8.26/3.57	YES
10.06/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.06/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.06/4.06	
10.06/4.06	
10.06/4.06	H-Termination with start terms of the given HASKELL could be proven:
10.06/4.06	
10.06/4.06	(0) HASKELL
10.06/4.06	(1) BR [EQUIVALENT, 0 ms]
10.06/4.06	(2) HASKELL
10.06/4.06	(3) COR [EQUIVALENT, 0 ms]
10.06/4.06	(4) HASKELL
10.06/4.06	(5) Narrow [SOUND, 0 ms]
10.06/4.06	(6) QDP
10.06/4.06	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.06/4.06	(8) YES
10.06/4.06	
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(0)
10.06/4.06	Obligation:
10.06/4.06	mainModule Main 
10.06/4.06	module Main where {
10.06/4.06	import qualified Prelude;
10.06/4.06	data List a = Cons a (List a)  | Nil ;
10.06/4.06	
10.06/4.06	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
10.06/4.06	foldl f z Nil = z;
10.06/4.06	foldl f z (Cons x xs) = foldl f (f z x) xs;
10.06/4.06	
10.06/4.06	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
10.06/4.06	foldl1 f (Cons x xs) = foldl f x xs;
10.06/4.06	
10.06/4.06	}
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(1) BR (EQUIVALENT)
10.06/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(2)
10.06/4.06	Obligation:
10.06/4.06	mainModule Main 
10.06/4.06	module Main where {
10.06/4.06	import qualified Prelude;
10.06/4.06	data List a = Cons a (List a)  | Nil ;
10.06/4.06	
10.06/4.06	foldl :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
10.06/4.06	foldl f z Nil = z;
10.06/4.06	foldl f z (Cons x xs) = foldl f (f z x) xs;
10.06/4.06	
10.06/4.06	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
10.06/4.06	foldl1 f (Cons x xs) = foldl f x xs;
10.06/4.06	
10.06/4.06	}
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(3) COR (EQUIVALENT)
10.06/4.06	Cond Reductions:
10.06/4.06	The following Function with conditions
10.06/4.06	"undefined |Falseundefined;
10.06/4.06	"
10.06/4.06	is transformed to
10.06/4.06	"undefined  = undefined1;
10.06/4.06	"
10.06/4.06	"undefined0 True = undefined;
10.06/4.06	"
10.06/4.06	"undefined1  = undefined0 False;
10.06/4.06	"
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(4)
10.06/4.06	Obligation:
10.06/4.06	mainModule Main 
10.06/4.06	module Main where {
10.06/4.06	import qualified Prelude;
10.06/4.06	data List a = Cons a (List a)  | Nil ;
10.06/4.06	
10.06/4.06	foldl :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
10.06/4.06	foldl f z Nil = z;
10.06/4.06	foldl f z (Cons x xs) = foldl f (f z x) xs;
10.06/4.06	
10.06/4.06	foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
10.06/4.06	foldl1 f (Cons x xs) = foldl f x xs;
10.06/4.06	
10.06/4.06	}
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(5) Narrow (SOUND)
10.06/4.06	Haskell To QDPs
10.06/4.06	
10.06/4.06	digraph dp_graph {
10.06/4.06	node [outthreshold=100, inthreshold=100];1[label="foldl1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.06/4.06	3[label="foldl1 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.06/4.06	4[label="foldl1 vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];17[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
10.06/4.06	17 -> 5[label="",style="solid", color="burlywood", weight=3];
10.06/4.06	18[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
10.06/4.06	18 -> 6[label="",style="solid", color="burlywood", weight=3];
10.06/4.06	5[label="foldl1 vx3 (Cons vx40 vx41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.06/4.06	6[label="foldl1 vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.06/4.06	7[label="foldl vx3 vx40 vx41",fontsize=16,color="burlywood",shape="triangle"];19[label="vx41/Cons vx410 vx411",fontsize=10,color="white",style="solid",shape="box"];7 -> 19[label="",style="solid", color="burlywood", weight=9];
10.06/4.06	19 -> 9[label="",style="solid", color="burlywood", weight=3];
10.06/4.06	20[label="vx41/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 20[label="",style="solid", color="burlywood", weight=9];
10.06/4.06	20 -> 10[label="",style="solid", color="burlywood", weight=3];
10.06/4.06	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl vx3 vx40 (Cons vx410 vx411)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.06/4.06	10[label="foldl vx3 vx40 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.06/4.06	11 -> 7[label="",style="dashed", color="red", weight=0];
10.06/4.06	11[label="foldl vx3 (vx3 vx40 vx410) vx411",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
10.06/4.06	11 -> 14[label="",style="dashed", color="magenta", weight=3];
10.06/4.06	12[label="vx40",fontsize=16,color="green",shape="box"];13[label="vx411",fontsize=16,color="green",shape="box"];14[label="vx3 vx40 vx410",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
10.06/4.06	14 -> 16[label="",style="dashed", color="green", weight=3];
10.06/4.06	15[label="vx40",fontsize=16,color="green",shape="box"];16[label="vx410",fontsize=16,color="green",shape="box"];}
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(6)
10.06/4.06	Obligation:
10.06/4.06	Q DP problem:
10.06/4.06	The TRS P consists of the following rules:
10.06/4.06	
10.06/4.06	   new_foldl(vx3, Cons(vx410, vx411), h) -> new_foldl(vx3, vx411, h)
10.06/4.06	
10.06/4.06	R is empty.
10.06/4.06	Q is empty.
10.06/4.06	We have to consider all minimal (P,Q,R)-chains.
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(7) QDPSizeChangeProof (EQUIVALENT)
10.06/4.06	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. 
10.06/4.06	
10.06/4.06	From the DPs we obtained the following set of size-change graphs:
10.06/4.06	*new_foldl(vx3, Cons(vx410, vx411), h) -> new_foldl(vx3, vx411, h)
10.06/4.06	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.06/4.06	
10.06/4.06	
10.06/4.06	----------------------------------------
10.06/4.06	
10.06/4.06	(8)
10.06/4.06	YES
10.17/4.19	EOF