8.00/3.51	YES
9.52/4.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.52/4.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.52/4.02	
9.52/4.02	
9.52/4.02	H-Termination with start terms of the given HASKELL could be proven:
9.52/4.02	
9.52/4.02	(0) HASKELL
9.52/4.02	(1) BR [EQUIVALENT, 0 ms]
9.52/4.02	(2) HASKELL
9.52/4.02	(3) COR [EQUIVALENT, 0 ms]
9.52/4.02	(4) HASKELL
9.52/4.02	(5) Narrow [SOUND, 0 ms]
9.52/4.02	(6) QDP
9.52/4.02	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.52/4.02	(8) YES
9.52/4.02	
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(0)
9.52/4.02	Obligation:
9.52/4.02	mainModule Main 
9.52/4.02	module Main where {
9.52/4.02	import qualified Prelude;
9.52/4.02	data List a = Cons a (List a)  | Nil ;
9.52/4.02	
9.52/4.02	data MyBool = MyTrue  | MyFalse ;
9.52/4.02	
9.52/4.02	data Ordering = LT  | EQ  | GT ;
9.52/4.02	
9.52/4.02	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.52/4.02	foldl f z Nil = z;
9.52/4.02	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.52/4.02	
9.52/4.02	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.52/4.02	foldl1 f (Cons x xs) = foldl f x xs;
9.52/4.02	
9.52/4.02	ltEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.52/4.02	ltEsOrdering LT LT = MyTrue;
9.52/4.02	ltEsOrdering LT EQ = MyTrue;
9.52/4.02	ltEsOrdering LT GT = MyTrue;
9.52/4.02	ltEsOrdering EQ LT = MyFalse;
9.52/4.02	ltEsOrdering EQ EQ = MyTrue;
9.52/4.02	ltEsOrdering EQ GT = MyTrue;
9.52/4.02	ltEsOrdering GT LT = MyFalse;
9.52/4.02	ltEsOrdering GT EQ = MyFalse;
9.52/4.02	ltEsOrdering GT GT = MyTrue;
9.52/4.02	
9.52/4.02	min0 x y MyTrue = y;
9.52/4.02	
9.52/4.02	min1 x y MyTrue = x;
9.52/4.02	min1 x y MyFalse = min0 x y otherwise;
9.52/4.02	
9.52/4.02	min2 x y = min1 x y (ltEsOrdering x y);
9.52/4.02	
9.52/4.02	minOrdering :: Ordering  ->  Ordering  ->  Ordering;
9.52/4.02	minOrdering x y = min2 x y;
9.52/4.02	
9.52/4.02	minimumOrdering :: List Ordering  ->  Ordering;
9.52/4.02	minimumOrdering = foldl1 minOrdering;
9.52/4.02	
9.52/4.02	otherwise :: MyBool;
9.52/4.02	otherwise = MyTrue;
9.52/4.02	
9.52/4.02	}
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(1) BR (EQUIVALENT)
9.52/4.02	Replaced joker patterns by fresh variables and removed binding patterns.
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(2)
9.52/4.02	Obligation:
9.52/4.02	mainModule Main 
9.52/4.02	module Main where {
9.52/4.02	import qualified Prelude;
9.52/4.02	data List a = Cons a (List a)  | Nil ;
9.52/4.02	
9.52/4.02	data MyBool = MyTrue  | MyFalse ;
9.52/4.02	
9.52/4.02	data Ordering = LT  | EQ  | GT ;
9.52/4.02	
9.52/4.02	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.52/4.02	foldl f z Nil = z;
9.52/4.02	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.52/4.02	
9.52/4.02	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.52/4.02	foldl1 f (Cons x xs) = foldl f x xs;
9.52/4.02	
9.52/4.02	ltEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.52/4.02	ltEsOrdering LT LT = MyTrue;
9.52/4.02	ltEsOrdering LT EQ = MyTrue;
9.52/4.02	ltEsOrdering LT GT = MyTrue;
9.52/4.02	ltEsOrdering EQ LT = MyFalse;
9.52/4.02	ltEsOrdering EQ EQ = MyTrue;
9.52/4.02	ltEsOrdering EQ GT = MyTrue;
9.52/4.02	ltEsOrdering GT LT = MyFalse;
9.52/4.02	ltEsOrdering GT EQ = MyFalse;
9.52/4.02	ltEsOrdering GT GT = MyTrue;
9.52/4.02	
9.52/4.02	min0 x y MyTrue = y;
9.52/4.02	
9.52/4.02	min1 x y MyTrue = x;
9.52/4.02	min1 x y MyFalse = min0 x y otherwise;
9.52/4.02	
9.52/4.02	min2 x y = min1 x y (ltEsOrdering x y);
9.52/4.02	
9.52/4.02	minOrdering :: Ordering  ->  Ordering  ->  Ordering;
9.52/4.02	minOrdering x y = min2 x y;
9.52/4.02	
9.52/4.02	minimumOrdering :: List Ordering  ->  Ordering;
9.52/4.02	minimumOrdering = foldl1 minOrdering;
9.52/4.02	
9.52/4.02	otherwise :: MyBool;
9.52/4.02	otherwise = MyTrue;
9.52/4.02	
9.52/4.02	}
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(3) COR (EQUIVALENT)
9.52/4.02	Cond Reductions:
9.52/4.02	The following Function with conditions
9.52/4.02	"undefined |Falseundefined;
9.52/4.02	"
9.52/4.02	is transformed to
9.52/4.02	"undefined  = undefined1;
9.52/4.02	"
9.52/4.02	"undefined0 True = undefined;
9.52/4.02	"
9.52/4.02	"undefined1  = undefined0 False;
9.52/4.02	"
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(4)
9.52/4.02	Obligation:
9.52/4.02	mainModule Main 
9.52/4.02	module Main where {
9.52/4.02	import qualified Prelude;
9.52/4.02	data List a = Cons a (List a)  | Nil ;
9.52/4.02	
9.52/4.02	data MyBool = MyTrue  | MyFalse ;
9.52/4.02	
9.52/4.02	data Ordering = LT  | EQ  | GT ;
9.52/4.02	
9.52/4.02	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.52/4.02	foldl f z Nil = z;
9.52/4.02	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.52/4.02	
9.52/4.02	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.52/4.02	foldl1 f (Cons x xs) = foldl f x xs;
9.52/4.02	
