9.69/4.39	YES
11.78/4.96	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
11.78/4.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.78/4.96	
11.78/4.96	
11.78/4.96	H-Termination with start terms of the given HASKELL could be proven:
11.78/4.96	
11.78/4.96	(0) HASKELL
11.78/4.96	(1) CR [EQUIVALENT, 0 ms]
11.78/4.96	(2) HASKELL
11.78/4.96	(3) BR [EQUIVALENT, 0 ms]
11.78/4.96	(4) HASKELL
11.78/4.96	(5) COR [EQUIVALENT, 0 ms]
11.78/4.96	(6) HASKELL
11.78/4.96	(7) Narrow [EQUIVALENT, 0 ms]
11.78/4.96	(8) YES
11.78/4.96	
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(0)
11.78/4.96	Obligation:
11.78/4.96	mainModule Main 
11.78/4.96	module FiniteMap where {
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.78/4.96	
11.78/4.96	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
11.78/4.96	}
11.78/4.96	lookupFM :: Ord a => FiniteMap a b  ->  a  ->  Maybe b;
11.78/4.96	lookupFM EmptyFM key = Nothing;
11.78/4.96	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
11.78/4.96	 | key_to_find > key = lookupFM fm_r key_to_find
11.78/4.96	 | otherwise = Just elt;
11.78/4.96	
11.78/4.96	lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
11.78/4.96	lookupWithDefaultFM fm deflt key = case lookupFM fm key of { 
11.78/4.96	Nothing-> deflt; 
11.78/4.96	Just elt-> elt; 
11.78/4.96	 } ;
11.78/4.96	
11.78/4.96	}
11.78/4.96	module Maybe where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	module Main where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(1) CR (EQUIVALENT)
11.78/4.96	Case Reductions:
11.78/4.96	The following Case expression
11.78/4.96	"case lookupFM fm key of {
11.78/4.96	Nothing  -> deflt;
11.78/4.96	Just elt  -> elt}
11.78/4.96	"
11.78/4.96	is transformed to
11.78/4.96	"lookupWithDefaultFM0 deflt Nothing = deflt;
11.78/4.96	lookupWithDefaultFM0 deflt (Just elt) = elt;
11.78/4.96	"
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(2)
11.78/4.96	Obligation:
11.78/4.96	mainModule Main 
11.78/4.96	module FiniteMap where {
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.78/4.96	
11.78/4.96	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
11.78/4.96	}
11.78/4.96	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
11.78/4.96	lookupFM EmptyFM key = Nothing;
11.78/4.96	lookupFM (Branch key elt _ fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
11.78/4.96	 | key_to_find > key = lookupFM fm_r key_to_find
11.78/4.96	 | otherwise = Just elt;
11.78/4.96	
11.78/4.96	lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
11.78/4.96	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
11.78/4.96	
11.78/4.96	lookupWithDefaultFM0 deflt Nothing = deflt;
11.78/4.96	lookupWithDefaultFM0 deflt (Just elt) = elt;
11.78/4.96	
11.78/4.96	}
11.78/4.96	module Maybe where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	module Main where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(3) BR (EQUIVALENT)
11.78/4.96	Replaced joker patterns by fresh variables and removed binding patterns.
