8.02/3.59	YES
9.69/4.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.69/4.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.69/4.12	
9.69/4.12	
9.69/4.12	H-Termination with start terms of the given HASKELL could be proven:
9.69/4.12	
9.69/4.12	(0) HASKELL
9.69/4.12	(1) BR [EQUIVALENT, 0 ms]
9.69/4.12	(2) HASKELL
9.69/4.12	(3) COR [EQUIVALENT, 0 ms]
9.69/4.12	(4) HASKELL
9.69/4.12	(5) Narrow [SOUND, 0 ms]
9.69/4.12	(6) QDP
9.69/4.12	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.69/4.12	(8) YES
9.69/4.12	
9.69/4.12	
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(0)
9.69/4.12	Obligation:
9.69/4.12	mainModule Main 
9.69/4.12	module Main where {
9.69/4.12	import qualified Prelude;
9.69/4.12	data List a = Cons a (List a)  | Nil ;
9.69/4.12	
9.69/4.12	data MyBool = MyTrue  | MyFalse ;
9.69/4.12	
9.69/4.12	data Ordering = LT  | EQ  | GT ;
9.69/4.12	
9.69/4.12	data Tup0 = Tup0 ;
9.69/4.12	
9.69/4.12	compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
9.69/4.12	compareTup0 Tup0 Tup0 = EQ;
9.69/4.12	
9.69/4.12	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	esEsOrdering LT LT = MyTrue;
9.69/4.12	esEsOrdering LT EQ = MyFalse;
9.69/4.12	esEsOrdering LT GT = MyFalse;
9.69/4.12	esEsOrdering EQ LT = MyFalse;
9.69/4.12	esEsOrdering EQ EQ = MyTrue;
9.69/4.12	esEsOrdering EQ GT = MyFalse;
9.69/4.12	esEsOrdering GT LT = MyFalse;
9.69/4.12	esEsOrdering GT EQ = MyFalse;
9.69/4.12	esEsOrdering GT GT = MyTrue;
9.69/4.12	
9.69/4.12	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.69/4.12	foldl f z Nil = z;
9.69/4.12	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.69/4.12	
9.69/4.12	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.69/4.12	foldl1 f (Cons x xs) = foldl f x xs;
9.69/4.12	
9.69/4.12	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	fsEsOrdering x y = not (esEsOrdering x y);
9.69/4.12	
9.69/4.12	ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.69/4.12	ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;
9.69/4.12	
9.69/4.12	min0 x y MyTrue = y;
9.69/4.12	
9.69/4.12	min1 x y MyTrue = x;
9.69/4.12	min1 x y MyFalse = min0 x y otherwise;
9.69/4.12	
9.69/4.12	min2 x y = min1 x y (ltEsTup0 x y);
9.69/4.12	
9.69/4.12	minTup0 :: Tup0  ->  Tup0  ->  Tup0;
9.69/4.12	minTup0 x y = min2 x y;
9.69/4.12	
9.69/4.12	minimumTup0 :: List Tup0  ->  Tup0;
9.69/4.12	minimumTup0 = foldl1 minTup0;
9.69/4.12	
9.69/4.12	not :: MyBool  ->  MyBool;
9.69/4.12	not MyTrue = MyFalse;
9.69/4.12	not MyFalse = MyTrue;
9.69/4.12	
9.69/4.12	otherwise :: MyBool;
9.69/4.12	otherwise = MyTrue;
9.69/4.12	
9.69/4.12	}
9.69/4.12	
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(1) BR (EQUIVALENT)
9.69/4.12	Replaced joker patterns by fresh variables and removed binding patterns.
