7.56/3.58	YES
9.31/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.31/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.31/4.06	
9.31/4.06	
9.31/4.06	H-Termination with start terms of the given HASKELL could be proven:
9.31/4.06	
9.31/4.06	(0) HASKELL
9.31/4.06	(1) BR [EQUIVALENT, 0 ms]
9.31/4.06	(2) HASKELL
9.31/4.06	(3) COR [EQUIVALENT, 0 ms]
9.31/4.06	(4) HASKELL
9.31/4.06	(5) Narrow [SOUND, 0 ms]
9.31/4.06	(6) AND
9.31/4.06	    (7) QDP
9.31/4.06	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.31/4.06	        (9) YES
9.31/4.06	    (10) QDP
9.31/4.06	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.31/4.06	        (12) YES
9.31/4.06	
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(0)
9.31/4.06	Obligation:
9.31/4.06	mainModule Main 
9.31/4.06	module Main where {
9.31/4.06	import qualified Prelude;
9.31/4.06	data List a = Cons a (List a)  | Nil ;
9.31/4.06	
9.31/4.06	concat :: List (List a)  ->  List a;
9.31/4.06	concat = foldr psPs Nil;
9.31/4.06	
9.31/4.06	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.31/4.06	foldr f z Nil = z;
9.31/4.06	foldr f z (Cons x xs) = f x (foldr f z xs);
9.31/4.06	
9.31/4.06	psPs :: List a  ->  List a  ->  List a;
9.31/4.06	psPs Nil ys = ys;
9.31/4.06	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.31/4.06	
9.31/4.06	}
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(1) BR (EQUIVALENT)
9.31/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(2)
9.31/4.06	Obligation:
9.31/4.06	mainModule Main 
9.31/4.06	module Main where {
9.31/4.06	import qualified Prelude;
9.31/4.06	data List a = Cons a (List a)  | Nil ;
9.31/4.06	
9.31/4.06	concat :: List (List a)  ->  List a;
9.31/4.06	concat = foldr psPs Nil;
9.31/4.06	
9.31/4.06	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.31/4.06	foldr f z Nil = z;
9.31/4.06	foldr f z (Cons x xs) = f x (foldr f z xs);
9.31/4.06	
9.31/4.06	psPs :: List a  ->  List a  ->  List a;
9.31/4.06	psPs Nil ys = ys;
9.31/4.06	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.31/4.06	
9.31/4.06	}
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(3) COR (EQUIVALENT)
9.31/4.06	Cond Reductions:
9.31/4.06	The following Function with conditions
9.31/4.06	"undefined |Falseundefined;
9.31/4.06	"
9.31/4.06	is transformed to
9.31/4.06	"undefined  = undefined1;
9.31/4.06	"
9.31/4.06	"undefined0 True = undefined;
9.31/4.06	"
9.31/4.06	"undefined1  = undefined0 False;
9.31/4.06	"
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(4)
9.31/4.06	Obligation:
9.31/4.06	mainModule Main 
9.31/4.06	module Main where {
9.31/4.06	import qualified Prelude;
9.31/4.06	data List a = Cons a (List a)  | Nil ;
9.31/4.06	
9.31/4.06	concat :: List (List a)  ->  List a;
9.31/4.06	concat = foldr psPs Nil;
9.31/4.06	
9.31/4.06	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.31/4.06	foldr f z Nil = z;
9.31/4.06	foldr f z (Cons x xs) = f x (foldr f z xs);
9.31/4.06	
9.31/4.06	psPs :: List a  ->  List a  ->  List a;
9.31/4.06	psPs Nil ys = ys;
9.31/4.06	psPs (Cons x xs) ys = Cons x (psPs xs ys);
9.31/4.06	
9.31/4.06	}
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(5) Narrow (SOUND)
9.31/4.06	Haskell To QDPs
9.31/4.06	
9.31/4.06	digraph dp_graph {
9.31/4.06	node [outthreshold=100, inthreshold=100];1[label="concat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.31/4.06	3[label="concat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.31/4.06	4[label="foldr psPs Nil vx3",fontsize=16,color="burlywood",shape="triangle"];18[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
9.31/4.06	18 -> 5[label="",style="solid", color="burlywood", weight=3];
