src/Provers/splitter.ML
 author oheimb Fri Nov 07 18:05:25 1997 +0100 (1997-11-07 ago) changeset 4189 b8c7a6bc6c16 parent 3918 94e0fdcb7b91 child 4202 96876d71eef5 permissions -rw-r--r--
```     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 mk_case_split_tac_2 iffD order =
```
```    22 let
```
```    23
```
```    24
```
```    25 (************************************************************
```
```    26    Create lift-theorem "trlift" :
```
```    27
```
```    28    [| !! x. Q(x)==R(x) ; P(R) == C |] ==> P(Q)==C
```
```    29
```
```    30 *************************************************************)
```
```    31
```
```    32 val lift =
```
```    33   let val ct = read_cterm (#sign(rep_thm iffD))
```
```    34            ("[| !!x::'b::logic. Q(x) == R(x) |] ==> \
```
```    35             \P(%x. Q(x)) == P(%x. R(x))::'a::logic",propT)
```
```    36   in prove_goalw_cterm [] ct
```
```    37      (fn [prem] => [rewtac prem, rtac reflexive_thm 1])
```
```    38   end;
```
```    39
```
```    40 val trlift = lift RS transitive_thm;
```
```    41 val _ \$ (Var(P,PT)\$_) \$ _ = concl_of trlift;
```
```    42
```
```    43
```
```    44 (************************************************************************
```
```    45    Set up term for instantiation of P in the lift-theorem
```
```    46
```
```    47    Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
```
```    48    t     : lefthand side of meta-equality in subgoal
```
```    49            the lift theorem is applied to (see select)
```
```    50    pos   : "path" leading to abstraction, coded as a list
```
```    51    T     : type of body of P(...)
```
```    52    maxi  : maximum index of Vars
```
```    53 *************************************************************************)
```
```    54
```
```    55 fun mk_cntxt Ts t pos T maxi =
```
```    56   let fun var (t,i) = Var(("X",i),type_of1(Ts,t));
```
```    57       fun down [] t i = Bound 0
```
```    58         | down (p::ps) t i =
```
```    59             let val (h,ts) = strip_comb t
```
```    60                 val v1 = ListPair.map var (take(p,ts), i upto (i+p-1))
```
```    61                 val u::us = drop(p,ts)
```
```    62                 val v2 = ListPair.map var (us, (i+p) upto (i+length(ts)-2))
```
```    63       in list_comb(h,v1@[down ps u (i+length ts)]@v2) end;
```
```    64   in Abs("", T, down (rev pos) t maxi) end;
```
```    65
```
```    66
```
```    67 (************************************************************************
```
```    68    Set up term for instantiation of P in the split-theorem
```
```    69    P(...) == rhs
```
```    70
```
```    71    Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
```
```    72    t     : lefthand side of meta-equality in subgoal
```
```    73            the split theorem is applied to (see select)
```
```    74    T     : type of body of P(...)
```
```    75    tt    : the term  Const(..,..) \$ ...
```
```    76    maxi  : maximum index of Vars
```
```    77
```
```    78    lev   : abstraction level
```
```    79 *************************************************************************)
```
```    80
```
```    81 fun mk_cntxt_splitthm Ts t tt T maxi =
```
```    82   let fun down lev (Abs(v,T2,t)) = Abs(v,T2,down (lev+1) t)
```
```    83         | down lev (Bound i) = if i >= lev
```
```    84                                then Var(("X",maxi+i-lev),nth_elem(i-lev,Ts))
```
```    85                                else Bound i
```
```    86         | down lev t =
```
```    87             let val (h,ts) = strip_comb t
```
```    88                 val h2 = (case h of Bound _ => down lev h | _ => h)
```
```    89             in if incr_bv(lev,0,tt)=t
```
```    90                then
```
```    91                  Bound (lev)
```
```    92                else
```
```    93                  list_comb(h2,map (down lev) ts)
```
```    94             end;
```
```    95   in Abs("",T,down 0 t) end;
```
```    96
```
```    97
```
```    98 (* add all loose bound variables in t to list is *)
```
```    99 fun add_lbnos(is,t) = add_loose_bnos(t,0,is);
```
```   100
```
```   101 (* check if the innermost quantifier that needs to be removed
```
```   102    has a body of type T; otherwise the expansion thm will fail later on
```
```   103 *)
```
```   104 fun type_test(T,lbnos,apsns) =
```
```   105   let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns)
```
```   106   in T=U end;
```
```   107
```
```   108 (*************************************************************************
```
```   109    Create a "split_pack".
