8.38/3.61	YES
9.98/4.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.98/4.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.98/4.12	
9.98/4.12	
9.98/4.12	H-Termination with start terms of the given HASKELL could be proven:
9.98/4.12	
9.98/4.12	(0) HASKELL
9.98/4.12	(1) BR [EQUIVALENT, 0 ms]
9.98/4.12	(2) HASKELL
9.98/4.12	(3) COR [EQUIVALENT, 0 ms]
9.98/4.12	(4) HASKELL
9.98/4.12	(5) Narrow [SOUND, 0 ms]
9.98/4.12	(6) QDP
9.98/4.12	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.98/4.12	(8) YES
9.98/4.12	
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(0)
9.98/4.12	Obligation:
9.98/4.12	mainModule Main 
9.98/4.12	module Main where {
9.98/4.12	import qualified Prelude;
9.98/4.12	data List a = Cons a (List a)  | Nil ;
9.98/4.12	
9.98/4.12	data MyBool = MyTrue  | MyFalse ;
9.98/4.12	
9.98/4.12	data Tup2 a b = Tup2 a b ;
9.98/4.12	
9.98/4.12	otherwise :: MyBool;
9.98/4.12	otherwise = MyTrue;
9.98/4.12	
9.98/4.12	span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
9.98/4.12	span p Nil = span3 p Nil;
9.98/4.12	span p (Cons vv vw) = span2 p (Cons vv vw);
9.98/4.12	
9.98/4.12	span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);
9.98/4.12	
9.98/4.12	span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);
9.98/4.12	
9.98/4.12	span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
9.98/4.12	span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;
9.98/4.12	
9.98/4.12	span2Vu43 wx wy = span wx wy;
9.98/4.12	
9.98/4.12	span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Ys0 wx wy (Tup2 ys vx) = ys;
9.98/4.12	
9.98/4.12	span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Zs0 wx wy (Tup2 vy zs) = zs;
9.98/4.12	
9.98/4.12	span3 p Nil = Tup2 Nil Nil;
9.98/4.12	span3 wv ww = span2 wv ww;
9.98/4.12	
9.98/4.12	}
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(1) BR (EQUIVALENT)
9.98/4.12	Replaced joker patterns by fresh variables and removed binding patterns.
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(2)
9.98/4.12	Obligation:
9.98/4.12	mainModule Main 
9.98/4.12	module Main where {
9.98/4.12	import qualified Prelude;
9.98/4.12	data List a = Cons a (List a)  | Nil ;
9.98/4.12	
9.98/4.12	data MyBool = MyTrue  | MyFalse ;
9.98/4.12	
9.98/4.12	data Tup2 b a = Tup2 b a ;
9.98/4.12	
9.98/4.12	otherwise :: MyBool;
9.98/4.12	otherwise = MyTrue;
9.98/4.12	
9.98/4.12	span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
9.98/4.12	span p Nil = span3 p Nil;
9.98/4.12	span p (Cons vv vw) = span2 p (Cons vv vw);
9.98/4.12	
9.98/4.12	span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);
9.98/4.12	
9.98/4.12	span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);
9.98/4.12	
9.98/4.12	span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
9.98/4.12	span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;
9.98/4.12	
9.98/4.12	span2Vu43 wx wy = span wx wy;
9.98/4.12	
9.98/4.12	span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Ys0 wx wy (Tup2 ys vx) = ys;
9.98/4.12	
9.98/4.12	span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Zs0 wx wy (Tup2 vy zs) = zs;
9.98/4.12	
9.98/4.12	span3 p Nil = Tup2 Nil Nil;
9.98/4.12	span3 wv ww = span2 wv ww;
9.98/4.12	
9.98/4.12	}
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(3) COR (EQUIVALENT)
9.98/4.12	Cond Reductions:
9.98/4.12	The following Function with conditions
9.98/4.12	"undefined |Falseundefined;
9.98/4.12	"
9.98/4.12	is transformed to
9.98/4.12	"undefined  = undefined1;
9.98/4.12	"
9.98/4.12	"undefined0 True = undefined;
9.98/4.12	"
9.98/4.12	"undefined1  = undefined0 False;
9.98/4.12	"
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(4)
9.98/4.12	Obligation:
9.98/4.12	mainModule Main 
9.98/4.12	module Main where {
9.98/4.12	import qualified Prelude;
