author  wenzelm 
Wed, 04 Apr 2007 00:11:03 +0200  
changeset 22578  b0eb5652f210 
parent 21879  a3efbae45735 
child 22596  d0d2af4db18f 
permissions  rwrr 
4  1 
(* Title: Provers/splitter 
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ID: $Id$ 

3 
Author: Tobias Nipkow 

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Copyright 1995 TU Munich 
4  5 

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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  9 
*) 
10 

5304  11 
infix 4 addsplits delsplits; 
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signature SPLITTER_DATA = 

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sig 

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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" *) 
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end 
25 

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

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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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val split_add: attribute 
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val split_del: attribute 

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val split_modifiers : (Args.T list > (Method.modifier * Args.T list)) list 
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val setup: theory > theory 
5304  44 
end; 
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functor SplitterFun(Data: SPLITTER_DATA): SPLITTER = 

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struct 
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18545  49 
val Const (const_not, _) $ _ = 
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ObjectLogic.drop_judgment (the_context ()) 

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(#1 (Logic.dest_implies (Thm.prop_of Data.notnotD))); 

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val Const (const_or , _) $ _ $ _ = 
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ObjectLogic.drop_judgment (the_context ()) 

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(#1 (Logic.dest_implies (Thm.prop_of Data.disjE))); 

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val const_Trueprop = ObjectLogic.judgment_name (the_context ()); 

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fun split_format_err () = error "Wrong format for split rule"; 
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(* thm > (string * typ) * bool *) 
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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  68 

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(* thm list > (string * (typ * term * thm * typ * int) list) list *) 
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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 (cmap, thm) = 
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(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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let val info = (aT,lhs,thm,fastype_of t,length args) 
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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 
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end 
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 _ => split_format_err()) 
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 _ => split_format_err()) 
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in 
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Library.foldl 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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(* (int * int > order) > thm list > int > tactic * <split_posns> *) 
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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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*************************************************************) 
5304  102 

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val meta_iffD = Data.meta_eq_to_iff RS Data.iffD; (* (P == Q) ==> Q ==> P *) 
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val lift = 
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let val ct = read_cterm Pure.thy 
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("(!!x. (Q::('b::{})=>('c::{}))(x) == R(x)) ==> \ 

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\P(%x. Q(x)) == P(%x. R(x))::'a::{}",propT) 
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in OldGoals.prove_goalw_cterm [] ct 
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(fn [prem] => [rewtac prem, rtac reflexive_thm 1]) 
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end; 

4  112 

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val trlift = lift RS transitive_thm; 
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val _ $ (P $ _) $ _ = concl_of trlift; 
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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 (Library.take(p,ts), i upto (i+p1)) 
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val u::us = Library.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 
4232  153 
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: typ,_) = List.nth(apsns, Library.foldl Int.min (hd lbnos, tl lbnos)) 
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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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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  197 
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20664  199 
fun mk_split_pack (thm, T: typ, T', n, ts, apsns, pos, TB, t) = 
1064  200 
if n > length ts then [] 
201 
else let val lev = length apsns 

15570  202 
val lbnos = Library.foldl add_lbnos ([],Library.take(n,ts)) 
203 
val flbnos = List.filter (fn i => i < lev) lbnos 

4232  204 
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  208 
else [] 
1064  209 
end; 
0  210 

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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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(* Context.theory > Type.tyenv * (Term.typ * Term.typ) > Type.tyenv *) 
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fun typ_match sg (tyenv, TU) = (Sign.typ_match sg TU tyenv) 
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handle Type.TYPE_MATCH => raise MATCH 
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(* Context.theory > Term.typ list * Term.term * Term.term > bool *) 
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fun fomatch sg args = 
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let 
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(* Type.tyenv > Term.typ list * Term.term * Term.term > Type.tyenv *) 
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fun mtch tyinsts = fn 
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(Ts, Var(_,T), t) => 
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typ_match sg (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 sg (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 sg (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 sg (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; 
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end (* local *) 
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(* (string * (Term.typ * Term.term * Thm.thm * Term.typ * int) list) list > Context.theory > Term.typ list > Term.term > 
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(Thm.thm * (Term.typ * Term.typ * int list) list * int list * Term.typ * Term.term) list *) 
20237  259 
fun split_posns (cmap : (string * (typ * term * thm * typ * int) list) list) 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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let fun find [] = [] 
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 find ((gcT, pat, thm, T, n)::tups) = 
15570  273 
let val t2 = list_comb (h, Library.take (n, ts)) 
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in if Sign.typ_instance sg (cT, gcT) 
16935  275 
andalso fomatch sg (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 
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end 
17184  279 
in find (these (AList.lookup (op =) cmap c)) end 
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 _ => [] 
15570  281 
in snd(Library.foldl iter ((0, a), ts)) end 
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in posns Ts [] [] t end; 
0  283 

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fun nth_subgoal i thm = List.nth (prems_of thm, i1); 
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fun shorter ((_,ps,pos,_,_), (_,qs,qos,_,_)) = 
4519  287 
prod_ord (int_ord o pairself length) (order o pairself length) 
288 
((ps, pos), (qs, qos)); 

289 

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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  294 

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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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fun exported_split_posns cmap sg Ts t = 
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sort shorter (split_posns cmap sg Ts t); 
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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 
22578  325 
val cert = cterm_of (Thm.theory_of_thm state); 
17881  326 
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  329 

330 

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331 
(************************************************************* 
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332 
instantiate split theorem 
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333 

