7.65/3.52	YES
9.32/4.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.32/4.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.32/4.00	
9.32/4.00	
9.32/4.00	H-Termination with start terms of the given HASKELL could be proven:
9.32/4.00	
9.32/4.00	(0) HASKELL
9.32/4.00	(1) BR [EQUIVALENT, 0 ms]
9.32/4.00	(2) HASKELL
9.32/4.00	(3) COR [EQUIVALENT, 0 ms]
9.32/4.00	(4) HASKELL
9.32/4.00	(5) Narrow [SOUND, 0 ms]
9.32/4.00	(6) QDP
9.32/4.00	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.32/4.00	(8) YES
9.32/4.00	
9.32/4.00	
9.32/4.00	----------------------------------------
9.32/4.00	
9.32/4.00	(0)
9.32/4.00	Obligation:
9.32/4.00	mainModule Main 
9.32/4.00	module Main where {
9.32/4.00	import qualified Prelude;
9.32/4.00	data List a = Cons a (List a)  | Nil ;
9.32/4.00	
9.32/4.00	data MyBool = MyTrue  | MyFalse ;
9.32/4.00	
9.32/4.00	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.32/4.00	foldr f z Nil = z;
9.32/4.00	foldr f z (Cons x xs) = f x (foldr f z xs);
9.32/4.00	
9.32/4.00	or :: List MyBool  ->  MyBool;
9.32/4.00	or = foldr pePe MyFalse;
9.32/4.00	
9.32/4.00	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.32/4.00	pePe MyFalse x = x;
9.32/4.00	pePe MyTrue x = MyTrue;
9.32/4.00	
9.32/4.00	}
9.32/4.00	
9.32/4.00	----------------------------------------
9.32/4.00	
9.32/4.00	(1) BR (EQUIVALENT)
9.32/4.00	Replaced joker patterns by fresh variables and removed binding patterns.
9.32/4.00	----------------------------------------
9.32/4.00	
9.32/4.00	(2)
9.32/4.00	Obligation:
9.32/4.00	mainModule Main 
9.32/4.00	module Main where {
9.32/4.00	import qualified Prelude;
9.32/4.00	data List a = Cons a (List a)  | Nil ;
9.32/4.00	
9.32/4.00	data MyBool = MyTrue  | MyFalse ;
9.32/4.00	
9.32/4.00	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.32/4.00	foldr f z Nil = z;
9.32/4.00	foldr f z (Cons x xs) = f x (foldr f z xs);
9.32/4.00	
9.32/4.00	or :: List MyBool  ->  MyBool;
9.32/4.00	or = foldr pePe MyFalse;
9.32/4.00	
9.32/4.00	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.32/4.00	pePe MyFalse x = x;
9.32/4.00	pePe MyTrue x = MyTrue;
9.32/4.00	
9.32/4.00	}
9.32/4.00	
9.32/4.00	----------------------------------------
9.32/4.00	
9.32/4.00	(3) COR (EQUIVALENT)
9.32/4.00	Cond Reductions:
9.32/4.00	The following Function with conditions
9.32/4.00	"undefined |Falseundefined;
9.32/4.00	"
9.32/4.00	is transformed to
9.32/4.00	"undefined  = undefined1;
9.32/4.00	"
9.32/4.00	"undefined0 True = undefined;
9.32/4.00	"
9.32/4.00	"undefined1  = undefined0 False;
9.32/4.00	"
9.32/4.00	
9.32/4.00	----------------------------------------
9.32/4.00	
9.32/4.00	(4)
9.32/4.00	Obligation:
9.32/4.00	mainModule Main 
9.32/4.00	module Main where {
9.32/4.00	import qualified Prelude;
9.32/4.00	data List a = Cons a (List a)  | Nil ;
9.32/4.00	
9.32/4.00	data MyBool = MyTrue  | MyFalse ;
9.32/4.00	
9.32/4.00	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.32/4.00	foldr f z Nil = z;
9.32/4.00	foldr f z (Cons x xs) = f x (foldr f z xs);
9.32/4.00	
9.32/4.00	or :: List MyBool  ->  MyBool;
9.32/4.01	or = foldr pePe MyFalse;
9.32/4.01	
9.32/4.01	pePe :: MyBool  ->  MyBool  ->  MyBool;
9.32/4.01	pePe MyFalse x = x;
9.32/4.01	pePe MyTrue x = MyTrue;
9.32/4.01	
9.32/4.01	}
9.32/4.01	
9.32/4.01	----------------------------------------
9.32/4.01	
9.32/4.01	(5) Narrow (SOUND)
9.32/4.01	Haskell To QDPs
9.32/4.01	
9.32/4.01	digraph dp_graph {
9.32/4.01	node [outthreshold=100, inthreshold=100];1[label="or",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.32/4.01	3[label="or vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.32/4.01	4[label="foldr pePe MyFalse vx3",fontsize=16,color="burlywood",shape="triangle"];16[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
9.32/4.01	16 -> 5[label="",style="solid", color="burlywood", weight=3];
9.32/4.01	17[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
9.32/4.01	17 -> 6[label="",style="solid", color="burlywood", weight=3];
9.32/4.01	5[label="foldr pePe MyFalse (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.32/4.01	6[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.32/4.01	7 -> 9[label="",style="dashed", color="red", weight=0];
9.32/4.01	7[label="pePe vx30 (foldr pePe MyFalse vx31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.32/4.01	8[label="MyFalse",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.32/4.01	10[label="foldr pePe MyFalse vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.32/4.01	9[label="pePe vx30 vx4",fontsize=16,color="burlywood",shape="triangle"];18[label="vx30/MyTrue",fontsize=10,color="white",style="solid",shape="box"];9 -> 18[label="",style="solid", color="burlywood", weight=9];
9.32/4.01	18 -> 12[label="",style="solid", color="burlywood", weight=3];
9.32/4.01	19[label="vx30/MyFalse",fontsize=10,color="white",style="solid",shape="box"];9 -> 19[label="",style="solid", color="burlywood", weight=9];
9.32/4.01	19 -> 13[label="",style="solid", color="burlywood", weight=3];
9.32/4.01	11[label="vx31",fontsize=16,color="green",shape="box"];12[label="pePe MyTrue vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.32/4.01	13[label="pePe MyFalse vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.32/4.01	14[label="MyTrue",fontsize=16,color="green",shape="box"];15[label="vx4",fontsize=16,color="green",shape="box"];}
9.32/4.01	
9.32/4.01	----------------------------------------
9.32/4.01	
9.32/4.01	(6)
9.32/4.01	Obligation:
9.32/4.01	Q DP problem:
9.32/4.01	The TRS P consists of the following rules:
9.32/4.01	
9.32/4.01	   new_foldr(Cons(vx30, vx31)) -> new_foldr(vx31)
9.32/4.01	
9.32/4.01	R is empty.
9.32/4.01	Q is empty.
9.32/4.01	We have to consider all minimal (P,Q,R)-chains.
9.32/4.01	----------------------------------------
9.32/4.01	
9.32/4.01	(7) QDPSizeChangeProof (EQUIVALENT)
9.32/4.01	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. 
9.32/4.01	
9.32/4.01	From the DPs we obtained the following set of size-change graphs:
9.32/4.01	*new_foldr(Cons(vx30, vx31)) -> new_foldr(vx31)
9.32/4.01	The graph contains the following edges 1 > 1
9.32/4.01	
9.32/4.01	
9.32/4.01	----------------------------------------
9.32/4.01	
9.32/4.01	(8)
9.32/4.01	YES
9.32/4.05	EOF