```
```   110
```
```   111    thm   : the relevant split-theorem, i.e. P(...) == rhs , where P(...)
```
```   112            is of the form
```
```   113            P( Const(key,...) \$ t_1 \$ ... \$ t_n )      (e.g. key = "if")
```
```   114    T     : type of P(...)
```
```   115    n     : number of arguments expected by Const(key,...)
```
```   116    ts    : list of arguments actually found
```
```   117    apsns : list of tuples of the form (T,U,pos), one tuple for each
```
```   118            abstraction that is encountered on the way to the position where
```
```   119            Const(key, ...) \$ ...  occurs, where
```
```   120            T   : type of the variable bound by the abstraction
```
```   121            U   : type of the abstraction's body
```
```   122            pos : "path" leading to the body of the abstraction
```
```   123    pos   : "path" leading to the position where Const(key, ...) \$ ...  occurs.
```
```   124    TB    : type of  Const(key,...) \$ t_1 \$ ... \$ t_n
```
```   125    t     : the term Const(key,...) \$ t_1 \$ ... \$ t_n
```
```   126
```
```   127    A split pack is a tuple of the form
```
```   128    (thm, apsns, pos, TB)
```
```   129    Note : apsns is reversed, so that the outermost quantifier's position
```
```   130           comes first ! If the terms in ts don't contain variables bound
```
```   131           by other than meta-quantifiers, apsns is empty, because no further
```
```   132           lifting is required before applying the split-theorem.
```
```   133 ******************************************************************************)
```
```   134
```
```   135 fun mk_split_pack(thm,T,n,ts,apsns,pos,TB,t) =
```
```   136   if n > length ts then []
```
```   137   else let val lev = length apsns
```
```   138            val lbnos = foldl add_lbnos ([],take(n,ts))
```
```   139            val flbnos = filter (fn i => i < lev) lbnos
```
```   140            val tt = incr_bv(~lev,0,t)
```
```   141        in if null flbnos then [(thm,[],pos,TB,tt)]
```
```   142           else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)]
```
```   143                else []
```
```   144        end;
```
```   145
```
```   146
```
```   147 (****************************************************************************
```
```   148    Recursively scans term for occurences of Const(key,...) \$ ...
```
```   149    Returns a list of "split-packs" (one for each occurence of Const(key,...) )
```
```   150
```
```   151    cmap : association list of split-theorems that should be tried.
```
```   152           The elements have the format (key,(thm,T,n)) , where
```
```   153           key : the theorem's key constant ( Const(key,...) \$ ... )
```
```   154           thm : the theorem itself
```
```   155           T   : type of P( Const(key,...) \$ ... )
```
```   156           n   : number of arguments expected by Const(key,...)
```
```   157    Ts   : types of parameters
```
```   158    t    : the term to be scanned
```
```   159 ******************************************************************************)
```
```   160
```
```   161 fun split_posns cmap Ts t =
```
```   162   let fun posns Ts pos apsns (Abs(_,T,t)) =
```
```   163             let val U = fastype_of1(T::Ts,t)
```
```   164             in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end
```
```   165         | posns Ts pos apsns t =
```
```   166             let val (h,ts) = strip_comb t
```
```   167                 fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a);
```
```   168                 val a = case h of
```
```   169                   Const(c,_) =>
```
```   170                     (case assoc(cmap,c) of
```
```   171                        Some(thm,T,n) => mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts,t),t)
```
```   172                      | None => [])
```
```   173                 | _ => []
```
```   174              in snd(foldl iter ((0,a),ts)) end
```
```   175   in posns Ts [] [] t end;
```
```   176
```
```   177
```
```   178 fun nth_subgoal i thm = nth_elem(i-1,prems_of thm);
```
```   179
```
```   180 fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) =
```
```   181   let val ms = length ps and ns = length qs
```
```   182   in ms < ns orelse (ms = ns andalso order(length pos,length qos)) end;
```
```   183
```
```   184
```
```   185 (************************************************************
```
```   186    call split_posns with appropriate parameters
```
```   187 *************************************************************)
```
```   188
```
```   189 fun select cmap state i =
```
```   190   let val goali = nth_subgoal i state
```
```   191       val Ts = rev(map #2 (Logic.strip_params goali))
```
```   192       val _ \$ t \$ _ = Logic.strip_assums_concl goali;
```
```   193   in (Ts,t,sort shorter (split_posns cmap Ts t)) end;
```
```   194
```
```   195
```
```   196 (*************************************************************
```
```   197    instantiate lift theorem
```
```   198
```
```   199    if t is of the form
```
```   200    ... ( Const(...,...) \$ Abs( .... ) ) ...
