author  blanchet 
Mon, 15 Sep 2014 10:49:07 +0200  
changeset 58335  a5a3b576fcfb 
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child 58826  2ed2eaabe3df 
permissions  rwrr 
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(* 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 

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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 > 

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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 

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val split_asm_tac : thm list > int > tactic 

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val add_split: thm > Proof.context > Proof.context 
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val del_split: thm > Proof.context > Proof.context 
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val split_add: attribute 
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val split_del: attribute 

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

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

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val Const (const_or , _) $ _ $ _ = 
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Object_Logic.drop_judgment Data.thy 
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(#1 (Logic.dest_implies (Thm.prop_of Data.disjE))); 
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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(@{const_name Pure.eq}, _) $ (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) $ _ => 

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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 

75 
end 

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 _ => split_format_err()) 

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 _ => split_format_err()) 

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in 
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fold add_thm splits [] 
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end; 
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val abss = fold (Term.abs o pair ""); 
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(*  *) 
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(* mk_case_split_tac *) 
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(*  *) 
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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  97 

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val meta_iffD = Data.meta_eq_to_iff RS Data.iffD; (* (P == Q) ==> Q ==> P *) 
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22838  100 
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"] 
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(Syntax.read_prop_global Pure.thy "P(%x. Q(x)) == P(%x. R(x))") 

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(fn {context = ctxt, prems} => rewrite_goals_tac ctxt prems THEN rtac reflexive_thm 1) 
4  104 

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val _ $ _ $ (_ $ (_ $ abs_lift) $ _) = prop_of lift; 
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val trlift = lift RS transitive_thm; 
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(************************************************************************ 
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Set up term for instantiation of P in the lifttheorem 
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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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*************************************************************************) 
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fun mk_cntxt t pos T = 
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let 
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fun down [] t = (Bound 0, t) 
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 down (p :: ps) t = 
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let 
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val (h, ts) = strip_comb t 
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val (ts1, u :: ts2) = chop p ts 
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val (u1, u2) = down ps u 
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in 
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(list_comb (incr_boundvars 1 h, 
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map (incr_boundvars 1) ts1 @ u1 :: 
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map (incr_boundvars 1) ts2), 
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u2) 
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end; 
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val (u1, u2) = down (rev pos) t 
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in (Abs ("", T, u1), u2) 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 Envir.aeconv(incr_boundvars lev tt, t) then Bound lev 
4232  150 
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 *) 
33245  159 
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) = 
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let val (_, U: typ, _) = nth apsns (foldl1 Int.min lbnos) 
33029  166 
in T = U end; 
0  167 

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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  194 
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20664  196 
fun mk_split_pack (thm, T: typ, T', n, ts, apsns, pos, TB, t) = 
1064  197 
if n > length ts then [] 
198 
else let val lev = length apsns 

33955  199 
val lbnos = fold add_lbnos (take n ts) [] 
33317  200 
val flbnos = filter (fn i => i < lev) lbnos 
4232  201 
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  205 
else [] 
1064  206 
end; 
0  207 

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(**************************************************************************** 
58318  210 
Recursively scans term for occurrences of Const(key,...) $ ... 
211 
Returns a list of "splitpacks" (one for each occurrence 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  232 

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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  250 
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  261 
fun iter t (i, a) = (i+1, (posns Ts (i::pos) apsns t) @ a); 
262 
val a = 

263 
case h of 

264 
Const(c, cT) => 

265 
let fun find [] = [] 

266 
 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  271 
end 
272 
in find (these (AList.lookup (op =) cmap c)) end 

273 
 _ => [] 

274 
in snd (fold iter ts (0, a)) end 

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in posns Ts [] [] t end; 
0  276 

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fun shorter ((_,ps,pos,_,_), (_,qs,qos,_,_)) = 
4519  278 
prod_ord (int_ord o pairself length) (order o pairself length) 
279 
((ps, pos), (qs, qos)); 

280 

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(************************************************************ 
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call split_posns with appropriate parameters 
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*************************************************************) 
0  285 

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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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293 

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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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296 

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(************************************************************* 
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instantiate lift theorem 
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299 

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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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308 
the split theorem is applied to (see cmap) 
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309 
T,U,pos : see mk_split_pack 
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state : current proof state 
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i : no. of subgoal 
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**************************************************************) 
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313 

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fun inst_lift Ts t (T, U, pos) state i = 
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315 
let 
22578  316 
val cert = cterm_of (Thm.theory_of_thm state); 
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val (cntxt, u) = mk_cntxt t pos (T > U); 
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val trlift' = Thm.lift_rule (Thm.cprem_of state i) 
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(Thm.rename_boundvars abs_lift u trlift); 
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val (P, _) = strip_comb (fst (Logic.dest_equals 
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321 
(Logic.strip_assums_concl (Thm.prop_of trlift')))); 
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in cterm_instantiate [(cert P, cert (abss Ts cntxt))] trlift' 
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323 
end; 
0  324 

325 

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(************************************************************* 
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instantiate split theorem 
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328 

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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  332 
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  336 
i : number of subgoal 
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**************************************************************) 
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338 

4232  339 
fun inst_split Ts t tt thm TB state i = 
17881  340 
let 
18145  341 
val thm' = Thm.lift_rule (Thm.cprem_of state i) thm; 
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342 
val (P, _) = strip_comb (fst (Logic.dest_equals 
22596  343 
(Logic.strip_assums_concl (Thm.prop_of thm')))); 
22578  344 
val cert = cterm_of (Thm.theory_of_thm state); 
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val cntxt = mk_cntxt_splitthm t tt TB; 
33245  346 
in cterm_instantiate [(cert P, cert (abss Ts cntxt))] thm' 
4232  347 
end; 
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348 

