8.09/3.61	YES
9.60/4.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.60/4.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.60/4.08	
9.60/4.08	
9.60/4.08	H-Termination with start terms of the given HASKELL could be proven:
9.60/4.08	
9.60/4.08	(0) HASKELL
9.60/4.08	(1) BR [EQUIVALENT, 0 ms]
9.60/4.08	(2) HASKELL
9.60/4.08	(3) COR [EQUIVALENT, 0 ms]
9.60/4.08	(4) HASKELL
9.60/4.08	(5) Narrow [SOUND, 0 ms]
9.60/4.08	(6) QDP
9.60/4.08	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.60/4.08	(8) YES
9.60/4.08	
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(0)
9.60/4.08	Obligation:
9.60/4.08	mainModule Main 
9.60/4.08	module Main where {
9.60/4.08	import qualified Prelude;
9.60/4.08	}
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(1) BR (EQUIVALENT)
9.60/4.08	Replaced joker patterns by fresh variables and removed binding patterns.
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(2)
9.60/4.08	Obligation:
9.60/4.08	mainModule Main 
9.60/4.08	module Main where {
9.60/4.08	import qualified Prelude;
9.60/4.08	}
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(3) COR (EQUIVALENT)
9.60/4.08	Cond Reductions:
9.60/4.08	The following Function with conditions
9.60/4.08	"undefined |Falseundefined;
9.60/4.08	"
9.60/4.08	is transformed to
9.60/4.08	"undefined  = undefined1;
9.60/4.08	"
9.60/4.08	"undefined0 True = undefined;
9.60/4.08	"
9.60/4.08	"undefined1  = undefined0 False;
9.60/4.08	"
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(4)
9.60/4.08	Obligation:
9.60/4.08	mainModule Main 
9.60/4.08	module Main where {
9.60/4.08	import qualified Prelude;
9.60/4.08	}
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(5) Narrow (SOUND)
9.60/4.08	Haskell To QDPs
9.60/4.08	
9.60/4.08	digraph dp_graph {
9.60/4.08	node [outthreshold=100, inthreshold=100];1[label="odd",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.60/4.08	3[label="odd vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.60/4.08	4[label="not . even",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
9.60/4.08	5[label="not (even vx3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.60/4.08	6[label="not (primEvenInt vx3)",fontsize=16,color="burlywood",shape="box"];22[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 22[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	22 -> 7[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	23[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 23[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	23 -> 8[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	7[label="not (primEvenInt (Pos vx30))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.60/4.08	8[label="not (primEvenInt (Neg vx30))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.60/4.08	9[label="not (primEvenNat vx30)",fontsize=16,color="burlywood",shape="triangle"];24[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	24 -> 11[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	25[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	25 -> 12[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	10 -> 9[label="",style="dashed", color="red", weight=0];
9.60/4.08	10[label="not (primEvenNat vx30)",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
9.60/4.08	11[label="not (primEvenNat (Succ vx300))",fontsize=16,color="burlywood",shape="box"];26[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	26 -> 14[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	27[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 27[label="",style="solid", color="burlywood", weight=9];
9.60/4.08	27 -> 15[label="",style="solid", color="burlywood", weight=3];
9.60/4.08	12[label="not (primEvenNat Zero)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
9.60/4.08	13[label="vx30",fontsize=16,color="green",shape="box"];14[label="not (primEvenNat (Succ (Succ vx3000)))",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.60/4.08	15[label="not (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.60/4.08	16[label="not True",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
9.60/4.08	17 -> 9[label="",style="dashed", color="red", weight=0];
9.60/4.08	17[label="not (primEvenNat vx3000)",fontsize=16,color="magenta"];17 -> 20[label="",style="dashed", color="magenta", weight=3];
9.60/4.08	18[label="not False",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.60/4.08	19[label="False",fontsize=16,color="green",shape="box"];20[label="vx3000",fontsize=16,color="green",shape="box"];21[label="True",fontsize=16,color="green",shape="box"];}
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(6)
9.60/4.08	Obligation:
9.60/4.08	Q DP problem:
9.60/4.08	The TRS P consists of the following rules:
9.60/4.08	
9.60/4.08	   new_not(Succ(Succ(vx3000))) -> new_not(vx3000)
9.60/4.08	
9.60/4.08	R is empty.
9.60/4.08	Q is empty.
9.60/4.08	We have to consider all minimal (P,Q,R)-chains.
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(7) QDPSizeChangeProof (EQUIVALENT)
9.60/4.08	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. 
9.60/4.08	
9.60/4.08	From the DPs we obtained the following set of size-change graphs:
9.60/4.08	*new_not(Succ(Succ(vx3000))) -> new_not(vx3000)
9.60/4.08	The graph contains the following edges 1 > 1
9.60/4.08	
9.60/4.08	
9.60/4.08	----------------------------------------
9.60/4.08	
9.60/4.08	(8)
9.60/4.08	YES
9.78/4.12	EOF