8.71/3.75	YES
10.96/4.40	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.96/4.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.96/4.40	
10.96/4.40	
10.96/4.40	H-Termination with start terms of the given HASKELL could be proven:
10.96/4.40	
10.96/4.40	(0) HASKELL
10.96/4.40	(1) LR [EQUIVALENT, 0 ms]
10.96/4.40	(2) HASKELL
10.96/4.40	(3) CR [EQUIVALENT, 0 ms]
10.96/4.40	(4) HASKELL
10.96/4.40	(5) BR [EQUIVALENT, 0 ms]
10.96/4.40	(6) HASKELL
10.96/4.40	(7) COR [EQUIVALENT, 0 ms]
10.96/4.40	(8) HASKELL
10.96/4.40	(9) LetRed [EQUIVALENT, 10 ms]
10.96/4.40	(10) HASKELL
10.96/4.40	(11) Narrow [SOUND, 0 ms]
10.96/4.40	(12) AND
10.96/4.40	    (13) QDP
10.96/4.40	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/4.40	        (15) YES
10.96/4.40	    (16) QDP
10.96/4.40	        (17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/4.40	        (18) YES
10.96/4.40	    (19) QDP
10.96/4.40	        (20) TransformationProof [EQUIVALENT, 0 ms]
10.96/4.40	        (21) QDP
10.96/4.40	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/4.40	        (23) YES
10.96/4.40	    (24) QDP
10.96/4.40	        (25) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/4.40	        (26) YES
10.96/4.40	    (27) QDP
10.96/4.40	        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.96/4.40	        (29) YES
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(0)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(1) LR (EQUIVALENT)
10.96/4.40	Lambda Reductions:
10.96/4.40	The following Lambda expression
10.96/4.40	"\vu68->case vu68 of {
10.96/4.40	(cs@(_ : _),t)  -> (cs,t) : [];
10.96/4.40	_  -> []}
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"nonnull0 vu68 = case vu68 of {
10.96/4.40	(cs@(_ : _),t)  -> (cs,t) : [];
10.96/4.40	_  -> []}
10.96/4.40	;
10.96/4.40	"
10.96/4.40	The following Lambda expression
10.96/4.40	"\nd->n * radix + d"
10.96/4.40	is transformed to
10.96/4.40	"readInt0 radix n d = n * radix + d;
10.96/4.40	"
10.96/4.40	The following Lambda expression
10.96/4.40	"\vu77->case vu77 of {
10.96/4.40	(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
10.96/4.40	_  -> []}
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"readInt1 radix digToInt vu77 = case vu77 of {
10.96/4.40	(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
10.96/4.40	_  -> []}
10.96/4.40	;
10.96/4.40	"
10.96/4.40	The following Lambda expression
10.96/4.40	"\(_,zs)->zs"
10.96/4.40	is transformed to
10.96/4.40	"zs0 (_,zs) = zs;
10.96/4.40	"
10.96/4.40	The following Lambda expression
10.96/4.40	"\(ys,_)->ys"
10.96/4.40	is transformed to
10.96/4.40	"ys0 (ys,_) = ys;
10.96/4.40	"
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(2)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(3) CR (EQUIVALENT)
10.96/4.40	Case Reductions:
10.96/4.40	The following Case expression
10.96/4.40	"case vu68 of {
10.96/4.40	(cs@(_ : _),t)  -> (cs,t) : [];
10.96/4.40	_  -> []}
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"nonnull00 (cs@(_ : _),t) = (cs,t) : [];
10.96/4.40	nonnull00 _ = [];
10.96/4.40	"
10.96/4.40	The following Case expression
10.96/4.40	"case vu77 of {
10.96/4.40	(ds,r)  -> (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
10.96/4.40	_  -> []}
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"readInt10 radix digToInt (ds,r) = (foldl1 (readInt0 radix) (map (fromIntegral . digToInt) ds),r) : [];
10.96/4.40	readInt10 radix digToInt _ = [];
10.96/4.40	"
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(4)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(5) BR (EQUIVALENT)
10.96/4.40	Replaced joker patterns by fresh variables and removed binding patterns.
