   537  let 
   538  val divide = ipartition (gvvariant usednames)
   539  val tych = Thry.typecheck thy
   540  fun tych_binding(x,y) = (tych x, tych y)
   541  fun fail s = raise TFL_ERR{func = "mk_case", mesg = s}
   542  fun mk{rows=[],...} = fail"no rows"
   543    | mk{path=[], rows = [([], (thm, bindings))]} = 
   544                          R.IT_EXISTS (map tych_binding bindings) thm
   545    | mk{path = u::rstp, rows as (p::_, _)::_} =
   546      let val (pat_rectangle,rights) = ListPair.unzip rows
   547          val col0 = map hd pat_rectangle
   548          val pat_rectangle' = map tl pat_rectangle
   549      in 
   550      if (forall is_Free col0) (* column 0 is all variables *)
   551      then let val rights' = map (fn ((thm,theta),v) => (thm,theta@[(u,v)]))
   552                                 (ListPair.zip (rights, col0))
   553           in mk{path = rstp, rows = ListPair.zip (pat_rectangle', rights')}
   554           end
   555      else                     (* column 0 is all constructors *)
   556      let val Type (ty_name,_) = type_of p
   557      in
   558      case (ty_info ty_name)
   559      of None => fail("Not a known datatype: "^ty_name)
   560       | Some{constructors,nchotomy} =>
   561         let val thm' = R.ISPEC (tych u) nchotomy
   562             val disjuncts = S.strip_disj (concl thm')
   563             val subproblems = divide(constructors, rows)
   564             val groups      = map #group subproblems
   565             and new_formals = map #new_formals subproblems
   566             val existentials = ListPair.map alpha_ex_unroll
   567                                    (new_formals, disjuncts)
   568             val constraints = map #1 existentials
   569             val vexl = map #2 existentials
   570             fun expnd tm (pats,(th,b)) = (pats,(R.SUBS[R.ASSUME(tych tm)]th,b))
   571             val news = map (fn (nf,rows,c) => {path = nf@rstp, 
   572                                                rows = map (expnd c) rows})
   573                            (U.zip3 new_formals groups constraints)
   574             val recursive_thms = map mk news
   575             val build_exists = foldr
   576                                 (fn((x,t), th) => 
   577                                  R.CHOOSE (tych x, R.ASSUME (tych t)) th)
   578             val thms' = ListPair.map build_exists (vexl, recursive_thms)
   579             val same_concls = R.EVEN_ORS thms'
   580         in R.DISJ_CASESL thm' same_concls
   581         end 
   582      end end
   583  in mk
   584  end;
   585 
   586 
   587 fun complete_cases thy =
   588  let val tych = Thry.typecheck thy
   589      val ty_info = Thry.induct_info thy
   590  in fn pats =>
   591  let val names = foldr add_term_names (pats,[])
   592      val T = type_of (hd pats)
   593      val aname = Term.variant names "a"
   594      val vname = Term.variant (aname::names) "v"
   595      val a = Free (aname, T)
   596      val v = Free (vname, T)
   597      val a_eq_v = HOLogic.mk_eq(a,v)
   598      val ex_th0 = R.EXISTS (tych (S.mk_exists{Bvar=v,Body=a_eq_v}), tych a)
   599                            (R.REFL (tych a))
   600      val th0 = R.ASSUME (tych a_eq_v)
   601      val rows = map (fn x => ([x], (th0,[]))) pats
   602  in
   603  R.GEN (tych a) 
   604        (R.RIGHT_ASSOC
   605           (R.CHOOSE(tych v, ex_th0)
   606                 (mk_case ty_info (vname::aname::names)
   607 		 thy {path=[v], rows=rows})))
   608  end end;
   609 
   610 
   611 (*---------------------------------------------------------------------------
   612  * Constructing induction hypotheses: one for each recursive call.
   613  *
   614  * Note. R will never occur as a variable in the ind_clause, because
   615  * to do so, it would have to be from a nested definition, and we don't
   616  * allow nested defns to have R variable.
   617  *
   618  * Note. When the context is empty, there can be no local variables.
