8.27/3.59	YES
10.31/4.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.31/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.31/4.16	
10.31/4.16	
10.31/4.16	H-Termination with start terms of the given HASKELL could be proven:
10.31/4.16	
10.31/4.16	(0) HASKELL
10.31/4.16	(1) LR [EQUIVALENT, 0 ms]
10.31/4.16	(2) HASKELL
10.31/4.16	(3) BR [EQUIVALENT, 0 ms]
10.31/4.16	(4) HASKELL
10.31/4.16	(5) COR [EQUIVALENT, 0 ms]
10.31/4.16	(6) HASKELL
10.31/4.16	(7) LetRed [EQUIVALENT, 0 ms]
10.31/4.16	(8) HASKELL
10.31/4.16	(9) Narrow [SOUND, 0 ms]
10.31/4.16	(10) QDP
10.31/4.16	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.31/4.16	(12) YES
10.31/4.16	
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(0)
10.31/4.16	Obligation:
10.31/4.16	mainModule Main 
10.31/4.16	module Main where {
10.31/4.16	import qualified Prelude;
10.31/4.16	}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(1) LR (EQUIVALENT)
10.31/4.16	Lambda Reductions:
10.31/4.16	The following Lambda expression
10.31/4.16	"\(_,zs)->zs"
10.31/4.16	is transformed to
10.31/4.16	"zs0 (_,zs) = zs;
10.31/4.16	"
10.31/4.16	The following Lambda expression
10.31/4.16	"\(ys,_)->ys"
10.31/4.16	is transformed to
10.31/4.16	"ys0 (ys,_) = ys;
10.31/4.16	"
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(2)
10.31/4.16	Obligation:
10.31/4.16	mainModule Main 
10.31/4.16	module Main where {
10.31/4.16	import qualified Prelude;
10.31/4.16	}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(3) BR (EQUIVALENT)
10.31/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
10.31/4.16	
10.31/4.16	Binding Reductions:
10.31/4.16	The bind variable of the following binding Pattern
10.31/4.16	"xs@(vx : vy)"
10.31/4.16	is replaced by the following term
10.31/4.16	"vx : vy"
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(4)
10.31/4.16	Obligation:
10.31/4.16	mainModule Main 
10.31/4.16	module Main where {
10.31/4.16	import qualified Prelude;
10.31/4.16	}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(5) COR (EQUIVALENT)
10.31/4.16	Cond Reductions:
10.31/4.16	The following Function with conditions
10.31/4.16	"undefined |Falseundefined;
10.31/4.16	"
10.31/4.16	is transformed to
10.31/4.16	"undefined  = undefined1;
10.31/4.16	"
10.31/4.16	"undefined0 True = undefined;
10.31/4.16	"
10.31/4.16	"undefined1  = undefined0 False;
10.31/4.16	"
10.31/4.16	The following Function with conditions
10.31/4.16	"span p [] = ([],[]);
10.31/4.16	span p (vx : vy)|p vx(vx : ys,zs)|otherwise([],vx : vy) where {
10.31/4.16	vu43  = span p vy;
10.31/4.16	;
10.31/4.16	ys  = ys0 vu43;
10.31/4.16	;
10.31/4.16	ys0 (ys,wu) = ys;
10.31/4.16	;
10.31/4.16	zs  = zs0 vu43;
10.31/4.16	;
10.31/4.16	zs0 (vz,zs) = zs;
10.31/4.16	} 
10.31/4.16	;
10.31/4.16	"
10.31/4.16	is transformed to
10.31/4.16	"span p [] = span3 p [];
10.31/4.16	span p (vx : vy) = span2 p (vx : vy);
10.31/4.16	"
10.31/4.16	"span2 p (vx : vy) = span1 p vx vy (p vx) where {
10.31/4.16	span0 p vx vy True = ([],vx : vy);
10.31/4.16	;
10.31/4.16	span1 p vx vy True = (vx : ys,zs);
10.31/4.16	span1 p vx vy False = span0 p vx vy otherwise;
10.31/4.16	;
10.31/4.16	vu43  = span p vy;
10.31/4.16	;
10.31/4.16	ys  = ys0 vu43;
10.31/4.16	;
10.31/4.16	ys0 (ys,wu) = ys;
10.31/4.16	;
10.31/4.16	zs  = zs0 vu43;
10.31/4.16	;
10.31/4.16	zs0 (vz,zs) = zs;
10.31/4.16	} 
10.31/4.16	;
10.31/4.16	"
10.31/4.16	"span3 p [] = ([],[]);
10.31/4.16	span3 wx wy = span2 wx wy;
10.31/4.16	"
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(6)
10.31/4.16	Obligation:
10.31/4.16	mainModule Main 
10.31/4.16	module Main where {
10.31/4.16	import qualified Prelude;
10.31/4.16	}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(7) LetRed (EQUIVALENT)
10.31/4.16	Let/Where Reductions:
10.31/4.16	The bindings of the following Let/Where expression
10.31/4.16	"span1 p vx vy (p vx) where {
10.31/4.16	span0 p vx vy True = ([],vx : vy);
10.31/4.16	;
10.31/4.16	span1 p vx vy True = (vx : ys,zs);
10.31/4.16	span1 p vx vy False = span0 p vx vy otherwise;
10.31/4.16	;
10.31/4.16	vu43  = span p vy;
10.31/4.16	;
10.31/4.16	ys  = ys0 vu43;