10.96/4.40	
10.96/4.40	Binding Reductions:
10.96/4.40	The bind variable of the following binding Pattern
10.96/4.40	"cs@(vy : vz)"
10.96/4.40	is replaced by the following term
10.96/4.40	"vy : vz"
10.96/4.40	The bind variable of the following binding Pattern
10.96/4.40	"xs@(wx : wy)"
10.96/4.40	is replaced by the following term
10.96/4.40	"wx : wy"
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(6)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(7) COR (EQUIVALENT)
10.96/4.40	Cond Reductions:
10.96/4.40	The following Function with conditions
10.96/4.40	"undefined |Falseundefined;
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"undefined  = undefined1;
10.96/4.40	"
10.96/4.40	"undefined0 True = undefined;
10.96/4.40	"
10.96/4.40	"undefined1  = undefined0 False;
10.96/4.40	"
10.96/4.40	The following Function with conditions
10.96/4.40	"span p [] = ([],[]);
10.96/4.40	span p (wx : wy)|p wx(wx : ys,zs)|otherwise([],wx : wy) where {
10.96/4.40	vu43  = span p wy;
10.96/4.40	;
10.96/4.40	ys  = ys0 vu43;
10.96/4.40	;
10.96/4.40	ys0 (ys,xu) = ys;
10.96/4.40	;
10.96/4.40	zs  = zs0 vu43;
10.96/4.40	;
10.96/4.40	zs0 (wz,zs) = zs;
10.96/4.40	} 
10.96/4.40	;
10.96/4.40	"
10.96/4.40	is transformed to
10.96/4.40	"span p [] = span3 p [];
10.96/4.40	span p (wx : wy) = span2 p (wx : wy);
10.96/4.40	"
10.96/4.40	"span2 p (wx : wy) = span1 p wx wy (p wx) where {
10.96/4.40	span0 p wx wy True = ([],wx : wy);
10.96/4.40	;
10.96/4.40	span1 p wx wy True = (wx : ys,zs);
10.96/4.40	span1 p wx wy False = span0 p wx wy otherwise;
10.96/4.40	;
10.96/4.40	vu43  = span p wy;
10.96/4.40	;
10.96/4.40	ys  = ys0 vu43;
10.96/4.40	;
10.96/4.40	ys0 (ys,xu) = ys;
10.96/4.40	;
10.96/4.40	zs  = zs0 vu43;
10.96/4.40	;
10.96/4.40	zs0 (wz,zs) = zs;
10.96/4.40	} 
10.96/4.40	;
10.96/4.40	"
10.96/4.40	"span3 p [] = ([],[]);
10.96/4.40	span3 xx xy = span2 xx xy;
10.96/4.40	"
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(8)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(9) LetRed (EQUIVALENT)
10.96/4.40	Let/Where Reductions:
10.96/4.40	The bindings of the following Let/Where expression
10.96/4.40	"span1 p wx wy (p wx) where {
10.96/4.40	span0 p wx wy True = ([],wx : wy);
10.96/4.40	;
10.96/4.40	span1 p wx wy True = (wx : ys,zs);
10.96/4.40	span1 p wx wy False = span0 p wx wy otherwise;
10.96/4.40	;
10.96/4.40	vu43  = span p wy;
10.96/4.40	;
10.96/4.40	ys  = ys0 vu43;
10.96/4.40	;
10.96/4.40	ys0 (ys,xu) = ys;
10.96/4.40	;
10.96/4.40	zs  = zs0 vu43;
10.96/4.40	;
10.96/4.40	zs0 (wz,zs) = zs;
10.96/4.40	} 
10.96/4.40	"
10.96/4.40	are unpacked to the following functions on top level
10.96/4.40	"span2Span0 xz yu p wx wy True = ([],wx : wy);
10.96/4.40	"
10.96/4.40	"span2Ys xz yu = span2Ys0 xz yu (span2Vu43 xz yu);
10.96/4.40	"
10.96/4.40	"span2Vu43 xz yu = span xz yu;
10.96/4.40	"
10.96/4.40	"span2Ys0 xz yu (ys,xu) = ys;
10.96/4.40	"
10.96/4.40	"span2Zs0 xz yu (wz,zs) = zs;
10.96/4.40	"
10.96/4.40	"span2Span1 xz yu p wx wy True = (wx : span2Ys xz yu,span2Zs xz yu);
10.96/4.40	span2Span1 xz yu p wx wy False = span2Span0 xz yu p wx wy otherwise;
10.96/4.40	"
10.96/4.40	"span2Zs xz yu = span2Zs0 xz yu (span2Vu43 xz yu);
10.96/4.40	"
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(10)
10.96/4.40	Obligation:
10.96/4.40	mainModule Main 
10.96/4.40	module Main where {
10.96/4.40	import qualified Prelude;
10.96/4.40	}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(11) Narrow (SOUND)
10.96/4.40	Haskell To QDPs
10.96/4.40	
10.96/4.40	digraph dp_graph {
10.96/4.40	node [outthreshold=100, inthreshold=100];1[label="readInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.96/4.40	3[label="readInt yv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.96/4.40	4[label="readInt yv3 yv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
10.96/4.40	5[label="readInt yv3 yv4 yv5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