   619  *---------------------------------------------------------------------------*)
   620 
   621 local infix 5 ==>
   622       fun (tm1 ==> tm2) = S.mk_imp{ant = tm1, conseq = tm2}
   623 in
   624 fun build_ih f P (pat,TCs) = 
   625  let val globals = S.free_vars_lr pat
   626      fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
   627      fun dest_TC tm = 
   628          let val (cntxt,R_y_pat) = S.strip_imp(#2(S.strip_forall tm))
   629              val (R,y,_) = S.dest_relation R_y_pat
   630              val P_y = if (nested tm) then R_y_pat ==> P$y else P$y
   631          in case cntxt 
   632               of [] => (P_y, (tm,[]))
   633                | _  => let 
   634                     val imp = S.list_mk_conj cntxt ==> P_y
   635                     val lvs = gen_rems (op aconv) (S.free_vars_lr imp, globals)
   636                     val locals = #2(U.pluck (curry (op aconv) P) lvs) handle _ => lvs
   637                     in (S.list_mk_forall(locals,imp), (tm,locals)) end
   638          end
   639  in case TCs
   640     of [] => (S.list_mk_forall(globals, P$pat), [])
   641      |  _ => let val (ihs, TCs_locals) = ListPair.unzip(map dest_TC TCs)
   642                  val ind_clause = S.list_mk_conj ihs ==> P$pat
   643              in (S.list_mk_forall(globals,ind_clause), TCs_locals)
   644              end
   645  end
   646 end;
   647 
   648 
   649 
   650 (*---------------------------------------------------------------------------
   651  * This function makes good on the promise made in "build_ih: we prove
   652  * <something>.
   653  *
   654  * Input  is tm = "(!y. R y pat ==> P y) ==> P pat",  
   655  *           TCs = TC_1[pat] ... TC_n[pat]
   656  *           thm = ih1 /\ ... /\ ih_n |- ih[pat]
   657  *---------------------------------------------------------------------------*)
   658 fun prove_case f thy (tm,TCs_locals,thm) =
   659  let val tych = Thry.typecheck thy
   660      val antc = tych(#ant(S.dest_imp tm))
   661      val thm' = R.SPEC_ALL thm
   662      fun nested tm = is_some (S.find_term (curry (op aconv) f) tm)
   663      fun get_cntxt TC = tych(#ant(S.dest_imp(#2(S.strip_forall(concl TC)))))
   664      fun mk_ih ((TC,locals),th2,nested) =
   665          R.GENL (map tych locals)
   666             (if nested 
   667               then R.DISCH (get_cntxt TC) th2 handle _ => th2
   668                else if S.is_imp(concl TC) 
   669                      then R.IMP_TRANS TC th2 
   670                       else R.MP th2 TC)
   671  in 
   672  R.DISCH antc
   673  (if S.is_imp(concl thm') (* recursive calls in this clause *)
   674   then let val th1 = R.ASSUME antc
   675            val TCs = map #1 TCs_locals
   676            val ylist = map (#2 o S.dest_relation o #2 o S.strip_imp o 
   677                             #2 o S.strip_forall) TCs
   678            val TClist = map (fn(TC,lvs) => (R.SPEC_ALL(R.ASSUME(tych TC)),lvs))
   679                             TCs_locals
   680            val th2list = map (U.C R.SPEC th1 o tych) ylist
   681            val nlist = map nested TCs
   682            val triples = U.zip3 TClist th2list nlist
   683            val Pylist = map mk_ih triples
   684        in R.MP thm' (R.LIST_CONJ Pylist) end
   685   else thm')
   686  end;
   687 
   688 
   689 (*---------------------------------------------------------------------------
   690  *
   691  *         x = (v1,...,vn)  |- M[x]
   692  *    ---------------------------------------------
   693  *      ?v1 ... vn. x = (v1,...,vn) |- M[x]
   694  *
   695  *---------------------------------------------------------------------------*)
   696 fun LEFT_ABS_VSTRUCT tych thm = 
   697   let fun CHOOSER v (tm,thm) = 
   698         let val ex_tm = S.mk_exists{Bvar=v,Body=tm}
   699         in (ex_tm, R.CHOOSE(tych v, R.ASSUME (tych ex_tm)) thm)
   700         end
   701       val [veq] = filter (U.can S.dest_eq) (#1 (R.dest_thm thm))
   702       val {lhs,rhs} = S.dest_eq veq
   703       val L = S.free_vars_lr rhs
   704   in  #2 (U.itlist CHOOSER L (veq,thm))  end;
   705 
   706 
   707 (*----------------------------------------------------------------------------
   708  * Input : f, R,  and  [(pat1,TCs1),..., (patn,TCsn)]
   709  *
   710  * Instantiates WF_INDUCTION_THM, getting Sinduct and then tries to prove
   711  * recursion induction (Rinduct) by proving the antecedent of Sinduct from 
   712  * the antecedent of Rinduct.
