7.75/3.52	YES
9.23/3.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.23/3.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.23/3.94	
9.23/3.94	
9.23/3.94	H-Termination with start terms of the given HASKELL could be proven:
9.23/3.94	
9.23/3.94	(0) HASKELL
9.23/3.94	(1) BR [EQUIVALENT, 0 ms]
9.23/3.94	(2) HASKELL
9.23/3.94	(3) COR [EQUIVALENT, 0 ms]
9.23/3.94	(4) HASKELL
9.23/3.94	(5) Narrow [SOUND, 0 ms]
9.23/3.94	(6) QDP
9.23/3.94	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.23/3.94	(8) YES
9.23/3.94	
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(0)
9.23/3.94	Obligation:
9.23/3.94	mainModule Main 
9.23/3.94	module Main where {
9.23/3.94	import qualified Prelude;
9.23/3.94	}
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(1) BR (EQUIVALENT)
9.23/3.94	Replaced joker patterns by fresh variables and removed binding patterns.
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(2)
9.23/3.94	Obligation:
9.23/3.94	mainModule Main 
9.23/3.94	module Main where {
9.23/3.94	import qualified Prelude;
9.23/3.94	}
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(3) COR (EQUIVALENT)
9.23/3.94	Cond Reductions:
9.23/3.94	The following Function with conditions
9.23/3.94	"undefined |Falseundefined;
9.23/3.94	"
9.23/3.94	is transformed to
9.23/3.94	"undefined  = undefined1;
9.23/3.94	"
9.23/3.94	"undefined0 True = undefined;
9.23/3.94	"
9.23/3.94	"undefined1  = undefined0 False;
9.23/3.94	"
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(4)
9.23/3.94	Obligation:
9.23/3.94	mainModule Main 
9.23/3.94	module Main where {
9.23/3.94	import qualified Prelude;
9.23/3.94	}
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(5) Narrow (SOUND)
9.23/3.94	Haskell To QDPs
9.23/3.94	
9.23/3.94	digraph dp_graph {
9.23/3.94	node [outthreshold=100, inthreshold=100];1[label="all",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.23/3.94	3[label="all vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.23/3.94	4[label="all vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.23/3.94	5[label="and . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.23/3.94	6[label="and (map vx3 vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.23/3.94	7[label="foldr (&&) True (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];28[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.23/3.94	28 -> 8[label="",style="solid", color="burlywood", weight=3];
9.23/3.94	29[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 29[label="",style="solid", color="burlywood", weight=9];
9.23/3.94	29 -> 9[label="",style="solid", color="burlywood", weight=3];
9.23/3.94	8[label="foldr (&&) True (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.23/3.94	9[label="foldr (&&) True (map vx3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.23/3.94	10[label="foldr (&&) True (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.23/3.94	11[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.23/3.94	12 -> 17[label="",style="dashed", color="red", weight=0];
9.23/3.94	12[label="(&&) vx3 vx40 foldr (&&) True (map vx3 vx41)",fontsize=16,color="magenta"];12 -> 18[label="",style="dashed", color="magenta", weight=3];
9.23/3.94	12 -> 19[label="",style="dashed", color="magenta", weight=3];
9.23/3.94	13[label="True",fontsize=16,color="green",shape="box"];18[label="vx3 vx40",fontsize=16,color="green",shape="box"];18 -> 21[label="",style="dashed", color="green", weight=3];
9.23/3.94	19 -> 7[label="",style="dashed", color="red", weight=0];
9.23/3.94	19[label="foldr (&&) True (map vx3 vx41)",fontsize=16,color="magenta"];19 -> 22[label="",style="dashed", color="magenta", weight=3];
9.23/3.94	17[label="(&&) vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];30[label="vx6/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 30[label="",style="solid", color="burlywood", weight=9];
9.23/3.94	30 -> 23[label="",style="solid", color="burlywood", weight=3];
9.23/3.94	31[label="vx6/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 31[label="",style="solid", color="burlywood", weight=9];
9.23/3.94	31 -> 24[label="",style="solid", color="burlywood", weight=3];
9.23/3.94	21[label="vx40",fontsize=16,color="green",shape="box"];22[label="vx41",fontsize=16,color="green",shape="box"];23[label="(&&) False vx5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.23/3.94	24[label="(&&) True vx5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.23/3.94	26[label="False",fontsize=16,color="green",shape="box"];27[label="vx5",fontsize=16,color="green",shape="box"];}
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(6)
9.23/3.94	Obligation:
9.23/3.94	Q DP problem:
9.23/3.94	The TRS P consists of the following rules:
9.23/3.94	
9.23/3.94	   new_foldr(vx3, :(vx40, vx41), h) -> new_foldr(vx3, vx41, h)
9.23/3.94	
9.23/3.94	R is empty.
9.23/3.94	Q is empty.
9.23/3.94	We have to consider all minimal (P,Q,R)-chains.
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(7) QDPSizeChangeProof (EQUIVALENT)
9.23/3.94	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. 
9.23/3.94	
9.23/3.94	From the DPs we obtained the following set of size-change graphs:
9.23/3.94	*new_foldr(vx3, :(vx40, vx41), h) -> new_foldr(vx3, vx41, h)
9.23/3.94	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.23/3.94	
9.23/3.94	
9.23/3.94	----------------------------------------
9.23/3.94	
9.23/3.94	(8)
9.23/3.94	YES
9.30/4.00	EOF