10.96/4.40	6[label="readInt yv3 yv4 yv5 yv6",fontsize=16,color="black",shape="triangle"];6 -> 7[label="",style="solid", color="black", weight=3];
10.96/4.40	7[label="concatMap (readInt1 yv3 yv5) (nonnull yv4 yv6)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.96/4.40	8[label="concat . map (readInt1 yv3 yv5)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.96/4.40	9[label="concat (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10.96/4.40	10[label="foldr (++) [] (map (readInt1 yv3 yv5) (nonnull yv4 yv6))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
10.96/4.40	11[label="foldr (++) [] (map (readInt1 yv3 yv5) (concatMap nonnull0 (span yv4 yv6 : [])))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
10.96/4.40	12[label="foldr (++) [] (map (readInt1 yv3 yv5) (concat . map nonnull0))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
10.96/4.40	13[label="foldr (++) [] (map (readInt1 yv3 yv5) (concat (map nonnull0 (span yv4 yv6 : []))))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
10.96/4.40	14[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 (span yv4 yv6 : []))))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.96/4.40	15[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (nonnull0 (span yv4 yv6) : map nonnull0 [])))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.96/4.40	16[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull0 (span yv4 yv6) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
10.96/4.40	17[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 yv6) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="burlywood",shape="box"];278[label="yv6/yv60 : yv61",fontsize=10,color="white",style="solid",shape="box"];17 -> 278[label="",style="solid", color="burlywood", weight=9];
10.96/4.40	278 -> 18[label="",style="solid", color="burlywood", weight=3];
10.96/4.40	279[label="yv6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 279[label="",style="solid", color="burlywood", weight=9];
10.96/4.40	279 -> 19[label="",style="solid", color="burlywood", weight=3];
10.96/4.40	18[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 (yv60 : yv61)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
10.96/4.40	19[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
10.96/4.40	20[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2 yv4 (yv60 : yv61)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
10.96/4.40	21[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span3 yv4 []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
10.96/4.40	22 -> 24[label="",style="dashed", color="red", weight=0];
10.96/4.40	22[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 (yv4 yv60)) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];22 -> 25[label="",style="dashed", color="magenta", weight=3];
10.96/4.40	23[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 ([],[]) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
10.96/4.40	25[label="yv4 yv60",fontsize=16,color="green",shape="box"];25 -> 30[label="",style="dashed", color="green", weight=3];
10.96/4.40	24[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 yv7) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="burlywood",shape="triangle"];280[label="yv7/False",fontsize=10,color="white",style="solid",shape="box"];24 -> 280[label="",style="solid", color="burlywood", weight=9];
10.96/4.40	280 -> 28[label="",style="solid", color="burlywood", weight=3];
10.96/4.40	281[label="yv7/True",fontsize=10,color="white",style="solid",shape="box"];24 -> 281[label="",style="solid", color="burlywood", weight=9];
10.96/4.40	281 -> 29[label="",style="solid", color="burlywood", weight=3];
10.96/4.40	26[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) [] foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="triangle"];26 -> 31[label="",style="solid", color="black", weight=3];