   713  *---------------------------------------------------------------------------*)
   714 fun mk_induction thy f R pat_TCs_list =
   715 let val tych = Thry.typecheck thy
   716     val Sinduction = R.UNDISCH (R.ISPEC (tych R) Thms.WF_INDUCTION_THM)
   717     val (pats,TCsl) = ListPair.unzip pat_TCs_list
   718     val case_thm = complete_cases thy pats
   719     val domain = (type_of o hd) pats
   720     val Pname = Term.variant (foldr (foldr add_term_names) 
   721 			      (pats::TCsl, [])) "P"
   722     val P = Free(Pname, domain --> HOLogic.boolT)
   723     val Sinduct = R.SPEC (tych P) Sinduction
   724     val Sinduct_assumf = S.rand ((#ant o S.dest_imp o concl) Sinduct)
   725     val Rassums_TCl' = map (build_ih f P) pat_TCs_list
   726     val (Rassums,TCl') = ListPair.unzip Rassums_TCl'
   727     val Rinduct_assum = R.ASSUME (tych (S.list_mk_conj Rassums))
   728     val cases = map (fn pat => betapply (Sinduct_assumf, pat)) pats
   729     val tasks = U.zip3 cases TCl' (R.CONJUNCTS Rinduct_assum)
   730     val proved_cases = map (prove_case f thy) tasks
   731     val v = Free (variant (foldr add_term_names (map concl proved_cases, []))
   732 		    "v",
   733 		  domain)
   734     val vtyped = tych v
   735     val substs = map (R.SYM o R.ASSUME o tych o (curry HOLogic.mk_eq v)) pats
   736     val proved_cases1 = ListPair.map (fn (th,th') => R.SUBS[th]th') 
   737                           (substs, proved_cases)
   738     val abs_cases = map (LEFT_ABS_VSTRUCT tych) proved_cases1
   739     val dant = R.GEN vtyped (R.DISJ_CASESL (R.ISPEC vtyped case_thm) abs_cases)
   740     val dc = R.MP Sinduct dant
   741     val Parg_ty = type_of(#Bvar(S.dest_forall(concl dc)))
   742     val vars = map (gvvariant[Pname]) (S.strip_prod_type Parg_ty)
   743     val dc' = U.itlist (R.GEN o tych) vars
   744                        (R.SPEC (tych(S.mk_vstruct Parg_ty vars)) dc)
   745 in 
   746    R.GEN (tych P) (R.DISCH (tych(concl Rinduct_assum)) dc')
   747 end
   748 handle _ => raise TFL_ERR{func = "mk_induction", mesg = "failed derivation"};
   749 
   750 
   751 
   752 
   753 (*---------------------------------------------------------------------------
   754  * 
   755  *                        POST PROCESSING
   756  *
   757  *---------------------------------------------------------------------------*)
   758 
   759 
   760 fun simplify_induction thy hth ind = 
   761   let val tych = Thry.typecheck thy
   762       val (asl,_) = R.dest_thm ind
   763       val (_,tc_eq_tc') = R.dest_thm hth
   764       val tc = S.lhs tc_eq_tc'
   765       fun loop [] = ind
   766         | loop (asm::rst) = 
   767           if (U.can (Thry.match_term thy asm) tc)
   768           then R.UNDISCH
   769                  (R.MATCH_MP
   770                      (R.MATCH_MP Thms.simp_thm (R.DISCH (tych asm) ind)) 
   771                      hth)
   772          else loop rst
   773   in loop asl
   774 end;
   775 
   776 
   777 (*---------------------------------------------------------------------------
   778  * The termination condition is an antecedent to the rule, and an 
   779  * assumption to the theorem.