9.52/4.02	ltEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.52/4.02	ltEsOrdering LT LT = MyTrue;
9.52/4.02	ltEsOrdering LT EQ = MyTrue;
9.52/4.02	ltEsOrdering LT GT = MyTrue;
9.52/4.02	ltEsOrdering EQ LT = MyFalse;
9.52/4.02	ltEsOrdering EQ EQ = MyTrue;
9.52/4.02	ltEsOrdering EQ GT = MyTrue;
9.52/4.02	ltEsOrdering GT LT = MyFalse;
9.52/4.02	ltEsOrdering GT EQ = MyFalse;
9.52/4.02	ltEsOrdering GT GT = MyTrue;
9.52/4.02	
9.52/4.02	min0 x y MyTrue = y;
9.52/4.02	
9.52/4.02	min1 x y MyTrue = x;
9.52/4.02	min1 x y MyFalse = min0 x y otherwise;
9.52/4.02	
9.52/4.02	min2 x y = min1 x y (ltEsOrdering x y);
9.52/4.02	
9.52/4.02	minOrdering :: Ordering  ->  Ordering  ->  Ordering;
9.52/4.02	minOrdering x y = min2 x y;
9.52/4.02	
9.52/4.02	minimumOrdering :: List Ordering  ->  Ordering;
9.52/4.02	minimumOrdering = foldl1 minOrdering;
9.52/4.02	
9.52/4.02	otherwise :: MyBool;
9.52/4.02	otherwise = MyTrue;
9.52/4.02	
9.52/4.02	}
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(5) Narrow (SOUND)
9.52/4.02	Haskell To QDPs
9.52/4.02	
9.52/4.02	digraph dp_graph {
9.52/4.02	node [outthreshold=100, inthreshold=100];1[label="minimumOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.52/4.02	3[label="minimumOrdering vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.52/4.02	4[label="foldl1 minOrdering vx3",fontsize=16,color="burlywood",shape="box"];53[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 53[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	53 -> 5[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	54[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	54 -> 6[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	5[label="foldl1 minOrdering (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.52/4.02	6[label="foldl1 minOrdering Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.52/4.02	7[label="foldl minOrdering vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];55[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 55[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	55 -> 9[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	56[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 56[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	56 -> 10[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl minOrdering vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.52/4.02	10[label="foldl minOrdering vx30 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.52/4.02	11 -> 7[label="",style="dashed", color="red", weight=0];
9.52/4.02	11[label="foldl minOrdering (minOrdering vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
9.52/4.02	11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.52/4.02	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="minOrdering vx30 vx310",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.52/4.02	14[label="vx311",fontsize=16,color="green",shape="box"];15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.52/4.02	16[label="min1 vx30 vx310 (ltEsOrdering vx30 vx310)",fontsize=16,color="burlywood",shape="box"];57[label="vx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 57[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	57 -> 17[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	58[label="vx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 58[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	58 -> 18[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	59[label="vx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	59 -> 19[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	17[label="min1 LT vx310 (ltEsOrdering LT vx310)",fontsize=16,color="burlywood",shape="box"];60[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 60[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	60 -> 20[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	61[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	61 -> 21[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	62[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	62 -> 22[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	18[label="min1 EQ vx310 (ltEsOrdering EQ vx310)",fontsize=16,color="burlywood",shape="box"];63[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 63[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	63 -> 23[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	64[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 64[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	64 -> 24[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	65[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 65[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	65 -> 25[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	19[label="min1 GT vx310 (ltEsOrdering GT vx310)",fontsize=16,color="burlywood",shape="box"];66[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 66[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	66 -> 26[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	67[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 67[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	67 -> 27[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	68[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 68[label="",style="solid", color="burlywood", weight=9];
