src/Provers/splitter.ML
author oheimb
Thu May 14 16:50:09 1998 +0200 (1998-05-14)
changeset 4930 89271bc4e7ed
parent 4668 131989b78417
child 5304 c133f16febc7
permissions -rw-r--r--
extended addsplits and delsplits to handle also split rules for assumptions
extended const_of_split_thm, renamed it to split_thm_info
     1 (*  Title:      Provers/splitter
     2     ID:         $Id$
     3     Author:     Tobias Nipkow
     4     Copyright   1995  TU Munich
     5 
     6 Generic case-splitter, suitable for most logics.
     7 
     8 Use:
     9 
    10 val split_tac = mk_case_split_tac iffD;
    11 
    12 by(split_tac splits i);
    13 
    14 where splits = [P(elim(...)) == rhs, ...]
    15       iffD  = [| P <-> Q; Q |] ==> P (* is called iffD2 in HOL *)
    16 
    17 *)
    18 
    19 local
    20 
    21 fun split_format_err() = error("Wrong format for split rule");
    22 
    23 fun mk_case_split_tac_2 iffD order =
    24 let
    25 
    26 
    27 (************************************************************
    28    Create lift-theorem "trlift" :
    29 
    30    [| !! x. Q(x)==R(x) ; P(R) == C |] ==> P(Q)==C
    31 
    32 *************************************************************)
    33  
    34 val lift =
    35   let val ct = read_cterm (#sign(rep_thm iffD))
    36            ("[| !!x::'b::logic. Q(x) == R(x) |] ==> \
    37             \P(%x. Q(x)) == P(%x. R(x))::'a::logic",propT)
    38   in prove_goalw_cterm [] ct
    39      (fn [prem] => [rewtac prem, rtac reflexive_thm 1])
    40   end;
    41 
    42 val trlift = lift RS transitive_thm;
    43 val _ $ (Var(P,PT)$_) $ _ = concl_of trlift;
    44 
    45 
    46 (************************************************************************ 
    47    Set up term for instantiation of P in the lift-theorem
    48    
    49    Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
    50    t     : lefthand side of meta-equality in subgoal
    51            the lift theorem is applied to (see select)
    52    pos   : "path" leading to abstraction, coded as a list
    53    T     : type of body of P(...)
    54    maxi  : maximum index of Vars
    55 *************************************************************************)
    56 
    57 fun mk_cntxt Ts t pos T maxi =
    58   let fun var (t,i) = Var(("X",i),type_of1(Ts,t));
    59       fun down [] t i = Bound 0
    60         | down (p::ps) t i =
    61             let val (h,ts) = strip_comb t
    62                 val v1 = ListPair.map var (take(p,ts), i upto (i+p-1))
    63                 val u::us = drop(p,ts)
    64                 val v2 = ListPair.map var (us, (i+p) upto (i+length(ts)-2))
    65       in list_comb(h,v1@[down ps u (i+length ts)]@v2) end;
    66   in Abs("", T, down (rev pos) t maxi) end;
    67 
    68 
    69 (************************************************************************ 
    70    Set up term for instantiation of P in the split-theorem
    71    P(...) == rhs
    72 
    73    t     : lefthand side of meta-equality in subgoal
    74            the split theorem is applied to (see select)
    75    T     : type of body of P(...)
    76    tt    : the term  Const(key,..) $ ...
    77 *************************************************************************)
    78 
    79 fun mk_cntxt_splitthm t tt T =
    80   let fun repl lev t =
    81     if incr_boundvars lev tt = t then Bound lev
    82     else case t of
    83         (Abs (v, T2, t)) => Abs (v, T2, repl (lev+1) t)
    84       | (Bound i) => Bound (if i>=lev then i+1 else i)
    85       | (t1 $ t2) => (repl lev t1) $ (repl lev t2)
    86       | t => t
    87   in Abs("", T, repl 0 t) end;
    88 
    89 
    90 (* add all loose bound variables in t to list is *)
    91 fun add_lbnos(is,t) = add_loose_bnos(t,0,is);
    92 
    93 (* check if the innermost quantifier that needs to be removed
    94    has a body of type T; otherwise the expansion thm will fail later on
    95 *)
    96 fun type_test(T,lbnos,apsns) =
    97   let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns)
    98   in T=U end;
    99 
   100 (*************************************************************************
   101    Create a "split_pack".
   102 
   103    thm   : the relevant split-theorem, i.e. P(...) == rhs , where P(...)
   104            is of the form
   105            P( Const(key,...) $ t_1 $ ... $ t_n )      (e.g. key = "if")
   106    T     : type of P(...)
   107    n     : number of arguments expected by Const(key,...)
   108    ts    : list of arguments actually found
   109    apsns : list of tuples of the form (T,U,pos), one tuple for each
   110            abstraction that is encountered on the way to the position where 
   111            Const(key, ...) $ ...  occurs, where
   112            T   : type of the variable bound by the abstraction
   113            U   : type of the abstraction's body
   114            pos : "path" leading to the body of the abstraction
   115    pos   : "path" leading to the position where Const(key, ...) $ ...  occurs.