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(2)
9.69/4.12	Obligation:
9.69/4.12	mainModule Main 
9.69/4.12	module Main where {
9.69/4.12	import qualified Prelude;
9.69/4.12	data List a = Cons a (List a)  | Nil ;
9.69/4.12	
9.69/4.12	data MyBool = MyTrue  | MyFalse ;
9.69/4.12	
9.69/4.12	data Ordering = LT  | EQ  | GT ;
9.69/4.12	
9.69/4.12	data Tup0 = Tup0 ;
9.69/4.12	
9.69/4.12	compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
9.69/4.12	compareTup0 Tup0 Tup0 = EQ;
9.69/4.12	
9.69/4.12	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	esEsOrdering LT LT = MyTrue;
9.69/4.12	esEsOrdering LT EQ = MyFalse;
9.69/4.12	esEsOrdering LT GT = MyFalse;
9.69/4.12	esEsOrdering EQ LT = MyFalse;
9.69/4.12	esEsOrdering EQ EQ = MyTrue;
9.69/4.12	esEsOrdering EQ GT = MyFalse;
9.69/4.12	esEsOrdering GT LT = MyFalse;
9.69/4.12	esEsOrdering GT EQ = MyFalse;
9.69/4.12	esEsOrdering GT GT = MyTrue;
9.69/4.12	
9.69/4.12	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
9.69/4.12	foldl f z Nil = z;
9.69/4.12	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.69/4.12	
9.69/4.12	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.69/4.12	foldl1 f (Cons x xs) = foldl f x xs;
9.69/4.12	
9.69/4.12	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	fsEsOrdering x y = not (esEsOrdering x y);
9.69/4.12	
9.69/4.12	ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.69/4.12	ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;
9.69/4.12	
9.69/4.12	min0 x y MyTrue = y;
9.69/4.12	
9.69/4.12	min1 x y MyTrue = x;
9.69/4.12	min1 x y MyFalse = min0 x y otherwise;
9.69/4.12	
9.69/4.12	min2 x y = min1 x y (ltEsTup0 x y);
9.69/4.12	
9.69/4.12	minTup0 :: Tup0  ->  Tup0  ->  Tup0;
9.69/4.12	minTup0 x y = min2 x y;
9.69/4.12	
9.69/4.12	minimumTup0 :: List Tup0  ->  Tup0;
9.69/4.12	minimumTup0 = foldl1 minTup0;
9.69/4.12	
9.69/4.12	not :: MyBool  ->  MyBool;
9.69/4.12	not MyTrue = MyFalse;
9.69/4.12	not MyFalse = MyTrue;
9.69/4.12	
9.69/4.12	otherwise :: MyBool;
9.69/4.12	otherwise = MyTrue;
9.69/4.12	
9.69/4.12	}
9.69/4.12	
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(3) COR (EQUIVALENT)
9.69/4.12	Cond Reductions:
9.69/4.12	The following Function with conditions
9.69/4.12	"undefined |Falseundefined;
9.69/4.12	"
9.69/4.12	is transformed to
9.69/4.12	"undefined  = undefined1;
9.69/4.12	"
9.69/4.12	"undefined0 True = undefined;
9.69/4.12	"
9.69/4.12	"undefined1  = undefined0 False;
9.69/4.12	"
9.69/4.12	
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(4)
9.69/4.12	Obligation:
9.69/4.12	mainModule Main 
9.69/4.12	module Main where {
9.69/4.12	import qualified Prelude;
9.69/4.12	data List a = Cons a (List a)  | Nil ;
9.69/4.12	
9.69/4.12	data MyBool = MyTrue  | MyFalse ;
9.69/4.12	
9.69/4.12	data Ordering = LT  | EQ  | GT ;
9.69/4.12	
9.69/4.12	data Tup0 = Tup0 ;
9.69/4.12	
9.69/4.12	compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
9.69/4.12	compareTup0 Tup0 Tup0 = EQ;
9.69/4.12	
9.69/4.12	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	esEsOrdering LT LT = MyTrue;
9.69/4.12	esEsOrdering LT EQ = MyFalse;
9.69/4.12	esEsOrdering LT GT = MyFalse;
9.69/4.12	esEsOrdering EQ LT = MyFalse;
9.69/4.12	esEsOrdering EQ EQ = MyTrue;
9.69/4.12	esEsOrdering EQ GT = MyFalse;
9.69/4.12	esEsOrdering GT LT = MyFalse;
9.69/4.12	esEsOrdering GT EQ = MyFalse;
9.69/4.12	esEsOrdering GT GT = MyTrue;
9.69/4.12	
9.69/4.12	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
9.69/4.12	foldl f z Nil = z;
9.69/4.12	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.69/4.12	
9.69/4.12	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
9.69/4.12	foldl1 f (Cons x xs) = foldl f x xs;
9.69/4.12	
9.69/4.12	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.69/4.12	fsEsOrdering x y = not (esEsOrdering x y);
9.69/4.12	
9.69/4.12	ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
9.69/4.12	ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;
9.69/4.12	
9.69/4.12	min0 x y MyTrue = y;
9.69/4.12	
9.69/4.12	min1 x y MyTrue = x;
9.69/4.12	min1 x y MyFalse = min0 x y otherwise;
9.69/4.12	
9.69/4.12	min2 x y = min1 x y (ltEsTup0 x y);
9.69/4.12	
9.69/4.12	minTup0 :: Tup0  ->  Tup0  ->  Tup0;
9.69/4.12	minTup0 x y = min2 x y;
9.69/4.12	
9.69/4.12	minimumTup0 :: List Tup0  ->  Tup0;
9.69/4.12	minimumTup0 = foldl1 minTup0;
9.69/4.12	
9.69/4.12	not :: MyBool  ->  MyBool;
9.69/4.12	not MyTrue = MyFalse;
9.69/4.12	not MyFalse = MyTrue;
9.69/4.12	
9.69/4.12	otherwise :: MyBool;