10.96/4.40	30[label="yv60",fontsize=16,color="green",shape="box"];28[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 False) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.96/4.40	29[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span1 yv4 yv61 yv4 yv60 yv61 True) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
10.96/4.40	31[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];31 -> 34[label="",style="solid", color="black", weight=3];
10.96/4.40	32[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span0 yv4 yv61 yv4 yv60 yv61 otherwise) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3];
10.96/4.40	33[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3];
10.96/4.40	34[label="foldr (++) [] (map (readInt1 yv3 yv5) (foldr (++) [] []))",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
10.96/4.40	35[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 (span2Span0 yv4 yv61 yv4 yv60 yv61 True) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
10.96/4.40	36[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) ((yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : []) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
10.96/4.40	37[label="foldr (++) [] (map (readInt1 yv3 yv5) [])",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
10.96/4.40	38[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) nonnull00 ([],yv60 : yv61) foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3];
10.96/4.40	39[label="foldr (++) [] (map (readInt1 yv3 yv5) ((yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : [] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3];
10.96/4.40	40[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3];
10.96/4.40	41 -> 26[label="",style="dashed", color="red", weight=0];
10.96/4.40	41[label="foldr (++) [] (map (readInt1 yv3 yv5) ((++) [] foldr (++) [] (map nonnull0 [])))",fontsize=16,color="magenta"];42[label="foldr (++) [] (readInt1 yv3 yv5 (yv60 : span2Ys yv4 yv61,span2Zs yv4 yv61) : map (readInt1 yv3 yv5) ([] ++ foldr (++) [] (map nonnull0 [])))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
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10.96/4.40	273[label="primPlusNat yv190 yv1500",fontsize=16,color="magenta"];273 -> 276[label="",style="dashed", color="magenta", weight=3];
10.96/4.40	273 -> 277[label="",style="dashed", color="magenta", weight=3];
10.96/4.40	274[label="Succ yv300",fontsize=16,color="green",shape="box"];275[label="yv1400",fontsize=16,color="green",shape="box"];276[label="yv1500",fontsize=16,color="green",shape="box"];277[label="yv190",fontsize=16,color="green",shape="box"];}
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(12)
10.96/4.40	Complex Obligation (AND)
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(13)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_span2Zs0(yv4, yv610, yv611) -> new_span2Zs(yv4, yv611)
10.96/4.40	   new_span2Zs(yv4, :(yv610, yv611)) -> new_span2Zs0(yv4, yv610, yv611)
10.96/4.40	
10.96/4.40	R is empty.
10.96/4.40	Q is empty.
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(14) QDPSizeChangeProof (EQUIVALENT)
10.96/4.40	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/4.40	
10.96/4.40	From the DPs we obtained the following set of size-change graphs:
10.96/4.40	*new_span2Zs(yv4, :(yv610, yv611)) -> new_span2Zs0(yv4, yv610, yv611)
10.96/4.40	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
10.96/4.40	
10.96/4.40	
10.96/4.40	*new_span2Zs0(yv4, yv610, yv611) -> new_span2Zs(yv4, yv611)
10.96/4.40	The graph contains the following edges 1 >= 1, 3 >= 2
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(15)
10.96/4.40	YES
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(16)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_primMulNat(Succ(yv1400), Succ(yv300)) -> new_primMulNat(yv1400, Succ(yv300))
10.96/4.40	
10.96/4.40	R is empty.
10.96/4.40	Q is empty.