   116    TB    : type of  Const(key,...) $ t_1 $ ... $ t_n
   117    t     : the term Const(key,...) $ t_1 $ ... $ t_n
   118 
   119    A split pack is a tuple of the form
   120    (thm, apsns, pos, TB)
   121    Note : apsns is reversed, so that the outermost quantifier's position
   122           comes first ! If the terms in ts don't contain variables bound
   123           by other than meta-quantifiers, apsns is empty, because no further
   124           lifting is required before applying the split-theorem.
   125 ******************************************************************************) 
   126 
   127 fun mk_split_pack(thm,T,n,ts,apsns,pos,TB,t) =
   128   if n > length ts then []
   129   else let val lev = length apsns
   130            val lbnos = foldl add_lbnos ([],take(n,ts))
   131            val flbnos = filter (fn i => i < lev) lbnos
   132            val tt = incr_boundvars (~lev) t
   133        in if null flbnos then [(thm,[],pos,TB,tt)]
   134           else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)] 
   135                else []
   136        end;
   137 
   138 
   139 (****************************************************************************
   140    Recursively scans term for occurences of Const(key,...) $ ...
   141    Returns a list of "split-packs" (one for each occurence of Const(key,...) )
   142 
   143    cmap : association list of split-theorems that should be tried.
   144           The elements have the format (key,(thm,T,n)) , where
   145           key : the theorem's key constant ( Const(key,...) $ ... )
   146           thm : the theorem itself
   147           T   : type of P( Const(key,...) $ ... )
   148           n   : number of arguments expected by Const(key,...)
   149    Ts   : types of parameters
   150    t    : the term to be scanned
   151 ******************************************************************************)
   152 
   153 fun split_posns cmap Ts t =
   154   let fun posns Ts pos apsns (Abs(_,T,t)) =
   155             let val U = fastype_of1(T::Ts,t)
   156             in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end
   157         | posns Ts pos apsns t =
   158             let val (h,ts) = strip_comb t
   159                 fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a);
   160                 val a = case h of
   161                   Const(c,_) =>
   162                     (case assoc(cmap,c) of
   163                        Some(thm, T, n) =>
   164                          let val t2 = list_comb (h, take (n, ts)) in
   165                            mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts, t2),t2)
   166                          end
   167                      | None => [])
   168                 | _ => []
   169              in snd(foldl iter ((0,a),ts)) end
   170   in posns Ts [] [] t end;
   171 
   172 
   173 fun nth_subgoal i thm = nth_elem(i-1,prems_of thm);
   174 
   175 fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) =
   176   prod_ord (int_ord o pairself length) (order o pairself length)
   177     ((ps, pos), (qs, qos));
   178 
   179 
   180 
   181 (************************************************************
   182    call split_posns with appropriate parameters
   183 *************************************************************)
   184 
   185 fun select cmap state i =
   186   let val goali = nth_subgoal i state
   187       val Ts = rev(map #2 (Logic.strip_params goali))
   188       val _ $ t $ _ = Logic.strip_assums_concl goali;
   189   in (Ts,t, sort shorter (split_posns cmap Ts t)) end;
   190 
   191 
   192 (*************************************************************
   193    instantiate lift theorem
   194 
   195    if t is of the form
   196    ... ( Const(...,...) $ Abs( .... ) ) ...
   197    then
   198    P = %a.  ... ( Const(...,...) $ a ) ...
   199    where a has type T --> U
   200 
   201    Ts      : types of parameters
   202    t       : lefthand side of meta-equality in subgoal
   203              the split theorem is applied to (see cmap)
   204    T,U,pos : see mk_split_pack
   205    state   : current proof state
   206    lift    : the lift theorem
   207    i       : no. of subgoal
   208 **************************************************************)
   209 
   210 fun inst_lift Ts t (T,U,pos) state lift i =
   211   let val sg = #sign(rep_thm state)
   212       val tsig = #tsig(Sign.rep_sg sg)
   213       val cntxt = mk_cntxt Ts t pos (T-->U) (#maxidx(rep_thm lift))
   214       val cu = cterm_of sg cntxt
   215       val uT = #T(rep_cterm cu)
   216       val cP' = cterm_of sg (Var(P,uT))
   217       val ixnTs = Type.typ_match tsig ([],(PT,uT));
   218       val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
   219   in instantiate (ixncTs, [(cP',cu)]) lift end;
   220 
   221 
   222 (*************************************************************
   223    instantiate split theorem
   224 
   225    Ts    : types of parameters
   226    t     : lefthand side of meta-equality in subgoal
   227            the split theorem is applied to (see cmap)
   228    tt    : the term  Const(key,..) $ ...
