8.11/3.65	YES
9.54/4.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.54/4.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.54/4.15	
9.54/4.15	
9.54/4.15	H-Termination with start terms of the given HASKELL could be proven:
9.54/4.15	
9.54/4.15	(0) HASKELL
9.54/4.15	(1) BR [EQUIVALENT, 0 ms]
9.54/4.15	(2) HASKELL
9.54/4.15	(3) COR [EQUIVALENT, 0 ms]
9.54/4.15	(4) HASKELL
9.54/4.15	(5) Narrow [SOUND, 0 ms]
9.54/4.15	(6) QDP
9.54/4.15	(7) MRRProof [EQUIVALENT, 36 ms]
9.54/4.15	(8) QDP
9.54/4.15	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
9.54/4.15	(10) TRUE
9.54/4.15	
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(0)
9.54/4.15	Obligation:
9.54/4.15	mainModule Main 
9.54/4.15	module Main where {
9.54/4.15	import qualified Prelude;
9.54/4.15	data List a = Cons a (List a)  | Nil ;
9.54/4.15	
9.54/4.15	scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
9.54/4.15	scanr1 f Nil = Nil;
9.54/4.15	scanr1 f (Cons x Nil) = Cons x Nil;
9.54/4.15	scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);
9.54/4.15	
9.54/4.15	scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Q1 vw vx (Cons q vv) = q;
9.54/4.15	
9.54/4.15	scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Qs0 vw vx qs = qs;
9.54/4.15	
9.54/4.15	scanr1Vu41 vw vx = scanr1 vw vx;
9.54/4.15	
9.54/4.15	}
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(1) BR (EQUIVALENT)
9.54/4.15	Replaced joker patterns by fresh variables and removed binding patterns.
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(2)
9.54/4.15	Obligation:
9.54/4.15	mainModule Main 
9.54/4.15	module Main where {
9.54/4.15	import qualified Prelude;
9.54/4.15	data List a = Cons a (List a)  | Nil ;
9.54/4.15	
9.54/4.15	scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
9.54/4.15	scanr1 f Nil = Nil;
9.54/4.15	scanr1 f (Cons x Nil) = Cons x Nil;
9.54/4.15	scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);
9.54/4.15	
9.54/4.15	scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Q1 vw vx (Cons q vv) = q;
9.54/4.15	
9.54/4.15	scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Qs0 vw vx qs = qs;
9.54/4.15	
9.54/4.15	scanr1Vu41 vw vx = scanr1 vw vx;
9.54/4.15	
9.54/4.15	}
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(3) COR (EQUIVALENT)
9.54/4.15	Cond Reductions:
9.54/4.15	The following Function with conditions
9.54/4.15	"undefined |Falseundefined;
9.54/4.15	"
9.54/4.15	is transformed to
9.54/4.15	"undefined  = undefined1;
9.54/4.15	"
9.54/4.15	"undefined0 True = undefined;
9.54/4.15	"
9.54/4.15	"undefined1  = undefined0 False;
9.54/4.15	"
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(4)
9.54/4.15	Obligation:
9.54/4.15	mainModule Main 
9.54/4.15	module Main where {
9.54/4.15	import qualified Prelude;
9.54/4.15	data List a = Cons a (List a)  | Nil ;
9.54/4.15	
9.54/4.15	scanr1 :: (a  ->  a  ->  a)  ->  List a  ->  List a;
9.54/4.15	scanr1 f Nil = Nil;
9.54/4.15	scanr1 f (Cons x Nil) = Cons x Nil;
9.54/4.15	scanr1 f (Cons x xs) = Cons (f x (scanr1Q f xs)) (scanr1Qs f xs);
9.54/4.15	
9.54/4.15	scanr1Q vw vx = scanr1Q1 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Q1 vw vx (Cons q vv) = q;
9.54/4.15	
9.54/4.15	scanr1Qs vw vx = scanr1Qs0 vw vx (scanr1Vu41 vw vx);
9.54/4.15	
9.54/4.15	scanr1Qs0 vw vx qs = qs;
9.54/4.15	
9.54/4.15	scanr1Vu41 vw vx = scanr1 vw vx;
9.54/4.15	
9.54/4.15	}
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(5) Narrow (SOUND)
9.54/4.15	Haskell To QDPs
9.54/4.15	
9.54/4.15	digraph dp_graph {
9.54/4.15	node [outthreshold=100, inthreshold=100];1[label="scanr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.54/4.15	3[label="scanr1 wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.54/4.15	4[label="scanr1 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];29[label="wu4/Cons wu40 wu41",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	29 -> 5[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	30[label="wu4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	30 -> 6[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	5[label="scanr1 wu3 (Cons wu40 wu41)",fontsize=16,color="burlywood",shape="box"];31[label="wu41/Cons wu410 wu411",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	31 -> 7[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	32[label="wu41/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 32[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	32 -> 8[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	6[label="scanr1 wu3 Nil",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.54/4.15	7[label="scanr1 wu3 (Cons wu40 (Cons wu410 wu411))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.54/4.15	8[label="scanr1 wu3 (Cons wu40 Nil)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.54/4.15	9[label="Nil",fontsize=16,color="green",shape="box"];10[label="Cons (wu3 wu40 (scanr1Q wu3 (Cons wu410 wu411))) (scanr1Qs wu3 (Cons wu410 wu411))",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
