10.34/4.37	YES
12.28/4.96	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.28/4.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.28/4.96	
12.28/4.96	
12.28/4.96	H-Termination with start terms of the given HASKELL could be proven:
12.28/4.96	
12.28/4.96	(0) HASKELL
12.28/4.96	(1) IFR [EQUIVALENT, 0 ms]
12.28/4.96	(2) HASKELL
12.28/4.96	(3) BR [EQUIVALENT, 0 ms]
12.28/4.96	(4) HASKELL
12.28/4.96	(5) COR [EQUIVALENT, 31 ms]
12.28/4.96	(6) HASKELL
12.28/4.96	(7) Narrow [SOUND, 0 ms]
12.28/4.96	(8) AND
12.28/4.96	    (9) QDP
12.28/4.96	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
12.28/4.96	        (11) AND
12.28/4.96	            (12) QDP
12.28/4.96	                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.28/4.96	                (14) YES
12.28/4.96	            (15) QDP
12.28/4.96	                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.28/4.96	                (17) YES
12.28/4.96	    (18) QDP
12.28/4.96	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.28/4.96	        (20) YES
12.28/4.96	
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(0)
12.28/4.96	Obligation:
12.28/4.96	mainModule Main 
12.28/4.96	module Maybe where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	module List where {
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	infix 5 \\;
12.28/4.96	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.28/4.96	(\\) = foldl (flip delete);
12.28/4.96	
12.28/4.96	delete :: Eq a => a  ->  [a]  ->  [a];
12.28/4.96	delete = deleteBy (==);
12.28/4.96	
12.28/4.96	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.28/4.96	deleteBy _ _ [] = [];
12.28/4.96	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.28/4.96	
12.28/4.96	}
12.28/4.96	module Main where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(1) IFR (EQUIVALENT)
12.28/4.96	If Reductions:
12.28/4.96	The following If expression
12.28/4.96	"if eq x y then ys else y : deleteBy eq x ys"
12.28/4.96	is transformed to
12.28/4.96	"deleteBy0 ys y eq x True = ys;
12.28/4.96	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.28/4.96	"
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(2)
12.28/4.96	Obligation:
12.28/4.96	mainModule Main 
12.28/4.96	module Maybe where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	module List where {
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	infix 5 \\;
12.28/4.96	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.28/4.96	(\\) = foldl (flip delete);
12.28/4.96	
12.28/4.96	delete :: Eq a => a  ->  [a]  ->  [a];
12.28/4.96	delete = deleteBy (==);
12.28/4.96	
12.28/4.96	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.28/4.96	deleteBy _ _ [] = [];
12.28/4.96	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.28/4.96	
12.28/4.96	deleteBy0 ys y eq x True = ys;
12.28/4.96	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.28/4.96	
12.28/4.96	}
12.28/4.96	module Main where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(3) BR (EQUIVALENT)
12.28/4.96	Replaced joker patterns by fresh variables and removed binding patterns.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(4)
12.28/4.96	Obligation:
12.28/4.96	mainModule Main 
12.28/4.96	module Maybe where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	module List where {
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	infix 5 \\;
12.28/4.96	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.28/4.96	(\\) = foldl (flip delete);
12.28/4.96	
12.28/4.96	delete :: Eq a => a  ->  [a]  ->  [a];
12.28/4.96	delete = deleteBy (==);
12.28/4.96	
12.28/4.96	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.28/4.96	deleteBy vy vz [] = [];
12.28/4.96	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.28/4.96	
12.28/4.96	deleteBy0 ys y eq x True = ys;
12.28/4.96	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.28/4.96	
12.28/4.96	}
12.28/4.96	module Main where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(5) COR (EQUIVALENT)
12.28/4.96	Cond Reductions:
12.28/4.96	The following Function with conditions
12.28/4.96	"undefined |Falseundefined;
12.28/4.96	"
12.28/4.96	is transformed to
12.28/4.96	"undefined  = undefined1;
12.28/4.96	"
12.28/4.96	"undefined0 True = undefined;
12.28/4.96	"
12.28/4.96	"undefined1  = undefined0 False;
12.28/4.96	"
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(6)
12.28/4.96	Obligation:
12.28/4.96	mainModule Main 
12.28/4.96	module Maybe where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	module List where {
12.28/4.96	import qualified Main;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	infix 5 \\;
12.28/4.96	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.28/4.96	(\\) = foldl (flip delete);
12.28/4.96	
12.28/4.96	delete :: Eq a => a  ->  [a]  ->  [a];
12.28/4.96	delete = deleteBy (==);
12.28/4.96	
12.28/4.96	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.28/4.96	deleteBy vy vz [] = [];
12.28/4.96	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.28/4.96	
12.28/4.96	deleteBy0 ys y eq x True = ys;
12.28/4.96	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.28/4.96	
12.28/4.96	}
12.28/4.96	module Main where {
12.28/4.96	import qualified List;
12.28/4.96	import qualified Maybe;
12.28/4.96	import qualified Prelude;
12.28/4.96	}
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(7) Narrow (SOUND)
12.28/4.96	Haskell To QDPs
12.28/4.96	
12.28/4.96	digraph dp_graph {
