7.69/3.54	YES
9.18/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.18/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.18/3.98	
9.18/3.98	
9.18/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.18/3.98	
9.18/3.98	(0) HASKELL
9.18/3.98	(1) BR [EQUIVALENT, 0 ms]
9.18/3.98	(2) HASKELL
9.18/3.98	(3) COR [EQUIVALENT, 0 ms]
9.18/3.98	(4) HASKELL
9.18/3.98	(5) Narrow [SOUND, 0 ms]
9.18/3.98	(6) AND
9.18/3.98	    (7) QDP
9.18/3.98	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.18/3.98	        (9) YES
9.18/3.98	    (10) QDP
9.18/3.98	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.18/3.98	        (12) YES
9.18/3.98	
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(0)
9.18/3.98	Obligation:
9.18/3.98	mainModule Main 
9.18/3.98	module Main where {
9.18/3.98	import qualified Prelude;
9.18/3.98	}
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(1) BR (EQUIVALENT)
9.18/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(2)
9.18/3.98	Obligation:
9.18/3.98	mainModule Main 
9.18/3.98	module Main where {
9.18/3.98	import qualified Prelude;
9.18/3.98	}
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(3) COR (EQUIVALENT)
9.18/3.98	Cond Reductions:
9.18/3.98	The following Function with conditions
9.18/3.98	"undefined |Falseundefined;
9.18/3.98	"
9.18/3.98	is transformed to
9.18/3.98	"undefined  = undefined1;
9.18/3.98	"
9.18/3.98	"undefined0 True = undefined;
9.18/3.98	"
9.18/3.98	"undefined1  = undefined0 False;
9.18/3.98	"
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(4)
9.18/3.98	Obligation:
9.18/3.98	mainModule Main 
9.18/3.98	module Main where {
9.18/3.98	import qualified Prelude;
9.18/3.98	}
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(5) Narrow (SOUND)
9.18/3.98	Haskell To QDPs
9.18/3.98	
9.18/3.98	digraph dp_graph {
9.18/3.98	node [outthreshold=100, inthreshold=100];1[label="elem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.18/3.98	3[label="elem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.18/3.98	4[label="elem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.18/3.98	5[label="any . (==)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.18/3.98	6[label="any ((==) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.18/3.98	7[label="or . map ((==) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.18/3.98	8[label="or (map ((==) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.18/3.98	9[label="foldr (||) False (map ((==) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];37[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 37[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	37 -> 10[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	38[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 38[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	38 -> 11[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	10[label="foldr (||) False (map ((==) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.18/3.98	11[label="foldr (||) False (map ((==) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.18/3.98	12[label="foldr (||) False (((==) vx3 vx40) : map ((==) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.18/3.98	13[label="foldr (||) False []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.18/3.98	14 -> 16[label="",style="dashed", color="red", weight=0];
9.18/3.98	14[label="(||) (==) vx3 vx40 foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.18/3.98	15[label="False",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.18/3.98	17[label="foldr (||) False (map ((==) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.18/3.98	16[label="(||) (==) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.18/3.98	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(||) primEqChar vx3 vx40 vx5",fontsize=16,color="burlywood",shape="box"];39[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];19 -> 39[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	39 -> 20[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	20[label="(||) primEqChar (Char vx30) vx40 vx5",fontsize=16,color="burlywood",shape="box"];40[label="vx40/Char vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 40[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	40 -> 21[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	21[label="(||) primEqChar (Char vx30) (Char vx400) vx5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.18/3.98	22[label="(||) primEqNat vx30 vx400 vx5",fontsize=16,color="burlywood",shape="triangle"];41[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];22 -> 41[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	41 -> 23[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	42[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 42[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	42 -> 24[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	23[label="(||) primEqNat (Succ vx300) vx400 vx5",fontsize=16,color="burlywood",shape="box"];43[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 43[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	43 -> 25[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	44[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 44[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	44 -> 26[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	24[label="(||) primEqNat Zero vx400 vx5",fontsize=16,color="burlywood",shape="box"];45[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 45[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	45 -> 27[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	46[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 46[label="",style="solid", color="burlywood", weight=9];
9.18/3.98	46 -> 28[label="",style="solid", color="burlywood", weight=3];
9.18/3.98	25[label="(||) primEqNat (Succ vx300) (Succ vx4000) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
9.18/3.98	26[label="(||) primEqNat (Succ vx300) Zero vx5",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
9.18/3.98	27[label="(||) primEqNat Zero (Succ vx4000) vx5",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
9.18/3.98	28[label="(||) primEqNat Zero Zero vx5",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
9.18/3.98	29 -> 22[label="",style="dashed", color="red", weight=0];
9.18/3.98	29[label="(||) primEqNat vx300 vx4000 vx5",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
9.18/3.98	29 -> 34[label="",style="dashed", color="magenta", weight=3];
9.18/3.98	30[label="(||) False vx5",fontsize=16,color="black",shape="triangle"];30 -> 35[label="",style="solid", color="black", weight=3];
9.18/3.98	31 -> 30[label="",style="dashed", color="red", weight=0];
9.18/3.98	31[label="(||) False vx5",fontsize=16,color="magenta"];32[label="(||) True vx5",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
9.18/3.98	33[label="vx4000",fontsize=16,color="green",shape="box"];34[label="vx300",fontsize=16,color="green",shape="box"];35[label="vx5",fontsize=16,color="green",shape="box"];36[label="True",fontsize=16,color="green",shape="box"];}
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(6)
9.18/3.98	Complex Obligation (AND)
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(7)
9.18/3.98	Obligation:
9.18/3.98	Q DP problem:
9.18/3.98	The TRS P consists of the following rules:
9.18/3.98	
9.18/3.98	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.18/3.98	
9.18/3.98	R is empty.
9.18/3.98	Q is empty.
9.18/3.98	We have to consider all minimal (P,Q,R)-chains.
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(8) QDPSizeChangeProof (EQUIVALENT)
9.18/3.98	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. 
9.18/3.98	
9.18/3.98	From the DPs we obtained the following set of size-change graphs:
9.18/3.98	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.18/3.98	The graph contains the following edges 1 >= 1, 2 > 2
9.18/3.98	
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(9)
9.18/3.98	YES
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(10)
9.18/3.98	Obligation:
9.18/3.98	Q DP problem:
9.18/3.98	The TRS P consists of the following rules:
9.18/3.98	
9.18/3.98	   new_pePe(Succ(vx300), Succ(vx4000), vx5) -> new_pePe(vx300, vx4000, vx5)
9.18/3.98	
9.18/3.98	R is empty.
9.18/3.98	Q is empty.
9.18/3.98	We have to consider all minimal (P,Q,R)-chains.
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(11) QDPSizeChangeProof (EQUIVALENT)
9.18/3.98	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. 
9.18/3.98	
9.18/3.98	From the DPs we obtained the following set of size-change graphs:
9.18/3.98	*new_pePe(Succ(vx300), Succ(vx4000), vx5) -> new_pePe(vx300, vx4000, vx5)
9.18/3.98	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
9.18/3.98	
9.18/3.98	
9.18/3.98	----------------------------------------
9.18/3.98	
9.18/3.98	(12)
9.18/3.98	YES
9.52/4.08	EOF