7.66/3.57	YES
9.41/4.07	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.41/4.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.41/4.07	
9.41/4.07	
9.41/4.07	H-Termination with start terms of the given HASKELL could be proven:
9.41/4.07	
9.41/4.07	(0) HASKELL
9.41/4.07	(1) BR [EQUIVALENT, 0 ms]
9.41/4.07	(2) HASKELL
9.41/4.07	(3) COR [EQUIVALENT, 0 ms]
9.41/4.07	(4) HASKELL
9.41/4.07	(5) Narrow [SOUND, 0 ms]
9.41/4.07	(6) QDP
9.41/4.07	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.41/4.07	(8) YES
9.41/4.07	
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(0)
9.41/4.07	Obligation:
9.41/4.07	mainModule Main 
9.41/4.07	module Main where {
9.41/4.07	import qualified Prelude;
9.41/4.07	data List a = Cons a (List a)  | Nil ;
9.41/4.07	
9.41/4.07	map :: (a  ->  b)  ->  List a  ->  List b;
9.41/4.07	map f Nil = Nil;
9.41/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.41/4.07	
9.41/4.07	}
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(1) BR (EQUIVALENT)
9.41/4.07	Replaced joker patterns by fresh variables and removed binding patterns.
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(2)
9.41/4.07	Obligation:
9.41/4.07	mainModule Main 
9.41/4.07	module Main where {
9.41/4.07	import qualified Prelude;
9.41/4.07	data List a = Cons a (List a)  | Nil ;
9.41/4.07	
9.41/4.07	map :: (b  ->  a)  ->  List b  ->  List a;
9.41/4.07	map f Nil = Nil;
9.41/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.41/4.07	
9.41/4.07	}
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(3) COR (EQUIVALENT)
9.41/4.07	Cond Reductions:
9.41/4.07	The following Function with conditions
9.41/4.07	"undefined |Falseundefined;
9.41/4.07	"
9.41/4.07	is transformed to
9.41/4.07	"undefined  = undefined1;
9.41/4.07	"
9.41/4.07	"undefined0 True = undefined;
9.41/4.07	"
9.41/4.07	"undefined1  = undefined0 False;
9.41/4.07	"
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(4)
9.41/4.07	Obligation:
9.41/4.07	mainModule Main 
9.41/4.07	module Main where {
9.41/4.07	import qualified Prelude;
9.41/4.07	data List a = Cons a (List a)  | Nil ;
9.41/4.07	
9.41/4.07	map :: (b  ->  a)  ->  List b  ->  List a;
9.41/4.07	map f Nil = Nil;
9.41/4.07	map f (Cons x xs) = Cons (f x) (map f xs);
9.41/4.07	
9.41/4.07	}
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(5) Narrow (SOUND)
9.41/4.07	Haskell To QDPs
9.41/4.07	
9.41/4.07	digraph dp_graph {
9.41/4.07	node [outthreshold=100, inthreshold=100];1[label="map",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.41/4.07	3[label="map vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.41/4.07	4[label="map vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];13[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 13[label="",style="solid", color="burlywood", weight=9];
9.41/4.07	13 -> 5[label="",style="solid", color="burlywood", weight=3];
9.41/4.07	14[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 14[label="",style="solid", color="burlywood", weight=9];
9.41/4.07	14 -> 6[label="",style="solid", color="burlywood", weight=3];
9.41/4.07	5[label="map vx3 (Cons vx40 vx41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.41/4.07	6[label="map vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.41/4.07	7[label="Cons (vx3 vx40) (map vx3 vx41)",fontsize=16,color="green",shape="box"];7 -> 9[label="",style="dashed", color="green", weight=3];
9.41/4.07	7 -> 10[label="",style="dashed", color="green", weight=3];
9.41/4.07	8[label="Nil",fontsize=16,color="green",shape="box"];9[label="vx3 vx40",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
9.41/4.07	10 -> 4[label="",style="dashed", color="red", weight=0];
9.41/4.07	10[label="map vx3 vx41",fontsize=16,color="magenta"];10 -> 12[label="",style="dashed", color="magenta", weight=3];
9.41/4.07	11[label="vx40",fontsize=16,color="green",shape="box"];12[label="vx41",fontsize=16,color="green",shape="box"];}
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(6)
9.41/4.07	Obligation:
9.41/4.07	Q DP problem:
9.41/4.07	The TRS P consists of the following rules:
9.41/4.07	
9.41/4.07	   new_map(vx3, Cons(vx40, vx41), h, ba) -> new_map(vx3, vx41, h, ba)
9.41/4.07	
9.41/4.07	R is empty.
9.41/4.07	Q is empty.
9.41/4.07	We have to consider all minimal (P,Q,R)-chains.
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(7) QDPSizeChangeProof (EQUIVALENT)
9.41/4.07	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. 
9.41/4.07	
9.41/4.07	From the DPs we obtained the following set of size-change graphs:
9.41/4.07	*new_map(vx3, Cons(vx40, vx41), h, ba) -> new_map(vx3, vx41, h, ba)
9.41/4.07	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.41/4.07	
9.41/4.07	
9.41/4.07	----------------------------------------
9.41/4.07	
9.41/4.07	(8)
9.41/4.07	YES
9.59/4.18	EOF