12.28/4.96	node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.28/4.96	3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.28/4.96	4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.28/4.96	5[label="foldl (flip List.delete) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];38[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 38[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	38 -> 6[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	39[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 39[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	39 -> 7[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.28/4.96	7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.28/4.96	8 -> 5[label="",style="dashed", color="red", weight=0];
12.28/4.96	8[label="foldl (flip List.delete) (flip List.delete wu3 wu40) wu41",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	8 -> 11[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	9[label="wu3",fontsize=16,color="green",shape="box"];10[label="flip List.delete wu3 wu40",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
12.28/4.96	11[label="wu41",fontsize=16,color="green",shape="box"];12[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.28/4.96	13[label="List.deleteBy (==) wu40 wu3",fontsize=16,color="burlywood",shape="triangle"];40[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];13 -> 40[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	40 -> 14[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	41[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	41 -> 15[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.28/4.96	15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.28/4.96	16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="burlywood",shape="box"];42[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 42[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	42 -> 18[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	43[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 43[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	43 -> 19[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	17[label="[]",fontsize=16,color="green",shape="box"];18[label="List.deleteBy0 wu31 wu30 (==) False ((==) False wu30)",fontsize=16,color="burlywood",shape="box"];44[label="wu30/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 44[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	44 -> 20[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	45[label="wu30/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 45[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	45 -> 21[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	19[label="List.deleteBy0 wu31 wu30 (==) True ((==) True wu30)",fontsize=16,color="burlywood",shape="box"];46[label="wu30/False",fontsize=10,color="white",style="solid",shape="box"];19 -> 46[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	46 -> 22[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	47[label="wu30/True",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
12.28/4.96	47 -> 23[label="",style="solid", color="burlywood", weight=3];
12.28/4.96	20[label="List.deleteBy0 wu31 False (==) False ((==) False False)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.28/4.96	21[label="List.deleteBy0 wu31 True (==) False ((==) False True)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.28/4.96	22[label="List.deleteBy0 wu31 False (==) True ((==) True False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.28/4.96	23[label="List.deleteBy0 wu31 True (==) True ((==) True True)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.28/4.96	24[label="List.deleteBy0 wu31 False (==) False True",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.28/4.96	25[label="List.deleteBy0 wu31 True (==) False False",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.28/4.96	26[label="List.deleteBy0 wu31 False (==) True False",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.28/4.96	27[label="List.deleteBy0 wu31 True (==) True True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.28/4.96	28[label="wu31",fontsize=16,color="green",shape="box"];29[label="True : List.deleteBy (==) False wu31",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="green", weight=3];
12.28/4.96	30[label="False : List.deleteBy (==) True wu31",fontsize=16,color="green",shape="box"];30 -> 33[label="",style="dashed", color="green", weight=3];
12.28/4.96	31[label="wu31",fontsize=16,color="green",shape="box"];32 -> 13[label="",style="dashed", color="red", weight=0];
12.28/4.96	32[label="List.deleteBy (==) False wu31",fontsize=16,color="magenta"];32 -> 34[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	32 -> 35[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	33 -> 13[label="",style="dashed", color="red", weight=0];
12.28/4.96	33[label="List.deleteBy (==) True wu31",fontsize=16,color="magenta"];33 -> 36[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	33 -> 37[label="",style="dashed", color="magenta", weight=3];
12.28/4.96	34[label="wu31",fontsize=16,color="green",shape="box"];35[label="False",fontsize=16,color="green",shape="box"];36[label="wu31",fontsize=16,color="green",shape="box"];37[label="True",fontsize=16,color="green",shape="box"];}
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(8)
12.28/4.96	Complex Obligation (AND)
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(9)
12.28/4.96	Obligation:
12.28/4.96	Q DP problem:
12.28/4.96	The TRS P consists of the following rules:
12.28/4.96	
12.28/4.96	   new_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)
12.28/4.96	   new_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)
12.28/4.96	
12.28/4.96	R is empty.