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349 

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350 
(***************************************************************************** 
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351 
The splittactic 
17881  352 

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353 
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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355 
*****************************************************************************) 
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356 

0  357 
fun split_tac [] i = no_tac 
358 
 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 = compose_tac (false, inst_lift Ts t p st i, 2) 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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363 
in case splits of 
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364 
[] => no_tac state 
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 (thm, apsns, pos, TB, tt)::_ => 
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366 
(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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371 
end 
17881  372 
in COND (has_fewer_prems i) no_tac 
5304  373 
(rtac meta_iffD i THEN lift_split_tac) 
0  374 
end; 
375 

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in (split_tac, exported_split_posns) end; (* mk_case_split_tac *) 
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377 

5304  378 

33242  379 
val (split_tac, split_posns) = mk_case_split_tac int_ord; 
4189  380 

33242  381 
val (split_inside_tac, _) = mk_case_split_tac (rev_order o int_ord); 
5304  382 

4189  383 

384 
(***************************************************************************** 

385 
The splittactic for premises 

17881  386 

4189  387 
splits : list of splittheorems to be tried 
5304  388 
****************************************************************************) 
33242  389 
fun split_asm_tac [] = K no_tac 
17881  390 
 split_asm_tac splits = 
5304  391 

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let val cname_list = map (fst o fst o split_thm_info) splits; 
17881  393 
fun tac (t,i) = 
20664  394 
let val n = find_index (exists_Const (member (op =) cname_list o #1)) 
17881  395 
(Logic.strip_assums_hyp t); 
56245  396 
fun first_prem_is_disj (Const (@{const_name Pure.imp}, _) $ (Const (c, _) 
18545  397 
$ (Const (s, _) $ _ $ _ )) $ _ ) = c = const_Trueprop andalso s = const_or 
56245  398 
 first_prem_is_disj (Const(@{const_name Pure.all},_)$Abs(_,_,t)) = 
17881  399 
first_prem_is_disj t 
400 
 first_prem_is_disj _ = false; 

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(* does not work properly if the split variable is bound by a quantifier *) 
17881  402 
fun flat_prems_tac i = SUBGOAL (fn (t,i) => 
403 
(if first_prem_is_disj t 

404 
then EVERY[etac Data.disjE i,rotate_tac ~1 i, 

405 
rotate_tac ~1 (i+1), 

406 
flat_prems_tac (i+1)] 

407 
else all_tac) 

408 
THEN REPEAT (eresolve_tac [Data.conjE,Data.exE] i) 

409 
THEN REPEAT (dresolve_tac [Data.notnotD] i)) i; 

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in if n<0 then no_tac else (DETERM (EVERY' 
17881  411 
[rotate_tac n, etac Data.contrapos2, 
412 
split_tac splits, 

413 
rotate_tac ~1, etac Data.contrapos, rotate_tac ~1, 

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414 
flat_prems_tac] i)) 
17881  415 
end; 
4189  416 
in SUBGOAL tac 
417 
end; 

418 

10652  419 
fun gen_split_tac [] = K no_tac 
420 
 gen_split_tac (split::splits) = 

421 
let val (_,asm) = split_thm_info split 

422 
in (if asm then split_asm_tac else split_tac) [split] ORELSE' 

423 
gen_split_tac splits 

424 
end; 

8468  425 

18688  426 

8468  427 
(** declare split rules **) 
428 

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(* add_split / del_split *) 
8468  430 

33242  431 
fun string_of_typ (Type (s, Ts)) = 
432 
(if null Ts then "" else enclose "(" ")" (commas (map string_of_typ Ts))) ^ s 

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 string_of_typ _ = "_"; 
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434 

17881  435 
fun split_name (name, T) asm = "split " ^ 
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(if asm then "asm " else "") ^ name ^ " :: " ^ string_of_typ T; 
4189  437 

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438 
fun add_split split ctxt = 
33242  439 
let 
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440 
val (name, asm) = split_thm_info split 
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441 
val tac = (if asm then split_asm_tac else split_tac) [split] 
52037  442 
in Simplifier.addloop (ctxt, (split_name name asm, K tac)) end; 
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443 

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fun del_split split ctxt = 
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let val (name, asm) = split_thm_info split 
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446 
in Simplifier.delloop (ctxt, split_name name asm) end; 
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447 

8468  448 

449 
(* attributes *) 

450 

451 
val splitN = "split"; 

452 

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val split_add = Simplifier.attrib add_split; 
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val split_del = Simplifier.attrib del_split; 
8634  455 

456 

9703  457 
(* methods *) 
8468  458 

459 
val split_modifiers = 

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[Args.$$$ splitN  Args.colon >> K (Method.modifier split_add @{here}), 
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Args.$$$ splitN  Args.add  Args.colon >> K (Method.modifier split_add @{here}), 
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Args.$$$ splitN  Args.del  Args.colon >> K (Method.modifier split_del @{here})]; 
8468  463 

464 

18688  465 
(* theory setup *) 
8468  466 

9703  467 
val setup = 
33242  468 
Attrib.setup @{binding split} 
469 
(Attrib.add_del split_add split_del) "declare case split rule" #> 

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Method.setup @{binding split} 
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(Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (CHANGED_PROP o gen_split_tac ths)))) 
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472 
"apply case split rule"; 
4189  473 

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474 
end; 