7.67/3.50	YES
9.23/3.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.23/3.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.23/3.98	
9.23/3.98	
9.23/3.98	H-Termination with start terms of the given HASKELL could be proven:
9.23/3.98	
9.23/3.98	(0) HASKELL
9.23/3.98	(1) BR [EQUIVALENT, 0 ms]
9.23/3.98	(2) HASKELL
9.23/3.98	(3) COR [EQUIVALENT, 0 ms]
9.23/3.98	(4) HASKELL
9.23/3.98	(5) Narrow [SOUND, 0 ms]
9.23/3.98	(6) QDP
9.23/3.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.23/3.98	(8) YES
9.23/3.98	
9.23/3.98	
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(0)
9.23/3.98	Obligation:
9.23/3.98	mainModule Main 
9.23/3.98	module Main where {
9.23/3.98	import qualified Prelude;
9.23/3.98	}
9.23/3.98	
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(1) BR (EQUIVALENT)
9.23/3.98	Replaced joker patterns by fresh variables and removed binding patterns.
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(2)
9.23/3.98	Obligation:
9.23/3.98	mainModule Main 
9.23/3.98	module Main where {
9.23/3.98	import qualified Prelude;
9.23/3.98	}
9.23/3.98	
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(3) COR (EQUIVALENT)
9.23/3.98	Cond Reductions:
9.23/3.98	The following Function with conditions
9.23/3.98	"undefined |Falseundefined;
9.23/3.98	"
9.23/3.98	is transformed to
9.23/3.98	"undefined  = undefined1;
9.23/3.98	"
9.23/3.98	"undefined0 True = undefined;
9.23/3.98	"
9.23/3.98	"undefined1  = undefined0 False;
9.23/3.98	"
9.23/3.98	
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(4)
9.23/3.98	Obligation:
9.23/3.98	mainModule Main 
9.23/3.98	module Main where {
9.23/3.98	import qualified Prelude;
9.23/3.98	}
9.23/3.98	
9.23/3.98	----------------------------------------
9.23/3.98	
9.23/3.98	(5) Narrow (SOUND)
9.23/3.98	Haskell To QDPs
9.23/3.98	
9.23/3.98	digraph dp_graph {
9.23/3.98	node [outthreshold=100, inthreshold=100];1[label="compare",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.23/3.98	3[label="compare vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.23/3.98	4[label="compare vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.23/3.98	5[label="primCmpInt vx3 vx4",fontsize=16,color="burlywood",shape="box"];66[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 66[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	66 -> 6[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	67[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 67[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	67 -> 7[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	6[label="primCmpInt (Pos vx30) vx4",fontsize=16,color="burlywood",shape="box"];68[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];6 -> 68[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	68 -> 8[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	69[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];6 -> 69[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	69 -> 9[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	7[label="primCmpInt (Neg vx30) vx4",fontsize=16,color="burlywood",shape="box"];70[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 70[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	70 -> 10[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	71[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 71[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	71 -> 11[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	8[label="primCmpInt (Pos (Succ vx300)) vx4",fontsize=16,color="burlywood",shape="box"];72[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];8 -> 72[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	72 -> 12[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	73[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];8 -> 73[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	73 -> 13[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	9[label="primCmpInt (Pos Zero) vx4",fontsize=16,color="burlywood",shape="box"];74[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 74[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	74 -> 14[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	75[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];9 -> 75[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	75 -> 15[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	10[label="primCmpInt (Neg (Succ vx300)) vx4",fontsize=16,color="burlywood",shape="box"];76[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 76[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	76 -> 16[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	77[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];10 -> 77[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	77 -> 17[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	11[label="primCmpInt (Neg Zero) vx4",fontsize=16,color="burlywood",shape="box"];78[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 78[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	78 -> 18[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	79[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];11 -> 79[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	79 -> 19[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	12[label="primCmpInt (Pos (Succ vx300)) (Pos vx40)",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
