7.84/3.58	YES
9.44/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.44/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.44/4.06	
9.44/4.06	
9.44/4.06	H-Termination with start terms of the given HASKELL could be proven:
9.44/4.06	
9.44/4.06	(0) HASKELL
9.44/4.06	(1) BR [EQUIVALENT, 0 ms]
9.44/4.06	(2) HASKELL
9.44/4.06	(3) COR [EQUIVALENT, 0 ms]
9.44/4.06	(4) HASKELL
9.44/4.06	(5) Narrow [SOUND, 0 ms]
9.44/4.06	(6) QDP
9.44/4.06	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.44/4.06	(8) YES
9.44/4.06	
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(0)
9.44/4.06	Obligation:
9.44/4.06	mainModule Main 
9.44/4.06	module Main where {
9.44/4.06	import qualified Prelude;
9.44/4.06	data List a = Cons a (List a)  | Nil ;
9.44/4.06	
9.44/4.06	last :: List a  ->  a;
9.44/4.06	last (Cons x Nil) = x;
9.44/4.06	last (Cons vv xs) = last xs;
9.44/4.06	
9.44/4.06	}
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(1) BR (EQUIVALENT)
9.44/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(2)
9.44/4.06	Obligation:
9.44/4.06	mainModule Main 
9.44/4.06	module Main where {
9.44/4.06	import qualified Prelude;
9.44/4.06	data List a = Cons a (List a)  | Nil ;
9.44/4.06	
9.44/4.06	last :: List a  ->  a;
9.44/4.06	last (Cons x Nil) = x;
9.44/4.06	last (Cons vv xs) = last xs;
9.44/4.06	
9.44/4.06	}
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(3) COR (EQUIVALENT)
9.44/4.06	Cond Reductions:
9.44/4.06	The following Function with conditions
9.44/4.06	"undefined |Falseundefined;
9.44/4.06	"
9.44/4.06	is transformed to
9.44/4.06	"undefined  = undefined1;
9.44/4.06	"
9.44/4.06	"undefined0 True = undefined;
9.44/4.06	"
9.44/4.06	"undefined1  = undefined0 False;
9.44/4.06	"
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(4)
9.44/4.06	Obligation:
9.44/4.06	mainModule Main 
9.44/4.06	module Main where {
9.44/4.06	import qualified Prelude;
9.44/4.06	data List a = Cons a (List a)  | Nil ;
9.44/4.06	
9.44/4.06	last :: List a  ->  a;
9.44/4.06	last (Cons x Nil) = x;
9.44/4.06	last (Cons vv xs) = last xs;
9.44/4.06	
9.44/4.06	}
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(5) Narrow (SOUND)
9.44/4.06	Haskell To QDPs
9.44/4.06	
9.44/4.06	digraph dp_graph {
9.44/4.06	node [outthreshold=100, inthreshold=100];1[label="last",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.44/4.06	3[label="last vy3",fontsize=16,color="burlywood",shape="triangle"];12[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];3 -> 12[label="",style="solid", color="burlywood", weight=9];
9.44/4.06	12 -> 4[label="",style="solid", color="burlywood", weight=3];
9.44/4.06	13[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];3 -> 13[label="",style="solid", color="burlywood", weight=9];
9.44/4.06	13 -> 5[label="",style="solid", color="burlywood", weight=3];
9.44/4.06	4[label="last (Cons vy30 vy31)",fontsize=16,color="burlywood",shape="box"];14[label="vy31/Cons vy310 vy311",fontsize=10,color="white",style="solid",shape="box"];4 -> 14[label="",style="solid", color="burlywood", weight=9];
9.44/4.06	14 -> 6[label="",style="solid", color="burlywood", weight=3];
9.44/4.06	15[label="vy31/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
9.44/4.06	15 -> 7[label="",style="solid", color="burlywood", weight=3];
9.44/4.06	5[label="last Nil",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
9.44/4.06	6[label="last (Cons vy30 (Cons vy310 vy311))",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.44/4.06	7[label="last (Cons vy30 Nil)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.44/4.06	8[label="error []",fontsize=16,color="red",shape="box"];9 -> 3[label="",style="dashed", color="red", weight=0];
9.44/4.06	9[label="last (Cons vy310 vy311)",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3];
9.44/4.06	10[label="vy30",fontsize=16,color="green",shape="box"];11[label="Cons vy310 vy311",fontsize=16,color="green",shape="box"];}
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(6)
9.44/4.06	Obligation:
9.44/4.06	Q DP problem:
9.44/4.06	The TRS P consists of the following rules:
9.44/4.06	
9.44/4.06	   new_last(Cons(vy30, Cons(vy310, vy311)), h) -> new_last(Cons(vy310, vy311), h)
9.44/4.06	
9.44/4.06	R is empty.
9.44/4.06	Q is empty.
9.44/4.06	We have to consider all minimal (P,Q,R)-chains.
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(7) QDPSizeChangeProof (EQUIVALENT)
9.44/4.06	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. 
9.44/4.06	
9.44/4.06	From the DPs we obtained the following set of size-change graphs:
9.44/4.06	*new_last(Cons(vy30, Cons(vy310, vy311)), h) -> new_last(Cons(vy310, vy311), h)
9.44/4.06	The graph contains the following edges 1 > 1, 2 >= 2
9.44/4.06	
9.44/4.06	
9.44/4.06	----------------------------------------
9.44/4.06	
9.44/4.06	(8)
9.44/4.06	YES
9.60/4.10	EOF