author  berghofe 
Fri, 01 Oct 1999 10:23:13 +0200  
changeset 7672  c092e67d12f8 
parent 6130  30b84ad2131d 
child 8468  d99902232df8 
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 

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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 

5553  14 
val mk_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" *) 

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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 = 

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struct 

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type simpset = Data.Simplifier.simpset; 

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

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

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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_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) 

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 _ => 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(%x. R x) == C ] ==> P (%x. Q x) == 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 
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(fn [prem] => [rewtac prem, rtac reflexive_thm 1]) 

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end; 

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val trlift = lift RS transitive_thm; 
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val _ $ (P $ _) $ _ = concl_of trlift; 
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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 aconv t then Bound lev 
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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 abstraction 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; 
0  134 

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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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T' : type of term to be scanned 
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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, tt) 
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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, T', n, ts, apsns, pos, TB, t) = 
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if n > length ts then [] 
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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 
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if T = T' then [(thm,[],pos,TB,tt)] else [] 
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else if type_test(T,flbnos,apsns) then [(thm, rev apsns,pos,TB,tt)] 
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else [] 
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end; 
0  174 

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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 sg Ts t = 
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let 
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val T' = fastype_of1 (Ts, t); 
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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 
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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, cT) => 
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(case assoc(cmap, c) of 
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Some(gcT, thm, T, n) => 
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if Type.typ_instance(Sign.tsig_of sg, cT, gcT) 
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then 
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let val t2 = list_comb (h, take (n, ts)) 
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in mk_split_pack (thm, T, T', n, ts, apsns, pos, type_of1 (Ts, t2), t2) 
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end 
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else [] 
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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  214 

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0  216 
fun nth_subgoal i thm = nth_elem(i1,prems_of thm); 
217 

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fun shorter((_,ps,pos,_,_),(_,qs,qos,_,_)) = 
4519  219 
prod_ord (int_ord o pairself length) (order o pairself length) 
220 
((ps, pos), (qs, qos)); 

221 

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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  227 

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fun select cmap state i = 
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let val sg = #sign(rep_thm state) 
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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 sg Ts t)) end; 
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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 i = 
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let 
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val cert = cterm_of (sign_of_thm state); 
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val cntxt = mk_cntxt Ts t pos (T > U) (#maxidx(rep_thm trlift)); 
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in cterm_instantiate [(cert P, cert cntxt)] trlift 
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end; 
0  260 

261 

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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  268 
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  272 
i : number of subgoal 
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**************************************************************) 
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4232  275 
fun inst_split Ts t tt thm TB state i = 
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let 
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val thm' = Thm.lift_rule (state, i) thm; 
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val (P, _) = strip_comb (fst (Logic.dest_equals 
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(Logic.strip_assums_concl (#prop (rep_thm thm'))))); 
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val cert = cterm_of (sign_of_thm state); 
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val cntxt = mk_cntxt_splitthm t tt TB; 
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val abss = foldl (fn (t, T) => Abs ("", T, t)); 
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in cterm_instantiate [(cert P, cert (abss (cntxt, Ts)))] thm' 
4232  284 
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  294 
fun split_tac [] i = no_tac 
295 
 split_tac splits i = 

5553  296 
let val splits = map Data.mk_eq splits; 
5304  297 
fun const(thm) = 
3918  298 
(case concl_of thm of _$(t as _$lhs)$_ => 
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(case strip_comb lhs of (Const(a,aT),args) => 
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(a,(aT,thm,fastype_of t,length args)) 
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 _ => split_format_err()) 
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 _ => split_format_err()) 
0  303 
val cmap = map const splits; 
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fun lift_tac Ts t p st = rtac (inst_lift Ts t p st i) i st 
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fun lift_split_tac state = 
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let val (Ts, t, splits) = select cmap state i 
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in case splits of 
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[] => no_tac state 
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 (thm, apsns, pos, TB, tt)::_ => 
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(case apsns of 
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[] => compose_tac (false, inst_split Ts t tt thm TB state i, 0) i state 
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 p::_ => EVERY [lift_tac Ts t p, 
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rtac reflexive_thm (i+1), 
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lift_split_tac] state) 
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315 
end 
3537  316 
in COND (has_fewer_prems i) no_tac 
5304  317 
(rtac meta_iffD i THEN lift_split_tac) 
0  318 
end; 
319 

320 
in split_tac end; 

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321 

5304  322 

323 
val split_tac = mk_case_split_tac int_ord; 

4189  324 

5304  325 
val split_inside_tac = mk_case_split_tac (rev_order o int_ord); 
326 

4189  327 

328 
(***************************************************************************** 

329 
The splittactic for premises 

330 

331 
splits : list of splittheorems to be tried 

5304  332 
****************************************************************************) 
4202  333 
fun split_asm_tac [] = K no_tac 
334 
 split_asm_tac splits = 

5304  335 

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let val cname_list = map (fst o split_thm_info) splits; 
4189  337 
fun is_case (a,_) = a mem cname_list; 
338 
fun tac (t,i) = 

339 
let val n = find_index (exists_Const is_case) 

340 
(Logic.strip_assums_hyp t); 

341 
fun first_prem_is_disj (Const ("==>", _) $ (Const ("Trueprop", _) 

5304  342 
$ (Const (s, _) $ _ $ _ )) $ _ ) = (s=const_or) 
4202  343 
 first_prem_is_disj (Const("all",_)$Abs(_,_,t)) = 
344 
first_prem_is_disj t 

4189  345 
 first_prem_is_disj _ = false; 
5437  346 
(* does not work properly if the split variable is bound by a quantfier *) 
4202  347 
fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
5304  348 
(if first_prem_is_disj t 
349 
then EVERY[etac Data.disjE i,rotate_tac ~1 i, 

350 
rotate_tac ~1 (i+1), 

351 
flat_prems_tac (i+1)] 

352 
else all_tac) 

353 
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i) 

354 
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i; 

4189  355 
in if n<0 then no_tac else DETERM (EVERY' 
5304  356 
[rotate_tac n, etac Data.contrapos2, 
4189  357 
split_tac splits, 
5304  358 
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1, 
4202  359 
flat_prems_tac] i) 
4189  360 
end; 
361 
in SUBGOAL tac 

362 
end; 

363 

5304  364 
fun split_name name asm = "split " ^ name ^ (if asm then " asm" else ""); 
4189  365 

5304  366 
fun ss addsplits splits = 
367 
let fun addsplit (ss,split) = 

368 
let val (name,asm) = split_thm_info split 

369 
in Data.Simplifier.addloop(ss,(split_name name asm, 

370 
(if asm then split_asm_tac else split_tac) [split])) end 

371 
in foldl addsplit (ss,splits) end; 

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5304  373 
fun ss delsplits splits = 
374 
let fun delsplit(ss,split) = 

375 
let val (name,asm) = split_thm_info split 

376 
in Data.Simplifier.delloop(ss,split_name name asm) 

377 
end in foldl delsplit (ss,splits) end; 

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378 

5304  379 
fun Addsplits splits = (Data.Simplifier.simpset_ref() := 
380 
Data.Simplifier.simpset() addsplits splits); 

381 
fun Delsplits splits = (Data.Simplifier.simpset_ref() := 

382 
Data.Simplifier.simpset() delsplits splits); 

4189  383 

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384 
end; 