8.70/3.63	YES
10.29/4.09	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
10.29/4.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.29/4.09	
10.29/4.09	
10.29/4.09	H-Termination with start terms of the given HASKELL could be proven:
10.29/4.09	
10.29/4.09	(0) HASKELL
10.29/4.09	(1) BR [EQUIVALENT, 0 ms]
10.29/4.09	(2) HASKELL
10.29/4.09	(3) COR [EQUIVALENT, 0 ms]
10.29/4.09	(4) HASKELL
10.29/4.09	(5) Narrow [SOUND, 0 ms]
10.29/4.09	(6) QDP
10.29/4.09	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.29/4.09	(8) YES
10.29/4.09	
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(0)
10.29/4.09	Obligation:
10.29/4.09	mainModule Main 
10.29/4.09	module Main where {
10.29/4.09	import qualified Prelude;
10.29/4.09	}
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(1) BR (EQUIVALENT)
10.29/4.09	Replaced joker patterns by fresh variables and removed binding patterns.
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(2)
10.29/4.09	Obligation:
10.29/4.09	mainModule Main 
10.29/4.09	module Main where {
10.29/4.09	import qualified Prelude;
10.29/4.09	}
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(3) COR (EQUIVALENT)
10.29/4.09	Cond Reductions:
10.29/4.09	The following Function with conditions
10.29/4.09	"undefined |Falseundefined;
10.29/4.09	"
10.29/4.09	is transformed to
10.29/4.09	"undefined  = undefined1;
10.29/4.09	"
10.29/4.09	"undefined0 True = undefined;
10.29/4.09	"
10.29/4.09	"undefined1  = undefined0 False;
10.29/4.09	"
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(4)
10.29/4.09	Obligation:
10.29/4.09	mainModule Main 
10.29/4.09	module Main where {
10.29/4.09	import qualified Prelude;
10.29/4.09	}
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(5) Narrow (SOUND)
10.29/4.09	Haskell To QDPs
10.29/4.09	
10.29/4.09	digraph dp_graph {
10.29/4.09	node [outthreshold=100, inthreshold=100];1[label="(<=)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.29/4.09	3[label="(<=) vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.29/4.09	4[label="(<=) vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.29/4.09	5[label="compare vx3 vx4 /= GT",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.29/4.09	6[label="not (compare vx3 vx4 == GT)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.29/4.09	7[label="not (primCmpInt vx3 vx4 == GT)",fontsize=16,color="burlywood",shape="box"];73[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 73[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	73 -> 8[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	74[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];7 -> 74[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	74 -> 9[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	8[label="not (primCmpInt (Pos vx30) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];75[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];8 -> 75[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	75 -> 10[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	76[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 76[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	76 -> 11[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	9[label="not (primCmpInt (Neg vx30) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];77[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	77 -> 12[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	78[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 78[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	78 -> 13[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	10[label="not (primCmpInt (Pos (Succ vx300)) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];79[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 79[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	79 -> 14[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	80[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 80[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	80 -> 15[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	11[label="not (primCmpInt (Pos Zero) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];81[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	81 -> 16[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	82[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 82[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	82 -> 17[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	12[label="not (primCmpInt (Neg (Succ vx300)) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];83[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	83 -> 18[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	84[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];12 -> 84[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	84 -> 19[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	13[label="not (primCmpInt (Neg Zero) vx4 == GT)",fontsize=16,color="burlywood",shape="box"];85[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 85[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	85 -> 20[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	86[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	86 -> 21[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	14[label="not (primCmpInt (Pos (Succ vx300)) (Pos vx40) == GT)",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
10.29/4.09	15[label="not (primCmpInt (Pos (Succ vx300)) (Neg vx40) == GT)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
