7.60/3.48	YES
9.19/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.19/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.19/3.98	
9.19/3.98	
9.19/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.19/3.98	
9.19/3.98	(0) HASKELL
9.19/3.98	(1) BR [EQUIVALENT, 0 ms]
9.19/3.98	(2) HASKELL
9.19/3.98	(3) COR [EQUIVALENT, 0 ms]
9.19/3.98	(4) HASKELL
9.19/3.98	(5) Narrow [SOUND, 0 ms]
9.19/3.98	(6) QDP
9.19/3.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.19/3.98	(8) YES
9.19/3.98	
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(0)
9.19/3.98	Obligation:
9.19/3.98	mainModule Main 
9.19/3.98	module Main where {
9.19/3.98	import qualified Prelude;
9.19/3.98	data List a = Cons a (List a)  | Nil ;
9.19/3.98	
9.19/3.98	data Tup3 a c b = Tup3 a c b ;
9.19/3.98	
9.19/3.98	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
9.19/3.98	foldr f z Nil = z;
9.19/3.98	foldr f z (Cons x xs) = f x (foldr f z xs);
9.19/3.98	
9.19/3.98	unzip3 :: List (Tup3 b a c)  ->  Tup3 (List b) (List a) (List c);
9.19/3.98	unzip3 = foldr unzip30 (Tup3 Nil Nil Nil);
9.19/3.98	
9.19/3.98	unzip30 (Tup3 a b c) vv = Tup3 (Cons a (unzip300 vv)) (Cons b (unzip301 vv)) (Cons c (unzip302 vv));
9.19/3.98	
9.19/3.98	unzip300 (Tup3 as bs cs) = as;
9.19/3.98	
9.19/3.98	unzip301 (Tup3 as bs cs) = bs;
9.19/3.98	
9.19/3.98	unzip302 (Tup3 as bs cs) = cs;
9.19/3.98	
9.19/3.98	}
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(1) BR (EQUIVALENT)
9.19/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(2)
9.19/3.98	Obligation:
9.19/3.98	mainModule Main 
9.19/3.98	module Main where {
9.19/3.98	import qualified Prelude;
9.19/3.98	data List a = Cons a (List a)  | Nil ;
9.19/3.98	
9.19/3.98	data Tup3 a c b = Tup3 a c b ;
9.19/3.98	
9.19/3.98	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.19/3.98	foldr f z Nil = z;
9.19/3.98	foldr f z (Cons x xs) = f x (foldr f z xs);
9.19/3.98	
9.19/3.98	unzip3 :: List (Tup3 a b c)  ->  Tup3 (List a) (List b) (List c);
9.19/3.98	unzip3 = foldr unzip30 (Tup3 Nil Nil Nil);
9.19/3.98	
9.19/3.98	unzip30 (Tup3 a b c) vv = Tup3 (Cons a (unzip300 vv)) (Cons b (unzip301 vv)) (Cons c (unzip302 vv));
9.19/3.98	
9.19/3.98	unzip300 (Tup3 as bs cs) = as;
9.19/3.98	
9.19/3.98	unzip301 (Tup3 as bs cs) = bs;
9.19/3.98	
9.19/3.98	unzip302 (Tup3 as bs cs) = cs;
9.19/3.98	
9.19/3.98	}
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(3) COR (EQUIVALENT)
9.19/3.98	Cond Reductions:
9.19/3.98	The following Function with conditions
9.19/3.98	"undefined |Falseundefined;
9.19/3.98	"
9.19/3.98	is transformed to
9.19/3.98	"undefined  = undefined1;
9.19/3.98	"
9.19/3.98	"undefined0 True = undefined;
9.19/3.98	"
9.19/3.98	"undefined1  = undefined0 False;
9.19/3.98	"
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(4)
9.19/3.98	Obligation:
9.19/3.98	mainModule Main 
9.19/3.98	module Main where {
9.19/3.98	import qualified Prelude;
9.19/3.98	data List a = Cons a (List a)  | Nil ;
9.19/3.98	
9.19/3.98	data Tup3 b c a = Tup3 b c a ;
9.19/3.98	
9.19/3.98	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
9.19/3.98	foldr f z Nil = z;
9.19/3.98	foldr f z (Cons x xs) = f x (foldr f z xs);
9.19/3.98	
9.19/3.98	unzip3 :: List (Tup3 b c a)  ->  Tup3 (List b) (List c) (List a);
9.19/3.98	unzip3 = foldr unzip30 (Tup3 Nil Nil Nil);
9.19/3.98	
9.19/3.98	unzip30 (Tup3 a b c) vv = Tup3 (Cons a (unzip300 vv)) (Cons b (unzip301 vv)) (Cons c (unzip302 vv));
9.19/3.98	
9.19/3.98	unzip300 (Tup3 as bs cs) = as;
9.19/3.98	
9.19/3.98	unzip301 (Tup3 as bs cs) = bs;
9.19/3.98	
9.19/3.98	unzip302 (Tup3 as bs cs) = cs;
9.19/3.98	
9.19/3.98	}
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(5) Narrow (SOUND)
9.19/3.98	Haskell To QDPs
9.19/3.98	
9.19/3.98	digraph dp_graph {