```
```   201    then
```
```   202    P = %a.  ... ( Const(...,...) \$ a ) ...
```
```   203    where a has type T --> U
```
```   204
```
```   205    Ts      : types of parameters
```
```   206    t       : lefthand side of meta-equality in subgoal
```
```   207              the split theorem is applied to (see cmap)
```
```   208    T,U,pos : see mk_split_pack
```
```   209    state   : current proof state
```
```   210    lift    : the lift theorem
```
```   211    i       : no. of subgoal
```
```   212 **************************************************************)
```
```   213
```
```   214 fun inst_lift Ts t (T,U,pos) state lift i =
```
```   215   let val sg = #sign(rep_thm state)
```
```   216       val tsig = #tsig(Sign.rep_sg sg)
```
```   217       val cntxt = mk_cntxt Ts t pos (T-->U) (#maxidx(rep_thm lift))
```
```   218       val cu = cterm_of sg cntxt
```
```   219       val uT = #T(rep_cterm cu)
```
```   220       val cP' = cterm_of sg (Var(P,uT))
```
```   221       val ixnTs = Type.typ_match tsig ([],(PT,uT));
```
```   222       val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
```
```   223   in instantiate (ixncTs, [(cP',cu)]) lift end;
```
```   224
```
```   225
```
```   226 (*************************************************************
```
```   227    instantiate split theorem
```
```   228
```
```   229    Ts    : types of parameters
```
```   230    t     : lefthand side of meta-equality in subgoal
```
```   231            the split theorem is applied to (see cmap)
```
```   232    pos   : "path" to the body of P(...)
```
```   233    thm   : the split theorem
```
```   234    TB    : type of body of P(...)
```
```   235    state : current proof state
```
```   236 **************************************************************)
```
```   237
```
```   238 fun inst_split Ts t tt thm TB state =
```
```   239   let val _\$((Var(P2,PT2))\$_)\$_ = concl_of thm
```
```   240       val sg = #sign(rep_thm state)
```
```   241       val tsig = #tsig(Sign.rep_sg sg)
```
```   242       val cntxt = mk_cntxt_splitthm Ts t tt TB (#maxidx(rep_thm thm))
```
```   243       val cu = cterm_of sg cntxt
```
```   244       val uT = #T(rep_cterm cu)
```
```   245       val cP' = cterm_of sg (Var(P2,uT))
```
```   246       val ixnTs = Type.typ_match tsig ([],(PT2,uT));
```
```   247       val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
```
```   248   in instantiate (ixncTs, [(cP',cu)]) thm end;
```
```   249
```
```   250
```
```   251 (*****************************************************************************
```
```   252    The split-tactic
```
```   253
```
```   254    splits : list of split-theorems to be tried
```
```   255    i      : number of subgoal the tactic should be applied to
```
```   256 *****************************************************************************)
```
```   257
```
```   258 fun split_tac [] i = no_tac
```
```   259   | split_tac splits i =
```
```   260   let fun const(thm) =
```
```   261             (case concl_of thm of _\$(t as _\$lhs)\$_ =>
```
```   262                (case strip_comb lhs of (Const(a,_),args) =>
```
```   263                   (a,(thm,fastype_of t,length args))
```
```   264                 | _ => error("Wrong format for split rule"))
```
```   265              | _ => error("Wrong format for split rule"))
```
```   266       val cmap = map const splits;
```
```   267       fun lift_tac Ts t p st = (rtac (inst_lift Ts t p st trlift i) i) st
```
```   268       fun lift_split_tac st = st |>
