8.04/3.54	YES
9.34/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.34/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.34/3.98	
9.34/3.98	
9.34/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.34/3.98	
9.34/3.98	(0) HASKELL
9.34/3.98	(1) BR [EQUIVALENT, 0 ms]
9.34/3.98	(2) HASKELL
9.34/3.98	(3) COR [EQUIVALENT, 0 ms]
9.34/3.98	(4) HASKELL
9.34/3.98	(5) Narrow [SOUND, 0 ms]
9.34/3.98	(6) QDP
9.34/3.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.34/3.98	(8) YES
9.34/3.98	
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(0)
9.34/3.98	Obligation:
9.34/3.98	mainModule Main 
9.34/3.98	module Main where {
9.34/3.98	import qualified Prelude;
9.34/3.98	}
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(1) BR (EQUIVALENT)
9.34/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(2)
9.34/3.98	Obligation:
9.34/3.98	mainModule Main 
9.34/3.98	module Main where {
9.34/3.98	import qualified Prelude;
9.34/3.98	}
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(3) COR (EQUIVALENT)
9.34/3.98	Cond Reductions:
9.34/3.98	The following Function with conditions
9.34/3.98	"undefined |Falseundefined;
9.34/3.98	"
9.34/3.98	is transformed to
9.34/3.98	"undefined  = undefined1;
9.34/3.98	"
9.34/3.98	"undefined0 True = undefined;
9.34/3.98	"
9.34/3.98	"undefined1  = undefined0 False;
9.34/3.98	"
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(4)
9.34/3.98	Obligation:
9.34/3.98	mainModule Main 
9.34/3.98	module Main where {
9.34/3.98	import qualified Prelude;
9.34/3.98	}
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(5) Narrow (SOUND)
9.34/3.98	Haskell To QDPs
9.34/3.98	
9.34/3.98	digraph dp_graph {
9.34/3.98	node [outthreshold=100, inthreshold=100];1[label="init",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.34/3.98	3[label="init vx3",fontsize=16,color="burlywood",shape="triangle"];13[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 13[label="",style="solid", color="burlywood", weight=9];
9.34/3.98	13 -> 4[label="",style="solid", color="burlywood", weight=3];
9.34/3.98	14[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 14[label="",style="solid", color="burlywood", weight=9];
9.34/3.98	14 -> 5[label="",style="solid", color="burlywood", weight=3];
9.34/3.98	4[label="init (vx30 : vx31)",fontsize=16,color="burlywood",shape="box"];15[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
9.34/3.98	15 -> 6[label="",style="solid", color="burlywood", weight=3];
9.34/3.98	16[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 16[label="",style="solid", color="burlywood", weight=9];
9.34/3.98	16 -> 7[label="",style="solid", color="burlywood", weight=3];
9.34/3.98	5[label="init []",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
9.34/3.98	6[label="init (vx30 : vx310 : vx311)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.34/3.98	7[label="init (vx30 : [])",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.34/3.98	8[label="error []",fontsize=16,color="red",shape="box"];9[label="vx30 : init (vx310 : vx311)",fontsize=16,color="green",shape="box"];9 -> 11[label="",style="dashed", color="green", weight=3];
9.34/3.98	10[label="[]",fontsize=16,color="green",shape="box"];11 -> 3[label="",style="dashed", color="red", weight=0];
9.34/3.98	11[label="init (vx310 : vx311)",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3];
9.34/3.98	12[label="vx310 : vx311",fontsize=16,color="green",shape="box"];}
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(6)
9.34/3.98	Obligation:
9.34/3.98	Q DP problem:
9.34/3.98	The TRS P consists of the following rules:
9.34/3.98	
9.34/3.98	   new_init(:(vx30, :(vx310, vx311)), h) -> new_init(:(vx310, vx311), h)
9.34/3.98	
9.34/3.98	R is empty.
9.34/3.98	Q is empty.
9.34/3.98	We have to consider all minimal (P,Q,R)-chains.
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(7) QDPSizeChangeProof (EQUIVALENT)
9.34/3.98	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. 
9.34/3.98	
9.34/3.98	From the DPs we obtained the following set of size-change graphs:
9.34/3.98	*new_init(:(vx30, :(vx310, vx311)), h) -> new_init(:(vx310, vx311), h)
9.34/3.98	The graph contains the following edges 1 > 1, 2 >= 2
9.34/3.98	
9.34/3.98	
9.34/3.98	----------------------------------------
9.34/3.98	
9.34/3.98	(8)
9.34/3.98	YES
9.62/4.03	EOF