10.08/5.24	YES
12.26/5.81	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.26/5.81	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.26/5.81	
12.26/5.81	
12.26/5.81	H-Termination with start terms of the given HASKELL could be proven:
12.26/5.81	
12.26/5.81	(0) HASKELL
12.26/5.81	(1) BR [EQUIVALENT, 0 ms]
12.26/5.81	(2) HASKELL
12.26/5.81	(3) COR [EQUIVALENT, 21 ms]
12.26/5.81	(4) HASKELL
12.26/5.81	(5) LetRed [EQUIVALENT, 0 ms]
12.26/5.81	(6) HASKELL
12.26/5.81	(7) Narrow [SOUND, 0 ms]
12.26/5.81	(8) QDP
12.26/5.81	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.26/5.81	(10) YES
12.26/5.81	
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(0)
12.26/5.81	Obligation:
12.26/5.81	mainModule Main 
12.26/5.81	module Maybe where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	module List where {
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	nub :: Eq a => [a]  ->  [a];
12.26/5.81	nub l = nub' l [] where { 
12.26/5.81	nub' [] _ = [];
12.26/5.81	nub' (x : xs) ls  | x `elem` ls = nub' xs ls
12.26/5.81	 | otherwise = x : nub' xs (x : ls);
12.26/5.81	 };
12.26/5.81	
12.26/5.81	}
12.26/5.81	module Main where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(1) BR (EQUIVALENT)
12.26/5.81	Replaced joker patterns by fresh variables and removed binding patterns.
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(2)
12.26/5.81	Obligation:
12.26/5.81	mainModule Main 
12.26/5.81	module Maybe where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	module List where {
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	nub :: Eq a => [a]  ->  [a];
12.26/5.81	nub l = nub' l [] where { 
12.26/5.81	nub' [] vy = [];
12.26/5.81	nub' (x : xs) ls  | x `elem` ls = nub' xs ls
12.26/5.81	 | otherwise = x : nub' xs (x : ls);
12.26/5.81	 };
12.26/5.81	
12.26/5.81	}
12.26/5.81	module Main where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(3) COR (EQUIVALENT)
12.26/5.81	Cond Reductions:
12.26/5.81	The following Function with conditions
12.26/5.81	"undefined |Falseundefined;
12.26/5.81	"
12.26/5.81	is transformed to
12.26/5.81	"undefined  = undefined1;
12.26/5.81	"
12.26/5.81	"undefined0 True = undefined;
12.26/5.81	"
12.26/5.81	"undefined1  = undefined0 False;
12.26/5.81	"
12.26/5.81	The following Function with conditions
12.26/5.81	"nub' [] vy = [];
12.26/5.81	nub' (x : xs) ls|x `elem` lsnub' xs ls|otherwisex : nub' xs (x : ls);
12.26/5.81	"
12.26/5.81	is transformed to
12.26/5.81	"nub' [] vy = nub'3 [] vy;
12.26/5.81	nub' (x : xs) ls = nub'2 (x : xs) ls;
12.26/5.81	"
12.26/5.81	"nub'0 x xs ls True = x : nub' xs (x : ls);
12.26/5.81	"
12.26/5.81	"nub'1 x xs ls True = nub' xs ls;
12.26/5.81	nub'1 x xs ls False = nub'0 x xs ls otherwise;
12.26/5.81	"
12.26/5.81	"nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls);
12.26/5.81	"
12.26/5.81	"nub'3 [] vy = [];
12.26/5.81	nub'3 wv ww = nub'2 wv ww;
12.26/5.81	"
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(4)
12.26/5.81	Obligation:
12.26/5.81	mainModule Main 
12.26/5.81	module Maybe where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	module List where {
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	nub :: Eq a => [a]  ->  [a];
12.26/5.81	nub l = nub' l [] where { 
12.26/5.81	nub' [] vy = nub'3 [] vy;
12.26/5.81	nub' (x : xs) ls = nub'2 (x : xs) ls;
12.26/5.81	nub'0 x xs ls True = x : nub' xs (x : ls);
12.26/5.81	nub'1 x xs ls True = nub' xs ls;
12.26/5.81	nub'1 x xs ls False = nub'0 x xs ls otherwise;
12.26/5.81	nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls);
12.26/5.81	nub'3 [] vy = [];
12.26/5.81	nub'3 wv ww = nub'2 wv ww;
12.26/5.81	 };
12.26/5.81	
12.26/5.81	}
12.26/5.81	module Main where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(5) LetRed (EQUIVALENT)
12.26/5.81	Let/Where Reductions:
12.26/5.81	The bindings of the following Let/Where expression
12.26/5.81	"nub' l [] where {
12.26/5.81	nub' [] vy = nub'3 [] vy;
12.26/5.81	nub' (x : xs) ls = nub'2 (x : xs) ls;
12.26/5.81	;
12.26/5.81	nub'0 x xs ls True = x : nub' xs (x : ls);
12.26/5.81	;
12.26/5.81	nub'1 x xs ls True = nub' xs ls;
12.26/5.81	nub'1 x xs ls False = nub'0 x xs ls otherwise;
12.26/5.81	;
12.26/5.81	nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls);
12.26/5.81	;
12.26/5.81	nub'3 [] vy = [];
