7.88/3.56	YES
9.51/4.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.51/4.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.51/4.07	
9.51/4.07	
9.51/4.07	H-Termination with start terms of the given HASKELL could be proven:
9.51/4.07	
9.51/4.07	(0) HASKELL
9.51/4.07	(1) BR [EQUIVALENT, 0 ms]
9.51/4.07	(2) HASKELL
9.51/4.07	(3) COR [EQUIVALENT, 0 ms]
9.51/4.07	(4) HASKELL
9.51/4.07	(5) Narrow [SOUND, 0 ms]
9.51/4.07	(6) QDP
9.51/4.07	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.51/4.07	(8) YES
9.51/4.07	
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(0)
9.51/4.07	Obligation:
9.51/4.07	mainModule Main 
9.51/4.07	module Main where {
9.51/4.07	import qualified Prelude;
9.51/4.07	data List a = Cons a (List a)  | Nil ;
9.51/4.07	
9.51/4.07	data MyBool = MyTrue  | MyFalse ;
9.51/4.07	
9.51/4.07	data Ordering = LT  | EQ  | GT ;
9.51/4.07	
9.51/4.07	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.51/4.07	any p = pt or (map p);
9.51/4.07	
9.51/4.07	elemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.51/4.07	elemOrdering = pt any esEsOrdering;
9.51/4.07	
9.51/4.07	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.51/4.07	esEsOrdering LT LT = MyTrue;
9.51/4.07	esEsOrdering LT EQ = MyFalse;
9.51/4.07	esEsOrdering LT GT = MyFalse;
9.51/4.07	esEsOrdering EQ LT = MyFalse;
9.51/4.07	esEsOrdering EQ EQ = MyTrue;
9.51/4.07	esEsOrdering EQ GT = MyFalse;
9.51/4.07	esEsOrdering GT LT = MyFalse;
9.51/4.07	esEsOrdering GT EQ = MyFalse;
9.51/4.07	esEsOrdering GT GT = MyTrue;
9.51/4.07	
9.51/4.07	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.51/4.07	foldr f z Nil = z;
9.51/4.07	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.07	
9.51/4.07	map :: (b  ->  a)  ->  List b  ->  List a;
9.51/4.07	map f Nil = Nil;
9.51/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.07	
9.51/4.07	or :: List MyBool  ->  MyBool;
9.51/4.07	or = foldr pePe MyFalse;
9.51/4.07	
9.51/4.07	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.51/4.07	pePe MyFalse x = x;
9.51/4.07	pePe MyTrue x = MyTrue;
9.51/4.07	
9.51/4.07	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.51/4.07	pt f g x = f (g x);
9.51/4.07	
9.51/4.07	}
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(1) BR (EQUIVALENT)
9.51/4.07	Replaced joker patterns by fresh variables and removed binding patterns.
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(2)
9.51/4.07	Obligation:
9.51/4.07	mainModule Main 
9.51/4.07	module Main where {
9.51/4.07	import qualified Prelude;
9.51/4.07	data List a = Cons a (List a)  | Nil ;
9.51/4.07	
9.51/4.07	data MyBool = MyTrue  | MyFalse ;
9.51/4.07	
9.51/4.07	data Ordering = LT  | EQ  | GT ;
9.51/4.07	
9.51/4.07	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.51/4.07	any p = pt or (map p);
9.51/4.07	
9.51/4.07	elemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.51/4.07	elemOrdering = pt any esEsOrdering;
9.51/4.07	
9.51/4.07	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.51/4.07	esEsOrdering LT LT = MyTrue;
9.51/4.07	esEsOrdering LT EQ = MyFalse;
9.51/4.07	esEsOrdering LT GT = MyFalse;
9.51/4.07	esEsOrdering EQ LT = MyFalse;
9.51/4.07	esEsOrdering EQ EQ = MyTrue;
9.51/4.07	esEsOrdering EQ GT = MyFalse;
9.51/4.07	esEsOrdering GT LT = MyFalse;
9.51/4.07	esEsOrdering GT EQ = MyFalse;
9.51/4.07	esEsOrdering GT GT = MyTrue;
9.51/4.07	
9.51/4.07	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.51/4.07	foldr f z Nil = z;
9.51/4.07	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.07	
9.51/4.07	map :: (b  ->  a)  ->  List b  ->  List a;
9.51/4.07	map f Nil = Nil;
9.51/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.07	
9.51/4.07	or :: List MyBool  ->  MyBool;
9.51/4.07	or = foldr pePe MyFalse;
9.51/4.07	
9.51/4.07	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.51/4.07	pePe MyFalse x = x;
9.51/4.07	pePe MyTrue x = MyTrue;
9.51/4.07	
9.51/4.07	pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
9.51/4.07	pt f g x = f (g x);
9.51/4.07	
9.51/4.07	}
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(3) COR (EQUIVALENT)
9.51/4.07	Cond Reductions:
9.51/4.07	The following Function with conditions
9.51/4.07	"undefined |Falseundefined;
9.51/4.07	"
9.51/4.07	is transformed to
9.51/4.07	"undefined  = undefined1;
9.51/4.07	"
9.51/4.07	"undefined0 True = undefined;
9.51/4.07	"
9.51/4.07	"undefined1  = undefined0 False;
9.51/4.07	"
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(4)
9.51/4.07	Obligation:
9.51/4.07	mainModule Main 
9.51/4.07	module Main where {