9.31/4.06	19[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
9.31/4.06	19 -> 6[label="",style="solid", color="burlywood", weight=3];
9.31/4.06	5[label="foldr psPs Nil (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.31/4.06	6[label="foldr psPs Nil Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.31/4.06	7 -> 9[label="",style="dashed", color="red", weight=0];
9.31/4.06	7[label="psPs vx30 (foldr psPs Nil vx31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.31/4.06	8[label="Nil",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.31/4.06	10[label="foldr psPs Nil vx31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.31/4.06	9[label="psPs vx30 vx4",fontsize=16,color="burlywood",shape="triangle"];20[label="vx30/Cons vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];9 -> 20[label="",style="solid", color="burlywood", weight=9];
9.31/4.06	20 -> 12[label="",style="solid", color="burlywood", weight=3];
9.31/4.06	21[label="vx30/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 21[label="",style="solid", color="burlywood", weight=9];
9.31/4.06	21 -> 13[label="",style="solid", color="burlywood", weight=3];
9.31/4.06	11[label="vx31",fontsize=16,color="green",shape="box"];12[label="psPs (Cons vx300 vx301) vx4",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.31/4.06	13[label="psPs Nil vx4",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.31/4.06	14[label="Cons vx300 (psPs vx301 vx4)",fontsize=16,color="green",shape="box"];14 -> 16[label="",style="dashed", color="green", weight=3];
9.31/4.06	15[label="vx4",fontsize=16,color="green",shape="box"];16 -> 9[label="",style="dashed", color="red", weight=0];
9.31/4.06	16[label="psPs vx301 vx4",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
9.31/4.06	17[label="vx301",fontsize=16,color="green",shape="box"];}
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(6)
9.31/4.06	Complex Obligation (AND)
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(7)
9.31/4.06	Obligation:
9.31/4.06	Q DP problem:
9.31/4.06	The TRS P consists of the following rules:
9.31/4.06	
9.31/4.06	   new_psPs(Cons(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)
9.31/4.06	
9.31/4.06	R is empty.
9.31/4.06	Q is empty.
9.31/4.06	We have to consider all minimal (P,Q,R)-chains.
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(8) QDPSizeChangeProof (EQUIVALENT)
9.31/4.06	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. 
9.31/4.06	
9.31/4.06	From the DPs we obtained the following set of size-change graphs:
9.31/4.06	*new_psPs(Cons(vx300, vx301), vx4, h) -> new_psPs(vx301, vx4, h)
9.31/4.06	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.31/4.06	
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(9)
9.31/4.06	YES
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(10)
9.31/4.06	Obligation:
9.31/4.06	Q DP problem:
9.31/4.06	The TRS P consists of the following rules:
9.31/4.06	
9.31/4.06	   new_foldr(Cons(vx30, vx31), h) -> new_foldr(vx31, h)
9.31/4.06	
9.31/4.06	R is empty.
9.31/4.06	Q is empty.
9.31/4.06	We have to consider all minimal (P,Q,R)-chains.
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(11) QDPSizeChangeProof (EQUIVALENT)
9.31/4.06	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. 
9.31/4.06	
9.31/4.06	From the DPs we obtained the following set of size-change graphs:
9.31/4.06	*new_foldr(Cons(vx30, vx31), h) -> new_foldr(vx31, h)
9.31/4.06	The graph contains the following edges 1 > 1, 2 >= 2
9.31/4.06	
9.31/4.06	
9.31/4.06	----------------------------------------
9.31/4.06	
9.31/4.06	(12)
9.31/4.06	YES
9.36/4.10	EOF