7.55/3.53	YES
8.88/3.98	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
8.88/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.88/3.98	
8.88/3.98	
8.88/3.98	H-Termination with start terms of the given HASKELL could be proven:
8.88/3.98	
8.88/3.98	(0) HASKELL
8.88/3.98	(1) BR [EQUIVALENT, 0 ms]
8.88/3.98	(2) HASKELL
8.88/3.98	(3) COR [EQUIVALENT, 0 ms]
8.88/3.98	(4) HASKELL
8.88/3.98	(5) Narrow [SOUND, 0 ms]
8.88/3.98	(6) QDP
8.88/3.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
8.88/3.98	(8) YES
8.88/3.98	
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(0)
8.88/3.98	Obligation:
8.88/3.98	mainModule Main 
8.88/3.98	module Main where {
8.88/3.98	import qualified Prelude;
8.88/3.98	}
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(1) BR (EQUIVALENT)
8.88/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(2)
8.88/3.98	Obligation:
8.88/3.98	mainModule Main 
8.88/3.98	module Main where {
8.88/3.98	import qualified Prelude;
8.88/3.98	}
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(3) COR (EQUIVALENT)
8.88/3.98	Cond Reductions:
8.88/3.98	The following Function with conditions
8.88/3.98	"undefined |Falseundefined;
8.88/3.98	"
8.88/3.98	is transformed to
8.88/3.98	"undefined  = undefined1;
8.88/3.98	"
8.88/3.98	"undefined0 True = undefined;
8.88/3.98	"
8.88/3.98	"undefined1  = undefined0 False;
8.88/3.98	"
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(4)
8.88/3.98	Obligation:
8.88/3.98	mainModule Main 
8.88/3.98	module Main where {
8.88/3.98	import qualified Prelude;
8.88/3.98	}
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(5) Narrow (SOUND)
8.88/3.98	Haskell To QDPs
8.88/3.98	
8.88/3.98	digraph dp_graph {
8.88/3.98	node [outthreshold=100, inthreshold=100];1[label="foldr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
8.88/3.98	3[label="foldr1 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
8.88/3.98	4[label="foldr1 vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];15[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
8.88/3.98	15 -> 5[label="",style="solid", color="burlywood", weight=3];
8.88/3.98	16[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
8.88/3.98	16 -> 6[label="",style="solid", color="burlywood", weight=3];
8.88/3.98	5[label="foldr1 vx3 (vx40 : vx41)",fontsize=16,color="burlywood",shape="box"];17[label="vx41/vx410 : vx411",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
8.88/3.98	17 -> 7[label="",style="solid", color="burlywood", weight=3];
8.88/3.98	18[label="vx41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
8.88/3.98	18 -> 8[label="",style="solid", color="burlywood", weight=3];
8.88/3.98	6[label="foldr1 vx3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
8.88/3.98	7[label="foldr1 vx3 (vx40 : vx410 : vx411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8.88/3.98	8[label="foldr1 vx3 (vx40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
8.88/3.98	9[label="error []",fontsize=16,color="red",shape="box"];10[label="vx3 vx40 (foldr1 vx3 (vx410 : vx411))",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
8.88/3.98	10 -> 13[label="",style="dashed", color="green", weight=3];
8.88/3.98	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];
8.88/3.98	13[label="foldr1 vx3 (vx410 : vx411)",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
8.88/3.98	14[label="vx410 : vx411",fontsize=16,color="green",shape="box"];}
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(6)
8.88/3.98	Obligation:
8.88/3.98	Q DP problem:
8.88/3.98	The TRS P consists of the following rules:
8.88/3.98	
8.88/3.98	   new_foldr1(vx3, :(vx40, :(vx410, vx411)), h) -> new_foldr1(vx3, :(vx410, vx411), h)
8.88/3.98	
8.88/3.98	R is empty.
8.88/3.98	Q is empty.
8.88/3.98	We have to consider all minimal (P,Q,R)-chains.
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(7) QDPSizeChangeProof (EQUIVALENT)
8.88/3.98	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. 
8.88/3.98	
8.88/3.98	From the DPs we obtained the following set of size-change graphs:
8.88/3.98	*new_foldr1(vx3, :(vx40, :(vx410, vx411)), h) -> new_foldr1(vx3, :(vx410, vx411), h)
8.88/3.98	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
8.88/3.98	
8.88/3.98	
8.88/3.98	----------------------------------------
8.88/3.98	
8.88/3.98	(8)
8.88/3.98	YES
9.12/4.02	EOF