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(4)
11.78/4.96	Obligation:
11.78/4.96	mainModule Main 
11.78/4.96	module FiniteMap where {
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.78/4.96	
11.78/4.96	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
11.78/4.96	}
11.78/4.96	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
11.78/4.96	lookupFM EmptyFM key = Nothing;
11.78/4.96	lookupFM (Branch key elt vy fm_l fm_r) key_to_find  | key_to_find < key = lookupFM fm_l key_to_find
11.78/4.96	 | key_to_find > key = lookupFM fm_r key_to_find
11.78/4.96	 | otherwise = Just elt;
11.78/4.96	
11.78/4.96	lookupWithDefaultFM :: Ord b => FiniteMap b a  ->  a  ->  b  ->  a;
11.78/4.96	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
11.78/4.96	
11.78/4.96	lookupWithDefaultFM0 deflt Nothing = deflt;
11.78/4.96	lookupWithDefaultFM0 deflt (Just elt) = elt;
11.78/4.96	
11.78/4.96	}
11.78/4.96	module Maybe where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	module Main where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(5) COR (EQUIVALENT)
11.78/4.96	Cond Reductions:
11.78/4.96	The following Function with conditions
11.78/4.96	"undefined |Falseundefined;
11.78/4.96	"
11.78/4.96	is transformed to
11.78/4.96	"undefined  = undefined1;
11.78/4.96	"
11.78/4.96	"undefined0 True = undefined;
11.78/4.96	"
11.78/4.96	"undefined1  = undefined0 False;
11.78/4.96	"
11.78/4.96	The following Function with conditions
11.78/4.96	"lookupFM EmptyFM key = Nothing;
11.78/4.96	lookupFM (Branch key elt vy fm_l fm_r) key_to_find|key_to_find < keylookupFM fm_l key_to_find|key_to_find > keylookupFM fm_r key_to_find|otherwiseJust elt;
11.78/4.96	"
11.78/4.96	is transformed to
11.78/4.96	"lookupFM EmptyFM key = lookupFM4 EmptyFM key;
11.78/4.96	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
11.78/4.96	"
11.78/4.96	"lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
11.78/4.96	lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);
11.78/4.96	"
11.78/4.96	"lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
11.78/4.96	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
11.78/4.96	"
11.78/4.96	"lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
11.78/4.96	"
11.78/4.96	"lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);
11.78/4.96	"
11.78/4.96	"lookupFM4 EmptyFM key = Nothing;
11.78/4.96	lookupFM4 wv ww = lookupFM3 wv ww;
11.78/4.96	"
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(6)
11.78/4.96	Obligation:
11.78/4.96	mainModule Main 
11.78/4.96	module FiniteMap where {
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.78/4.96	
11.78/4.96	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
11.78/4.96	}
11.78/4.96	lookupFM :: Ord b => FiniteMap b a  ->  b  ->  Maybe a;
11.78/4.96	lookupFM EmptyFM key = lookupFM4 EmptyFM key;
11.78/4.96	lookupFM (Branch key elt vy fm_l fm_r) key_to_find = lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find;
11.78/4.96	
11.78/4.96	lookupFM0 key elt vy fm_l fm_r key_to_find True = Just elt;
11.78/4.96	
11.78/4.96	lookupFM1 key elt vy fm_l fm_r key_to_find True = lookupFM fm_r key_to_find;
11.78/4.96	lookupFM1 key elt vy fm_l fm_r key_to_find False = lookupFM0 key elt vy fm_l fm_r key_to_find otherwise;
11.78/4.96	
11.78/4.96	lookupFM2 key elt vy fm_l fm_r key_to_find True = lookupFM fm_l key_to_find;
11.78/4.96	lookupFM2 key elt vy fm_l fm_r key_to_find False = lookupFM1 key elt vy fm_l fm_r key_to_find (key_to_find > key);
11.78/4.96	
11.78/4.96	lookupFM3 (Branch key elt vy fm_l fm_r) key_to_find = lookupFM2 key elt vy fm_l fm_r key_to_find (key_to_find < key);
11.78/4.96	
11.78/4.96	lookupFM4 EmptyFM key = Nothing;
11.78/4.96	lookupFM4 wv ww = lookupFM3 wv ww;
11.78/4.96	
11.78/4.96	lookupWithDefaultFM :: Ord a => FiniteMap a b  ->  b  ->  a  ->  b;
11.78/4.96	lookupWithDefaultFM fm deflt key = lookupWithDefaultFM0 deflt (lookupFM fm key);
11.78/4.96	
11.78/4.96	lookupWithDefaultFM0 deflt Nothing = deflt;
11.78/4.96	lookupWithDefaultFM0 deflt (Just elt) = elt;
11.78/4.96	
11.78/4.96	}
11.78/4.96	module Maybe where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Main;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	module Main where {
11.78/4.96	import qualified FiniteMap;
11.78/4.96	import qualified Maybe;
11.78/4.96	import qualified Prelude;
11.78/4.96	}