10.31/4.16	;
10.31/4.16	ys0 (ys,wu) = ys;
10.31/4.16	;
10.31/4.16	zs  = zs0 vu43;
10.31/4.16	;
10.31/4.16	zs0 (vz,zs) = zs;
10.31/4.16	} 
10.31/4.16	"
10.31/4.16	are unpacked to the following functions on top level
10.31/4.16	"span2Zs wz xu = span2Zs0 wz xu (span2Vu43 wz xu);
10.31/4.16	"
10.31/4.16	"span2Span1 wz xu p vx vy True = (vx : span2Ys wz xu,span2Zs wz xu);
10.31/4.16	span2Span1 wz xu p vx vy False = span2Span0 wz xu p vx vy otherwise;
10.31/4.16	"
10.31/4.16	"span2Ys wz xu = span2Ys0 wz xu (span2Vu43 wz xu);
10.31/4.16	"
10.31/4.16	"span2Zs0 wz xu (vz,zs) = zs;
10.31/4.16	"
10.31/4.16	"span2Ys0 wz xu (ys,wu) = ys;
10.31/4.16	"
10.31/4.16	"span2Vu43 wz xu = span wz xu;
10.31/4.16	"
10.31/4.16	"span2Span0 wz xu p vx vy True = ([],vx : vy);
10.31/4.16	"
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(8)
10.31/4.16	Obligation:
10.31/4.16	mainModule Main 
10.31/4.16	module Main where {
10.31/4.16	import qualified Prelude;
10.31/4.16	}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(9) Narrow (SOUND)
10.31/4.16	Haskell To QDPs
10.31/4.16	
10.31/4.16	digraph dp_graph {
10.31/4.16	node [outthreshold=100, inthreshold=100];1[label="break",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.31/4.16	3[label="break xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.31/4.16	4[label="break xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.31/4.16	5[label="span (not . xv3) xv4",fontsize=16,color="burlywood",shape="triangle"];43[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];5 -> 43[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	43 -> 6[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	44[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	44 -> 7[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	6[label="span (not . xv3) (xv40 : xv41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.31/4.16	7[label="span (not . xv3) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.31/4.16	8[label="span2 (not . xv3) (xv40 : xv41)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
10.31/4.16	9[label="span3 (not . xv3) []",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.31/4.16	10[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not . xv3)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.31/4.16	11[label="([],[])",fontsize=16,color="green",shape="box"];12 -> 13[label="",style="dashed", color="red", weight=0];
10.31/4.16	12[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not (xv3 xv40))",fontsize=16,color="magenta"];12 -> 14[label="",style="dashed", color="magenta", weight=3];
10.31/4.16	14[label="xv3 xv40",fontsize=16,color="green",shape="box"];14 -> 18[label="",style="dashed", color="green", weight=3];
10.31/4.16	13[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not xv5)",fontsize=16,color="burlywood",shape="triangle"];45[label="xv5/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 45[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	45 -> 16[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	46[label="xv5/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 46[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	46 -> 17[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	18[label="xv40",fontsize=16,color="green",shape="box"];16[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not False)",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
10.31/4.16	17[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not True)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
10.31/4.16	19[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
10.31/4.16	20[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 False",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
10.31/4.16	21[label="(xv40 : span2Ys (not . xv3) xv41,span2Zs (not . xv3) xv41)",fontsize=16,color="green",shape="box"];21 -> 23[label="",style="dashed", color="green", weight=3];
10.31/4.16	21 -> 24[label="",style="dashed", color="green", weight=3];
10.31/4.16	22[label="span2Span0 (not . xv3) xv41 (not . xv3) xv40 xv41 otherwise",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