9.69/4.12	otherwise = MyTrue;
9.69/4.12	
9.69/4.12	}
9.69/4.12	
9.69/4.12	----------------------------------------
9.69/4.12	
9.69/4.12	(5) Narrow (SOUND)
9.69/4.12	Haskell To QDPs
9.69/4.12	
9.69/4.12	digraph dp_graph {
9.69/4.12	node [outthreshold=100, inthreshold=100];1[label="minimumTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.69/4.12	3[label="minimumTup0 vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.69/4.12	4[label="foldl1 minTup0 vx3",fontsize=16,color="burlywood",shape="box"];25[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
9.69/4.12	25 -> 5[label="",style="solid", color="burlywood", weight=3];
9.69/4.12	26[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
9.69/4.12	26 -> 6[label="",style="solid", color="burlywood", weight=3];
9.69/4.12	5[label="foldl1 minTup0 (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.69/4.12	6[label="foldl1 minTup0 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.69/4.12	7[label="foldl minTup0 vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];27[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 27[label="",style="solid", color="burlywood", weight=9];
10.09/4.12	27 -> 9[label="",style="solid", color="burlywood", weight=3];
10.09/4.12	28[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
10.09/4.12	28 -> 10[label="",style="solid", color="burlywood", weight=3];
10.09/4.12	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl minTup0 vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.09/4.12	10[label="foldl minTup0 vx30 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.09/4.12	11 -> 7[label="",style="dashed", color="red", weight=0];
10.09/4.12	11[label="foldl minTup0 (minTup0 vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
10.09/4.12	11 -> 14[label="",style="dashed", color="magenta", weight=3];
10.09/4.12	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="minTup0 vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.09/4.12	15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.09/4.12	16[label="min1 vx30 vx310 (ltEsTup0 vx30 vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
10.09/4.12	17[label="min1 vx30 vx310 (fsEsOrdering (compareTup0 vx30 vx310) GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
10.09/4.12	18[label="min1 vx30 vx310 (not (esEsOrdering (compareTup0 vx30 vx310) GT))",fontsize=16,color="burlywood",shape="box"];29[label="vx30/Tup0",fontsize=10,color="white",style="solid",shape="box"];18 -> 29[label="",style="solid", color="burlywood", weight=9];
10.09/4.12	29 -> 19[label="",style="solid", color="burlywood", weight=3];
10.09/4.12	19[label="min1 Tup0 vx310 (not (esEsOrdering (compareTup0 Tup0 vx310) GT))",fontsize=16,color="burlywood",shape="box"];30[label="vx310/Tup0",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
10.09/4.12	30 -> 20[label="",style="solid", color="burlywood", weight=3];
10.09/4.12	20[label="min1 Tup0 Tup0 (not (esEsOrdering (compareTup0 Tup0 Tup0) GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
10.09/4.12	21[label="min1 Tup0 Tup0 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
10.09/4.12	22[label="min1 Tup0 Tup0 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
10.09/4.12	23[label="min1 Tup0 Tup0 MyTrue",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
10.09/4.12	24[label="Tup0",fontsize=16,color="green",shape="box"];}
10.09/4.12	
10.09/4.12	----------------------------------------
10.09/4.12	
10.09/4.12	(6)
10.09/4.12	Obligation:
10.09/4.12	Q DP problem:
10.09/4.12	The TRS P consists of the following rules:
10.09/4.12	
10.09/4.12	   new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
10.09/4.12	
10.09/4.12	The TRS R consists of the following rules:
10.09/4.12	
10.09/4.12	   new_min1(Tup0, Tup0) -> Tup0
10.09/4.12	
10.09/4.12	The set Q consists of the following terms:
10.09/4.12	
10.09/4.12	   new_min1(Tup0, Tup0)
10.09/4.12	
10.09/4.12	We have to consider all minimal (P,Q,R)-chains.
10.09/4.12	----------------------------------------
10.09/4.12	
10.09/4.12	(7) QDPSizeChangeProof (EQUIVALENT)
10.09/4.12	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. 
10.09/4.12	
10.09/4.12	From the DPs we obtained the following set of size-change graphs:
10.09/4.12	*new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
10.09/4.12	The graph contains the following edges 2 > 2
10.09/4.12	
10.09/4.12	
10.09/4.12	----------------------------------------
10.09/4.12	
10.09/4.12	(8)
10.09/4.12	YES
10.09/4.18	EOF