author  oheimb 
Wed, 12 Aug 1998 16:21:18 +0200  
changeset 5304  c133f16febc7 
parent 4930  89271bc4e7ed 
child 5437  f68b9d225942 
permissions  rwrr 
4  1 
(* Title: Provers/splitter 
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ID: $Id$ 

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Author: Tobias Nipkow 

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Copyright 1995 TU Munich 
4  5 

6 
Generic casesplitter, suitable for most logics. 

0  7 
*) 
8 

5304  9 
infix 4 addsplits delsplits; 
10 

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signature SPLITTER_DATA = 

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sig 

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structure Simplifier: SIMPLIFIER 

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val mk_meta_eq : thm > thm 

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val meta_eq_to_iff: thm (* "x == y ==> x = y" *) 

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val iffD : thm (* "[ P = Q; Q ] ==> P" *) 

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val disjE : thm (* "[ P  Q; P ==> R; Q ==> R ] ==> R" *) 

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val conjE : thm (* "[ P & Q; [ P; Q ] ==> R ] ==> R" *) 

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val exE : thm (* "[ x. P x; !!x. P x ==> Q ] ==> Q" *) 

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val contrapos : thm (* "[ ~ Q; P ==> Q ] ==> ~ P" *) 

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val contrapos2 : thm (* "[ Q; ~ P ==> ~ Q ] ==> P" *) 

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val notnotD : thm (* "~ ~ P ==> P" *) 

23 
end 

24 

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signature SPLITTER = 

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sig 

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type simpset 

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val split_tac : thm list > int > tactic 

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val split_inside_tac: thm list > int > tactic 

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val split_asm_tac : thm list > int > tactic 

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val addsplits : simpset * thm list > simpset 

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val delsplits : simpset * thm list > simpset 

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val Addsplits : thm list > unit 

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val Delsplits : thm list > unit 

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end; 

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functor SplitterFun(Data: SPLITTER_DATA): SPLITTER = 

38 
struct 

39 

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type simpset = Data.Simplifier.simpset; 

41 

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val Const ("==>", _) $ (Const ("Trueprop", _) $ 

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(Const (const_not, _) $ _ )) $ _ = #prop (rep_thm(Data.notnotD)); 

44 

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val Const ("==>", _) $ (Const ("Trueprop", _) $ 

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(Const (const_or , _) $ _ $ _)) $ _ = #prop (rep_thm(Data.disjE)); 

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fun split_format_err() = error("Wrong format for split rule"); 
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fun split_thm_info thm = case concl_of (Data.mk_meta_eq thm) of 
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Const("==", _)$(Var _$t)$c => 

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(case strip_comb t of 

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(Const(a,_),_) => (a,case c of (Const(s,_)$_)=>s=const_not_=> false) 

54 
 _ => split_format_err()) 

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 _ => split_format_err(); 

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fun mk_case_split_tac order = 

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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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val meta_iffD = Data.meta_eq_to_iff RS Data.iffD; 

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val lift = 
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let val ct = read_cterm (#sign(rep_thm Data.iffD)) 
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("[ !!x::'b::logic. Q(x) == R(x) ] ==> \ 
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\P(%x. Q(x)) == P(%x. R(x))::'a::logic",propT) 
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in prove_goalw_cterm [] ct 
74 
(fn [prem] => [rewtac prem, rtac reflexive_thm 1]) 

75 
end; 

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0  77 
val trlift = lift RS transitive_thm; 
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val _ $ (Var(P,PT)$_) $ _ = concl_of trlift; 

79 

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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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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(key,..) $ ... 
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*************************************************************************) 
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fun mk_cntxt_splitthm t tt T = 
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let fun repl lev t = 

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if incr_boundvars lev tt = t then Bound lev 

117 
else case t of 

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(Abs (v, T2, t)) => Abs (v, T2, repl (lev+1) t) 

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 (Bound i) => Bound (if i>=lev then i+1 else i) 

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 (t1 $ t2) => (repl lev t1) $ (repl lev t2) 

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 t => t 

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in Abs("", T, repl 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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(* check if the innermost quantifier that needs to be removed 
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has a body of type T; otherwise the expansion thm will fail later on 

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*) 

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fun type_test(T,lbnos,apsns) = 

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let val (_,U,_) = nth_elem(foldl Int.min (hd lbnos, tl lbnos), apsns) 
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in T=U end; 
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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  163 
if n > length ts then [] 
164 
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_boundvars (~lev) t 
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in if null flbnos then [(thm,[],pos,TB,tt)] 
2143  169 
else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)] 
170 
else [] 

1064  171 
end; 
0  172 

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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 
4232  198 
Some(thm, T, n) => 
199 
let val t2 = list_comb (h, take (n, ts)) in 

200 
mk_split_pack(thm,T,n,ts,apsns,pos,type_of1(Ts, t2),t2) 

201 
end 

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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  206 

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0  208 
fun nth_subgoal i thm = nth_elem(i1,prems_of thm); 
209 

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fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) = 
4519  211 
prod_ord (int_ord o pairself length) (order o pairself length) 
212 
((ps, pos), (qs, qos)); 

213 

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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  219 

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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; 
4519  224 
in (Ts,t, sort shorter (split_posns cmap Ts t)) end; 
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225 

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(************************************************************* 
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instantiate lift theorem 
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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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fun inst_lift Ts t (T,U,pos) state lift i = 
0  246 
let val sg = #sign(rep_thm state) 
247 
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  249 
val cu = cterm_of sg cntxt 
250 
val uT = #T(rep_cterm cu) 