10.31/4.16	23[label="span2Ys (not . xv3) xv41",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
10.31/4.16	24[label="span2Zs (not . xv3) xv41",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
10.31/4.16	25[label="span2Span0 (not . xv3) xv41 (not . xv3) xv40 xv41 True",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
10.31/4.16	26 -> 31[label="",style="dashed", color="red", weight=0];
10.31/4.16	26[label="span2Ys0 (not . xv3) xv41 (span2Vu43 (not . xv3) xv41)",fontsize=16,color="magenta"];26 -> 32[label="",style="dashed", color="magenta", weight=3];
10.31/4.16	27 -> 36[label="",style="dashed", color="red", weight=0];
10.31/4.16	27[label="span2Zs0 (not . xv3) xv41 (span2Vu43 (not . xv3) xv41)",fontsize=16,color="magenta"];27 -> 37[label="",style="dashed", color="magenta", weight=3];
10.31/4.16	28[label="([],xv40 : xv41)",fontsize=16,color="green",shape="box"];32[label="span2Vu43 (not . xv3) xv41",fontsize=16,color="black",shape="triangle"];32 -> 34[label="",style="solid", color="black", weight=3];
10.31/4.16	31[label="span2Ys0 (not . xv3) xv41 xv6",fontsize=16,color="burlywood",shape="triangle"];47[label="xv6/(xv60,xv61)",fontsize=10,color="white",style="solid",shape="box"];31 -> 47[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	47 -> 35[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	37 -> 32[label="",style="dashed", color="red", weight=0];
10.31/4.16	37[label="span2Vu43 (not . xv3) xv41",fontsize=16,color="magenta"];36[label="span2Zs0 (not . xv3) xv41 xv7",fontsize=16,color="burlywood",shape="triangle"];48[label="xv7/(xv70,xv71)",fontsize=10,color="white",style="solid",shape="box"];36 -> 48[label="",style="solid", color="burlywood", weight=9];
10.31/4.16	48 -> 39[label="",style="solid", color="burlywood", weight=3];
10.31/4.16	34 -> 5[label="",style="dashed", color="red", weight=0];
10.31/4.16	34[label="span (not . xv3) xv41",fontsize=16,color="magenta"];34 -> 40[label="",style="dashed", color="magenta", weight=3];
10.31/4.16	35[label="span2Ys0 (not . xv3) xv41 (xv60,xv61)",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3];
10.31/4.16	39[label="span2Zs0 (not . xv3) xv41 (xv70,xv71)",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3];
10.31/4.16	40[label="xv41",fontsize=16,color="green",shape="box"];41[label="xv60",fontsize=16,color="green",shape="box"];42[label="xv71",fontsize=16,color="green",shape="box"];}
10.31/4.16	
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(10)
10.31/4.16	Obligation:
10.31/4.16	Q DP problem:
10.31/4.16	The TRS P consists of the following rules:
10.31/4.16	
10.31/4.16	   new_span(xv3, :(xv40, xv41), h) -> new_span2Span1(xv3, xv41, xv40, h)
10.31/4.16	   new_span2Vu43(xv3, xv41, h) -> new_span(xv3, xv41, h)
10.31/4.16	   new_span2Span1(xv3, xv41, xv40, h) -> new_span(xv3, xv41, h)
10.31/4.16	   new_span2Span1(xv3, xv41, xv40, h) -> new_span2Vu43(xv3, xv41, h)
10.31/4.16	
10.31/4.16	R is empty.
10.31/4.16	Q is empty.
10.31/4.16	We have to consider all minimal (P,Q,R)-chains.
10.31/4.16	----------------------------------------
10.31/4.16	
10.31/4.16	(11) QDPSizeChangeProof (EQUIVALENT)
10.31/4.16	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. 
10.31/4.16	
10.31/4.16	From the DPs we obtained the following set of size-change graphs:
10.31/4.17	*new_span2Span1(xv3, xv41, xv40, h) -> new_span(xv3, xv41, h)
10.31/4.17	The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
10.31/4.17	
10.31/4.17	
10.31/4.17	*new_span2Span1(xv3, xv41, xv40, h) -> new_span2Vu43(xv3, xv41, h)
10.31/4.17	The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
10.31/4.17	
10.31/4.17	
10.31/4.17	*new_span2Vu43(xv3, xv41, h) -> new_span(xv3, xv41, h)
10.31/4.17	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
10.31/4.17	
10.31/4.17	
10.31/4.17	*new_span(xv3, :(xv40, xv41), h) -> new_span2Span1(xv3, xv41, xv40, h)
10.31/4.17	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
10.31/4.17	
10.31/4.17	
10.31/4.17	----------------------------------------
10.31/4.17	
10.31/4.17	(12)
10.31/4.17	YES
10.31/4.20	EOF