9.98/4.12	data List a = Cons a (List a)  | Nil ;
9.98/4.12	
9.98/4.12	data MyBool = MyTrue  | MyFalse ;
9.98/4.12	
9.98/4.12	data Tup2 a b = Tup2 a b ;
9.98/4.12	
9.98/4.12	otherwise :: MyBool;
9.98/4.12	otherwise = MyTrue;
9.98/4.12	
9.98/4.12	span :: (a  ->  MyBool)  ->  List a  ->  Tup2 (List a) (List a);
9.98/4.12	span p Nil = span3 p Nil;
9.98/4.12	span p (Cons vv vw) = span2 p (Cons vv vw);
9.98/4.12	
9.98/4.12	span2 p (Cons vv vw) = span2Span1 p vw p vv vw (p vv);
9.98/4.12	
9.98/4.12	span2Span0 wx wy p vv vw MyTrue = Tup2 Nil (Cons vv vw);
9.98/4.12	
9.98/4.12	span2Span1 wx wy p vv vw MyTrue = Tup2 (Cons vv (span2Ys wx wy)) (span2Zs wx wy);
9.98/4.12	span2Span1 wx wy p vv vw MyFalse = span2Span0 wx wy p vv vw otherwise;
9.98/4.12	
9.98/4.12	span2Vu43 wx wy = span wx wy;
9.98/4.12	
9.98/4.12	span2Ys wx wy = span2Ys0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Ys0 wx wy (Tup2 ys vx) = ys;
9.98/4.12	
9.98/4.12	span2Zs wx wy = span2Zs0 wx wy (span2Vu43 wx wy);
9.98/4.12	
9.98/4.12	span2Zs0 wx wy (Tup2 vy zs) = zs;
9.98/4.12	
9.98/4.12	span3 p Nil = Tup2 Nil Nil;
9.98/4.12	span3 wv ww = span2 wv ww;
9.98/4.12	
9.98/4.12	}
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(5) Narrow (SOUND)
9.98/4.12	Haskell To QDPs
9.98/4.12	
9.98/4.12	digraph dp_graph {
9.98/4.12	node [outthreshold=100, inthreshold=100];1[label="span",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.98/4.12	3[label="span wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.98/4.12	4[label="span wz3 wz4",fontsize=16,color="burlywood",shape="triangle"];39[label="wz4/Cons wz40 wz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	39 -> 5[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	40[label="wz4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	40 -> 6[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	5[label="span wz3 (Cons wz40 wz41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.98/4.12	6[label="span wz3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.98/4.12	7[label="span2 wz3 (Cons wz40 wz41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.98/4.12	8[label="span3 wz3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.98/4.12	9 -> 11[label="",style="dashed", color="red", weight=0];
9.98/4.12	9[label="span2Span1 wz3 wz41 wz3 wz40 wz41 (wz3 wz40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9.98/4.12	10[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];12[label="wz3 wz40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
9.98/4.12	11[label="span2Span1 wz3 wz41 wz3 wz40 wz41 wz5",fontsize=16,color="burlywood",shape="triangle"];41[label="wz5/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 41[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	41 -> 14[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	42[label="wz5/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 42[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	42 -> 15[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	16[label="wz40",fontsize=16,color="green",shape="box"];14[label="span2Span1 wz3 wz41 wz3 wz40 wz41 MyTrue",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.98/4.12	15[label="span2Span1 wz3 wz41 wz3 wz40 wz41 MyFalse",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.98/4.12	17[label="Tup2 (Cons wz40 (span2Ys wz3 wz41)) (span2Zs wz3 wz41)",fontsize=16,color="green",shape="box"];17 -> 19[label="",style="dashed", color="green", weight=3];
9.98/4.12	17 -> 20[label="",style="dashed", color="green", weight=3];
9.98/4.12	18[label="span2Span0 wz3 wz41 wz3 wz40 wz41 otherwise",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.98/4.12	19[label="span2Ys wz3 wz41",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