12.26/5.81	nub'3 wv ww = nub'2 wv ww;
12.26/5.81	} 
12.26/5.81	"
12.26/5.81	are unpacked to the following functions on top level
12.26/5.81	"nubNub' [] vy = nubNub'3 [] vy;
12.26/5.81	nubNub' (x : xs) ls = nubNub'2 (x : xs) ls;
12.26/5.81	"
12.26/5.81	"nubNub'2 (x : xs) ls = nubNub'1 x xs ls (x `elem` ls);
12.26/5.81	"
12.26/5.81	"nubNub'0 x xs ls True = x : nubNub' xs (x : ls);
12.26/5.81	"
12.26/5.81	"nubNub'1 x xs ls True = nubNub' xs ls;
12.26/5.81	nubNub'1 x xs ls False = nubNub'0 x xs ls otherwise;
12.26/5.81	"
12.26/5.81	"nubNub'3 [] vy = [];
12.26/5.81	nubNub'3 wv ww = nubNub'2 wv ww;
12.26/5.81	"
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(6)
12.26/5.81	Obligation:
12.26/5.81	mainModule Main 
12.26/5.81	module Maybe where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	module List where {
12.26/5.81	import qualified Main;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	nub :: Eq a => [a]  ->  [a];
12.26/5.81	nub l = nubNub' l [];
12.26/5.81	
12.26/5.81	nubNub' [] vy = nubNub'3 [] vy;
12.26/5.81	nubNub' (x : xs) ls = nubNub'2 (x : xs) ls;
12.26/5.81	
12.26/5.81	nubNub'0 x xs ls True = x : nubNub' xs (x : ls);
12.26/5.81	
12.26/5.81	nubNub'1 x xs ls True = nubNub' xs ls;
12.26/5.81	nubNub'1 x xs ls False = nubNub'0 x xs ls otherwise;
12.26/5.81	
12.26/5.81	nubNub'2 (x : xs) ls = nubNub'1 x xs ls (x `elem` ls);
12.26/5.81	
12.26/5.81	nubNub'3 [] vy = [];
12.26/5.81	nubNub'3 wv ww = nubNub'2 wv ww;
12.26/5.81	
12.26/5.81	}
12.26/5.81	module Main where {
12.26/5.81	import qualified List;
12.26/5.81	import qualified Maybe;
12.26/5.81	import qualified Prelude;
12.26/5.81	}
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(7) Narrow (SOUND)
12.26/5.81	Haskell To QDPs
12.26/5.81	
12.26/5.81	digraph dp_graph {
12.26/5.81	node [outthreshold=100, inthreshold=100];1[label="List.nub",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.26/5.81	3[label="List.nub wx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
12.26/5.81	4[label="List.nubNub' wx3 []",fontsize=16,color="burlywood",shape="box"];42[label="wx3/wx30 : wx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 42[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	42 -> 5[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	43[label="wx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 43[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	43 -> 6[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	5[label="List.nubNub' (wx30 : wx31) []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.26/5.81	6[label="List.nubNub' [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.26/5.81	7[label="List.nubNub'2 (wx30 : wx31) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.26/5.81	8[label="List.nubNub'3 [] []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.26/5.81	9[label="List.nubNub'1 wx30 wx31 [] (wx30 `elem` [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.26/5.81	10[label="[]",fontsize=16,color="green",shape="box"];11[label="List.nubNub'1 wx30 wx31 [] (any . (==))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.26/5.81	12[label="List.nubNub'1 wx30 wx31 [] (any ((==) wx30) [])",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.26/5.81	13[label="List.nubNub'1 wx30 wx31 [] (or . map ((==) wx30))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
12.26/5.81	14[label="List.nubNub'1 wx30 wx31 [] (or (map ((==) wx30) []))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
12.26/5.81	15[label="List.nubNub'1 wx30 wx31 [] (foldr (||) False (map ((==) wx30) []))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
12.26/5.81	16[label="List.nubNub'1 wx30 wx31 [] (foldr (||) False [])",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
12.26/5.81	17[label="List.nubNub'1 wx30 wx31 [] False",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
12.26/5.81	18[label="List.nubNub'0 wx30 wx31 [] otherwise",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
12.26/5.81	19[label="List.nubNub'0 wx30 wx31 [] True",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
12.26/5.81	20[label="wx30 : List.nubNub' wx31 (wx30 : [])",fontsize=16,color="green",shape="box"];20 -> 21[label="",style="dashed", color="green", weight=3];