```
```   269             let val (Ts,t,splits) = select cmap st i
```
```   270             in case splits of
```
```   271                  [] => no_tac
```
```   272                | (thm,apsns,pos,TB,tt)::_ =>
```
```   273                    (case apsns of
```
```   274                       [] => (fn state => state |>
```
```   275 			           rtac (inst_split Ts t tt thm TB state) i)
```
```   276                     | p::_ => EVERY[lift_tac Ts t p,
```
```   277                                     rtac reflexive_thm (i+1),
```
```   278                                     lift_split_tac])
```
```   279             end
```
```   280   in COND (has_fewer_prems i) no_tac
```
```   281           (rtac iffD i THEN lift_split_tac)
```
```   282   end;
```
```   283
```
```   284 in split_tac end;
```
```   285
```
```   286
```
```   287 fun mk_case_split_prem_tac split_tac disjE conjE exE contrapos
```
```   288 			   contrapos2 notnotD =
```
```   289 let
```
```   290
```
```   291 (*****************************************************************************
```
```   292    The split-tactic for premises
```
```   293
```
```   294    splits : list of split-theorems to be tried
```
```   295    i      : number of subgoal the tactic should be applied to
```
```   296 *****************************************************************************)
```
```   297
```
```   298 fun split_prem_tac []     = K no_tac
```
```   299   | split_prem_tac splits =
```
```   300   let fun const thm =
```
```   301             (case concl_of thm of Const ("Trueprop",_)\$
```
```   302 				 (Const ("op =", _)\$(Var _\$t)\$_) =>
```
```   303                (case strip_comb t of (Const(a,_),_) => a
```
```   304                 | _ => error("Wrong format for split rule"))
```
```   305              | _ =>    error("Wrong format for split rule"))
```
```   306       val cname_list = map const splits;
```
```   307       fun is_case (a,_) = a mem cname_list;
```
```   308       fun tac (t,i) =
```
```   309 	  let val n = find_index (exists_Const is_case)
```
```   310 				 (Logic.strip_assums_hyp t);
```
```   311 	      fun first_prem_is_disj (Const ("==>", _) \$ (Const ("Trueprop", _)
```
```   312 				 \$ (Const ("op |", _) \$ _ \$ _ )) \$ _ ) = true
```
```   313 	      |   first_prem_is_disj _ = false;
```
```   314 	      fun flat_prems_tac j = SUBGOAL (fn (t,i) =>
```
```   315 				   (if first_prem_is_disj t
```
```   316 				    then EVERY[etac disjE i, rotate_tac ~1 i,
```
```   317 					       rotate_tac ~1  (i+1),
```
```   318 					       flat_prems_tac (i+1)]
```
```   319 				    else all_tac)
```
```   320 				   THEN REPEAT (eresolve_tac [conjE,exE] i)
```
```   321 				   THEN REPEAT (dresolve_tac [notnotD]   i)) j;
```
```   322 	  in if n<0 then no_tac else DETERM (EVERY'
```
```   323 		[rotate_tac n, etac contrapos2,
```
```   324 		 split_tac splits,
```
```   325 		 rotate_tac ~1, etac contrapos, rotate_tac ~1,
```
```   326 		 SELECT_GOAL (flat_prems_tac 1)] i)
```
```   327 	  end;
```
```   328   in SUBGOAL tac
```
```   329   end;
```
```   330
```
```   331 in split_prem_tac end;
```
```   332
```
```   333
```
```   334 in
```
```   335
```
```   336 fun mk_case_split_tac iffD = mk_case_split_tac_2 iffD (op <=) ;
```
```   337
```
```   338 fun mk_case_split_inside_tac iffD = mk_case_split_tac_2 iffD (op >=) ;
```
```   339
```
```   340 val mk_case_split_prem_tac = mk_case_split_prem_tac;
```
```   341
```
```   342 end;
```