7.72/4.24	YES
9.46/4.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.46/4.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.46/4.74	
9.46/4.74	
9.46/4.74	H-Termination with start terms of the given HASKELL could be proven:
9.46/4.74	
9.46/4.74	(0) HASKELL
9.46/4.74	(1) BR [EQUIVALENT, 0 ms]
9.46/4.74	(2) HASKELL
9.46/4.74	(3) COR [EQUIVALENT, 0 ms]
9.46/4.74	(4) HASKELL
9.46/4.74	(5) Narrow [SOUND, 0 ms]
9.46/4.74	(6) QDP
9.46/4.74	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.46/4.74	(8) YES
9.46/4.74	
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(0)
9.46/4.74	Obligation:
9.46/4.74	mainModule Main 
9.46/4.74	module Main where {
9.46/4.74	import qualified Prelude;
9.46/4.74	data List a = Cons a (List a)  | Nil ;
9.46/4.74	
9.46/4.74	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
9.46/4.74	foldl f z Nil = z;
9.46/4.74	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.46/4.74	
9.46/4.74	}
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(1) BR (EQUIVALENT)
9.46/4.74	Replaced joker patterns by fresh variables and removed binding patterns.
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(2)
9.46/4.74	Obligation:
9.46/4.74	mainModule Main 
9.46/4.74	module Main where {
9.46/4.74	import qualified Prelude;
9.46/4.74	data List a = Cons a (List a)  | Nil ;
9.46/4.74	
9.46/4.74	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
9.46/4.74	foldl f z Nil = z;
9.46/4.74	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.46/4.74	
9.46/4.74	}
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(3) COR (EQUIVALENT)
9.46/4.74	Cond Reductions:
9.46/4.74	The following Function with conditions
9.46/4.74	"undefined |Falseundefined;
9.46/4.74	"
9.46/4.74	is transformed to
9.46/4.74	"undefined  = undefined1;
9.46/4.74	"
9.46/4.74	"undefined0 True = undefined;
9.46/4.74	"
9.46/4.74	"undefined1  = undefined0 False;
9.46/4.74	"
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(4)
9.46/4.74	Obligation:
9.46/4.74	mainModule Main 
9.46/4.74	module Main where {
9.46/4.74	import qualified Prelude;
9.46/4.74	data List a = Cons a (List a)  | Nil ;
9.46/4.74	
9.46/4.74	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
9.46/4.74	foldl f z Nil = z;
9.46/4.74	foldl f z (Cons x xs) = foldl f (f z x) xs;
9.46/4.74	
9.46/4.74	}
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(5) Narrow (SOUND)
9.46/4.74	Haskell To QDPs
9.46/4.74	
9.46/4.74	digraph dp_graph {
9.46/4.74	node [outthreshold=100, inthreshold=100];1[label="foldl",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.46/4.74	3[label="foldl vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.46/4.74	4[label="foldl vx3 vx4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
9.46/4.74	5[label="foldl vx3 vx4 vx5",fontsize=16,color="burlywood",shape="triangle"];14[label="vx5/Cons vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
9.46/4.74	14 -> 6[label="",style="solid", color="burlywood", weight=3];
9.46/4.74	15[label="vx5/Nil",fontsize=10,color="white",style="solid",shape="box"];5 -> 15[label="",style="solid", color="burlywood", weight=9];
9.46/4.74	15 -> 7[label="",style="solid", color="burlywood", weight=3];
9.46/4.74	6[label="foldl vx3 vx4 (Cons vx50 vx51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.46/4.74	7[label="foldl vx3 vx4 Nil",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.46/4.74	8 -> 5[label="",style="dashed", color="red", weight=0];
9.46/4.74	8[label="foldl vx3 (vx3 vx4 vx50) vx51",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
9.46/4.74	8 -> 11[label="",style="dashed", color="magenta", weight=3];
9.46/4.74	9[label="vx4",fontsize=16,color="green",shape="box"];10[label="vx51",fontsize=16,color="green",shape="box"];11[label="vx3 vx4 vx50",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
9.46/4.74	11 -> 13[label="",style="dashed", color="green", weight=3];
9.46/4.74	12[label="vx4",fontsize=16,color="green",shape="box"];13[label="vx50",fontsize=16,color="green",shape="box"];}
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(6)
9.46/4.74	Obligation:
9.46/4.74	Q DP problem:
9.46/4.74	The TRS P consists of the following rules:
9.46/4.74	
9.46/4.74	   new_foldl(vx3, Cons(vx50, vx51), h, ba) -> new_foldl(vx3, vx51, h, ba)
9.46/4.74	
9.46/4.74	R is empty.
9.46/4.74	Q is empty.
9.46/4.74	We have to consider all minimal (P,Q,R)-chains.
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(7) QDPSizeChangeProof (EQUIVALENT)
9.46/4.74	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. 
9.46/4.74	
9.46/4.74	From the DPs we obtained the following set of size-change graphs:
9.46/4.74	*new_foldl(vx3, Cons(vx50, vx51), h, ba) -> new_foldl(vx3, vx51, h, ba)
9.46/4.74	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.46/4.74	
9.46/4.74	
9.46/4.74	----------------------------------------
9.46/4.74	
9.46/4.74	(8)
9.46/4.74	YES
9.47/4.79	EOF