10.15/4.29	YES
12.79/4.97	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.79/4.97	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.79/4.97	
12.79/4.97	
12.79/4.97	H-Termination with start terms of the given HASKELL could be proven:
12.79/4.97	
12.79/4.97	(0) HASKELL
12.79/4.97	(1) LR [EQUIVALENT, 0 ms]
12.79/4.97	(2) HASKELL
12.79/4.97	(3) CR [EQUIVALENT, 0 ms]
12.79/4.97	(4) HASKELL
12.79/4.97	(5) IFR [EQUIVALENT, 0 ms]
12.79/4.97	(6) HASKELL
12.79/4.97	(7) BR [EQUIVALENT, 0 ms]
12.79/4.97	(8) HASKELL
12.79/4.97	(9) COR [EQUIVALENT, 0 ms]
12.79/4.97	(10) HASKELL
12.79/4.97	(11) Narrow [SOUND, 0 ms]
12.79/4.97	(12) QDP
12.79/4.97	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.79/4.97	(14) YES
12.79/4.97	
12.79/4.97	
12.79/4.97	----------------------------------------
12.79/4.97	
12.79/4.97	(0)
12.79/4.97	Obligation:
12.79/4.97	mainModule Main 
12.79/4.97	module Maybe where {
12.79/4.97	import qualified List;
12.79/4.97	import qualified Main;
12.79/4.97	import qualified Prelude;
12.79/4.97	listToMaybe :: [a]  ->  Maybe a;
12.79/4.97	listToMaybe [] = Nothing;
12.79/4.97	listToMaybe (a : _) = Just a;
12.79/4.97	
12.79/4.97	}
12.79/4.97	module List where {
12.79/4.97	import qualified Main;
12.79/4.97	import qualified Maybe;
12.79/4.97	import qualified Prelude;
12.79/4.97	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.97	find p = Maybe.listToMaybe . filter p;
12.79/4.97	
12.79/4.97	}
12.79/4.97	module Main where {
12.79/4.97	import qualified List;
12.79/4.97	import qualified Maybe;
12.79/4.97	import qualified Prelude;
12.79/4.97	}
12.79/4.97	
12.79/4.97	----------------------------------------
12.79/4.97	
12.79/4.97	(1) LR (EQUIVALENT)
12.79/4.97	Lambda Reductions:
12.79/4.97	The following Lambda expression
12.79/4.97	"\vu39->case vu39 of {
12.79/4.97	x  -> if p x then x : [] else [];
12.79/4.97	_  -> []}
12.79/4.97	"
12.79/4.97	is transformed to
12.79/4.97	"filter0 p vu39 = case vu39 of {
12.79/4.97	x  -> if p x then x : [] else [];
12.79/4.97	_  -> []}
12.79/4.97	;
12.79/4.97	"
12.79/4.97	
12.79/4.97	----------------------------------------
12.79/4.97	
12.79/4.97	(2)
12.79/4.97	Obligation:
12.79/4.97	mainModule Main 
12.79/4.97	module Maybe where {
12.79/4.97	import qualified List;
12.79/4.97	import qualified Main;
12.79/4.97	import qualified Prelude;
12.79/4.97	listToMaybe :: [a]  ->  Maybe a;
12.79/4.97	listToMaybe [] = Nothing;
12.79/4.97	listToMaybe (a : _) = Just a;
12.79/4.97	
12.79/4.97	}
12.79/4.97	module List where {
12.79/4.97	import qualified Main;
12.79/4.97	import qualified Maybe;
12.79/4.97	import qualified Prelude;
12.79/4.97	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.97	find p = Maybe.listToMaybe . filter p;
12.79/4.97	
12.79/4.97	}
12.79/4.97	module Main where {
12.79/4.97	import qualified List;
12.79/4.97	import qualified Maybe;
12.79/4.97	import qualified Prelude;
12.79/4.97	}
12.79/4.97	
12.79/4.97	----------------------------------------
12.79/4.97	
12.79/4.97	(3) CR (EQUIVALENT)
12.79/4.97	Case Reductions:
12.79/4.97	The following Case expression
12.79/4.97	"case vu39 of {
12.79/4.97	x  -> if p x then x : [] else [];
12.79/4.97	_  -> []}
12.79/4.97	"
12.79/4.97	is transformed to
12.79/4.98	"filter00 p x = if p x then x : [] else [];
12.79/4.98	filter00 p _ = [];
12.79/4.98	"
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(4)
12.79/4.98	Obligation:
12.79/4.98	mainModule Main 
12.79/4.98	module Maybe where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Prelude;
12.79/4.98	listToMaybe :: [a]  ->  Maybe a;
12.79/4.98	listToMaybe [] = Nothing;
12.79/4.98	listToMaybe (a : _) = Just a;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module List where {
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.98	find p = Maybe.listToMaybe . filter p;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module Main where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	}
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(5) IFR (EQUIVALENT)
12.79/4.98	If Reductions:
12.79/4.98	The following If expression
