8.01/3.95	YES
9.88/4.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.88/4.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.88/4.48	
9.88/4.48	
9.88/4.48	H-Termination with start terms of the given HASKELL could be proven:
9.88/4.48	
9.88/4.48	(0) HASKELL
9.88/4.48	(1) BR [EQUIVALENT, 0 ms]
9.88/4.48	(2) HASKELL
9.88/4.48	(3) COR [EQUIVALENT, 0 ms]
9.88/4.48	(4) HASKELL
9.88/4.48	(5) Narrow [SOUND, 0 ms]
9.88/4.48	(6) QDP
9.88/4.48	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.88/4.48	(8) YES
9.88/4.48	
9.88/4.48	
9.88/4.48	----------------------------------------
9.88/4.48	
9.88/4.48	(0)
9.88/4.48	Obligation:
9.88/4.48	mainModule Main 
9.88/4.48	module Main where {
9.88/4.48	import qualified Prelude;
9.88/4.48	data List a = Cons a (List a)  | Nil ;
9.88/4.48	
9.88/4.48	data MyBool = MyTrue  | MyFalse ;
9.88/4.48	
9.88/4.48	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.88/4.48	any p = pt or (map p);
9.88/4.48	
9.88/4.48	elemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
9.88/4.48	elemMyBool = pt any esEsMyBool;
9.88/4.48	
9.88/4.48	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.48	esEsMyBool MyFalse MyFalse = MyTrue;
9.88/4.48	esEsMyBool MyFalse MyTrue = MyFalse;
9.88/4.48	esEsMyBool MyTrue MyFalse = MyFalse;
9.88/4.48	esEsMyBool MyTrue MyTrue = MyTrue;
9.88/4.48	
9.88/4.48	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.88/4.48	foldr f z Nil = z;
9.88/4.48	foldr f z (Cons x xs) = f x (foldr f z xs);
9.88/4.48	
9.88/4.48	map :: (a  ->  b)  ->  List a  ->  List b;
9.88/4.48	map f Nil = Nil;
9.88/4.48	map f (Cons x xs) = Cons (f x) (map f xs);
9.88/4.48	
9.88/4.48	or :: List MyBool  ->  MyBool;
9.88/4.48	or = foldr pePe MyFalse;
9.88/4.48	
9.88/4.48	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.48	pePe MyFalse x = x;
9.88/4.48	pePe MyTrue x = MyTrue;
9.88/4.48	
9.88/4.48	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.88/4.48	pt f g x = f (g x);
9.88/4.48	
9.88/4.48	}
9.88/4.48	
9.88/4.48	----------------------------------------
9.88/4.48	
9.88/4.48	(1) BR (EQUIVALENT)
9.88/4.48	Replaced joker patterns by fresh variables and removed binding patterns.
