8.00/3.51	YES
9.57/4.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.57/4.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.57/4.01	
9.57/4.01	
9.57/4.01	H-Termination with start terms of the given HASKELL could be proven:
9.57/4.01	
9.57/4.01	(0) HASKELL
9.57/4.01	(1) BR [EQUIVALENT, 0 ms]
9.57/4.01	(2) HASKELL
9.57/4.01	(3) COR [EQUIVALENT, 0 ms]
9.57/4.01	(4) HASKELL
9.57/4.01	(5) Narrow [SOUND, 0 ms]
9.57/4.01	(6) QDP
9.57/4.01	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.57/4.01	(8) YES
9.57/4.01	
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(0)
9.57/4.01	Obligation:
9.57/4.01	mainModule Main 
9.57/4.01	module Main where {
9.57/4.01	import qualified Prelude;
9.57/4.01	data List a = Cons a (List a)  | Nil ;
9.57/4.01	
9.57/4.01	data Tup2 a b = Tup2 a b ;
9.57/4.01	
9.57/4.01	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.57/4.01	foldr f z Nil = z;
9.57/4.01	foldr f z (Cons x xs) = f x (foldr f z xs);
9.57/4.01	
9.57/4.01	unzip :: List (Tup2 b a)  ->  Tup2 (List b) (List a);
9.57/4.01	unzip = foldr unzip0 (Tup2 Nil Nil);
9.57/4.01	
9.57/4.01	unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));
9.57/4.01	
9.57/4.01	unzip00 (Tup2 as bs) = as;
9.57/4.01	
9.57/4.01	unzip01 (Tup2 as bs) = bs;
9.57/4.01	
9.57/4.01	}
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(1) BR (EQUIVALENT)
9.57/4.01	Replaced joker patterns by fresh variables and removed binding patterns.
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(2)
9.57/4.01	Obligation:
9.57/4.01	mainModule Main 
9.57/4.01	module Main where {
9.57/4.01	import qualified Prelude;
9.57/4.01	data List a = Cons a (List a)  | Nil ;
9.57/4.01	
9.57/4.01	data Tup2 a b = Tup2 a b ;
9.57/4.01	
9.57/4.01	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.57/4.01	foldr f z Nil = z;
9.57/4.01	foldr f z (Cons x xs) = f x (foldr f z xs);
9.57/4.01	
9.57/4.01	unzip :: List (Tup2 a b)  ->  Tup2 (List a) (List b);
9.57/4.01	unzip = foldr unzip0 (Tup2 Nil Nil);
9.57/4.01	
9.57/4.01	unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));
9.57/4.01	
9.57/4.01	unzip00 (Tup2 as bs) = as;
9.57/4.01	
9.57/4.01	unzip01 (Tup2 as bs) = bs;
9.57/4.01	
9.57/4.01	}
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(3) COR (EQUIVALENT)
9.57/4.01	Cond Reductions:
9.57/4.01	The following Function with conditions
9.57/4.01	"undefined |Falseundefined;
9.57/4.01	"
9.57/4.01	is transformed to
9.57/4.01	"undefined  = undefined1;
9.57/4.01	"
9.57/4.01	"undefined0 True = undefined;
9.57/4.01	"
9.57/4.01	"undefined1  = undefined0 False;
9.57/4.01	"
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(4)
9.57/4.01	Obligation:
9.57/4.01	mainModule Main 
9.57/4.01	module Main where {
9.57/4.01	import qualified Prelude;
9.57/4.01	data List a = Cons a (List a)  | Nil ;
9.57/4.01	
9.57/4.01	data Tup2 b a = Tup2 b a ;
9.57/4.01	
9.57/4.01	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.57/4.01	foldr f z Nil = z;
9.57/4.01	foldr f z (Cons x xs) = f x (foldr f z xs);
9.57/4.01	
9.57/4.01	unzip :: List (Tup2 a b)  ->  Tup2 (List a) (List b);
9.57/4.01	unzip = foldr unzip0 (Tup2 Nil Nil);
9.57/4.01	
9.57/4.01	unzip0 (Tup2 a b) vv = Tup2 (Cons a (unzip00 vv)) (Cons b (unzip01 vv));
9.57/4.01	
9.57/4.01	unzip00 (Tup2 as bs) = as;
9.57/4.01	
9.57/4.01	unzip01 (Tup2 as bs) = bs;
9.57/4.01	
9.57/4.01	}
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(5) Narrow (SOUND)
9.57/4.01	Haskell To QDPs
9.57/4.01	
9.57/4.01	digraph dp_graph {