9.98/4.12	20[label="span2Zs wz3 wz41",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
9.98/4.12	21[label="span2Span0 wz3 wz41 wz3 wz40 wz41 MyTrue",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
9.98/4.12	22 -> 27[label="",style="dashed", color="red", weight=0];
9.98/4.12	22[label="span2Ys0 wz3 wz41 (span2Vu43 wz3 wz41)",fontsize=16,color="magenta"];22 -> 28[label="",style="dashed", color="magenta", weight=3];
9.98/4.12	23 -> 32[label="",style="dashed", color="red", weight=0];
9.98/4.12	23[label="span2Zs0 wz3 wz41 (span2Vu43 wz3 wz41)",fontsize=16,color="magenta"];23 -> 33[label="",style="dashed", color="magenta", weight=3];
9.98/4.12	24[label="Tup2 Nil (Cons wz40 wz41)",fontsize=16,color="green",shape="box"];28[label="span2Vu43 wz3 wz41",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
9.98/4.12	27[label="span2Ys0 wz3 wz41 wz6",fontsize=16,color="burlywood",shape="triangle"];43[label="wz6/Tup2 wz60 wz61",fontsize=10,color="white",style="solid",shape="box"];27 -> 43[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	43 -> 31[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	33 -> 28[label="",style="dashed", color="red", weight=0];
9.98/4.12	33[label="span2Vu43 wz3 wz41",fontsize=16,color="magenta"];32[label="span2Zs0 wz3 wz41 wz7",fontsize=16,color="burlywood",shape="triangle"];44[label="wz7/Tup2 wz70 wz71",fontsize=10,color="white",style="solid",shape="box"];32 -> 44[label="",style="solid", color="burlywood", weight=9];
9.98/4.12	44 -> 35[label="",style="solid", color="burlywood", weight=3];
9.98/4.12	30 -> 4[label="",style="dashed", color="red", weight=0];
9.98/4.12	30[label="span wz3 wz41",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3];
9.98/4.12	31[label="span2Ys0 wz3 wz41 (Tup2 wz60 wz61)",fontsize=16,color="black",shape="box"];31 -> 37[label="",style="solid", color="black", weight=3];
9.98/4.12	35[label="span2Zs0 wz3 wz41 (Tup2 wz70 wz71)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
9.98/4.12	36[label="wz41",fontsize=16,color="green",shape="box"];37[label="wz60",fontsize=16,color="green",shape="box"];38[label="wz71",fontsize=16,color="green",shape="box"];}
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(6)
9.98/4.12	Obligation:
9.98/4.12	Q DP problem:
9.98/4.12	The TRS P consists of the following rules:
9.98/4.12	
9.98/4.12	   new_span2Vu43(wz3, wz41, h) -> new_span(wz3, wz41, h)
9.98/4.12	   new_span(wz3, Cons(wz40, wz41), h) -> new_span2Span1(wz3, wz41, wz40, h)
9.98/4.12	   new_span2Span1(wz3, wz41, wz40, h) -> new_span(wz3, wz41, h)
9.98/4.12	   new_span2Span1(wz3, wz41, wz40, h) -> new_span2Vu43(wz3, wz41, h)
9.98/4.12	
9.98/4.12	R is empty.
9.98/4.12	Q is empty.
9.98/4.12	We have to consider all minimal (P,Q,R)-chains.
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(7) QDPSizeChangeProof (EQUIVALENT)
9.98/4.12	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. 
9.98/4.12	
9.98/4.12	From the DPs we obtained the following set of size-change graphs:
9.98/4.12	*new_span(wz3, Cons(wz40, wz41), h) -> new_span2Span1(wz3, wz41, wz40, h)
9.98/4.12	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4
9.98/4.12	
9.98/4.12	
9.98/4.12	*new_span2Span1(wz3, wz41, wz40, h) -> new_span2Vu43(wz3, wz41, h)
9.98/4.12	The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
9.98/4.12	
9.98/4.12	
9.98/4.12	*new_span2Span1(wz3, wz41, wz40, h) -> new_span(wz3, wz41, h)
9.98/4.12	The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
9.98/4.12	
9.98/4.12	
9.98/4.12	*new_span2Vu43(wz3, wz41, h) -> new_span(wz3, wz41, h)
9.98/4.12	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
9.98/4.12	
9.98/4.12	
9.98/4.12	----------------------------------------
9.98/4.12	
9.98/4.12	(8)
9.98/4.12	YES
10.35/4.15	EOF