   780  *---------------------------------------------------------------------------*)
   781 fun elim_tc tcthm (rule,induction) = 
   782    (R.MP rule tcthm, R.PROVE_HYP tcthm induction)
   783 
   784 
   785 fun postprocess{WFtac, terminator, simplifier} theory {rules,induction,TCs} =
   786 let val tych = Thry.typecheck theory
   787 
   788    (*---------------------------------------------------------------------
   789     * Attempt to eliminate WF condition. It's the only assumption of rules
   790     *---------------------------------------------------------------------*)
   791    val (rules1,induction1)  = 
   792        let val thm = R.prove(tych(HOLogic.mk_Trueprop 
   793 				  (hd(#1(R.dest_thm rules)))),
   794 			     WFtac)
   795        in (R.PROVE_HYP thm rules,  R.PROVE_HYP thm induction)
   796        end handle _ => (rules,induction)
   797 
   798    (*----------------------------------------------------------------------
   799     * The termination condition (tc) is simplified to |- tc = tc' (there
   800     * might not be a change!) and then 3 attempts are made:
   801     *
   802     *   1. if |- tc = T, then eliminate it with eqT; otherwise,
   803     *   2. apply the terminator to tc'. If |- tc' = T then eliminate; else
   804     *   3. replace tc by tc' in both the rules and the induction theorem.
   805     *---------------------------------------------------------------------*)
   806    fun simplify_tc tc (r,ind) =
   807        let val tc_eq = simplifier (tych tc)
   808        in 
   809        elim_tc (R.MATCH_MP Thms.eqT tc_eq) (r,ind)
   810        handle _ => 
   811         (elim_tc (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
   812 		  (R.prove(tych(HOLogic.mk_Trueprop(S.rhs(concl tc_eq))), 
   813 			   terminator)))
   814                  (r,ind)
   815          handle _ => 
   816           (R.UNDISCH(R.MATCH_MP (R.MATCH_MP Thms.simp_thm r) tc_eq), 
   817            simplify_induction theory tc_eq ind))
   818        end
   819 
   820    (*----------------------------------------------------------------------
   821     * Nested termination conditions are harder to get at, since they are
   822     * left embedded in the body of the function (and in induction 
   823     * theorem hypotheses). Our "solution" is to simplify them, and try to 
   824     * prove termination, but leave the application of the resulting theorem 
   825     * to a higher level. So things go much as in "simplify_tc": the 
   826     * termination condition (tc) is simplified to |- tc = tc' (there might 
   827     * not be a change) and then 2 attempts are made:
   828     *
   829     *   1. if |- tc = T, then return |- tc; otherwise,
   830     *   2. apply the terminator to tc'. If |- tc' = T then return |- tc; else
   831     *   3. return |- tc = tc'
   832     *---------------------------------------------------------------------*)
   833    fun simplify_nested_tc tc =
   834       let val tc_eq = simplifier (tych (#2 (S.strip_forall tc)))
   835       in
   836       R.GEN_ALL
   837        (R.MATCH_MP Thms.eqT tc_eq
   838         handle _
   839         => (R.MATCH_MP(R.MATCH_MP Thms.rev_eq_mp tc_eq)
   840                       (R.prove(tych(HOLogic.mk_Trueprop (S.rhs(concl tc_eq))),
   841 			       terminator))
   842             handle _ => tc_eq))
   843       end
   844 
   845    (*-------------------------------------------------------------------
   846     * Attempt to simplify the termination conditions in each rule and 
   847     * in the induction theorem.
   848     *-------------------------------------------------------------------*)
   849    fun strip_imp tm = if S.is_neg tm then ([],tm) else S.strip_imp tm
   850    fun loop ([],extras,R,ind) = (rev R, ind, extras)
   851      | loop ((r,ftcs)::rst, nthms, R, ind) =
   852         let val tcs = #1(strip_imp (concl r))
   853             val extra_tcs = gen_rems (op aconv) (ftcs, tcs)
   854             val extra_tc_thms = map simplify_nested_tc extra_tcs
   855             val (r1,ind1) = U.rev_itlist simplify_tc tcs (r,ind)
   856             val r2 = R.FILTER_DISCH_ALL(not o S.is_WFR) r1
   857         in loop(rst, nthms@extra_tc_thms, r2::R, ind1)
   858         end
   859    val rules_tcs = ListPair.zip (R.CONJUNCTS rules1, TCs)
   860    val (rules2,ind2,extras) = loop(rules_tcs,[],[],induction1)
   861 in
   862   {induction = ind2, rules = R.LIST_CONJ rules2, nested_tcs = extras}
   863 end;
   864 
   865 end; (* TFL *)