author  wenzelm 
Wed, 23 Nov 2011 22:59:39 +0100  
changeset 45620  f2a587696afb 
parent 42367  577d85fb5862 
child 51717  9e7d1c139569 
permissions  rwrr 
32174  1 
(* Title: Provers/splitter.ML 
4  2 
Author: Tobias Nipkow 
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Copyright 1995 TU Munich 
4  4 

5 
Generic casesplitter, suitable for most logics. 

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Deals with equalities of the form ?P(f args) = ... 
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where "f args" must be a firstorder term without duplicate variables. 
0  8 
*) 
9 

5304  10 
signature SPLITTER_DATA = 
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sig 

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val thy : theory 
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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 (* "[ EX 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" *) 
5304  22 
end 
23 

24 
signature SPLITTER = 

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sig 

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(* somewhat more internal functions *) 
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val cmap_of_split_thms: thm list > (string * (typ * term * thm * typ * int) list) list 
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val split_posns: (string * (typ * term * thm * typ * int) list) list > 

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theory > typ list > term > (thm * (typ * typ * int list) list * int list * typ * term) list 

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(* first argument is a "cmap", returns a list of "split packs" *) 

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(* the "real" interface, providing a number of tactics *) 
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val split_tac : thm list > int > tactic 
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val split_inside_tac: thm list > int > tactic 

34 
val split_asm_tac : thm list > int > tactic 

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val add_split: thm > simpset > simpset 
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val del_split: thm > simpset > simpset 
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val split_add: attribute 
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val split_del: attribute 

30513  39 
val split_modifiers : Method.modifier parser list 
18708  40 
val setup: theory > theory 
5304  41 
end; 
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functor Splitter(Data: SPLITTER_DATA): SPLITTER = 
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struct 
5304  45 

18545  46 
val Const (const_not, _) $ _ = 
35625  47 
Object_Logic.drop_judgment Data.thy 
18545  48 
(#1 (Logic.dest_implies (Thm.prop_of Data.notnotD))); 
5304  49 

18545  50 
val Const (const_or , _) $ _ $ _ = 
35625  51 
Object_Logic.drop_judgment Data.thy 
18545  52 
(#1 (Logic.dest_implies (Thm.prop_of Data.disjE))); 
53 

35625  54 
val const_Trueprop = Object_Logic.judgment_name Data.thy; 
18545  55 

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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 => (case strip_comb t of 
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(Const p, _) => (p, case c of (Const (s, _) $ _) => s = const_not  _ => false) 
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 _ => split_format_err ()) 
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 _ => split_format_err (); 
5304  64 

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fun cmap_of_split_thms thms = 
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let 
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val splits = map Data.mk_eq thms 
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fun add_thm thm cmap = 
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(case concl_of thm of _ $ (t as _ $ lhs) $ _ => 

70 
(case strip_comb lhs of (Const(a,aT),args) => 

71 
let val info = (aT,lhs,thm,fastype_of t,length args) 

72 
in case AList.lookup (op =) cmap a of 

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SOME infos => AList.update (op =) (a, info::infos) cmap 

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 NONE => (a,[info])::cmap 

75 
end 

76 
 _ => split_format_err()) 

77 
 _ => split_format_err()) 

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in 
33242  79 
fold add_thm splits [] 
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end; 
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(*  *) 
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(* mk_case_split_tac *) 
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(*  *) 
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5304  86 
fun mk_case_split_tac order = 
0  87 
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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*************************************************************) 
5304  95 

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val meta_iffD = Data.meta_eq_to_iff RS Data.iffD; (* (P == Q) ==> Q ==> P *) 
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22838  98 
val lift = Goal.prove_global Pure.thy ["P", "Q", "R"] 
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[Syntax.read_prop_global Pure.thy "!!x :: 'b. Q(x) == R(x) :: 'c"] 
100 
(Syntax.read_prop_global Pure.thy "P(%x. Q(x)) == P(%x. R(x))") 

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(fn {prems, ...} => rewrite_goals_tac prems THEN rtac reflexive_thm 1) 
4  102 

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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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4232  140 
fun mk_cntxt_splitthm t tt T = 
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let fun repl lev t = 

29548  142 
if Pattern.aeconv(incr_boundvars lev tt, t) then Bound lev 
4232  143 
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 

148 
in Abs("", T, repl 0 t) end; 

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(* add all loose bound variables in t to list is *) 
33245  152 
fun add_lbnos t is = 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) = 
42364  158 
let val (_, U: typ, _) = nth apsns (foldl1 Int.min lbnos) 
33029  159 
in T = U end; 
0  160 

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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. 
17881  187 
******************************************************************************) 
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20664  189 
fun mk_split_pack (thm, T: typ, T', n, ts, apsns, pos, TB, t) = 
1064  190 
if n > length ts then [] 
191 
else let val lev = length apsns 

33955  192 
val lbnos = fold add_lbnos (take n ts) [] 
33317  193 
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)] 
2143  198 
else [] 
1064  199 
end; 
0  200 