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(17) QDPSizeChangeProof (EQUIVALENT)
10.96/4.40	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/4.40	
10.96/4.40	From the DPs we obtained the following set of size-change graphs:
10.96/4.40	*new_primMulNat(Succ(yv1400), Succ(yv300)) -> new_primMulNat(yv1400, Succ(yv300))
10.96/4.40	The graph contains the following edges 1 > 1, 2 >= 2
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(18)
10.96/4.40	YES
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(19)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) -> new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611)
10.96/4.40	   new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
10.96/4.40	   new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
10.96/4.40	
10.96/4.40	The TRS R consists of the following rules:
10.96/4.40	
10.96/4.40	   new_primPlusInt0(yv140, yv30, Neg(yv150)) -> new_primMinusNat0(new_primMulNat0(yv140, yv30), yv150)
10.96/4.40	   new_primPlusNat0(Succ(yv190), Zero) -> Succ(yv190)
10.96/4.40	   new_primPlusNat0(Zero, Succ(yv1500)) -> Succ(yv1500)
10.96/4.40	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.96/4.40	   new_primMulNat0(Zero, Zero) -> Zero
10.96/4.40	   new_primPlusNat0(Zero, Zero) -> Zero
10.96/4.40	   new_primMinusNat0(Zero, Succ(yv1500)) -> Neg(Succ(yv1500))
10.96/4.40	   new_primPlusInt(yv140, yv30, Neg(yv150)) -> Neg(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
10.96/4.40	   new_primMulNat0(Succ(yv1400), Zero) -> Zero
10.96/4.40	   new_primMulNat0(Zero, Succ(yv300)) -> Zero
10.96/4.40	   new_readInt0(Neg(yv30), Neg(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
10.96/4.40	   new_primPlusInt(yv140, yv30, Pos(yv150)) -> new_primMinusNat0(yv150, new_primMulNat0(yv140, yv30))
10.96/4.40	   new_readInt0(Pos(yv30), Pos(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
10.96/4.40	   new_primMulNat0(Succ(yv1400), Succ(yv300)) -> new_primPlusNat0(new_primMulNat0(yv1400, Succ(yv300)), Succ(yv300))
10.96/4.40	   new_readInt0(Neg(yv30), Pos(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
10.96/4.40	   new_readInt0(Pos(yv30), Neg(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
10.96/4.40	   new_primPlusNat0(Succ(yv190), Succ(yv1500)) -> Succ(Succ(new_primPlusNat0(yv190, yv1500)))
10.96/4.40	   new_primMinusNat0(Succ(yv180), Succ(yv1500)) -> new_primMinusNat0(yv180, yv1500)
10.96/4.40	   new_primPlusInt0(yv140, yv30, Pos(yv150)) -> Pos(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
10.96/4.40	   new_primMinusNat0(Succ(yv180), Zero) -> Pos(Succ(yv180))
10.96/4.40	
10.96/4.40	The set Q consists of the following terms:
10.96/4.40	
10.96/4.40	   new_primPlusNat0(Zero, Succ(x0))
10.96/4.40	   new_primMulNat0(Zero, Zero)
10.96/4.40	   new_readInt0(Pos(x0), Pos(x1), x2)
10.96/4.40	   new_primMulNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primPlusInt0(x0, x1, Pos(x2))
10.96/4.40	   new_primPlusInt(x0, x1, Pos(x2))
10.96/4.40	   new_primPlusInt0(x0, x1, Neg(x2))
10.96/4.40	   new_primMinusNat0(Zero, Zero)
10.96/4.40	   new_primPlusNat0(Succ(x0), Zero)
10.96/4.40	   new_readInt0(Neg(x0), Neg(x1), x2)
10.96/4.40	   new_primMinusNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primMinusNat0(Zero, Succ(x0))
10.96/4.40	   new_readInt0(Neg(x0), Pos(x1), x2)