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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  337 
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  341 
i : number of subgoal 
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**************************************************************) 
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4232  344 
fun inst_split Ts t tt thm TB state i = 
17881  345 
let 
18145  346 
val thm' = Thm.lift_rule (Thm.cprem_of 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'))))); 
22578  349 
val cert = cterm_of (Thm.theory_of_thm state); 
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val cntxt = mk_cntxt_splitthm t tt TB; 
15570  351 
val abss = Library.foldl (fn (t, T) => Abs ("", T, t)); 
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in cterm_instantiate [(cert P, cert (abss (cntxt, Ts)))] thm' 
4232  353 
end; 
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354 

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355 

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(***************************************************************************** 
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The splittactic 
17881  358 

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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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(* thm list > int > tactic *) 
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0  365 
fun split_tac [] i = no_tac 
366 
 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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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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end 
17881  380 
in COND (has_fewer_prems i) no_tac 
5304  381 
(rtac meta_iffD i THEN lift_split_tac) 
0  382 
end; 
383 

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in (split_tac, exported_split_posns) end; (* mk_case_split_tac *) 
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5304  386 

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val (split_tac, split_posns) = mk_case_split_tac int_ord; 
4189  388 

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val (split_inside_tac, _) = mk_case_split_tac (rev_order o int_ord); 
5304  390 

4189  391 

392 
(***************************************************************************** 

393 
The splittactic for premises 

17881  394 

4189  395 
splits : list of splittheorems to be tried 
5304  396 
****************************************************************************) 
4202  397 
fun split_asm_tac [] = K no_tac 
17881  398 
 split_asm_tac splits = 
5304  399 

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let val cname_list = map (fst o fst o split_thm_info) splits; 
17881  401 
fun tac (t,i) = 
20664  402 
let val n = find_index (exists_Const (member (op =) cname_list o #1)) 
17881  403 
(Logic.strip_assums_hyp t); 
18545  404 
fun first_prem_is_disj (Const ("==>", _) $ (Const (c, _) 
405 
$ (Const (s, _) $ _ $ _ )) $ _ ) = c = const_Trueprop andalso s = const_or 

17881  406 
 first_prem_is_disj (Const("all",_)$Abs(_,_,t)) = 
407 
first_prem_is_disj t 

408 
 first_prem_is_disj _ = false; 

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(* does not work properly if the split variable is bound by a quantifier *) 
17881  410 
fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
411 
(if first_prem_is_disj t 

412 
then EVERY[etac Data.disjE i,rotate_tac ~1 i, 

413 
rotate_tac ~1 (i+1), 

414 
flat_prems_tac (i+1)] 

415 
else all_tac) 

416 
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i) 

417 
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i; 

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in if n<0 then no_tac else (DETERM (EVERY' 
17881  419 
[rotate_tac n, etac Data.contrapos2, 
420 
split_tac splits, 

421 
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1, 

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422 
flat_prems_tac] i)) 
17881  423 
end; 
4189  424 
in SUBGOAL tac 
425 
end; 

426 

10652  427 
fun gen_split_tac [] = K no_tac 
428 
 gen_split_tac (split::splits) = 

429 
let val (_,asm) = split_thm_info split 

430 
in (if asm then split_asm_tac else split_tac) [split] ORELSE' 

431 
gen_split_tac splits 

432 
end; 

8468  433 

18688  434 

8468  435 
(** declare split rules **) 
436 

437 
(* addsplits / delsplits *) 

438 

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fun string_of_typ (Type (s, Ts)) = (if null Ts then "" 
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else enclose "(" ")" (commas (map string_of_typ Ts))) ^ s 
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 string_of_typ _ = "_"; 
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17881  443 
fun split_name (name, T) asm = "split " ^ 
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(if asm then "asm " else "") ^ name ^ " :: " ^ string_of_typ T; 
4189  445 

5304  446 
fun ss addsplits splits = 
447 
let fun addsplit (ss,split) = 

448 
let val (name,asm) = split_thm_info split 

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in Simplifier.addloop (ss, (split_name name asm, 
17881  450 
(if asm then split_asm_tac else split_tac) [split])) end 
15570  451 
in Library.foldl addsplit (ss,splits) end; 
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5304  453 
fun ss delsplits splits = 
454 
let fun delsplit(ss,split) = 

455 
let val (name,asm) = split_thm_info split 

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in Simplifier.delloop (ss, split_name name asm) 
15570  457 
end in Library.foldl delsplit (ss,splits) end; 
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458 

17881  459 
fun Addsplits splits = (change_simpset (fn ss => ss addsplits splits)); 
460 
fun Delsplits splits = (change_simpset (fn ss => ss delsplits splits)); 

8468  461 

462 

463 
(* attributes *) 

464 

465 
val splitN = "split"; 

466 

18688  467 
val split_add = Simplifier.attrib (op addsplits); 
468 
val split_del = Simplifier.attrib (op delsplits); 

8634  469 

470 

9703  471 
(* methods *) 
8468  472 

473 
val split_modifiers = 

18728  474 
[Args.$$$ splitN  Args.colon >> K ((I, split_add): Method.modifier), 
475 
Args.$$$ splitN  Args.add  Args.colon >> K (I, split_add), 

476 
Args.$$$ splitN  Args.del  Args.colon >> K (I, split_del)]; 

8468  477 

18688  478 
fun split_meth src = 
18988  479 
Method.syntax Attrib.thms src 
21879  480 
#> (fn (ths, _) => Method.SIMPLE_METHOD' (CHANGED_PROP o gen_split_tac ths)); 
9703  481 

8468  482 

18688  483 
(* theory setup *) 
8468  484 

9703  485 
val setup = 
18708  486 
(Attrib.add_attributes 
18728  487 
[(splitN, Attrib.add_del_args split_add split_del, "declaration of case split rule")] #> 
18708  488 
Method.add_methods [(splitN, split_meth, "apply case split rule")]); 
4189  489 

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490 
end; 