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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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(* Simplified firstorder matching; 
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assumes that all Vars in the pattern are distinct; 
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see Pure/pattern.ML for the full version; 
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*) 
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local 
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exception MATCH 
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in 
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fun typ_match thy (tyenv, TU) = Sign.typ_match thy TU tyenv 
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handle Type.TYPE_MATCH => raise MATCH; 
33242  225 

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fun fomatch thy args = 
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let 
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fun mtch tyinsts = fn 
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(Ts, Var(_,T), t) => 
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typ_match thy (tyinsts, (T, fastype_of1(Ts,t))) 
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 (_, Free (a,T), Free (b,U)) => 
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if a=b then typ_match thy (tyinsts,(T,U)) else raise MATCH 
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 (_, Const (a,T), Const (b,U)) => 
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if a=b then typ_match thy (tyinsts,(T,U)) else raise MATCH 
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 (_, Bound i, Bound j) => 
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if i=j then tyinsts else raise MATCH 
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 (Ts, Abs(_,T,t), Abs(_,U,u)) => 
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mtch (typ_match thy (tyinsts,(T,U))) (U::Ts,t,u) 
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 (Ts, f$t, g$u) => 
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mtch (mtch tyinsts (Ts,f,g)) (Ts, t, u) 
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 _ => raise MATCH 
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in (mtch Vartab.empty args; true) handle MATCH => false end; 
33242  243 
end; 
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fun split_posns (cmap : (string * (typ * term * thm * typ * int) list) list) thy 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 
33245  254 
fun iter t (i, a) = (i+1, (posns Ts (i::pos) apsns t) @ a); 
255 
val a = 

256 
case h of 

257 
Const(c, cT) => 

258 
let fun find [] = [] 

259 
 find ((gcT, pat, thm, T, n)::tups) = 

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let val t2 = list_comb (h, take n ts) in 
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if Sign.typ_instance thy (cT, gcT) andalso fomatch thy (Ts, pat, t2) 
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then mk_split_pack(thm,T,T',n,ts,apsns,pos,type_of1(Ts,t2),t2) 
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else find tups 
33245  264 
end 
265 
in find (these (AList.lookup (op =) cmap c)) end 

266 
 _ => [] 

267 
in snd (fold iter ts (0, a)) end 

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in posns Ts [] [] t end; 
0  269 

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fun shorter ((_,ps,pos,_,_), (_,qs,qos,_,_)) = 
4519  271 
prod_ord (int_ord o pairself length) (order o pairself length) 
272 
((ps, pos), (qs, qos)); 

273 

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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  278 

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fun select cmap state i = 
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let 
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val thy = Thm.theory_of_thm state 
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val goal = term_of (Thm.cprem_of state i); 
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val Ts = rev (map #2 (Logic.strip_params goal)); 
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val _ $ t $ _ = Logic.strip_assums_concl goal; 
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in (Ts, t, sort shorter (split_posns cmap thy Ts t)) end; 
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286 

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fun exported_split_posns cmap thy Ts t = 
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sort shorter (split_posns cmap thy Ts t); 
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(************************************************************* 
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instantiate lift theorem 
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292 

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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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307 

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fun inst_lift Ts t (T, U, pos) state i = 
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309 
let 
22578  310 
val cert = cterm_of (Thm.theory_of_thm state); 
22596  311 
val cntxt = mk_cntxt Ts t pos (T > U) (Thm.maxidx_of trlift); 
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in cterm_instantiate [(cert P, cert cntxt)] trlift 
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end; 
0  314 

315 

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(************************************************************* 
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instantiate split theorem 
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318 

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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  322 
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  326 
i : number of subgoal 
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**************************************************************) 
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328 

4232  329 
fun inst_split Ts t tt thm TB state i = 
17881  330 
let 
18145  331 
val thm' = Thm.lift_rule (Thm.cprem_of state i) thm; 
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332 
val (P, _) = strip_comb (fst (Logic.dest_equals 
22596  333 
(Logic.strip_assums_concl (Thm.prop_of thm')))); 
22578  334 
val cert = cterm_of (Thm.theory_of_thm state); 
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335 
val cntxt = mk_cntxt_splitthm t tt TB; 
33245  336 
val abss = fold (fn T => fn t => Abs ("", T, t)); 
337 
in cterm_instantiate [(cert P, cert (abss Ts cntxt))] thm' 

4232  338 
end; 
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339 

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340 

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(***************************************************************************** 
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The splittactic 
17881  343 

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splits : list of splittheorems to be tried 
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345 
i : number of subgoal the tactic should be applied to 
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346 
*****************************************************************************) 
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347 

