8.00/3.52	YES
9.57/3.95	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.57/3.95	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.57/3.95	
9.57/3.95	
9.57/3.95	H-Termination with start terms of the given HASKELL could be proven:
9.57/3.95	
9.57/3.95	(0) HASKELL
9.57/3.95	(1) BR [EQUIVALENT, 0 ms]
9.57/3.95	(2) HASKELL
9.57/3.95	(3) COR [EQUIVALENT, 0 ms]
9.57/3.95	(4) HASKELL
9.57/3.95	(5) Narrow [SOUND, 0 ms]
9.57/3.95	(6) QDP
9.57/3.95	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.57/3.95	(8) YES
9.57/3.95	
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(0)
9.57/3.95	Obligation:
9.57/3.95	mainModule Main 
9.57/3.95	module Main where {
9.57/3.95	import qualified Prelude;
9.57/3.95	}
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(1) BR (EQUIVALENT)
9.57/3.95	Replaced joker patterns by fresh variables and removed binding patterns.
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(2)
9.57/3.95	Obligation:
9.57/3.95	mainModule Main 
9.57/3.95	module Main where {
9.57/3.95	import qualified Prelude;
9.57/3.95	}
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(3) COR (EQUIVALENT)
9.57/3.95	Cond Reductions:
9.57/3.95	The following Function with conditions
9.57/3.95	"undefined |Falseundefined;
9.57/3.95	"
9.57/3.95	is transformed to
9.57/3.95	"undefined  = undefined1;
9.57/3.95	"
9.57/3.95	"undefined0 True = undefined;
9.57/3.95	"
9.57/3.95	"undefined1  = undefined0 False;
9.57/3.95	"
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(4)
9.57/3.95	Obligation:
9.57/3.95	mainModule Main 
9.57/3.95	module Main where {
9.57/3.95	import qualified Prelude;
9.57/3.95	}
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(5) Narrow (SOUND)
9.57/3.95	Haskell To QDPs
9.57/3.95	
9.57/3.95	digraph dp_graph {
9.57/3.95	node [outthreshold=100, inthreshold=100];1[label="(==)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.57/3.95	3[label="(==) vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.57/3.95	4[label="(==) vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.57/3.95	5[label="primEqInt vz3 vz4",fontsize=16,color="burlywood",shape="box"];60[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 60[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	60 -> 6[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	61[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];5 -> 61[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	61 -> 7[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	6[label="primEqInt (Pos vz30) vz4",fontsize=16,color="burlywood",shape="box"];62[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];6 -> 62[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	62 -> 8[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	63[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	63 -> 9[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	7[label="primEqInt (Neg vz30) vz4",fontsize=16,color="burlywood",shape="box"];64[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];7 -> 64[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	64 -> 10[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	65[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	65 -> 11[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	8[label="primEqInt (Pos (Succ vz300)) vz4",fontsize=16,color="burlywood",shape="box"];66[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];8 -> 66[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	66 -> 12[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	67[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];8 -> 67[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	67 -> 13[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	9[label="primEqInt (Pos Zero) vz4",fontsize=16,color="burlywood",shape="box"];68[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 68[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	68 -> 14[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	69[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 69[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	69 -> 15[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	10[label="primEqInt (Neg (Succ vz300)) vz4",fontsize=16,color="burlywood",shape="box"];70[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 70[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	70 -> 16[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	71[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 71[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	71 -> 17[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	11[label="primEqInt (Neg Zero) vz4",fontsize=16,color="burlywood",shape="box"];72[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 72[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	72 -> 18[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	73[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 73[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	73 -> 19[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	12[label="primEqInt (Pos (Succ vz300)) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];74[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	74 -> 20[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	75[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 75[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	75 -> 21[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	13[label="primEqInt (Pos (Succ vz300)) (Neg vz40)",fontsize=16,color="black",shape="box"];13 -> 22[label="",style="solid", color="black", weight=3];
9.57/3.95	14[label="primEqInt (Pos Zero) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];76[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];14 -> 76[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	76 -> 23[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	77[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 77[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	77 -> 24[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	15[label="primEqInt (Pos Zero) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];78[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];15 -> 78[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	78 -> 25[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	79[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 79[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	79 -> 26[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	16[label="primEqInt (Neg (Succ vz300)) (Pos vz40)",fontsize=16,color="black",shape="box"];16 -> 27[label="",style="solid", color="black", weight=3];