12.28/4.96	Q is empty.
12.28/4.96	We have to consider all minimal (P,Q,R)-chains.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(10) DependencyGraphProof (EQUIVALENT)
12.28/4.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(11)
12.28/4.96	Complex Obligation (AND)
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(12)
12.28/4.96	Obligation:
12.28/4.96	Q DP problem:
12.28/4.96	The TRS P consists of the following rules:
12.28/4.96	
12.28/4.96	   new_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)
12.28/4.96	
12.28/4.96	R is empty.
12.28/4.96	Q is empty.
12.28/4.96	We have to consider all minimal (P,Q,R)-chains.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(13) QDPSizeChangeProof (EQUIVALENT)
12.28/4.96	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. 
12.28/4.96	
12.28/4.96	From the DPs we obtained the following set of size-change graphs:
12.28/4.96	*new_deleteBy(False, :(True, wu31)) -> new_deleteBy(False, wu31)
12.28/4.96	The graph contains the following edges 1 >= 1, 2 > 2
12.28/4.96	
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(14)
12.28/4.96	YES
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(15)
12.28/4.96	Obligation:
12.28/4.96	Q DP problem:
12.28/4.96	The TRS P consists of the following rules:
12.28/4.96	
12.28/4.96	   new_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)
12.28/4.96	
12.28/4.96	R is empty.
12.28/4.96	Q is empty.
12.28/4.96	We have to consider all minimal (P,Q,R)-chains.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(16) QDPSizeChangeProof (EQUIVALENT)
12.28/4.96	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. 
12.28/4.96	
12.28/4.96	From the DPs we obtained the following set of size-change graphs:
12.28/4.96	*new_deleteBy(True, :(False, wu31)) -> new_deleteBy(True, wu31)
12.28/4.96	The graph contains the following edges 1 >= 1, 2 > 2
12.28/4.96	
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(17)
12.28/4.96	YES
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(18)
12.28/4.96	Obligation:
12.28/4.96	Q DP problem:
12.28/4.96	The TRS P consists of the following rules:
12.28/4.96	
12.28/4.96	   new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)
12.28/4.96	
12.28/4.96	The TRS R consists of the following rules:
12.28/4.96	
12.28/4.96	   new_deleteBy0(wu40, []) -> []
12.28/4.96	   new_deleteBy0(True, :(False, wu31)) -> :(False, new_deleteBy0(True, wu31))
12.28/4.96	   new_deleteBy0(True, :(True, wu31)) -> wu31
12.28/4.96	   new_deleteBy0(False, :(True, wu31)) -> :(True, new_deleteBy0(False, wu31))
12.28/4.96	   new_deleteBy0(False, :(False, wu31)) -> wu31
12.28/4.96	
12.28/4.96	The set Q consists of the following terms:
12.28/4.96	
12.28/4.96	   new_deleteBy0(False, :(True, x0))
12.28/4.96	   new_deleteBy0(x0, [])
12.28/4.96	   new_deleteBy0(True, :(False, x0))
12.28/4.96	   new_deleteBy0(True, :(True, x0))
12.28/4.96	   new_deleteBy0(False, :(False, x0))
12.28/4.96	
12.28/4.96	We have to consider all minimal (P,Q,R)-chains.
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(19) QDPSizeChangeProof (EQUIVALENT)
12.28/4.96	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. 
12.28/4.96	
12.28/4.96	From the DPs we obtained the following set of size-change graphs:
12.28/4.96	*new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy0(wu40, wu3), wu41)
12.28/4.96	The graph contains the following edges 2 > 2
12.28/4.96	
12.28/4.96	
12.28/4.96	----------------------------------------
12.28/4.96	
12.28/4.96	(20)
12.28/4.96	YES
12.62/4.99	EOF