src/Provers/splitter.ML
 author wenzelm Wed, 15 Oct 1997 15:12:59 +0200 changeset 3872 a5839ecee7b8 parent 3835 9a5a4e123859 child 3918 94e0fdcb7b91 permissions -rw-r--r--
tuned; prepare ext;
```
(*  Title:      Provers/splitter
ID:         \$Id\$
Author:     Tobias Nipkow

Generic case-splitter, suitable for most logics.

Use:

val split_tac = mk_case_split_tac iffD;

by(case_split_tac splits i);

where splits = [P(elim(...)) == rhs, ...]
iffD  = [| P <-> Q; Q |] ==> P (* is called iffD2 in HOL *)

*)

local

fun mk_case_split_tac_2 iffD order =
let

(************************************************************
Create lift-theorem "trlift" :

[| !! x. Q(x)==R(x) ; P(R) == C |] ==> P(Q)==C

*************************************************************)

val lift =
let val ct = read_cterm (#sign(rep_thm iffD))
("[| !!x::'b::logic. Q(x) == R(x) |] ==> \
\P(%x. Q(x)) == P(%x. R(x))::'a::logic",propT)
in prove_goalw_cterm [] ct
(fn [prem] => [rewtac prem, rtac reflexive_thm 1])
end;

val trlift = lift RS transitive_thm;
val _ \$ (Var(P,PT)\$_) \$ _ = concl_of trlift;

(************************************************************************
Set up term for instantiation of P in the lift-theorem

Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
t     : lefthand side of meta-equality in subgoal
the lift theorem is applied to (see select)
pos   : "path" leading to abstraction, coded as a list
T     : type of body of P(...)
maxi  : maximum index of Vars
*************************************************************************)

fun mk_cntxt Ts t pos T maxi =
let fun var (t,i) = Var(("X",i),type_of1(Ts,t));
fun down [] t i = Bound 0
| down (p::ps) t i =
let val (h,ts) = strip_comb t
val v1 = ListPair.map var (take(p,ts), i upto (i+p-1))
val u::us = drop(p,ts)
val v2 = ListPair.map var (us, (i+p) upto (i+length(ts)-2))
in list_comb(h,v1@[down ps u (i+length ts)]@v2) end;
in Abs("", T, down (rev pos) t maxi) end;

(************************************************************************
Set up term for instantiation of P in the split-theorem
P(...) == rhs

Ts    : types of parameters (i.e. variables bound by meta-quantifiers)
t     : lefthand side of meta-equality in subgoal
the split theorem is applied to (see select)
T     : type of body of P(...)
tt    : the term  Const(..,..) \$ ...
maxi  : maximum index of Vars

lev   : abstraction level
*************************************************************************)

fun mk_cntxt_splitthm Ts t tt T maxi =
let fun down lev (Abs(v,T2,t)) = Abs(v,T2,down (lev+1) t)
| down lev (Bound i) = if i >= lev
then Var(("X",maxi+i-lev),nth_elem(i-lev,Ts))
else Bound i
| down lev t =
let val (h,ts) = strip_comb t
val h2 = (case h of Bound _ => down lev h | _ => h)
in if incr_bv(lev,0,tt)=t
then
Bound (lev)
else
list_comb(h2,map (down lev) ts)
end;
in Abs("",T,down 0 t) end;

(* add all loose bound variables in t to list is *)

(* check if the innermost quantifier that needs to be removed
has a body of type T; otherwise the expansion thm will fail later on
*)
fun type_test(T,lbnos,apsns) =
let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns)
in T=U end;

(*************************************************************************
Create a "split_pack".

thm   : the relevant split-theorem, i.e. P(...) == rhs , where P(...)
is of the form
P( Const(key,...) \$ t_1 \$ ... \$ t_n )      (e.g. key = "if")
T     : type of P(...)
n     : number of arguments expected by Const(key,...)
ts    : list of arguments actually found
apsns : list of tuples of the form (T,U,pos), one tuple for each
abstraction that is encountered on the way to the position where
Const(key, ...) \$ ...  occurs, where
T   : type of the variable bound by the abstraction
U   : type of the abstraction's body
pos : "path" leading to the body of the abstraction
pos   : "path" leading to the position where Const(key, ...) \$ ...  occurs.
TB    : type of  Const(key,...) \$ t_1 \$ ... \$ t_n
t     : the term Const(key,...) \$ t_1 \$ ... \$ t_n

A split pack is a tuple of the form
(thm, apsns, pos, TB)
Note : apsns is reversed, so that the outermost quantifier's position
comes first ! If the terms in ts don't contain variables bound
by other than meta-quantifiers, apsns is empty, because no further
lifting is required before applying the split-theorem.
******************************************************************************)

fun mk_split_pack(thm,T,n,ts,apsns,pos,TB,t) =
if n > length ts then []
else let val lev = length apsns
val lbnos = foldl add_lbnos ([],take(n,ts))
val flbnos = filter (fn i => i < lev) lbnos
val tt = incr_bv(~lev,0,t)
in if null flbnos then [(thm,[],pos,TB,tt)]
else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)]
else []
end;

(****************************************************************************
Recursively scans term for occurences of Const(key,...) \$ ...