12.26/5.81	21[label="List.nubNub' wx31 (wx30 : [])",fontsize=16,color="burlywood",shape="triangle"];44[label="wx31/wx310 : wx311",fontsize=10,color="white",style="solid",shape="box"];21 -> 44[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	44 -> 22[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	45[label="wx31/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 45[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	45 -> 23[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	22[label="List.nubNub' (wx310 : wx311) (wx30 : [])",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
12.26/5.81	23[label="List.nubNub' [] (wx30 : [])",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
12.26/5.81	24[label="List.nubNub'2 (wx310 : wx311) (wx30 : [])",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
12.26/5.81	25[label="List.nubNub'3 [] (wx30 : [])",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
12.26/5.81	26[label="List.nubNub'1 wx310 wx311 (wx30 : []) (wx310 `elem` wx30 : [])",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
12.26/5.81	27[label="[]",fontsize=16,color="green",shape="box"];28[label="List.nubNub'1 wx310 wx311 (wx30 : []) (any . (==))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
12.26/5.81	29[label="List.nubNub'1 wx310 wx311 (wx30 : []) (any ((==) wx310) (wx30 : []))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
12.26/5.81	30[label="List.nubNub'1 wx310 wx311 (wx30 : []) (or . map ((==) wx310))",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
12.26/5.81	31[label="List.nubNub'1 wx310 wx311 (wx30 : []) (or (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
12.26/5.81	32[label="List.nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3];
12.26/5.81	33[label="List.nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (((==) wx310 wx30) : map ((==) wx310) []))",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
12.26/5.81	34[label="List.nubNub'1 wx310 wx311 (wx30 : []) ((||) (==) wx310 wx30 foldr (||) False (map ((==) wx310) []))",fontsize=16,color="burlywood",shape="box"];46[label="wx310/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 46[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	46 -> 35[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	35[label="List.nubNub'1 () wx311 (wx30 : []) ((||) (==) () wx30 foldr (||) False (map ((==) ()) []))",fontsize=16,color="burlywood",shape="box"];47[label="wx30/()",fontsize=10,color="white",style="solid",shape="box"];35 -> 47[label="",style="solid", color="burlywood", weight=9];
12.26/5.81	47 -> 36[label="",style="solid", color="burlywood", weight=3];
12.26/5.81	36[label="List.nubNub'1 () wx311 (() : []) ((||) (==) () () foldr (||) False (map ((==) ()) []))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
12.26/5.81	37[label="List.nubNub'1 () wx311 (() : []) ((||) True foldr (||) False (map ((==) ()) []))",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
12.26/5.81	38[label="List.nubNub'1 () wx311 (() : []) True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
12.26/5.81	39 -> 21[label="",style="dashed", color="red", weight=0];
12.26/5.81	39[label="List.nubNub' wx311 (() : [])",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
12.26/5.81	39 -> 41[label="",style="dashed", color="magenta", weight=3];
12.26/5.81	40[label="()",fontsize=16,color="green",shape="box"];41[label="wx311",fontsize=16,color="green",shape="box"];}
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(8)
12.26/5.81	Obligation:
12.26/5.81	Q DP problem:
12.26/5.81	The TRS P consists of the following rules:
12.26/5.81	
12.26/5.81	   new_nubNub'(:(@0, wx311), @0) -> new_nubNub'(wx311, @0)
12.26/5.81	
12.26/5.81	R is empty.
12.26/5.81	Q is empty.
12.26/5.81	We have to consider all minimal (P,Q,R)-chains.
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(9) QDPSizeChangeProof (EQUIVALENT)
12.26/5.81	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. 
12.26/5.81	
12.26/5.81	From the DPs we obtained the following set of size-change graphs:
12.26/5.81	*new_nubNub'(:(@0, wx311), @0) -> new_nubNub'(wx311, @0)
12.26/5.81	The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2
12.26/5.81	
12.26/5.81	
12.26/5.81	----------------------------------------
12.26/5.81	
12.26/5.81	(10)
12.26/5.81	YES
12.40/5.85	EOF