9.19/3.98	node [outthreshold=100, inthreshold=100];1[label="unzip3",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.19/3.98	3[label="unzip3 vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.19/3.98	4[label="foldr unzip30 (Tup3 Nil Nil Nil) vy3",fontsize=16,color="burlywood",shape="triangle"];23[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	23 -> 5[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	24[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 24[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	24 -> 6[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	5[label="foldr unzip30 (Tup3 Nil Nil Nil) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.19/3.98	6[label="foldr unzip30 (Tup3 Nil Nil Nil) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.19/3.98	7 -> 9[label="",style="dashed", color="red", weight=0];
9.19/3.98	7[label="unzip30 vy30 (foldr unzip30 (Tup3 Nil Nil Nil) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
9.19/3.98	8[label="Tup3 Nil Nil Nil",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
9.19/3.98	10[label="foldr unzip30 (Tup3 Nil Nil Nil) vy31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9.19/3.98	9[label="unzip30 vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];25[label="vy30/Tup3 vy300 vy301 vy302",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	25 -> 12[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	11[label="vy31",fontsize=16,color="green",shape="box"];12[label="unzip30 (Tup3 vy300 vy301 vy302) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
9.19/3.98	13[label="Tup3 (Cons vy300 (unzip300 vy4)) (Cons vy301 (unzip301 vy4)) (Cons vy302 (unzip302 vy4))",fontsize=16,color="green",shape="box"];13 -> 14[label="",style="dashed", color="green", weight=3];
9.19/3.98	13 -> 15[label="",style="dashed", color="green", weight=3];
9.19/3.98	13 -> 16[label="",style="dashed", color="green", weight=3];
9.19/3.98	14[label="unzip300 vy4",fontsize=16,color="burlywood",shape="box"];26[label="vy4/Tup3 vy40 vy41 vy42",fontsize=10,color="white",style="solid",shape="box"];14 -> 26[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	26 -> 17[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	15[label="unzip301 vy4",fontsize=16,color="burlywood",shape="box"];27[label="vy4/Tup3 vy40 vy41 vy42",fontsize=10,color="white",style="solid",shape="box"];15 -> 27[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	27 -> 18[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	16[label="unzip302 vy4",fontsize=16,color="burlywood",shape="box"];28[label="vy4/Tup3 vy40 vy41 vy42",fontsize=10,color="white",style="solid",shape="box"];16 -> 28[label="",style="solid", color="burlywood", weight=9];
9.19/3.98	28 -> 19[label="",style="solid", color="burlywood", weight=3];
9.19/3.98	17[label="unzip300 (Tup3 vy40 vy41 vy42)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
9.19/3.98	18[label="unzip301 (Tup3 vy40 vy41 vy42)",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.19/3.98	19[label="unzip302 (Tup3 vy40 vy41 vy42)",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
9.19/3.98	20[label="vy40",fontsize=16,color="green",shape="box"];21[label="vy41",fontsize=16,color="green",shape="box"];22[label="vy42",fontsize=16,color="green",shape="box"];}
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(6)
9.19/3.98	Obligation:
9.19/3.98	Q DP problem:
9.19/3.98	The TRS P consists of the following rules:
9.19/3.98	
9.19/3.98	   new_foldr(Cons(vy30, vy31), h, ba, bb) -> new_foldr(vy31, h, ba, bb)
9.19/3.98	
9.19/3.98	R is empty.
9.19/3.98	Q is empty.
9.19/3.98	We have to consider all minimal (P,Q,R)-chains.
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(7) QDPSizeChangeProof (EQUIVALENT)
9.19/3.98	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. 
9.19/3.98	
9.19/3.98	From the DPs we obtained the following set of size-change graphs:
9.19/3.98	*new_foldr(Cons(vy30, vy31), h, ba, bb) -> new_foldr(vy31, h, ba, bb)
9.19/3.98	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
9.19/3.98	
9.19/3.98	
9.19/3.98	----------------------------------------
9.19/3.98	
9.19/3.98	(8)
9.19/3.98	YES
9.50/4.02	EOF