7.58/3.56	YES
9.23/4.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.23/4.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.23/4.09	
9.23/4.09	
9.23/4.09	H-Termination with start terms of the given HASKELL could be proven:
9.23/4.09	
9.23/4.09	(0) HASKELL
9.23/4.09	(1) BR [EQUIVALENT, 0 ms]
9.23/4.09	(2) HASKELL
9.23/4.09	(3) COR [EQUIVALENT, 0 ms]
9.23/4.09	(4) HASKELL
9.23/4.09	(5) Narrow [SOUND, 0 ms]
9.23/4.09	(6) QDP
9.23/4.09	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.23/4.09	(8) YES
9.23/4.09	
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(0)
9.23/4.09	Obligation:
9.23/4.09	mainModule Main 
9.23/4.09	module Main where {
9.23/4.09	import qualified Prelude;
9.23/4.09	data List a = Cons a (List a)  | Nil ;
9.23/4.09	
9.23/4.09	data MyBool = MyTrue  | MyFalse ;
9.23/4.09	
9.23/4.09	and :: List MyBool  ->  MyBool;
9.23/4.09	and = foldr asAs MyTrue;
9.23/4.09	
9.23/4.09	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.23/4.09	asAs MyFalse x = MyFalse;
9.23/4.09	asAs MyTrue x = x;
9.23/4.09	
9.23/4.09	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.23/4.09	foldr f z Nil = z;
9.23/4.09	foldr f z (Cons x xs) = f x (foldr f z xs);
9.23/4.09	
9.23/4.09	}
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(1) BR (EQUIVALENT)
9.23/4.09	Replaced joker patterns by fresh variables and removed binding patterns.
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(2)
9.23/4.09	Obligation:
9.23/4.09	mainModule Main 
9.23/4.09	module Main where {
9.23/4.09	import qualified Prelude;
9.23/4.09	data List a = Cons a (List a)  | Nil ;
9.23/4.09	
9.23/4.09	data MyBool = MyTrue  | MyFalse ;
9.23/4.09	
9.23/4.09	and :: List MyBool  ->  MyBool;
9.23/4.09	and = foldr asAs MyTrue;
9.23/4.09	
9.23/4.09	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.23/4.09	asAs MyFalse x = MyFalse;
9.23/4.09	asAs MyTrue x = x;
9.23/4.09	
9.23/4.09	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.23/4.09	foldr f z Nil = z;
9.23/4.09	foldr f z (Cons x xs) = f x (foldr f z xs);
9.23/4.09	
9.23/4.09	}
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(3) COR (EQUIVALENT)
9.23/4.09	Cond Reductions:
9.23/4.09	The following Function with conditions
9.23/4.09	"undefined |Falseundefined;
9.23/4.09	"
9.23/4.09	is transformed to
9.23/4.09	"undefined  = undefined1;
9.23/4.09	"
9.23/4.09	"undefined0 True = undefined;
9.23/4.09	"
9.23/4.09	"undefined1  = undefined0 False;
9.23/4.09	"
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(4)
9.23/4.09	Obligation:
9.23/4.09	mainModule Main 
9.23/4.09	module Main where {
9.23/4.09	import qualified Prelude;
9.23/4.09	data List a = Cons a (List a)  | Nil ;
9.23/4.09	
9.23/4.09	data MyBool = MyTrue  | MyFalse ;
9.23/4.09	
9.23/4.09	and :: List MyBool  ->  MyBool;
9.23/4.09	and = foldr asAs MyTrue;
9.23/4.09	
9.23/4.09	asAs :: MyBool  ->  MyBool  ->  MyBool;
9.23/4.09	asAs MyFalse x = MyFalse;
9.23/4.09	asAs MyTrue x = x;
9.23/4.09	
9.23/4.09	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.23/4.09	foldr f z Nil = z;
9.23/4.09	foldr f z (Cons x xs) = f x (foldr f z xs);
9.23/4.09	
9.23/4.09	}
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(5) Narrow (SOUND)
9.23/4.09	Haskell To QDPs
9.23/4.09	
9.23/4.09	digraph dp_graph {
9.23/4.09	node [outthreshold=100, inthreshold=100];1[label="and",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.23/4.09	3[label="and vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.23/4.09	4[label="foldr asAs MyTrue 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];
9.23/4.09	16 -> 5[label="",style="solid", color="burlywood", weight=3];
9.23/4.09	17[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
9.23/4.09	17 -> 6[label="",style="solid", color="burlywood", weight=3];
9.23/4.09	5[label="foldr asAs MyTrue (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.23/4.09	6[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.23/4.09	7 -> 9[label="",style="dashed", color="red", weight=0];
9.23/4.09	7[label="asAs vx30 (foldr asAs MyTrue vx31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.23/4.09	8[label="MyTrue",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.23/4.09	10[label="foldr asAs MyTrue vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.23/4.09	9[label="asAs 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];
9.23/4.09	18 -> 12[label="",style="solid", color="burlywood", weight=3];
9.23/4.09	19[label="vx30/MyFalse",fontsize=10,color="white",style="solid",shape="box"];9 -> 19[label="",style="solid", color="burlywood", weight=9];
9.23/4.09	19 -> 13[label="",style="solid", color="burlywood", weight=3];
9.23/4.09	11[label="vx31",fontsize=16,color="green",shape="box"];12[label="asAs MyTrue vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.23/4.09	13[label="asAs MyFalse vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.23/4.09	14[label="vx4",fontsize=16,color="green",shape="box"];15[label="MyFalse",fontsize=16,color="green",shape="box"];}
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(6)
9.23/4.09	Obligation:
9.23/4.09	Q DP problem:
9.23/4.09	The TRS P consists of the following rules:
9.23/4.09	
9.23/4.09	   new_foldr(Cons(vx30, vx31)) -> new_foldr(vx31)
9.23/4.09	
9.23/4.09	R is empty.
9.23/4.09	Q is empty.
9.23/4.09	We have to consider all minimal (P,Q,R)-chains.
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(7) QDPSizeChangeProof (EQUIVALENT)
9.23/4.09	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. 
9.23/4.09	
9.23/4.09	From the DPs we obtained the following set of size-change graphs:
9.23/4.09	*new_foldr(Cons(vx30, vx31)) -> new_foldr(vx31)
9.23/4.09	The graph contains the following edges 1 > 1
9.23/4.09	
9.23/4.09	
9.23/4.09	----------------------------------------
9.23/4.09	
9.23/4.09	(8)
9.23/4.09	YES
9.50/4.23	EOF