author  paulson 
Thu, 28 Nov 1996 10:44:24 +0100  
changeset 2266  82aef6857c5b 
parent 2143  093bbe6d333b 
child 3537  79ac9b475621 
permissions  rwrr 
4  1 
(* Title: Provers/splitter 
2 
ID: $Id$ 

3 
Author: Tobias Nipkow 

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4 
Copyright 1995 TU Munich 
4  5 

6 
Generic casesplitter, suitable for most logics. 

7 

0  8 
Use: 
9 

10 
val split_tac = mk_case_split_tac iffD; 

11 

12 
by(case_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 

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local 
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fun mk_case_split_tac_2 iffD order = 
0  22 
let 
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(************************************************************ 
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Create lifttheorem "trlift" : 
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[ !! x. Q(x)==R(x) ; P(R) == C ] ==> P(Q)==C 
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*************************************************************) 
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943  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; 

4  39 

0  40 
val trlift = lift RS transitive_thm; 
41 
val _ $ (Var(P,PT)$_) $ _ = concl_of trlift; 

42 

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(************************************************************************ 
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Set up term for instantiation of P in the lifttheorem 
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Ts : types of parameters (i.e. variables bound by metaquantifiers) 
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t : lefthand side of metaequality in subgoal 
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the lift theorem is applied to (see select) 
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pos : "path" leading to abstraction, coded as a list 
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T : type of body of P(...) 
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maxi : maximum index of Vars 
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*************************************************************************) 
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fun mk_cntxt Ts t pos T maxi = 
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let fun var (t,i) = Var(("X",i),type_of1(Ts,t)); 
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fun down [] t i = Bound 0 
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 down (p::ps) t i = 
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let val (h,ts) = strip_comb t 
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val v1 = ListPair.map var (take(p,ts), i upto (i+p1)) 
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val u::us = drop(p,ts) 
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val v2 = ListPair.map var (us, (i+p) upto (i+length(ts)2)) 
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in list_comb(h,v1@[down ps u (i+length ts)]@v2) end; 
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in Abs("", T, down (rev pos) t maxi) end; 
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(************************************************************************ 
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Set up term for instantiation of P in the splittheorem 
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P(...) == rhs 
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Ts : types of parameters (i.e. variables bound by metaquantifiers) 
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t : lefthand side of metaequality in subgoal 
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the split theorem is applied to (see select) 
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T : type of body of P(...) 
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tt : the term Const(..,..) $ ... 
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maxi : maximum index of Vars 
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lev : abstraction level 
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*************************************************************************) 
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fun mk_cntxt_splitthm Ts t tt T maxi = 
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let fun down lev (Abs(v,T2,t)) = Abs(v,T2,down (lev+1) t) 
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 down lev (Bound i) = if i >= lev 
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then Var(("X",maxi+ilev),nth_elem(ilev,Ts)) 
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else Bound i 
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 down lev t = 
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let val (h,ts) = strip_comb t 
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val h2 = (case h of Bound _ => down lev h  _ => h) 
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in if incr_bv(lev,0,tt)=t 
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then 
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Bound (lev) 
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else 
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list_comb(h2,map (down lev) ts) 
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end; 
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in Abs("",T,down 0 t) end; 
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(* add all loose bound variables in t to list is *) 
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fun add_lbnos(is,t) = add_loose_bnos(t,0,is); 
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1064  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) = 

2143  105 
let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns) 
1064  106 
in T=U end; 
0  107 

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(************************************************************************* 
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Create a "split_pack". 
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thm : the relevant splittheorem, i.e. P(...) == rhs , where P(...) 
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is of the form 
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P( Const(key,...) $ t_1 $ ... $ t_n ) (e.g. key = "if") 
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T : type of P(...) 
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n : number of arguments expected by Const(key,...) 
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ts : list of arguments actually found 
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apsns : list of tuples of the form (T,U,pos), one tuple for each 
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abstraction that is encountered on the way to the position where 
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Const(key, ...) $ ... occurs, where 
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T : type of the variable bound by the abstraction 
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U : type of the abstraction's body 
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pos : "path" leading to the body of the abstraction 
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pos : "path" leading to the position where Const(key, ...) $ ... occurs. 
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TB : type of Const(key,...) $ t_1 $ ... $ t_n 
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t : the term Const(key,...) $ t_1 $ ... $ t_n 
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A split pack is a tuple of the form 
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(thm, apsns, pos, TB) 
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Note : apsns is reversed, so that the outermost quantifier's position 
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comes first ! If the terms in ts don't contain variables bound 
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by other than metaquantifiers, apsns is empty, because no further 
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lifting is required before applying the splittheorem. 
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******************************************************************************) 
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fun mk_split_pack(thm,T,n,ts,apsns,pos,TB,t) = 
1064  136 
if n > length ts then [] 
137 
else let val lev = length apsns 

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val lbnos = foldl add_lbnos ([],take(n,ts)) 
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val flbnos = filter (fn i => i < lev) lbnos 
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val tt = incr_bv(~lev,0,t) 
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in if null flbnos then [(thm,[],pos,TB,tt)] 
2143  142 
else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)] 
143 
else [] 