9.54/4.15	10 -> 13[label="",style="dashed", color="green", weight=3];
9.54/4.15	11[label="Cons wu40 Nil",fontsize=16,color="green",shape="box"];12[label="wu3 wu40 (scanr1Q wu3 (Cons wu410 wu411))",fontsize=16,color="green",shape="box"];12 -> 14[label="",style="dashed", color="green", weight=3];
9.54/4.15	12 -> 15[label="",style="dashed", color="green", weight=3];
9.54/4.15	13[label="scanr1Qs wu3 (Cons wu410 wu411)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
9.54/4.15	14[label="wu40",fontsize=16,color="green",shape="box"];15[label="scanr1Q wu3 (Cons wu410 wu411)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
9.54/4.15	16[label="scanr1Qs0 wu3 (Cons wu410 wu411) (scanr1Vu41 wu3 (Cons wu410 wu411))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
9.54/4.15	17 -> 21[label="",style="dashed", color="red", weight=0];
9.54/4.15	17[label="scanr1Q1 wu3 (Cons wu410 wu411) (scanr1Vu41 wu3 (Cons wu410 wu411))",fontsize=16,color="magenta"];17 -> 22[label="",style="dashed", color="magenta", weight=3];
9.54/4.15	18[label="scanr1Vu41 wu3 (Cons wu410 wu411)",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
9.54/4.15	22 -> 18[label="",style="dashed", color="red", weight=0];
9.54/4.15	22[label="scanr1Vu41 wu3 (Cons wu410 wu411)",fontsize=16,color="magenta"];21[label="scanr1Q1 wu3 (Cons wu410 wu411) wu5",fontsize=16,color="burlywood",shape="triangle"];33[label="wu5/Cons wu50 wu51",fontsize=10,color="white",style="solid",shape="box"];21 -> 33[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	33 -> 24[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	34[label="wu5/Nil",fontsize=10,color="white",style="solid",shape="box"];21 -> 34[label="",style="solid", color="burlywood", weight=9];
9.54/4.15	34 -> 25[label="",style="solid", color="burlywood", weight=3];
9.54/4.15	20 -> 4[label="",style="dashed", color="red", weight=0];
9.54/4.15	20[label="scanr1 wu3 (Cons wu410 wu411)",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3];
9.54/4.15	24[label="scanr1Q1 wu3 (Cons wu410 wu411) (Cons wu50 wu51)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.54/4.15	25[label="scanr1Q1 wu3 (Cons wu410 wu411) Nil",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
9.54/4.15	26[label="Cons wu410 wu411",fontsize=16,color="green",shape="box"];27[label="wu50",fontsize=16,color="green",shape="box"];28[label="error []",fontsize=16,color="red",shape="box"];}
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(6)
9.54/4.15	Obligation:
9.54/4.15	Q DP problem:
9.54/4.15	The TRS P consists of the following rules:
9.54/4.15	
9.54/4.15	   new_scanr1(wu3, Cons(wu40, Cons(wu410, wu411)), h) -> new_scanr1(wu3, Cons(wu410, wu411), h)
9.54/4.15	   new_scanr1(wu3, Cons(wu40, Cons(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h)
9.54/4.15	   new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, Cons(wu410, wu411), h)
9.54/4.15	
9.54/4.15	R is empty.
9.54/4.15	Q is empty.
9.54/4.15	We have to consider all minimal (P,Q,R)-chains.
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(7) MRRProof (EQUIVALENT)
9.54/4.15	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
9.54/4.15	
9.54/4.15	Strictly oriented dependency pairs:
9.54/4.15	
9.54/4.15	   new_scanr1(wu3, Cons(wu40, Cons(wu410, wu411)), h) -> new_scanr1(wu3, Cons(wu410, wu411), h)
9.54/4.15	   new_scanr1(wu3, Cons(wu40, Cons(wu410, wu411)), h) -> new_scanr1Vu41(wu3, wu410, wu411, h)
9.54/4.15	
9.54/4.15	
9.54/4.15	Used ordering: Polynomial interpretation [POLO]:
9.54/4.15	
9.54/4.15	   POL(Cons(x_1, x_2)) = 1 + x_1 + 2*x_2
9.54/4.15	   POL(new_scanr1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
9.54/4.15	   POL(new_scanr1Vu41(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 + 2*x_3 + x_4
9.54/4.15	
9.54/4.15	
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(8)
9.54/4.15	Obligation:
9.54/4.15	Q DP problem:
9.54/4.15	The TRS P consists of the following rules:
9.54/4.15	
9.54/4.15	   new_scanr1Vu41(wu3, wu410, wu411, h) -> new_scanr1(wu3, Cons(wu410, wu411), h)
9.54/4.15	
9.54/4.15	R is empty.
9.54/4.15	Q is empty.
9.54/4.15	We have to consider all minimal (P,Q,R)-chains.
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(9) DependencyGraphProof (EQUIVALENT)
9.54/4.15	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
9.54/4.15	----------------------------------------
9.54/4.15	
9.54/4.15	(10)
9.54/4.15	TRUE
9.90/4.20	EOF