   229    thm   : the split theorem
   230    TB    : type of body of P(...)
   231    state : current proof state
   232    i     : number of subgoal
   233 **************************************************************)
   234 
   235 fun inst_split Ts t tt thm TB state i =
   236   let val _ $ ((Var (P2, PT2)) $ _) $ _ = concl_of thm;
   237       val sg = #sign(rep_thm state)
   238       val tsig = #tsig(Sign.rep_sg sg)
   239       val cntxt = mk_cntxt_splitthm t tt TB;
   240       val T = fastype_of1 (Ts, cntxt);
   241       val ixnTs = Type.typ_match tsig ([],(PT2, T))
   242       val abss = foldl (fn (t, T) => Abs ("", T, t))
   243   in
   244     term_lift_inst_rule (state, i, ixnTs, [((P2, T), abss (cntxt, Ts))], thm)
   245   end;
   246 
   247 (*****************************************************************************
   248    The split-tactic
   249    
   250    splits : list of split-theorems to be tried
   251    i      : number of subgoal the tactic should be applied to
   252 *****************************************************************************)
   253 
   254 fun split_tac [] i = no_tac
   255   | split_tac splits i =
   256   let fun const(thm) =
   257             (case concl_of thm of _$(t as _$lhs)$_ =>
   258                (case strip_comb lhs of (Const(a,_),args) =>
   259                   (a,(thm,fastype_of t,length args))
   260                 | _ => split_format_err())
   261              | _ => split_format_err())
   262       val cmap = map const splits;
   263       fun lift_tac Ts t p st = (rtac (inst_lift Ts t p st trlift i) i) st
   264       fun lift_split_tac st = st |>
   265             let val (Ts,t,splits) = select cmap st i
   266             in case splits of
   267                  [] => no_tac
   268                | (thm,apsns,pos,TB,tt)::_ =>
   269                    (case apsns of
   270                       [] => (fn state => state |>
   271 			           compose_tac (false, inst_split Ts t tt thm TB state i, 0) i)
   272                     | p::_ => EVERY[lift_tac Ts t p,
   273                                     rtac reflexive_thm (i+1),
   274                                     lift_split_tac])
   275             end
   276   in COND (has_fewer_prems i) no_tac 
   277           (rtac iffD i THEN lift_split_tac)
   278   end;
   279 
   280 in split_tac end;
   281 
   282 (* FIXME: this junk is only FOL/HOL specific and should therefore not go here!*)
   283 (* split_thm_info is used in FOL/simpdata.ML and HOL/simpdata.ML *)
   284 fun split_thm_info thm =
   285   (case concl_of thm of
   286      Const("Trueprop",_) $ (Const("op =", _)$(Var _$t)$c) =>
   287         (case strip_comb t of
   288            (Const(a,_),_) => (a,case c of (Const("Not",_)$_)=> true |_=> false)
   289          | _              => split_format_err())
   290    | _ => split_format_err());
   291 
   292 fun mk_case_split_asm_tac split_tac 
   293 			  (disjE,conjE,exE,contrapos,contrapos2,notnotD) = 
   294 let
   295 
   296 (*****************************************************************************
   297    The split-tactic for premises
   298    
   299    splits : list of split-theorems to be tried
   300    i      : number of subgoal the tactic should be applied to
   301 *****************************************************************************)
   302 
   303 fun split_asm_tac []     = K no_tac
   304   | split_asm_tac splits = 
   305   let val cname_list = map (fst o split_thm_info) splits;
   306       fun is_case (a,_) = a mem cname_list;
   307       fun tac (t,i) = 
   308 	  let val n = find_index (exists_Const is_case) 
   309 				 (Logic.strip_assums_hyp t);
   310 	      fun first_prem_is_disj (Const ("==>", _) $ (Const ("Trueprop", _)
   311 				 $ (Const ("op |", _) $ _ $ _ )) $ _ ) = true
   312 	      |   first_prem_is_disj (Const("all",_)$Abs(_,_,t)) = 
   313 					first_prem_is_disj t
   314 	      |   first_prem_is_disj _ = false;
   315 	      fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
   316 				   (if first_prem_is_disj t
   317 				    then EVERY[etac disjE i, rotate_tac ~1 i,
   318 					       rotate_tac ~1  (i+1),
   319 					       flat_prems_tac (i+1)]
   320 				    else all_tac) 
   321 				   THEN REPEAT (eresolve_tac [conjE,exE] i)
   322 				   THEN REPEAT (dresolve_tac [notnotD]   i)) i;
   323 	  in if n<0 then no_tac else DETERM (EVERY'
   324 		[rotate_tac n, etac contrapos2,
   325 		 split_tac splits, 
   326 		 rotate_tac ~1, etac contrapos, rotate_tac ~1, 
   327 		 flat_prems_tac] i)
   328 	  end;
   329   in SUBGOAL tac
   330   end;
   331 
   332 in split_asm_tac end;
   333 
   334 
   335 in
   336 
   337 val split_thm_info = split_thm_info;
   338 
   339 fun mk_case_split_tac iffD = mk_case_split_tac_2 iffD int_ord;
   340 
   341 fun mk_case_split_inside_tac iffD = mk_case_split_tac_2 iffD (rev_order o int_ord);
   342 
   343 val mk_case_split_asm_tac = mk_case_split_asm_tac;
   344 
   345 end;