7.89/3.55	YES
9.31/4.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.31/4.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.31/4.01	
9.31/4.01	
9.31/4.01	H-Termination with start terms of the given HASKELL could be proven:
9.31/4.01	
9.31/4.01	(0) HASKELL
9.31/4.01	(1) BR [EQUIVALENT, 0 ms]
9.31/4.01	(2) HASKELL
9.31/4.01	(3) COR [EQUIVALENT, 0 ms]
9.31/4.01	(4) HASKELL
9.31/4.01	(5) Narrow [SOUND, 0 ms]
9.31/4.01	(6) QDP
9.31/4.01	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.31/4.01	(8) YES
9.31/4.01	
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(0)
9.31/4.01	Obligation:
9.31/4.01	mainModule Main 
9.31/4.01	module Main where {
9.31/4.01	import qualified Prelude;
9.31/4.01	data List a = Cons a (List a)  | Nil ;
9.31/4.01	
9.31/4.01	foldr1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.31/4.01	foldr1 f (Cons x Nil) = x;
9.31/4.01	foldr1 f (Cons x xs) = f x (foldr1 f xs);
9.31/4.01	
9.31/4.01	}
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(1) BR (EQUIVALENT)
9.31/4.01	Replaced joker patterns by fresh variables and removed binding patterns.
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(2)
9.31/4.01	Obligation:
9.31/4.01	mainModule Main 
9.31/4.01	module Main where {
9.31/4.01	import qualified Prelude;
9.31/4.01	data List a = Cons a (List a)  | Nil ;
9.31/4.01	
9.31/4.01	foldr1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.31/4.01	foldr1 f (Cons x Nil) = x;
9.31/4.01	foldr1 f (Cons x xs) = f x (foldr1 f xs);
9.31/4.01	
9.31/4.01	}
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(3) COR (EQUIVALENT)
9.31/4.01	Cond Reductions:
9.31/4.01	The following Function with conditions
9.31/4.01	"undefined |Falseundefined;
9.31/4.01	"
9.31/4.01	is transformed to
9.31/4.01	"undefined  = undefined1;
9.31/4.01	"
9.31/4.01	"undefined0 True = undefined;
9.31/4.01	"
9.31/4.01	"undefined1  = undefined0 False;
9.31/4.01	"
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(4)
9.31/4.01	Obligation:
9.31/4.01	mainModule Main 
9.31/4.01	module Main where {
9.31/4.01	import qualified Prelude;
9.31/4.01	data List a = Cons a (List a)  | Nil ;
9.31/4.01	
9.31/4.01	foldr1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.31/4.01	foldr1 f (Cons x Nil) = x;
9.31/4.01	foldr1 f (Cons x xs) = f x (foldr1 f xs);
9.31/4.01	
9.31/4.01	}
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(5) Narrow (SOUND)
9.31/4.01	Haskell To QDPs
9.31/4.01	
9.31/4.01	digraph dp_graph {
9.31/4.01	node [outthreshold=100, inthreshold=100];1[label="foldr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.31/4.01	3[label="foldr1 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.31/4.01	4[label="foldr1 vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];15[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
9.31/4.01	15 -> 5[label="",style="solid", color="burlywood", weight=3];
9.31/4.01	16[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
9.31/4.01	16 -> 6[label="",style="solid", color="burlywood", weight=3];
9.31/4.01	5[label="foldr1 vx3 (Cons vx40 vx41)",fontsize=16,color="burlywood",shape="box"];17[label="vx41/Cons vx410 vx411",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
9.31/4.01	17 -> 7[label="",style="solid", color="burlywood", weight=3];
9.31/4.01	18[label="vx41/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
9.31/4.01	18 -> 8[label="",style="solid", color="burlywood", weight=3];
9.31/4.01	6[label="foldr1 vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.31/4.01	7[label="foldr1 vx3 (Cons vx40 (Cons vx410 vx411))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.31/4.01	8[label="foldr1 vx3 (Cons vx40 Nil)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9.31/4.01	9[label="error []",fontsize=16,color="red",shape="box"];10[label="vx3 vx40 (foldr1 vx3 (Cons vx410 vx411))",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
9.31/4.01	10 -> 13[label="",style="dashed", color="green", weight=3];
9.31/4.01	11[label="vx40",fontsize=16,color="green",shape="box"];12[label="vx40",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
9.31/4.01	13[label="foldr1 vx3 (Cons vx410 vx411)",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
9.31/4.01	14[label="Cons vx410 vx411",fontsize=16,color="green",shape="box"];}
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(6)
9.31/4.01	Obligation:
9.31/4.01	Q DP problem:
9.31/4.01	The TRS P consists of the following rules:
9.31/4.01	
9.31/4.01	   new_foldr1(vx3, Cons(vx40, Cons(vx410, vx411)), h) -> new_foldr1(vx3, Cons(vx410, vx411), h)
9.31/4.01	
9.31/4.01	R is empty.
9.31/4.01	Q is empty.
9.31/4.01	We have to consider all minimal (P,Q,R)-chains.
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(7) QDPSizeChangeProof (EQUIVALENT)
9.31/4.01	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. 
9.31/4.01	
9.31/4.01	From the DPs we obtained the following set of size-change graphs:
9.31/4.01	*new_foldr1(vx3, Cons(vx40, Cons(vx410, vx411)), h) -> new_foldr1(vx3, Cons(vx410, vx411), h)
9.31/4.01	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.31/4.01	
9.31/4.01	
9.31/4.01	----------------------------------------
9.31/4.01	
9.31/4.01	(8)
9.31/4.01	YES
9.52/4.04	EOF