10.17/4.34	YES
12.42/4.90	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.42/4.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.42/4.90	
12.42/4.90	
12.42/4.90	H-Termination with start terms of the given HASKELL could be proven:
12.42/4.90	
12.42/4.90	(0) HASKELL
12.42/4.90	(1) IFR [EQUIVALENT, 0 ms]
12.42/4.90	(2) HASKELL
12.42/4.90	(3) BR [EQUIVALENT, 0 ms]
12.42/4.90	(4) HASKELL
12.42/4.90	(5) COR [EQUIVALENT, 26 ms]
12.42/4.90	(6) HASKELL
12.42/4.90	(7) Narrow [EQUIVALENT, 0 ms]
12.42/4.90	(8) YES
12.42/4.90	
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(0)
12.42/4.90	Obligation:
12.42/4.90	mainModule Main 
12.42/4.90	module Maybe where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	module List where {
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	delete :: Eq a => a  ->  [a]  ->  [a];
12.42/4.90	delete = deleteBy (==);
12.42/4.90	
12.42/4.90	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.42/4.90	deleteBy _ _ [] = [];
12.42/4.90	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.42/4.90	
12.42/4.90	}
12.42/4.90	module Main where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(1) IFR (EQUIVALENT)
12.42/4.90	If Reductions:
12.42/4.90	The following If expression
12.42/4.90	"if eq x y then ys else y : deleteBy eq x ys"
12.42/4.90	is transformed to
12.42/4.90	"deleteBy0 ys y eq x True = ys;
12.42/4.90	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.42/4.90	"
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(2)
12.42/4.90	Obligation:
12.42/4.90	mainModule Main 
12.42/4.90	module Maybe where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	module List where {
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	delete :: Eq a => a  ->  [a]  ->  [a];
12.42/4.90	delete = deleteBy (==);
12.42/4.90	
12.42/4.90	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.42/4.90	deleteBy _ _ [] = [];
12.42/4.90	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.42/4.90	
12.42/4.90	deleteBy0 ys y eq x True = ys;
12.42/4.90	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.42/4.90	
12.42/4.90	}
12.42/4.90	module Main where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(3) BR (EQUIVALENT)
12.42/4.90	Replaced joker patterns by fresh variables and removed binding patterns.
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(4)
12.42/4.90	Obligation:
12.42/4.90	mainModule Main 
12.42/4.90	module Maybe where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	module List where {
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	delete :: Eq a => a  ->  [a]  ->  [a];
12.42/4.90	delete = deleteBy (==);
12.42/4.90	
12.42/4.90	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.42/4.90	deleteBy vy vz [] = [];
12.42/4.90	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.42/4.90	
12.42/4.90	deleteBy0 ys y eq x True = ys;
12.42/4.90	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.42/4.90	
12.42/4.90	}
12.42/4.90	module Main where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(5) COR (EQUIVALENT)
12.42/4.90	Cond Reductions:
12.42/4.90	The following Function with conditions
12.42/4.90	"undefined |Falseundefined;
12.42/4.90	"
12.42/4.90	is transformed to
12.42/4.90	"undefined  = undefined1;
12.42/4.90	"
12.42/4.90	"undefined0 True = undefined;
12.42/4.90	"
12.42/4.90	"undefined1  = undefined0 False;
12.42/4.90	"
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(6)
12.42/4.90	Obligation:
12.42/4.90	mainModule Main 
12.42/4.90	module Maybe where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	module List where {
12.42/4.90	import qualified Main;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	delete :: Eq a => a  ->  [a]  ->  [a];
12.42/4.90	delete = deleteBy (==);
12.42/4.90	
12.42/4.90	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.42/4.90	deleteBy vy vz [] = [];
12.42/4.90	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.42/4.90	
12.42/4.90	deleteBy0 ys y eq x True = ys;
12.42/4.90	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.42/4.90	
12.42/4.90	}
12.42/4.90	module Main where {
12.42/4.90	import qualified List;
12.42/4.90	import qualified Maybe;
12.42/4.90	import qualified Prelude;
12.42/4.90	}
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(7) Narrow (EQUIVALENT)
12.42/4.90	Haskell To QDPs
12.42/4.90	
12.42/4.90	digraph dp_graph {
12.42/4.90	node [outthreshold=100, inthreshold=100];1[label="List.delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.42/4.90	3[label="List.delete wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.42/4.90	4[label="List.delete wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.42/4.90	5[label="List.deleteBy (==) wu3 wu4",fontsize=16,color="burlywood",shape="box"];14[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
12.42/4.90	14 -> 6[label="",style="solid", color="burlywood", weight=3];
12.42/4.90	15[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 15[label="",style="solid", color="burlywood", weight=9];
12.42/4.90	15 -> 7[label="",style="solid", color="burlywood", weight=3];
12.42/4.90	6[label="List.deleteBy (==) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.42/4.90	7[label="List.deleteBy (==) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.42/4.90	8[label="List.deleteBy0 wu41 wu40 (==) wu3 ((==) wu3 wu40)",fontsize=16,color="burlywood",shape="box"];16[label="wu3/()",fontsize=10,color="white",style="solid",shape="box"];8 -> 16[label="",style="solid", color="burlywood", weight=9];
12.42/4.90	16 -> 10[label="",style="solid", color="burlywood", weight=3];
12.42/4.90	9[label="[]",fontsize=16,color="green",shape="box"];10[label="List.deleteBy0 wu41 wu40 (==) () ((==) () wu40)",fontsize=16,color="burlywood",shape="box"];17[label="wu40/()",fontsize=10,color="white",style="solid",shape="box"];10 -> 17[label="",style="solid", color="burlywood", weight=9];
12.42/4.90	17 -> 11[label="",style="solid", color="burlywood", weight=3];
12.42/4.90	11[label="List.deleteBy0 wu41 () (==) () ((==) () ())",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.42/4.90	12[label="List.deleteBy0 wu41 () (==) () True",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.42/4.90	13[label="wu41",fontsize=16,color="green",shape="box"];}
12.42/4.90	
12.42/4.90	----------------------------------------
12.42/4.90	
12.42/4.90	(8)
12.42/4.90	YES
12.59/4.98	EOF