12.79/4.98	"if p x then x : [] else []"
12.79/4.98	is transformed to
12.79/4.98	"filter000 x True = x : [];
12.79/4.98	filter000 x False = [];
12.79/4.98	"
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(6)
12.79/4.98	Obligation:
12.79/4.98	mainModule Main 
12.79/4.98	module Maybe where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Prelude;
12.79/4.98	listToMaybe :: [a]  ->  Maybe a;
12.79/4.98	listToMaybe [] = Nothing;
12.79/4.98	listToMaybe (a : _) = Just a;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module List where {
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.98	find p = Maybe.listToMaybe . filter p;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module Main where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	}
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(7) BR (EQUIVALENT)
12.79/4.98	Replaced joker patterns by fresh variables and removed binding patterns.
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(8)
12.79/4.98	Obligation:
12.79/4.98	mainModule Main 
12.79/4.98	module Maybe where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Prelude;
12.79/4.98	listToMaybe :: [a]  ->  Maybe a;
12.79/4.98	listToMaybe [] = Nothing;
12.79/4.98	listToMaybe (a : vz) = Just a;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module List where {
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.98	find p = Maybe.listToMaybe . filter p;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module Main where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	}
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(9) COR (EQUIVALENT)
12.79/4.98	Cond Reductions:
12.79/4.98	The following Function with conditions
12.79/4.98	"undefined |Falseundefined;
12.79/4.98	"
12.79/4.98	is transformed to
12.79/4.98	"undefined  = undefined1;
12.79/4.98	"
12.79/4.98	"undefined0 True = undefined;
12.79/4.98	"
12.79/4.98	"undefined1  = undefined0 False;
12.79/4.98	"
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(10)
12.79/4.98	Obligation:
12.79/4.98	mainModule Main 
12.79/4.98	module Maybe where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Prelude;
12.79/4.98	listToMaybe :: [a]  ->  Maybe a;
12.79/4.98	listToMaybe [] = Nothing;
12.79/4.98	listToMaybe (a : vz) = Just a;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module List where {
12.79/4.98	import qualified Main;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	find :: (a  ->  Bool)  ->  [a]  ->  Maybe a;
12.79/4.98	find p = Maybe.listToMaybe . filter p;
12.79/4.98	
12.79/4.98	}
12.79/4.98	module Main where {
12.79/4.98	import qualified List;
12.79/4.98	import qualified Maybe;
12.79/4.98	import qualified Prelude;
12.79/4.98	}
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(11) Narrow (SOUND)
12.79/4.98	Haskell To QDPs
12.79/4.98	
12.79/4.98	digraph dp_graph {
12.79/4.98	node [outthreshold=100, inthreshold=100];1[label="List.find",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.79/4.98	3[label="List.find wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.79/4.98	4[label="List.find wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.79/4.98	5[label="Maybe.listToMaybe . filter wu3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
12.79/4.98	6[label="Maybe.listToMaybe (filter wu3 wu4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
12.79/4.98	7[label="Maybe.listToMaybe (concatMap (filter0 wu3) wu4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
12.79/4.98	8[label="Maybe.listToMaybe (concat . map (filter0 wu3))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
12.79/4.98	9[label="Maybe.listToMaybe (concat (map (filter0 wu3) wu4))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
12.79/4.98	10[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu4))",fontsize=16,color="burlywood",shape="triangle"];32[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];10 -> 32[label="",style="solid", color="burlywood", weight=9];
12.79/4.98	32 -> 11[label="",style="solid", color="burlywood", weight=3];