10.96/4.40	   new_readInt0(Pos(x0), Neg(x1), x2)
10.96/4.40	   new_primPlusInt(x0, x1, Neg(x2))
10.96/4.40	   new_primPlusNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primMulNat0(Zero, Succ(x0))
10.96/4.40	   new_primMinusNat0(Succ(x0), Zero)
10.96/4.40	   new_primPlusNat0(Zero, Zero)
10.96/4.40	   new_primMulNat0(Succ(x0), Zero)
10.96/4.40	
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(20) TransformationProof (EQUIVALENT)
10.96/4.40	By instantiating [LPAR04] the rule new_foldl(yv3, yv14, yv5, yv4, yv610, yv611, yv12) -> new_foldl0(yv3, yv14, new_pt(yv5, yv610), yv5, yv4, yv611) we obtained the following new rules [LPAR04]:
10.96/4.40	
10.96/4.40	   (new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6),new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6))
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(21)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
10.96/4.40	   new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
10.96/4.40	   new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6)
10.96/4.40	
10.96/4.40	The TRS R consists of the following rules:
10.96/4.40	
10.96/4.40	   new_primPlusInt0(yv140, yv30, Neg(yv150)) -> new_primMinusNat0(new_primMulNat0(yv140, yv30), yv150)
10.96/4.40	   new_primPlusNat0(Succ(yv190), Zero) -> Succ(yv190)
10.96/4.40	   new_primPlusNat0(Zero, Succ(yv1500)) -> Succ(yv1500)
10.96/4.40	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.96/4.40	   new_primMulNat0(Zero, Zero) -> Zero
10.96/4.40	   new_primPlusNat0(Zero, Zero) -> Zero
10.96/4.40	   new_primMinusNat0(Zero, Succ(yv1500)) -> Neg(Succ(yv1500))
10.96/4.40	   new_primPlusInt(yv140, yv30, Neg(yv150)) -> Neg(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
10.96/4.40	   new_primMulNat0(Succ(yv1400), Zero) -> Zero
10.96/4.40	   new_primMulNat0(Zero, Succ(yv300)) -> Zero
10.96/4.40	   new_readInt0(Neg(yv30), Neg(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
10.96/4.40	   new_primPlusInt(yv140, yv30, Pos(yv150)) -> new_primMinusNat0(yv150, new_primMulNat0(yv140, yv30))
10.96/4.40	   new_readInt0(Pos(yv30), Pos(yv140), yv15) -> new_primPlusInt0(yv140, yv30, yv15)
10.96/4.40	   new_primMulNat0(Succ(yv1400), Succ(yv300)) -> new_primPlusNat0(new_primMulNat0(yv1400, Succ(yv300)), Succ(yv300))
10.96/4.40	   new_readInt0(Neg(yv30), Pos(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
10.96/4.40	   new_readInt0(Pos(yv30), Neg(yv140), yv15) -> new_primPlusInt(yv140, yv30, yv15)
10.96/4.40	   new_primPlusNat0(Succ(yv190), Succ(yv1500)) -> Succ(Succ(new_primPlusNat0(yv190, yv1500)))
10.96/4.40	   new_primMinusNat0(Succ(yv180), Succ(yv1500)) -> new_primMinusNat0(yv180, yv1500)
10.96/4.40	   new_primPlusInt0(yv140, yv30, Pos(yv150)) -> Pos(new_primPlusNat0(new_primMulNat0(yv140, yv30), yv150))
10.96/4.40	   new_primMinusNat0(Succ(yv180), Zero) -> Pos(Succ(yv180))
10.96/4.40	
10.96/4.40	The set Q consists of the following terms:
10.96/4.40	
10.96/4.40	   new_primPlusNat0(Zero, Succ(x0))
10.96/4.40	   new_primMulNat0(Zero, Zero)
10.96/4.40	   new_readInt0(Pos(x0), Pos(x1), x2)
10.96/4.40	   new_primMulNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primPlusInt0(x0, x1, Pos(x2))
10.96/4.40	   new_primPlusInt(x0, x1, Pos(x2))
10.96/4.40	   new_primPlusInt0(x0, x1, Neg(x2))