10.29/4.09	16[label="not (primCmpInt (Pos Zero) (Pos vx40) == GT)",fontsize=16,color="burlywood",shape="box"];87[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 87[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	87 -> 24[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	88[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 88[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	88 -> 25[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	17[label="not (primCmpInt (Pos Zero) (Neg vx40) == GT)",fontsize=16,color="burlywood",shape="box"];89[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];17 -> 89[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	89 -> 26[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	90[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 90[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	90 -> 27[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	18[label="not (primCmpInt (Neg (Succ vx300)) (Pos vx40) == GT)",fontsize=16,color="black",shape="box"];18 -> 28[label="",style="solid", color="black", weight=3];
10.29/4.09	19[label="not (primCmpInt (Neg (Succ vx300)) (Neg vx40) == GT)",fontsize=16,color="black",shape="box"];19 -> 29[label="",style="solid", color="black", weight=3];
10.29/4.09	20[label="not (primCmpInt (Neg Zero) (Pos vx40) == GT)",fontsize=16,color="burlywood",shape="box"];91[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 91[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	91 -> 30[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	92[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 92[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	92 -> 31[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	21[label="not (primCmpInt (Neg Zero) (Neg vx40) == GT)",fontsize=16,color="burlywood",shape="box"];93[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];21 -> 93[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	93 -> 32[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	94[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 94[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	94 -> 33[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	22[label="not (primCmpNat (Succ vx300) vx40 == GT)",fontsize=16,color="burlywood",shape="triangle"];95[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];22 -> 95[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	95 -> 34[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	96[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 96[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	96 -> 35[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	23[label="not (GT == GT)",fontsize=16,color="black",shape="triangle"];23 -> 36[label="",style="solid", color="black", weight=3];
10.29/4.09	24[label="not (primCmpInt (Pos Zero) (Pos (Succ vx400)) == GT)",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
10.29/4.09	25[label="not (primCmpInt (Pos Zero) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
10.29/4.09	26[label="not (primCmpInt (Pos Zero) (Neg (Succ vx400)) == GT)",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
10.29/4.09	27[label="not (primCmpInt (Pos Zero) (Neg Zero) == GT)",fontsize=16,color="black",shape="box"];27 -> 40[label="",style="solid", color="black", weight=3];
10.29/4.09	28[label="not (LT == GT)",fontsize=16,color="black",shape="triangle"];28 -> 41[label="",style="solid", color="black", weight=3];
10.29/4.09	29[label="not (primCmpNat vx40 (Succ vx300) == GT)",fontsize=16,color="burlywood",shape="triangle"];97[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	97 -> 42[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	98[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	98 -> 43[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	30[label="not (primCmpInt (Neg Zero) (Pos (Succ vx400)) == GT)",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
10.29/4.09	31[label="not (primCmpInt (Neg Zero) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
10.29/4.09	32[label="not (primCmpInt (Neg Zero) (Neg (Succ vx400)) == GT)",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
10.29/4.09	33[label="not (primCmpInt (Neg Zero) (Neg Zero) == GT)",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
10.29/4.09	34[label="not (primCmpNat (Succ vx300) (Succ vx400) == GT)",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
10.29/4.09	35[label="not (primCmpNat (Succ vx300) Zero == GT)",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
10.29/4.09	36[label="not True",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
10.29/4.09	37 -> 29[label="",style="dashed", color="red", weight=0];
10.29/4.09	37[label="not (primCmpNat Zero (Succ vx400) == GT)",fontsize=16,color="magenta"];37 -> 51[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	37 -> 52[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	38[label="not (EQ == GT)",fontsize=16,color="black",shape="triangle"];38 -> 53[label="",style="solid", color="black", weight=3];
10.29/4.09	39 -> 23[label="",style="dashed", color="red", weight=0];
10.29/4.09	39[label="not (GT == GT)",fontsize=16,color="magenta"];40 -> 38[label="",style="dashed", color="red", weight=0];
10.29/4.09	40[label="not (EQ == GT)",fontsize=16,color="magenta"];41[label="not False",fontsize=16,color="black",shape="triangle"];41 -> 54[label="",style="solid", color="black", weight=3];
10.29/4.09	42[label="not (primCmpNat (Succ vx400) (Succ vx300) == GT)",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