9.57/3.95	17[label="primEqInt (Neg (Succ vz300)) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];80[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];17 -> 80[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	80 -> 28[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	81[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 81[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	81 -> 29[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	18[label="primEqInt (Neg Zero) (Pos vz40)",fontsize=16,color="burlywood",shape="box"];82[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];18 -> 82[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	82 -> 30[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	83[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 83[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	83 -> 31[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	19[label="primEqInt (Neg Zero) (Neg vz40)",fontsize=16,color="burlywood",shape="box"];84[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];19 -> 84[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	84 -> 32[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	85[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 85[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	85 -> 33[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	20[label="primEqInt (Pos (Succ vz300)) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];20 -> 34[label="",style="solid", color="black", weight=3];
9.57/3.95	21[label="primEqInt (Pos (Succ vz300)) (Pos Zero)",fontsize=16,color="black",shape="box"];21 -> 35[label="",style="solid", color="black", weight=3];
9.57/3.95	22[label="False",fontsize=16,color="green",shape="box"];23[label="primEqInt (Pos Zero) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];23 -> 36[label="",style="solid", color="black", weight=3];
9.57/3.95	24[label="primEqInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 37[label="",style="solid", color="black", weight=3];
9.57/3.95	25[label="primEqInt (Pos Zero) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];25 -> 38[label="",style="solid", color="black", weight=3];
9.57/3.95	26[label="primEqInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 39[label="",style="solid", color="black", weight=3];
9.57/3.95	27[label="False",fontsize=16,color="green",shape="box"];28[label="primEqInt (Neg (Succ vz300)) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];28 -> 40[label="",style="solid", color="black", weight=3];
9.57/3.95	29[label="primEqInt (Neg (Succ vz300)) (Neg Zero)",fontsize=16,color="black",shape="box"];29 -> 41[label="",style="solid", color="black", weight=3];
9.57/3.95	30[label="primEqInt (Neg Zero) (Pos (Succ vz400))",fontsize=16,color="black",shape="box"];30 -> 42[label="",style="solid", color="black", weight=3];
9.57/3.95	31[label="primEqInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3];
9.57/3.95	32[label="primEqInt (Neg Zero) (Neg (Succ vz400))",fontsize=16,color="black",shape="box"];32 -> 44[label="",style="solid", color="black", weight=3];
9.57/3.95	33[label="primEqInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];33 -> 45[label="",style="solid", color="black", weight=3];
9.57/3.95	34[label="primEqNat vz300 vz400",fontsize=16,color="burlywood",shape="triangle"];86[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];34 -> 86[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	86 -> 46[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	87[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 87[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	87 -> 47[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	35[label="False",fontsize=16,color="green",shape="box"];36[label="False",fontsize=16,color="green",shape="box"];37[label="True",fontsize=16,color="green",shape="box"];38[label="False",fontsize=16,color="green",shape="box"];39[label="True",fontsize=16,color="green",shape="box"];40 -> 34[label="",style="dashed", color="red", weight=0];
9.57/3.95	40[label="primEqNat vz300 vz400",fontsize=16,color="magenta"];40 -> 48[label="",style="dashed", color="magenta", weight=3];
9.57/3.95	40 -> 49[label="",style="dashed", color="magenta", weight=3];
9.57/3.95	41[label="False",fontsize=16,color="green",shape="box"];42[label="False",fontsize=16,color="green",shape="box"];43[label="True",fontsize=16,color="green",shape="box"];44[label="False",fontsize=16,color="green",shape="box"];45[label="True",fontsize=16,color="green",shape="box"];46[label="primEqNat (Succ vz3000) vz400",fontsize=16,color="burlywood",shape="box"];88[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];46 -> 88[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	88 -> 50[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	89[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 89[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	89 -> 51[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	47[label="primEqNat Zero vz400",fontsize=16,color="burlywood",shape="box"];90[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];47 -> 90[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	90 -> 52[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	91[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 91[label="",style="solid", color="burlywood", weight=9];
9.57/3.95	91 -> 53[label="",style="solid", color="burlywood", weight=3];
9.57/3.95	48[label="vz300",fontsize=16,color="green",shape="box"];49[label="vz400",fontsize=16,color="green",shape="box"];50[label="primEqNat (Succ vz3000) (Succ vz4000)",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3];
9.57/3.95	51[label="primEqNat (Succ vz3000) Zero",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3];
9.57/3.95	52[label="primEqNat Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
9.57/3.95	53[label="primEqNat Zero Zero",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3];
9.57/3.95	54 -> 34[label="",style="dashed", color="red", weight=0];
9.57/3.95	54[label="primEqNat vz3000 vz4000",fontsize=16,color="magenta"];54 -> 58[label="",style="dashed", color="magenta", weight=3];
9.57/3.95	54 -> 59[label="",style="dashed", color="magenta", weight=3];
9.57/3.95	55[label="False",fontsize=16,color="green",shape="box"];56[label="False",fontsize=16,color="green",shape="box"];57[label="True",fontsize=16,color="green",shape="box"];58[label="vz3000",fontsize=16,color="green",shape="box"];59[label="vz4000",fontsize=16,color="green",shape="box"];}
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(6)
9.57/3.95	Obligation:
9.57/3.95	Q DP problem:
9.57/3.95	The TRS P consists of the following rules:
9.57/3.95	
9.57/3.95	   new_primEqNat(Succ(vz3000), Succ(vz4000)) -> new_primEqNat(vz3000, vz4000)
9.57/3.95	
9.57/3.95	R is empty.
9.57/3.95	Q is empty.
9.57/3.95	We have to consider all minimal (P,Q,R)-chains.
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(7) QDPSizeChangeProof (EQUIVALENT)
9.57/3.95	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. 
9.57/3.95	
9.57/3.95	From the DPs we obtained the following set of size-change graphs:
9.57/3.95	*new_primEqNat(Succ(vz3000), Succ(vz4000)) -> new_primEqNat(vz3000, vz4000)
9.57/3.95	The graph contains the following edges 1 > 1, 2 > 2
9.57/3.95	
9.57/3.95	
9.57/3.95	----------------------------------------
9.57/3.95	
9.57/3.95	(8)
9.57/3.95	YES
9.57/3.98	EOF