9.52/4.02	68 -> 28[label="",style="solid", color="burlywood", weight=3];
9.52/4.02	20[label="min1 LT LT (ltEsOrdering LT LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
9.52/4.02	21[label="min1 LT EQ (ltEsOrdering LT EQ)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
9.52/4.02	22[label="min1 LT GT (ltEsOrdering LT GT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
9.52/4.02	23[label="min1 EQ LT (ltEsOrdering EQ LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.52/4.02	24[label="min1 EQ EQ (ltEsOrdering EQ EQ)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.52/4.02	25[label="min1 EQ GT (ltEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.52/4.02	26[label="min1 GT LT (ltEsOrdering GT LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.52/4.02	27[label="min1 GT EQ (ltEsOrdering GT EQ)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.52/4.02	28[label="min1 GT GT (ltEsOrdering GT GT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.52/4.02	29[label="min1 LT LT MyTrue",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.52/4.02	30[label="min1 LT EQ MyTrue",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.52/4.02	31[label="min1 LT GT MyTrue",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
9.52/4.02	32[label="min1 EQ LT MyFalse",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
9.52/4.02	33[label="min1 EQ EQ MyTrue",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
9.52/4.02	34[label="min1 EQ GT MyTrue",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
9.52/4.02	35[label="min1 GT LT MyFalse",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
9.52/4.02	36[label="min1 GT EQ MyFalse",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
9.52/4.02	37[label="min1 GT GT MyTrue",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
9.52/4.02	38[label="LT",fontsize=16,color="green",shape="box"];39[label="LT",fontsize=16,color="green",shape="box"];40[label="LT",fontsize=16,color="green",shape="box"];41[label="min0 EQ LT otherwise",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
9.52/4.02	42[label="EQ",fontsize=16,color="green",shape="box"];43[label="EQ",fontsize=16,color="green",shape="box"];44[label="min0 GT LT otherwise",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
9.52/4.02	45[label="min0 GT EQ otherwise",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
9.52/4.02	46[label="GT",fontsize=16,color="green",shape="box"];47[label="min0 EQ LT MyTrue",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
9.52/4.02	48[label="min0 GT LT MyTrue",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
9.52/4.02	49[label="min0 GT EQ MyTrue",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
9.52/4.02	50[label="LT",fontsize=16,color="green",shape="box"];51[label="LT",fontsize=16,color="green",shape="box"];52[label="EQ",fontsize=16,color="green",shape="box"];}
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(6)
9.52/4.02	Obligation:
9.52/4.02	Q DP problem:
9.52/4.02	The TRS P consists of the following rules:
9.52/4.02	
9.52/4.02	   new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.52/4.02	
9.52/4.02	The TRS R consists of the following rules:
9.52/4.02	
9.52/4.02	   new_min1(LT, GT) -> LT
9.52/4.02	   new_min1(GT, LT) -> LT
9.52/4.02	   new_min1(GT, GT) -> GT
9.52/4.02	   new_min1(EQ, EQ) -> EQ
9.52/4.02	   new_min1(LT, EQ) -> LT
9.52/4.02	   new_min1(EQ, LT) -> LT
9.52/4.02	   new_min1(EQ, GT) -> EQ
9.52/4.02	   new_min1(GT, EQ) -> EQ
9.52/4.02	   new_min1(LT, LT) -> LT
9.52/4.02	
9.52/4.02	The set Q consists of the following terms:
9.52/4.02	
9.52/4.02	   new_min1(LT, EQ)
9.52/4.02	   new_min1(EQ, LT)
9.52/4.02	   new_min1(EQ, GT)
9.52/4.02	   new_min1(GT, EQ)
9.52/4.02	   new_min1(LT, GT)
9.52/4.02	   new_min1(GT, LT)
9.52/4.02	   new_min1(EQ, EQ)
9.52/4.02	   new_min1(GT, GT)
9.52/4.02	   new_min1(LT, LT)
9.52/4.02	
9.52/4.02	We have to consider all minimal (P,Q,R)-chains.
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(7) QDPSizeChangeProof (EQUIVALENT)
9.52/4.02	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. 
9.52/4.02	
9.52/4.02	From the DPs we obtained the following set of size-change graphs:
9.52/4.02	*new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.52/4.02	The graph contains the following edges 2 > 2
9.52/4.02	
9.52/4.02	
9.52/4.02	----------------------------------------
9.52/4.02	
9.52/4.02	(8)
9.52/4.02	YES
9.87/4.06	EOF