10.29/4.09	43[label="not (primCmpNat Zero (Succ vx300) == GT)",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
10.29/4.09	44 -> 28[label="",style="dashed", color="red", weight=0];
10.29/4.09	44[label="not (LT == GT)",fontsize=16,color="magenta"];45 -> 38[label="",style="dashed", color="red", weight=0];
10.29/4.09	45[label="not (EQ == GT)",fontsize=16,color="magenta"];46 -> 22[label="",style="dashed", color="red", weight=0];
10.29/4.09	46[label="not (primCmpNat (Succ vx400) Zero == GT)",fontsize=16,color="magenta"];46 -> 57[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	46 -> 58[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	47 -> 38[label="",style="dashed", color="red", weight=0];
10.29/4.09	47[label="not (EQ == GT)",fontsize=16,color="magenta"];48[label="not (primCmpNat vx300 vx400 == GT)",fontsize=16,color="burlywood",shape="triangle"];99[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 99[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	99 -> 59[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	100[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 100[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	100 -> 60[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	49 -> 23[label="",style="dashed", color="red", weight=0];
10.29/4.09	49[label="not (GT == GT)",fontsize=16,color="magenta"];50[label="False",fontsize=16,color="green",shape="box"];51[label="vx400",fontsize=16,color="green",shape="box"];52[label="Zero",fontsize=16,color="green",shape="box"];53 -> 41[label="",style="dashed", color="red", weight=0];
10.29/4.09	53[label="not False",fontsize=16,color="magenta"];54[label="True",fontsize=16,color="green",shape="box"];55 -> 48[label="",style="dashed", color="red", weight=0];
10.29/4.09	55[label="not (primCmpNat vx400 vx300 == GT)",fontsize=16,color="magenta"];55 -> 61[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	55 -> 62[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	56 -> 28[label="",style="dashed", color="red", weight=0];
10.29/4.09	56[label="not (LT == GT)",fontsize=16,color="magenta"];57[label="vx400",fontsize=16,color="green",shape="box"];58[label="Zero",fontsize=16,color="green",shape="box"];59[label="not (primCmpNat (Succ vx3000) vx400 == GT)",fontsize=16,color="burlywood",shape="box"];101[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];59 -> 101[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	101 -> 63[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	102[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 102[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	102 -> 64[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	60[label="not (primCmpNat Zero vx400 == GT)",fontsize=16,color="burlywood",shape="box"];103[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];60 -> 103[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	103 -> 65[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	104[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 104[label="",style="solid", color="burlywood", weight=9];
10.29/4.09	104 -> 66[label="",style="solid", color="burlywood", weight=3];
10.29/4.09	61[label="vx300",fontsize=16,color="green",shape="box"];62[label="vx400",fontsize=16,color="green",shape="box"];63[label="not (primCmpNat (Succ vx3000) (Succ vx4000) == GT)",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3];
10.29/4.09	64[label="not (primCmpNat (Succ vx3000) Zero == GT)",fontsize=16,color="black",shape="box"];64 -> 68[label="",style="solid", color="black", weight=3];
10.29/4.09	65[label="not (primCmpNat Zero (Succ vx4000) == GT)",fontsize=16,color="black",shape="box"];65 -> 69[label="",style="solid", color="black", weight=3];
10.29/4.09	66[label="not (primCmpNat Zero Zero == GT)",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
10.29/4.09	67 -> 48[label="",style="dashed", color="red", weight=0];
10.29/4.09	67[label="not (primCmpNat vx3000 vx4000 == GT)",fontsize=16,color="magenta"];67 -> 71[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	67 -> 72[label="",style="dashed", color="magenta", weight=3];
10.29/4.09	68 -> 23[label="",style="dashed", color="red", weight=0];
10.29/4.09	68[label="not (GT == GT)",fontsize=16,color="magenta"];69 -> 28[label="",style="dashed", color="red", weight=0];
10.29/4.09	69[label="not (LT == GT)",fontsize=16,color="magenta"];70 -> 38[label="",style="dashed", color="red", weight=0];
10.29/4.09	70[label="not (EQ == GT)",fontsize=16,color="magenta"];71[label="vx4000",fontsize=16,color="green",shape="box"];72[label="vx3000",fontsize=16,color="green",shape="box"];}
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(6)
10.29/4.09	Obligation:
10.29/4.09	Q DP problem:
10.29/4.09	The TRS P consists of the following rules:
10.29/4.09	
10.29/4.09	   new_not(Succ(vx3000), Succ(vx4000)) -> new_not(vx3000, vx4000)
10.29/4.09	
10.29/4.09	R is empty.
10.29/4.09	Q is empty.
10.29/4.09	We have to consider all minimal (P,Q,R)-chains.
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(7) QDPSizeChangeProof (EQUIVALENT)
10.29/4.09	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. 
10.29/4.09	
10.29/4.09	From the DPs we obtained the following set of size-change graphs:
10.29/4.09	*new_not(Succ(vx3000), Succ(vx4000)) -> new_not(vx3000, vx4000)
10.29/4.09	The graph contains the following edges 1 > 1, 2 > 2
10.29/4.09	
10.29/4.09	
10.29/4.09	----------------------------------------
10.29/4.09	
10.29/4.09	(8)
10.29/4.09	YES
10.29/4.12	EOF