9.57/4.01	node [outthreshold=100, inthreshold=100];1[label="unzip",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.57/4.01	3[label="unzip vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.57/4.01	4[label="foldr unzip0 (Tup2 Nil Nil) vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
9.57/4.01	20 -> 5[label="",style="solid", color="burlywood", weight=3];
9.57/4.01	21[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
9.57/4.01	21 -> 6[label="",style="solid", color="burlywood", weight=3];
9.57/4.01	5[label="foldr unzip0 (Tup2 Nil Nil) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.57/4.01	6[label="foldr unzip0 (Tup2 Nil Nil) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.57/4.01	7 -> 9[label="",style="dashed", color="red", weight=0];
9.57/4.01	7[label="unzip0 vy30 (foldr unzip0 (Tup2 Nil Nil) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.57/4.01	8[label="Tup2 Nil Nil",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.57/4.01	10[label="foldr unzip0 (Tup2 Nil Nil) vy31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.57/4.01	9[label="unzip0 vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy30/Tup2 vy300 vy301",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9];
9.57/4.01	22 -> 12[label="",style="solid", color="burlywood", weight=3];
9.57/4.01	11[label="vy31",fontsize=16,color="green",shape="box"];12[label="unzip0 (Tup2 vy300 vy301) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
9.57/4.01	13[label="Tup2 (Cons vy300 (unzip00 vy4)) (Cons vy301 (unzip01 vy4))",fontsize=16,color="green",shape="box"];13 -> 14[label="",style="dashed", color="green", weight=3];
9.57/4.01	13 -> 15[label="",style="dashed", color="green", weight=3];
9.57/4.01	14[label="unzip00 vy4",fontsize=16,color="burlywood",shape="box"];23[label="vy4/Tup2 vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];14 -> 23[label="",style="solid", color="burlywood", weight=9];
9.57/4.01	23 -> 16[label="",style="solid", color="burlywood", weight=3];
9.57/4.01	15[label="unzip01 vy4",fontsize=16,color="burlywood",shape="box"];24[label="vy4/Tup2 vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];15 -> 24[label="",style="solid", color="burlywood", weight=9];
9.57/4.01	24 -> 17[label="",style="solid", color="burlywood", weight=3];
9.57/4.01	16[label="unzip00 (Tup2 vy40 vy41)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
9.57/4.01	17[label="unzip01 (Tup2 vy40 vy41)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
9.57/4.01	18[label="vy40",fontsize=16,color="green",shape="box"];19[label="vy41",fontsize=16,color="green",shape="box"];}
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(6)
9.57/4.01	Obligation:
9.57/4.01	Q DP problem:
9.57/4.01	The TRS P consists of the following rules:
9.57/4.01	
9.57/4.01	   new_foldr(Cons(vy30, vy31), h, ba) -> new_foldr(vy31, h, ba)
9.57/4.01	
9.57/4.01	R is empty.
9.57/4.01	Q is empty.
9.57/4.01	We have to consider all minimal (P,Q,R)-chains.
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(7) QDPSizeChangeProof (EQUIVALENT)
9.57/4.01	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. 
9.57/4.01	
9.57/4.01	From the DPs we obtained the following set of size-change graphs:
9.57/4.01	*new_foldr(Cons(vy30, vy31), h, ba) -> new_foldr(vy31, h, ba)
9.57/4.01	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.57/4.01	
9.57/4.01	
9.57/4.01	----------------------------------------
9.57/4.01	
9.57/4.01	(8)
9.57/4.01	YES
9.92/4.12	EOF