10.96/4.40	   new_primMinusNat0(Zero, Zero)
10.96/4.40	   new_primPlusNat0(Succ(x0), Zero)
10.96/4.40	   new_readInt0(Neg(x0), Neg(x1), x2)
10.96/4.40	   new_primMinusNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primMinusNat0(Zero, Succ(x0))
10.96/4.40	   new_readInt0(Neg(x0), Pos(x1), x2)
10.96/4.40	   new_readInt0(Pos(x0), Neg(x1), x2)
10.96/4.40	   new_primPlusInt(x0, x1, Neg(x2))
10.96/4.40	   new_primPlusNat0(Succ(x0), Succ(x1))
10.96/4.40	   new_primMulNat0(Zero, Succ(x0))
10.96/4.40	   new_primMinusNat0(Succ(x0), Zero)
10.96/4.40	   new_primPlusNat0(Zero, Zero)
10.96/4.40	   new_primMulNat0(Succ(x0), Zero)
10.96/4.40	
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(22) QDPSizeChangeProof (EQUIVALENT)
10.96/4.40	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/4.40	
10.96/4.40	From the DPs we obtained the following set of size-change graphs:
10.96/4.40	*new_foldl(z0, y_0, z3, z4, z5, z6, new_span2Zs1(z4, z6)) -> new_foldl0(z0, y_0, new_pt(z3, z5), z3, z4, z6)
10.96/4.40	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5, 7 > 5, 6 >= 6, 7 > 6
10.96/4.40	
10.96/4.40	
10.96/4.40	*new_foldl0(yv3, yv14, yv15, yv5, yv4, :(yv6110, yv6111)) -> new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111)
10.96/4.40	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 6 > 7
10.96/4.40	
10.96/4.40	
10.96/4.40	*new_foldl1(yv3, yv14, yv15, yv5, yv4, yv6110, yv6111) -> new_foldl(yv3, new_readInt0(yv3, yv14, yv15), yv5, yv4, yv6110, yv6111, new_span2Zs1(yv4, yv6111))
10.96/4.40	The graph contains the following edges 1 >= 1, 4 >= 3, 5 >= 4, 6 >= 5, 7 >= 6
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(23)
10.96/4.40	YES
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(24)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_primMinusNat(Succ(yv180), Succ(yv1500)) -> new_primMinusNat(yv180, yv1500)
10.96/4.40	
10.96/4.40	R is empty.
10.96/4.40	Q is empty.
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(25) QDPSizeChangeProof (EQUIVALENT)
10.96/4.40	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/4.40	
10.96/4.40	From the DPs we obtained the following set of size-change graphs:
10.96/4.40	*new_primMinusNat(Succ(yv180), Succ(yv1500)) -> new_primMinusNat(yv180, yv1500)
10.96/4.40	The graph contains the following edges 1 > 1, 2 > 2
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(26)
10.96/4.40	YES
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(27)
10.96/4.40	Obligation:
10.96/4.40	Q DP problem:
10.96/4.40	The TRS P consists of the following rules:
10.96/4.40	
10.96/4.40	   new_primPlusNat(Succ(yv190), Succ(yv1500)) -> new_primPlusNat(yv190, yv1500)
10.96/4.40	
10.96/4.40	R is empty.
10.96/4.40	Q is empty.
10.96/4.40	We have to consider all minimal (P,Q,R)-chains.
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(28) QDPSizeChangeProof (EQUIVALENT)
10.96/4.40	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
10.96/4.40	
10.96/4.40	From the DPs we obtained the following set of size-change graphs:
10.96/4.40	*new_primPlusNat(Succ(yv190), Succ(yv1500)) -> new_primPlusNat(yv190, yv1500)
10.96/4.40	The graph contains the following edges 1 > 1, 2 > 2
10.96/4.40	
10.96/4.40	
10.96/4.40	----------------------------------------
10.96/4.40	
10.96/4.40	(29)
10.96/4.40	YES
11.23/4.45	EOF