9.51/4.07	import qualified Prelude;
9.51/4.07	data List a = Cons a (List a)  | Nil ;
9.51/4.07	
9.51/4.07	data MyBool = MyTrue  | MyFalse ;
9.51/4.07	
9.51/4.07	data Ordering = LT  | EQ  | GT ;
9.51/4.07	
9.51/4.07	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.51/4.07	any p = pt or (map p);
9.51/4.07	
9.51/4.07	elemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
9.51/4.07	elemOrdering = pt any esEsOrdering;
9.51/4.07	
9.51/4.07	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.51/4.07	esEsOrdering LT LT = MyTrue;
9.51/4.07	esEsOrdering LT EQ = MyFalse;
9.51/4.07	esEsOrdering LT GT = MyFalse;
9.51/4.07	esEsOrdering EQ LT = MyFalse;
9.51/4.07	esEsOrdering EQ EQ = MyTrue;
9.51/4.07	esEsOrdering EQ GT = MyFalse;
9.51/4.07	esEsOrdering GT LT = MyFalse;
9.51/4.07	esEsOrdering GT EQ = MyFalse;
9.51/4.07	esEsOrdering GT GT = MyTrue;
9.51/4.07	
9.51/4.07	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.51/4.07	foldr f z Nil = z;
9.51/4.07	foldr f z (Cons x xs) = f x (foldr f z xs);
9.51/4.07	
9.51/4.07	map :: (b  ->  a)  ->  List b  ->  List a;
9.51/4.07	map f Nil = Nil;
9.51/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.51/4.07	
9.51/4.07	or :: List MyBool  ->  MyBool;
9.51/4.07	or = foldr pePe MyFalse;
9.51/4.07	
9.51/4.07	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.51/4.07	pePe MyFalse x = x;
9.51/4.07	pePe MyTrue x = MyTrue;
9.51/4.07	
9.51/4.07	pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
9.51/4.07	pt f g x = f (g x);
9.51/4.07	
9.51/4.07	}
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(5) Narrow (SOUND)
9.51/4.07	Haskell To QDPs
9.51/4.07	
9.51/4.07	digraph dp_graph {
9.51/4.07	node [outthreshold=100, inthreshold=100];1[label="elemOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.51/4.07	3[label="elemOrdering vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.51/4.07	4[label="elemOrdering vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.51/4.07	5[label="pt any esEsOrdering vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.51/4.07	6[label="any (esEsOrdering vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.51/4.07	7[label="pt or (map (esEsOrdering vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.51/4.07	8[label="or (map (esEsOrdering vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.51/4.07	9[label="foldr pePe MyFalse (map (esEsOrdering vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];42[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 42[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	42 -> 10[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	43[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 43[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	43 -> 11[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	10[label="foldr pePe MyFalse (map (esEsOrdering vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.51/4.07	11[label="foldr pePe MyFalse (map (esEsOrdering vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.51/4.07	12[label="foldr pePe MyFalse (Cons (esEsOrdering vx3 vx40) (map (esEsOrdering vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.51/4.07	13[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.51/4.07	14 -> 16[label="",style="dashed", color="red", weight=0];
9.51/4.07	14[label="pePe (esEsOrdering vx3 vx40) (foldr pePe MyFalse (map (esEsOrdering vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.51/4.07	15[label="MyFalse",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.51/4.07	17[label="foldr pePe MyFalse (map (esEsOrdering vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.51/4.07	16[label="pePe (esEsOrdering vx3 vx40) vx5",fontsize=16,color="burlywood",shape="triangle"];44[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 44[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	44 -> 19[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	45[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 45[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	45 -> 20[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	46[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 46[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	46 -> 21[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="pePe (esEsOrdering LT vx40) vx5",fontsize=16,color="burlywood",shape="box"];47[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	47 -> 22[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	48[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 