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(7) Narrow (EQUIVALENT)
11.78/4.96	Haskell To QDPs
11.78/4.96	
11.78/4.96	digraph dp_graph {
11.78/4.96	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.lookupWithDefaultFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.78/4.96	3[label="FiniteMap.lookupWithDefaultFM wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.78/4.96	4[label="FiniteMap.lookupWithDefaultFM wx3 wx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.78/4.96	5[label="FiniteMap.lookupWithDefaultFM wx3 wx4 wx5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
11.78/4.96	6[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM wx3 wx5)",fontsize=16,color="burlywood",shape="box"];27[label="wx3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 27[label="",style="solid", color="burlywood", weight=9];
11.78/4.96	27 -> 7[label="",style="solid", color="burlywood", weight=3];
11.78/4.96	28[label="wx3/FiniteMap.Branch wx30 wx31 wx32 wx33 wx34",fontsize=10,color="white",style="solid",shape="box"];6 -> 28[label="",style="solid", color="burlywood", weight=9];
11.78/4.96	28 -> 8[label="",style="solid", color="burlywood", weight=3];
11.78/4.96	7[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM FiniteMap.EmptyFM wx5)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.78/4.96	8[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.78/4.96	9[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM4 FiniteMap.EmptyFM wx5)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
11.78/4.96	10[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM3 (FiniteMap.Branch wx30 wx31 wx32 wx33 wx34) wx5)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11.78/4.96	11[label="FiniteMap.lookupWithDefaultFM0 wx4 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.78/4.96	12[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (wx5 < wx30))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.78/4.96	13[label="wx4",fontsize=16,color="green",shape="box"];14[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 wx5 (compare wx5 wx30 == LT))",fontsize=16,color="burlywood",shape="box"];29[label="wx5/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 29[label="",style="solid", color="burlywood", weight=9];
11.78/4.96	29 -> 15[label="",style="solid", color="burlywood", weight=3];
11.78/4.96	15[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 wx30 wx31 wx32 wx33 wx34 () (compare () wx30 == LT))",fontsize=16,color="burlywood",shape="box"];30[label="wx30/()",fontsize=10,color="white",style="solid",shape="box"];15 -> 30[label="",style="solid", color="burlywood", weight=9];
11.78/4.96	30 -> 16[label="",style="solid", color="burlywood", weight=3];
11.78/4.96	16[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () (compare () () == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.78/4.96	17[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () (EQ == LT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
11.78/4.96	18[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM2 () wx31 wx32 wx33 wx34 () False)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
11.78/4.96	19[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (() > ()))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
11.78/4.96	20[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (compare () () == GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
11.78/4.96	21[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () (EQ == GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
11.78/4.96	22[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM1 () wx31 wx32 wx33 wx34 () False)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
11.78/4.96	23[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 () wx31 wx32 wx33 wx34 () otherwise)",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
11.78/4.96	24[label="FiniteMap.lookupWithDefaultFM0 wx4 (FiniteMap.lookupFM0 () wx31 wx32 wx33 wx34 () True)",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3];
11.78/4.96	25[label="FiniteMap.lookupWithDefaultFM0 wx4 (Just wx31)",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
11.78/4.96	26[label="wx31",fontsize=16,color="green",shape="box"];}
11.78/4.96	
11.78/4.96	----------------------------------------
11.78/4.96	
11.78/4.96	(8)
11.78/4.96	YES
11.80/5.00	EOF