12.79/4.98	33[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 33[label="",style="solid", color="burlywood", weight=9];
12.79/4.98	33 -> 12[label="",style="solid", color="burlywood", weight=3];
12.79/4.98	11[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) (wu40 : wu41)))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12.79/4.98	12[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) []))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
12.79/4.98	13[label="Maybe.listToMaybe (foldr (++) [] (filter0 wu3 wu40 : map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
12.79/4.98	14[label="Maybe.listToMaybe (foldr (++) [] [])",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.79/4.98	15[label="Maybe.listToMaybe ((++) filter0 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.79/4.98	16[label="Maybe.listToMaybe []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
12.79/4.98	17[label="Maybe.listToMaybe ((++) filter00 wu3 wu40 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
12.79/4.98	18[label="Nothing",fontsize=16,color="green",shape="box"];19 -> 20[label="",style="dashed", color="red", weight=0];
12.79/4.98	19[label="Maybe.listToMaybe ((++) filter000 wu40 (wu3 wu40) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
12.79/4.98	21[label="wu3 wu40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
12.79/4.98	20[label="Maybe.listToMaybe ((++) filter000 wu40 wu5 foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="burlywood",shape="triangle"];34[label="wu5/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 34[label="",style="solid", color="burlywood", weight=9];
12.79/4.98	34 -> 23[label="",style="solid", color="burlywood", weight=3];
12.79/4.98	35[label="wu5/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 35[label="",style="solid", color="burlywood", weight=9];
12.79/4.98	35 -> 24[label="",style="solid", color="burlywood", weight=3];
12.79/4.98	25[label="wu40",fontsize=16,color="green",shape="box"];23[label="Maybe.listToMaybe ((++) filter000 wu40 False foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
12.79/4.98	24[label="Maybe.listToMaybe ((++) filter000 wu40 True foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
12.79/4.98	26[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
12.79/4.98	27[label="Maybe.listToMaybe ((++) (wu40 : []) foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
12.79/4.98	28 -> 10[label="",style="dashed", color="red", weight=0];
12.79/4.98	28[label="Maybe.listToMaybe (foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
12.79/4.98	29[label="Maybe.listToMaybe (wu40 : [] ++ foldr (++) [] (map (filter0 wu3) wu41))",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
12.79/4.98	30[label="wu41",fontsize=16,color="green",shape="box"];31[label="Just wu40",fontsize=16,color="green",shape="box"];}
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(12)
12.79/4.98	Obligation:
12.79/4.98	Q DP problem:
12.79/4.98	The TRS P consists of the following rules:
12.79/4.98	
12.79/4.98	   new_listToMaybe0(wu3, :(wu40, wu41), ba) -> new_listToMaybe(wu40, wu3, wu41, ba)
12.79/4.98	   new_listToMaybe(wu40, wu3, wu41, ba) -> new_listToMaybe0(wu3, wu41, ba)
12.79/4.98	
12.79/4.98	R is empty.
12.79/4.98	Q is empty.
12.79/4.98	We have to consider all minimal (P,Q,R)-chains.
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(13) QDPSizeChangeProof (EQUIVALENT)
12.79/4.98	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. 
12.79/4.98	
12.79/4.98	From the DPs we obtained the following set of size-change graphs:
12.79/4.98	*new_listToMaybe(wu40, wu3, wu41, ba) -> new_listToMaybe0(wu3, wu41, ba)
12.79/4.98	The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3
12.79/4.98	
12.79/4.98	
12.79/4.98	*new_listToMaybe0(wu3, :(wu40, wu41), ba) -> new_listToMaybe(wu40, wu3, wu41, ba)
12.79/4.98	The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 3 >= 4
12.79/4.98	
12.79/4.98	
12.79/4.98	----------------------------------------
12.79/4.98	
12.79/4.98	(14)
12.79/4.98	YES
12.79/5.03	EOF