9.88/4.48	----------------------------------------
9.88/4.48	
9.88/4.48	(2)
9.88/4.48	Obligation:
9.88/4.48	mainModule Main 
9.88/4.48	module Main where {
9.88/4.48	import qualified Prelude;
9.88/4.48	data List a = Cons a (List a)  | Nil ;
9.88/4.48	
9.88/4.48	data MyBool = MyTrue  | MyFalse ;
9.88/4.48	
9.88/4.48	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.88/4.48	any p = pt or (map p);
9.88/4.48	
9.88/4.48	elemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
9.88/4.48	elemMyBool = pt any esEsMyBool;
9.88/4.48	
9.88/4.48	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.48	esEsMyBool MyFalse MyFalse = MyTrue;
9.88/4.48	esEsMyBool MyFalse MyTrue = MyFalse;
9.88/4.48	esEsMyBool MyTrue MyFalse = MyFalse;
9.88/4.48	esEsMyBool MyTrue MyTrue = MyTrue;
9.88/4.48	
9.88/4.48	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.88/4.48	foldr f z Nil = z;
9.88/4.48	foldr f z (Cons x xs) = f x (foldr f z xs);
9.88/4.48	
9.88/4.48	map :: (a  ->  b)  ->  List a  ->  List b;
9.88/4.48	map f Nil = Nil;
9.88/4.48	map f (Cons x xs) = Cons (f x) (map f xs);
9.88/4.48	
9.88/4.48	or :: List MyBool  ->  MyBool;
9.88/4.48	or = foldr pePe MyFalse;
9.88/4.48	
9.88/4.48	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.48	pePe MyFalse x = x;
9.88/4.48	pePe MyTrue x = MyTrue;
9.88/4.48	
9.88/4.48	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.88/4.48	pt f g x = f (g x);
9.88/4.48	
9.88/4.48	}
9.88/4.48	
9.88/4.48	----------------------------------------
9.88/4.48	
9.88/4.48	(3) COR (EQUIVALENT)
9.88/4.48	Cond Reductions:
9.88/4.48	The following Function with conditions
9.88/4.48	"undefined |Falseundefined;
9.88/4.48	"
9.88/4.48	is transformed to
9.88/4.48	"undefined  = undefined1;
9.88/4.48	"
9.88/4.48	"undefined0 True = undefined;
9.88/4.48	"
9.88/4.48	"undefined1  = undefined0 False;
9.88/4.48	"
9.88/4.48	
9.88/4.48	----------------------------------------
9.88/4.48	
9.88/4.48	(4)
9.88/4.48	Obligation:
9.88/4.48	mainModule Main 
9.88/4.48	module Main where {
9.88/4.48	import qualified Prelude;
9.88/4.48	data List a = Cons a (List a)  | Nil ;
9.88/4.48	
9.88/4.48	data MyBool = MyTrue  | MyFalse ;
9.88/4.48	
9.88/4.48	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
9.88/4.48	any p = pt or (map p);
9.88/4.48	
9.88/4.48	elemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
9.88/4.49	elemMyBool = pt any esEsMyBool;
9.88/4.49	
9.88/4.49	esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.49	esEsMyBool MyFalse MyFalse = MyTrue;
9.88/4.49	esEsMyBool MyFalse MyTrue = MyFalse;
9.88/4.49	esEsMyBool MyTrue MyFalse = MyFalse;
9.88/4.49	esEsMyBool MyTrue MyTrue = MyTrue;
9.88/4.49	
9.88/4.49	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.88/4.49	foldr f z Nil = z;
9.88/4.49	foldr f z (Cons x xs) = f x (foldr f z xs);
9.88/4.49	
9.88/4.49	map :: (a  ->  b)  ->  List a  ->  List b;
9.88/4.49	map f Nil = Nil;
9.88/4.49	map f (Cons x xs) = Cons (f x) (map f xs);
9.88/4.49	
9.88/4.49	or :: List MyBool  ->  MyBool;
9.88/4.49	or = foldr pePe MyFalse;
9.88/4.49	
9.88/4.49	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.88/4.49	pePe MyFalse x = x;
9.88/4.49	pePe MyTrue x = MyTrue;
9.88/4.49	
9.88/4.49	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.88/4.49	pt f g x = f (g x);
9.88/4.49	
9.88/4.49	}
9.88/4.49	
9.88/4.49	----------------------------------------
9.88/4.49	
9.88/4.49	(5) Narrow (SOUND)
9.88/4.49	Haskell To QDPs
9.88/4.49	
9.88/4.49	digraph dp_graph {
9.88/4.49	node [outthreshold=100, inthreshold=100];1[label="elemMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.88/4.49	3[label="elemMyBool vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.88/4.49	4[label="elemMyBool vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.88/4.49	5[label="pt any esEsMyBool vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.88/4.49	6[label="any (esEsMyBool vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.88/4.49	7[label="pt or (map (esEsMyBool vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.88/4.49	8[label="or (map (esEsMyBool vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.88/4.49	9[label="foldr pePe MyFalse (map (esEsMyBool vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];31[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 31[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	31 -> 