Returns a list of "split-packs" (one for each occurence of Const(key,...) )

cmap : association list of split-theorems that should be tried.
The elements have the format (key,(thm,T,n)) , where
key : the theorem's key constant ( Const(key,...) \$ ... )
thm : the theorem itself
T   : type of P( Const(key,...) \$ ... )
n   : number of arguments expected by Const(key,...)
Ts   : types of parameters
t    : the term to be scanned
******************************************************************************)

fun split_posns cmap Ts t =
let fun posns Ts pos apsns (Abs(_,T,t)) =
let val U = fastype_of1(T::Ts,t)
in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end
| posns Ts pos apsns t =
let val (h,ts) = strip_comb t
fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a);
val a = case h of
Const(c,_) =>
(case assoc(cmap,c) of
Some(thm,T,n) => mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts,t),t)
| None => [])
| _ => []
in snd(foldl iter ((0,a),ts)) end
in posns Ts [] [] t end;

fun nth_subgoal i thm = nth_elem(i-1,prems_of thm);

fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) =
let val ms = length ps and ns = length qs
in ms < ns orelse (ms = ns andalso order(length pos,length qos)) end;

(************************************************************
call split_posns with appropriate parameters
*************************************************************)

fun select cmap state i =
let val goali = nth_subgoal i state
val Ts = rev(map #2 (Logic.strip_params goali))
val _ \$ t \$ _ = Logic.strip_assums_concl goali;
in (Ts,t,sort shorter (split_posns cmap Ts t)) end;

(*************************************************************
instantiate lift theorem

if t is of the form
... ( Const(...,...) \$ Abs( .... ) ) ...
then
P = %a.  ... ( Const(...,...) \$ a ) ...
where a has type T --> U

Ts      : types of parameters
t       : lefthand side of meta-equality in subgoal
the split theorem is applied to (see cmap)
T,U,pos : see mk_split_pack
state   : current proof state
lift    : the lift theorem
i       : no. of subgoal
**************************************************************)

fun inst_lift Ts t (T,U,pos) state lift i =
let val sg = #sign(rep_thm state)
val tsig = #tsig(Sign.rep_sg sg)
val cntxt = mk_cntxt Ts t pos (T-->U) (#maxidx(rep_thm lift))
val cu = cterm_of sg cntxt
val uT = #T(rep_cterm cu)
val cP' = cterm_of sg (Var(P,uT))
val ixnTs = Type.typ_match tsig ([],(PT,uT));
val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
in instantiate (ixncTs, [(cP',cu)]) lift end;

(*************************************************************
instantiate split theorem

Ts    : types of parameters
t     : lefthand side of meta-equality in subgoal
the split theorem is applied to (see cmap)
pos   : "path" to the body of P(...)
thm   : the split theorem
TB    : type of body of P(...)
state : current proof state
**************************************************************)

fun inst_split Ts t tt thm TB state =
let val _\$((Var(P2,PT2))\$_)\$_ = concl_of thm
val sg = #sign(rep_thm state)
val tsig = #tsig(Sign.rep_sg sg)
val cntxt = mk_cntxt_splitthm Ts t tt TB (#maxidx(rep_thm thm))
val cu = cterm_of sg cntxt
val uT = #T(rep_cterm cu)
val cP' = cterm_of sg (Var(P2,uT))
val ixnTs = Type.typ_match tsig ([],(PT2,uT));
val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs;
in instantiate (ixncTs, [(cP',cu)]) thm end;

(*****************************************************************************
The split-tactic

splits : list of split-theorems to be tried
i      : number of subgoal the tactic should be applied to
*****************************************************************************)

fun split_tac [] i = no_tac
| split_tac splits i =
let fun const(thm) = let val _\$(t as _\$lhs)\$_ = concl_of thm
val (Const(a,_),args) = strip_comb lhs
in (a,(thm,fastype_of t,length args)) end
val cmap = map const splits;
fun lift_tac Ts t p st = (rtac (inst_lift Ts t p st trlift i) i) st
fun lift_split_tac st = st |>
let val (Ts,t,splits) = select cmap st i
in case splits of
[] => no_tac
| (thm,apsns,pos,TB,tt)::_ =>
(case apsns of
[] => (fn state => state |>
rtac (inst_split Ts t tt thm TB state) i)
| p::_ => EVERY[lift_tac Ts t p,
rtac reflexive_thm (i+1),
lift_split_tac])
end
in COND (has_fewer_prems i) no_tac
(rtac iffD i THEN lift_split_tac)
end;

in split_tac end;

in

fun mk_case_split_tac iffD = mk_case_split_tac_2 iffD (op <=) ;

fun mk_case_split_inside_tac iffD = mk_case_split_tac_2 iffD (op >=) ;

end;
```