9.23/3.98	13[label="primCmpInt (Pos (Succ vx300)) (Neg vx40)",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
9.23/3.98	14[label="primCmpInt (Pos Zero) (Pos vx40)",fontsize=16,color="burlywood",shape="box"];80[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	80 -> 22[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	81[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 81[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	81 -> 23[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	15[label="primCmpInt (Pos Zero) (Neg vx40)",fontsize=16,color="burlywood",shape="box"];82[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];15 -> 82[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	82 -> 24[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	83[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 83[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	83 -> 25[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	16[label="primCmpInt (Neg (Succ vx300)) (Pos vx40)",fontsize=16,color="black",shape="box"];16 -> 26[label="",style="solid", color="black", weight=3];
9.23/3.98	17[label="primCmpInt (Neg (Succ vx300)) (Neg vx40)",fontsize=16,color="black",shape="box"];17 -> 27[label="",style="solid", color="black", weight=3];
9.23/3.98	18[label="primCmpInt (Neg Zero) (Pos vx40)",fontsize=16,color="burlywood",shape="box"];84[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	84 -> 28[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	85[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 85[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	85 -> 29[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	19[label="primCmpInt (Neg Zero) (Neg vx40)",fontsize=16,color="burlywood",shape="box"];86[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	86 -> 30[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	87[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	87 -> 31[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	20[label="primCmpNat (Succ vx300) vx40",fontsize=16,color="burlywood",shape="triangle"];88[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];20 -> 88[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	88 -> 32[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	89[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 89[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	89 -> 33[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	21[label="GT",fontsize=16,color="green",shape="box"];22[label="primCmpInt (Pos Zero) (Pos (Succ vx400))",fontsize=16,color="black",shape="box"];22 -> 34[label="",style="solid", color="black", weight=3];
9.23/3.98	23[label="primCmpInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];23 -> 35[label="",style="solid", color="black", weight=3];
9.23/3.98	24[label="primCmpInt (Pos Zero) (Neg (Succ vx400))",fontsize=16,color="black",shape="box"];24 -> 36[label="",style="solid", color="black", weight=3];
9.23/3.98	25[label="primCmpInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];25 -> 37[label="",style="solid", color="black", weight=3];
9.23/3.98	26[label="LT",fontsize=16,color="green",shape="box"];27[label="primCmpNat vx40 (Succ vx300)",fontsize=16,color="burlywood",shape="triangle"];90[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];27 -> 90[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	90 -> 38[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	91[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 91[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	91 -> 39[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	28[label="primCmpInt (Neg Zero) (Pos (Succ vx400))",fontsize=16,color="black",shape="box"];28 -> 40[label="",style="solid", color="black", weight=3];
9.23/3.98	29[label="primCmpInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];29 -> 41[label="",style="solid", color="black", weight=3];
9.23/3.98	30[label="primCmpInt (Neg Zero) (Neg (Succ vx400))",fontsize=16,color="black",shape="box"];30 -> 42[label="",style="solid", color="black", weight=3];
9.23/3.98	31[label="primCmpInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3];
9.23/3.98	32[label="primCmpNat (Succ vx300) (Succ vx400)",fontsize=16,color="black",shape="box"];32 -> 44[label="",style="solid", color="black", weight=3];
9.23/3.98	33[label="primCmpNat (Succ vx300) Zero",fontsize=16,color="black",shape="box"];33 -> 45[label="",style="solid", color="black", weight=3];
9.23/3.98	34 -> 27[label="",style="dashed", color="red", weight=0];