1064  144 
end; 
0  145 

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(**************************************************************************** 
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Recursively scans term for occurences of Const(key,...) $ ... 
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Returns a list of "splitpacks" (one for each occurence of Const(key,...) ) 
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cmap : association list of splittheorems that should be tried. 
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The elements have the format (key,(thm,T,n)) , where 
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key : the theorem's key constant ( Const(key,...) $ ... ) 
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thm : the theorem itself 
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T : type of P( Const(key,...) $ ... ) 
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n : number of arguments expected by Const(key,...) 
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Ts : types of parameters 
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t : the term to be scanned 
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******************************************************************************) 
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fun split_posns cmap Ts t = 
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let fun posns Ts pos apsns (Abs(_,T,t)) = 
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let val U = fastype_of1(T::Ts,t) 
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in posns (T::Ts) (0::pos) ((T,U,pos)::apsns) t end 
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 posns Ts pos apsns t = 
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let val (h,ts) = strip_comb t 
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fun iter((i,a),t) = (i+1, (posns Ts (i::pos) apsns t) @ a); 
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val a = case h of 
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Const(c,_) => 
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(case assoc(cmap,c) of 
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Some(thm,T,n) => mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts,t),t) 
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 None => []) 
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 _ => [] 
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in snd(foldl iter ((0,a),ts)) end 
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in posns Ts [] [] t end; 
0  176 

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0  178 
fun nth_subgoal i thm = nth_elem(i1,prems_of thm); 
179 

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fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) = 
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let val ms = length ps and ns = length qs 
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in ms < ns orelse (ms = ns andalso order(length pos,length qos)) end; 
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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  188 

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fun select cmap state i = 
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let val goali = nth_subgoal i state 
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val Ts = rev(map #2 (Logic.strip_params goali)) 
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val _ $ t $ _ = Logic.strip_assums_concl goali; 
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in (Ts,t,sort shorter (split_posns cmap Ts t)) end; 
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(************************************************************* 
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instantiate lift theorem 
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198 

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if t is of the form 
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... ( Const(...,...) $ Abs( .... ) ) ... 
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then 
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P = %a. ... ( Const(...,...) $ a ) ... 
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where a has type T > U 
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Ts : types of parameters 
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t : lefthand side of metaequality in subgoal 
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the split theorem is applied to (see cmap) 
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T,U,pos : see mk_split_pack 
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state : current proof state 
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lift : the lift theorem 
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i : no. of subgoal 
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**************************************************************) 
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213 

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fun inst_lift Ts t (T,U,pos) state lift i = 
0  215 
let val sg = #sign(rep_thm state) 
216 
val tsig = #tsig(Sign.rep_sg sg) 

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val cntxt = mk_cntxt Ts t pos (T>U) (#maxidx(rep_thm lift)) 
231  218 
val cu = cterm_of sg cntxt 
219 
val uT = #T(rep_cterm cu) 

220 
val cP' = cterm_of sg (Var(P,uT)) 

0  221 
val ixnTs = Type.typ_match tsig ([],(PT,uT)); 
231  222 
val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs; 
0  223 
in instantiate (ixncTs, [(cP',cu)]) lift end; 
224 

225 

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(************************************************************* 
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instantiate split theorem 
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228 

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Ts : types of parameters 
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t : lefthand side of metaequality in subgoal 
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the split theorem is applied to (see cmap) 
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pos : "path" to the body of P(...) 
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thm : the split theorem 
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TB : type of body of P(...) 
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state : current proof state 
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**************************************************************) 
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237 

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fun inst_split Ts t tt thm TB state = 
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let val _$((Var(P2,PT2))$_)$_ = concl_of thm 
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val sg = #sign(rep_thm state) 
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val tsig = #tsig(Sign.rep_sg sg) 
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val cntxt = mk_cntxt_splitthm Ts t tt TB (#maxidx(rep_thm thm)) 
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val cu = cterm_of sg cntxt 
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val uT = #T(rep_cterm cu) 
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val cP' = cterm_of sg (Var(P2,uT)) 
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val ixnTs = Type.typ_match tsig ([],(PT2,uT)); 
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val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs; 
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in instantiate (ixncTs, [(cP',cu)]) thm end; 
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250 

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251 
(***************************************************************************** 
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252 
The splittactic 
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253 

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254 
splits : list of splittheorems to be tried 
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255 
i : number of subgoal the tactic should be applied to 
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256 
*****************************************************************************) 
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257 

0  258 
fun split_tac [] i = no_tac 
259 
 split_tac splits i = 

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let fun const(thm) = let val _$(t as _$lhs)$_ = concl_of thm 
0  261 
val (Const(a,_),args) = strip_comb lhs 
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262 
in (a,(thm,fastype_of t,length args)) end 
0  263 
val cmap = map const splits; 
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264 
fun lift Ts t p state = rtac (inst_lift Ts t p state trlift i) i 
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265 
fun lift_split state = 
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let val (Ts,t,splits) = select cmap state i 
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267 
in case splits of 
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[] => no_tac 
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 (thm,apsns,pos,TB,tt)::_ => 
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(case apsns of 
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[] => STATE(fn state => rtac (inst_split Ts t tt thm TB state) i) 
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 p::_ => EVERY[STATE(lift Ts t p), 
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273 
rtac reflexive_thm (i+1), 
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274 
STATE lift_split]) 
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275 
end 
0  276 
in STATE(fn thm => 
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if i <= nprems_of thm then rtac iffD i THEN STATE lift_split 
0  278 
else no_tac) 
279 
end; 

280 

281 
in split_tac end; 

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282 

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283 
in 
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284 

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285 
fun mk_case_split_tac iffD = mk_case_split_tac_2 iffD (op <=) ; 
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286 

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287 
fun mk_case_split_inside_tac iffD = mk_case_split_tac_2 iffD (op >=) ; 
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288 

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289 
end; 