251 
val cP' = cterm_of sg (Var(P,uT)) 

0  252 
val ixnTs = Type.typ_match tsig ([],(PT,uT)); 
231  253 
val ixncTs = map (fn (x,y) => (x,ctyp_of sg y)) ixnTs; 
0  254 
in instantiate (ixncTs, [(cP',cu)]) lift end; 
255 

256 

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(************************************************************* 
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instantiate split theorem 
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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) 
4232  263 
tt : the term Const(key,..) $ ... 
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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 
4232  267 
i : number of subgoal 
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**************************************************************) 
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4232  270 
fun inst_split Ts t tt thm TB state i = 
271 
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) 
4232  274 
val cntxt = mk_cntxt_splitthm t tt TB; 
4236  275 
val T = fastype_of1 (Ts, cntxt); 
4232  276 
val ixnTs = Type.typ_match tsig ([],(PT2, T)) 
277 
val abss = foldl (fn (t, T) => Abs ("", T, t)) 

278 
in 

279 
term_lift_inst_rule (state, i, ixnTs, [((P2, T), abss (cntxt, Ts))], thm) 

280 
end; 

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(***************************************************************************** 
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The splittactic 
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splits : list of splittheorems to be tried 
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i : number of subgoal the tactic should be applied to 
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*****************************************************************************) 
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0  289 
fun split_tac [] i = no_tac 
290 
 split_tac splits i = 

5304  291 
let val splits = map Data.mk_meta_eq splits; 
292 
fun const(thm) = 

3918  293 
(case concl_of thm of _$(t as _$lhs)$_ => 
294 
(case strip_comb lhs of (Const(a,_),args) => 

295 
(a,(thm,fastype_of t,length args)) 

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 _ => split_format_err()) 
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 _ => split_format_err()) 
0  298 
val cmap = map const splits; 
3537  299 
fun lift_tac Ts t p st = (rtac (inst_lift Ts t p st trlift i) i) st 
300 
fun lift_split_tac st = st > 

301 
let val (Ts,t,splits) = select cmap st i 

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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 
3537  306 
[] => (fn state => state > 
4232  307 
compose_tac (false, inst_split Ts t tt thm TB state i, 0) i) 
3537  308 
 p::_ => EVERY[lift_tac Ts t p, 
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rtac reflexive_thm (i+1), 
3537  310 
lift_split_tac]) 
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end 
3537  312 
in COND (has_fewer_prems i) no_tac 
5304  313 
(rtac meta_iffD i THEN lift_split_tac) 
0  314 
end; 
315 

316 
in split_tac end; 

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5304  318 

319 
val split_tac = mk_case_split_tac int_ord; 

4189  320 

5304  321 
val split_inside_tac = mk_case_split_tac (rev_order o int_ord); 
322 

4189  323 

324 
(***************************************************************************** 

325 
The splittactic for premises 

326 

327 
splits : list of splittheorems to be tried 

5304  328 
****************************************************************************) 
4202  329 
fun split_asm_tac [] = K no_tac 
330 
 split_asm_tac splits = 

5304  331 

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let val cname_list = map (fst o split_thm_info) splits; 
4189  333 
fun is_case (a,_) = a mem cname_list; 
334 
fun tac (t,i) = 

335 
let val n = find_index (exists_Const is_case) 

336 
(Logic.strip_assums_hyp t); 

337 
fun first_prem_is_disj (Const ("==>", _) $ (Const ("Trueprop", _) 

5304  338 
$ (Const (s, _) $ _ $ _ )) $ _ ) = (s=const_or) 
4202  339 
 first_prem_is_disj (Const("all",_)$Abs(_,_,t)) = 
340 
first_prem_is_disj t 

4189  341 
 first_prem_is_disj _ = false; 
4202  342 
fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
5304  343 
(if first_prem_is_disj t 
344 
then EVERY[etac Data.disjE i,rotate_tac ~1 i, 

345 
rotate_tac ~1 (i+1), 

346 
flat_prems_tac (i+1)] 

347 
else all_tac) 

348 
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i) 

349 
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i; 

4189  350 
in if n<0 then no_tac else DETERM (EVERY' 
5304  351 
[rotate_tac n, etac Data.contrapos2, 
4189  352 
split_tac splits, 
5304  353 
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1, 
4202  354 
flat_prems_tac] i) 
4189  355 
end; 
356 
in SUBGOAL tac 

357 
end; 

358 

5304  359 
fun split_name name asm = "split " ^ name ^ (if asm then " asm" else ""); 
4189  360 

5304  361 
fun ss addsplits splits = 
362 
let fun addsplit (ss,split) = 

363 
let val (name,asm) = split_thm_info split 

364 
in Data.Simplifier.addloop(ss,(split_name name asm, 

365 
(if asm then split_asm_tac else split_tac) [split])) end 

366 
in foldl addsplit (ss,splits) end; 

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5304  368 
fun ss delsplits splits = 
369 
let fun delsplit(ss,split) = 

370 
let val (name,asm) = split_thm_info split 

371 
in Data.Simplifier.delloop(ss,split_name name asm) 

372 
end in foldl delsplit (ss,splits) end; 

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373 

5304  374 
fun Addsplits splits = (Data.Simplifier.simpset_ref() := 
375 
Data.Simplifier.simpset() addsplits splits); 

376 
fun Delsplits splits = (Data.Simplifier.simpset_ref() := 

377 
Data.Simplifier.simpset() delsplits splits); 

4189  378 

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379 
end; 