9.23/3.98	34[label="primCmpNat Zero (Succ vx400)",fontsize=16,color="magenta"];34 -> 46[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	34 -> 47[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	35[label="EQ",fontsize=16,color="green",shape="box"];36[label="GT",fontsize=16,color="green",shape="box"];37[label="EQ",fontsize=16,color="green",shape="box"];38[label="primCmpNat (Succ vx400) (Succ vx300)",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3];
9.23/3.98	39[label="primCmpNat Zero (Succ vx300)",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3];
9.23/3.98	40[label="LT",fontsize=16,color="green",shape="box"];41[label="EQ",fontsize=16,color="green",shape="box"];42 -> 20[label="",style="dashed", color="red", weight=0];
9.23/3.98	42[label="primCmpNat (Succ vx400) Zero",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	42 -> 51[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	43[label="EQ",fontsize=16,color="green",shape="box"];44[label="primCmpNat vx300 vx400",fontsize=16,color="burlywood",shape="triangle"];92[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];44 -> 92[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	92 -> 52[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	93[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];44 -> 93[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	93 -> 53[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	45[label="GT",fontsize=16,color="green",shape="box"];46[label="Zero",fontsize=16,color="green",shape="box"];47[label="vx400",fontsize=16,color="green",shape="box"];48 -> 44[label="",style="dashed", color="red", weight=0];
9.23/3.98	48[label="primCmpNat vx400 vx300",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	48 -> 55[label="",style="dashed", color="magenta", weight=3];
9.23/3.98	49[label="LT",fontsize=16,color="green",shape="box"];50[label="vx400",fontsize=16,color="green",shape="box"];51[label="Zero",fontsize=16,color="green",shape="box"];52[label="primCmpNat (Succ vx3000) vx400",fontsize=16,color="burlywood",shape="box"];94[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];52 -> 94[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	94 -> 56[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	95[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 95[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	95 -> 57[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	53[label="primCmpNat Zero vx400",fontsize=16,color="burlywood",shape="box"];96[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];53 -> 96[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	96 -> 58[label="",style="solid", color="burlywood", weight=3];
9.23/3.98	97[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 97[label="",style="solid", color="burlywood", weight=9];
9.23/3.98	97 -> 59[label="",style="solid", color="burlywood", weight=3];
9.23/3.99	54[label="vx400",fontsize=16,color="green",shape="box"];55[label="vx300",fontsize=16,color="green",shape="box"];56[label="primCmpNat (Succ vx3000) (Succ vx4000)",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3];
9.23/3.99	57[label="primCmpNat (Succ vx3000) Zero",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3];
9.23/3.99	58[label="primCmpNat Zero (Succ vx4000)",fontsize=16,color="black",shape="box"];58 -> 62[label="",style="solid", color="black", weight=3];
9.23/3.99	59[label="primCmpNat Zero Zero",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3];
9.23/3.99	60 -> 44[label="",style="dashed", color="red", weight=0];
9.23/3.99	60[label="primCmpNat vx3000 vx4000",fontsize=16,color="magenta"];60 -> 64[label="",style="dashed", color="magenta", weight=3];
9.23/3.99	60 -> 65[label="",style="dashed", color="magenta", weight=3];
9.23/3.99	61[label="GT",fontsize=16,color="green",shape="box"];62[label="LT",fontsize=16,color="green",shape="box"];63[label="EQ",fontsize=16,color="green",shape="box"];64[label="vx3000",fontsize=16,color="green",shape="box"];65[label="vx4000",fontsize=16,color="green",shape="box"];}
9.23/3.99	
9.23/3.99	----------------------------------------
9.23/3.99	
9.23/3.99	(6)
9.23/3.99	Obligation:
9.23/3.99	Q DP problem:
9.23/3.99	The TRS P consists of the following rules:
9.23/3.99	
9.23/3.99	   new_primCmpNat(Succ(vx3000), Succ(vx4000)) -> new_primCmpNat(vx3000, vx4000)
9.23/3.99	
9.23/3.99	R is empty.
9.23/3.99	Q is empty.
9.23/3.99	We have to consider all minimal (P,Q,R)-chains.
9.23/3.99	----------------------------------------
9.23/3.99	
9.23/3.99	(7) QDPSizeChangeProof (EQUIVALENT)
9.23/3.99	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. 
9.23/3.99	
9.23/3.99	From the DPs we obtained the following set of size-change graphs:
9.23/3.99	*new_primCmpNat(Succ(vx3000), Succ(vx4000)) -> new_primCmpNat(vx3000, vx4000)
9.23/3.99	The graph contains the following edges 1 > 1, 2 > 2
9.23/3.99	
9.23/3.99	
9.23/3.99	----------------------------------------
9.23/3.99	
9.23/3.99	(8)
9.23/3.99	YES
9.23/4.02	EOF