48[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	48 -> 23[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	49[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 49[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	49 -> 24[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	20[label="pePe (esEsOrdering EQ vx40) vx5",fontsize=16,color="burlywood",shape="box"];50[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];20 -> 50[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	50 -> 25[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	51[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 51[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	51 -> 26[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	52[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 52[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	52 -> 27[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	21[label="pePe (esEsOrdering GT vx40) vx5",fontsize=16,color="burlywood",shape="box"];53[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];21 -> 53[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	53 -> 28[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	54[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];21 -> 54[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	54 -> 29[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	55[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];21 -> 55[label="",style="solid", color="burlywood", weight=9];
9.51/4.07	55 -> 30[label="",style="solid", color="burlywood", weight=3];
9.51/4.07	22[label="pePe (esEsOrdering LT LT) vx5",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
9.51/4.07	23[label="pePe (esEsOrdering LT EQ) vx5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.51/4.07	24[label="pePe (esEsOrdering LT GT) vx5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.51/4.07	25[label="pePe (esEsOrdering EQ LT) vx5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.51/4.07	26[label="pePe (esEsOrdering EQ EQ) vx5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.51/4.07	27[label="pePe (esEsOrdering EQ GT) vx5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.51/4.07	28[label="pePe (esEsOrdering GT LT) vx5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.51/4.07	29[label="pePe (esEsOrdering GT EQ) vx5",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.51/4.07	30[label="pePe (esEsOrdering GT GT) vx5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.51/4.07	31[label="pePe MyTrue vx5",fontsize=16,color="black",shape="triangle"];31 -> 40[label="",style="solid", color="black", weight=3];
9.51/4.07	32[label="pePe MyFalse vx5",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
9.51/4.07	33 -> 32[label="",style="dashed", color="red", weight=0];
9.51/4.07	33[label="pePe MyFalse vx5",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0];
9.51/4.07	34[label="pePe MyFalse vx5",fontsize=16,color="magenta"];35 -> 31[label="",style="dashed", color="red", weight=0];
9.51/4.07	35[label="pePe MyTrue vx5",fontsize=16,color="magenta"];36 -> 32[label="",style="dashed", color="red", weight=0];
9.51/4.07	36[label="pePe MyFalse vx5",fontsize=16,color="magenta"];37 -> 32[label="",style="dashed", color="red", weight=0];
9.51/4.07	37[label="pePe MyFalse vx5",fontsize=16,color="magenta"];38 -> 32[label="",style="dashed", color="red", weight=0];
9.51/4.07	38[label="pePe MyFalse vx5",fontsize=16,color="magenta"];39 -> 31[label="",style="dashed", color="red", weight=0];
9.51/4.07	39[label="pePe MyTrue vx5",fontsize=16,color="magenta"];40[label="MyTrue",fontsize=16,color="green",shape="box"];41[label="vx5",fontsize=16,color="green",shape="box"];}
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(6)
9.51/4.07	Obligation:
9.51/4.07	Q DP problem:
9.51/4.07	The TRS P consists of the following rules:
9.51/4.07	
9.51/4.07	   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.51/4.07	
9.51/4.07	R is empty.
9.51/4.07	Q is empty.
9.51/4.07	We have to consider all minimal (P,Q,R)-chains.
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(7) QDPSizeChangeProof (EQUIVALENT)
9.51/4.07	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. 
9.51/4.07	
9.51/4.07	From the DPs we obtained the following set of size-change graphs:
9.51/4.07	*new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.51/4.07	The graph contains the following edges 1 >= 1, 2 > 2
9.51/4.07	
9.51/4.07	
9.51/4.07	----------------------------------------
9.51/4.07	
9.51/4.07	(8)
9.51/4.07	YES
9.79/4.10	EOF