0  348 
fun split_tac [] i = no_tac 
349 
 split_tac splits i = 

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let val cmap = cmap_of_split_thms 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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352 
fun lift_split_tac state = 
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let val (Ts, t, splits) = select cmap state i 
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354 
in case splits of 
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355 
[] => no_tac state 
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356 
 (thm, apsns, pos, TB, tt)::_ => 
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357 
(case apsns of 
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358 
[] => compose_tac (false, inst_split Ts t tt thm TB state i, 0) i state 
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359 
 p::_ => EVERY [lift_tac Ts t p, 
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360 
rtac reflexive_thm (i+1), 
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361 
lift_split_tac] state) 
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362 
end 
17881  363 
in COND (has_fewer_prems i) no_tac 
5304  364 
(rtac meta_iffD i THEN lift_split_tac) 
0  365 
end; 
366 

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367 
in (split_tac, exported_split_posns) end; (* mk_case_split_tac *) 
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368 

5304  369 

33242  370 
val (split_tac, split_posns) = mk_case_split_tac int_ord; 
4189  371 

33242  372 
val (split_inside_tac, _) = mk_case_split_tac (rev_order o int_ord); 
5304  373 

4189  374 

375 
(***************************************************************************** 

376 
The splittactic for premises 

17881  377 

4189  378 
splits : list of splittheorems to be tried 
5304  379 
****************************************************************************) 
33242  380 
fun split_asm_tac [] = K no_tac 
17881  381 
 split_asm_tac splits = 
5304  382 

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let val cname_list = map (fst o fst o split_thm_info) splits; 
17881  384 
fun tac (t,i) = 
20664  385 
let val n = find_index (exists_Const (member (op =) cname_list o #1)) 
17881  386 
(Logic.strip_assums_hyp t); 
18545  387 
fun first_prem_is_disj (Const ("==>", _) $ (Const (c, _) 
388 
$ (Const (s, _) $ _ $ _ )) $ _ ) = c = const_Trueprop andalso s = const_or 

17881  389 
 first_prem_is_disj (Const("all",_)$Abs(_,_,t)) = 
390 
first_prem_is_disj t 

391 
 first_prem_is_disj _ = false; 

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392 
(* does not work properly if the split variable is bound by a quantifier *) 
17881  393 
fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
394 
(if first_prem_is_disj t 

395 
then EVERY[etac Data.disjE i,rotate_tac ~1 i, 

396 
rotate_tac ~1 (i+1), 

397 
flat_prems_tac (i+1)] 

398 
else all_tac) 

399 
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i) 

400 
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i; 

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401 
in if n<0 then no_tac else (DETERM (EVERY' 
17881  402 
[rotate_tac n, etac Data.contrapos2, 
403 
split_tac splits, 

404 
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1, 

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405 
flat_prems_tac] i)) 
17881  406 
end; 
4189  407 
in SUBGOAL tac 
408 
end; 

409 

10652  410 
fun gen_split_tac [] = K no_tac 
411 
 gen_split_tac (split::splits) = 

412 
let val (_,asm) = split_thm_info split 

413 
in (if asm then split_asm_tac else split_tac) [split] ORELSE' 

414 
gen_split_tac splits 

415 
end; 

8468  416 

18688  417 

8468  418 
(** declare split rules **) 
419 

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420 
(* add_split / del_split *) 
8468  421 

33242  422 
fun string_of_typ (Type (s, Ts)) = 
423 
(if null Ts then "" else enclose "(" ")" (commas (map string_of_typ Ts))) ^ s 

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424 
 string_of_typ _ = "_"; 
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425 

17881  426 
fun split_name (name, T) asm = "split " ^ 
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427 
(if asm then "asm " else "") ^ name ^ " :: " ^ string_of_typ T; 
4189  428 

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429 
fun add_split split ss = 
33242  430 
let 
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431 
val (name, asm) = split_thm_info split 
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432 
val tac = (if asm then split_asm_tac else split_tac) [split] 
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433 
in Simplifier.addloop (ss, (split_name name asm, tac)) end; 
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434 

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435 
fun del_split split ss = 
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436 
let val (name, asm) = split_thm_info split 
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437 
in Simplifier.delloop (ss, split_name name asm) end; 
1721
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438 

8468  439 

440 
(* attributes *) 

441 

442 
val splitN = "split"; 

443 

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444 
val split_add = Simplifier.attrib add_split; 
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445 
val split_del = Simplifier.attrib del_split; 
8634  446 

447 

9703  448 
(* methods *) 
8468  449 

450 
val split_modifiers = 

18728  451 
[Args.$$$ splitN  Args.colon >> K ((I, split_add): Method.modifier), 
452 
Args.$$$ splitN  Args.add  Args.colon >> K (I, split_add), 

453 
Args.$$$ splitN  Args.del  Args.colon >> K (I, split_del)]; 

8468  454 

455 

18688  456 
(* theory setup *) 
8468  457 

9703  458 
val setup = 
33242  459 
Attrib.setup @{binding split} 
460 
(Attrib.add_del split_add split_del) "declare case split rule" #> 

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461 
Method.setup @{binding split} 
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462 
(Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (CHANGED_PROP o gen_split_tac ths)))) 
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463 
"apply case split rule"; 
4189  464 

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465 
end; 