10[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	32[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 32[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	32 -> 11[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	10[label="foldr pePe MyFalse (map (esEsMyBool vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.88/4.49	11[label="foldr pePe MyFalse (map (esEsMyBool vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.88/4.49	12[label="foldr pePe MyFalse (Cons (esEsMyBool vx3 vx40) (map (esEsMyBool vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.88/4.49	13[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.88/4.49	14 -> 16[label="",style="dashed", color="red", weight=0];
9.88/4.49	14[label="pePe (esEsMyBool vx3 vx40) (foldr pePe MyFalse (map (esEsMyBool vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.88/4.49	15[label="MyFalse",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.88/4.49	17[label="foldr pePe MyFalse (map (esEsMyBool vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.88/4.49	16[label="pePe (esEsMyBool vx3 vx40) vx5",fontsize=16,color="burlywood",shape="triangle"];33[label="vx3/MyTrue",fontsize=10,color="white",style="solid",shape="box"];16 -> 33[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	33 -> 19[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	34[label="vx3/MyFalse",fontsize=10,color="white",style="solid",shape="box"];16 -> 34[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	34 -> 20[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="pePe (esEsMyBool MyTrue vx40) vx5",fontsize=16,color="burlywood",shape="box"];35[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];19 -> 35[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	35 -> 21[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	36[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];19 -> 36[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	36 -> 22[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	20[label="pePe (esEsMyBool MyFalse vx40) vx5",fontsize=16,color="burlywood",shape="box"];37[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];20 -> 37[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	37 -> 23[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	38[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];20 -> 38[label="",style="solid", color="burlywood", weight=9];
9.88/4.49	38 -> 24[label="",style="solid", color="burlywood", weight=3];
9.88/4.49	21[label="pePe (esEsMyBool MyTrue MyTrue) vx5",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.88/4.49	22[label="pePe (esEsMyBool MyTrue MyFalse) vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.88/4.49	23[label="pePe (esEsMyBool MyFalse MyTrue) vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.88/4.49	24[label="pePe (esEsMyBool MyFalse MyFalse) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.88/4.49	25[label="pePe MyTrue vx5",fontsize=16,color="black",shape="triangle"];25 -> 29[label="",style="solid", color="black", weight=3];
9.88/4.49	26[label="pePe MyFalse vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
9.88/4.49	27 -> 26[label="",style="dashed", color="red", weight=0];
9.88/4.49	27[label="pePe MyFalse vx5",fontsize=16,color="magenta"];28 -> 25[label="",style="dashed", color="red", weight=0];
9.88/4.49	28[label="pePe MyTrue vx5",fontsize=16,color="magenta"];29[label="MyTrue",fontsize=16,color="green",shape="box"];30[label="vx5",fontsize=16,color="green",shape="box"];}
9.88/4.49	
9.88/4.49	----------------------------------------
9.88/4.49	
9.88/4.49	(6)
9.88/4.49	Obligation:
9.88/4.49	Q DP problem:
9.88/4.49	The TRS P consists of the following rules:
9.88/4.49	
9.88/4.49	   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.88/4.49	
9.88/4.49	R is empty.
9.88/4.49	Q is empty.
9.88/4.49	We have to consider all minimal (P,Q,R)-chains.
9.88/4.49	----------------------------------------
9.88/4.49	
9.88/4.49	(7) QDPSizeChangeProof (EQUIVALENT)
9.88/4.49	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. 
9.88/4.49	
9.88/4.49	From the DPs we obtained the following set of size-change graphs:
9.88/4.49	*new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
9.88/4.49	The graph contains the following edges 1 >= 1, 2 > 2
9.88/4.49	
9.88/4.49	
9.88/4.49	----------------------------------------
9.88/4.49	
9.88/4.49	(8)
9.88/4.49	YES
10.04/4.54	EOF