8.85/3.80	YES
10.55/4.26	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.55/4.26	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.55/4.26	
10.55/4.26	
10.55/4.26	H-Termination with start terms of the given HASKELL could be proven:
10.55/4.26	
10.55/4.26	(0) HASKELL
10.55/4.26	(1) BR [EQUIVALENT, 0 ms]
10.55/4.26	(2) HASKELL
10.55/4.26	(3) COR [EQUIVALENT, 0 ms]
10.55/4.26	(4) HASKELL
10.55/4.26	(5) Narrow [SOUND, 0 ms]
10.55/4.26	(6) QDP
10.55/4.26	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
10.55/4.26	(8) AND
10.55/4.26	    (9) QDP
10.55/4.26	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.55/4.26	        (11) YES
10.55/4.26	    (12) QDP
10.55/4.26	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.55/4.26	        (14) YES
10.55/4.26	    (15) QDP
10.55/4.26	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.55/4.26	        (17) YES
10.55/4.26	    (18) QDP
10.55/4.26	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.55/4.26	        (20) YES
10.55/4.26	
10.55/4.26	
10.55/4.26	----------------------------------------
10.55/4.26	
10.55/4.26	(0)
10.55/4.26	Obligation:
10.55/4.26	mainModule Main 
10.55/4.26	module Main where {
10.55/4.26	import qualified Prelude;
10.55/4.26	}
10.55/4.26	
10.55/4.26	----------------------------------------
10.55/4.26	
10.55/4.26	(1) BR (EQUIVALENT)
10.55/4.26	Replaced joker patterns by fresh variables and removed binding patterns.
10.55/4.26	----------------------------------------
10.55/4.27	
10.55/4.27	(2)
10.55/4.27	Obligation:
10.55/4.27	mainModule Main 
10.55/4.27	module Main where {
10.55/4.27	import qualified Prelude;
10.55/4.27	}
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(3) COR (EQUIVALENT)
10.55/4.27	Cond Reductions:
10.55/4.27	The following Function with conditions
10.55/4.27	"lookup k [] = Nothing;
10.55/4.27	lookup k ((x,y) : xys)|k == xJust y|otherwiselookup k xys;
10.55/4.27	"
10.55/4.27	is transformed to
10.55/4.27	"lookup k [] = lookup3 k [];
10.55/4.27	lookup k ((x,y) : xys) = lookup2 k ((x,y) : xys);
10.55/4.27	"
10.55/4.27	"lookup1 k x y xys True = Just y;
10.55/4.27	lookup1 k x y xys False = lookup0 k x y xys otherwise;
10.55/4.27	"
10.55/4.27	"lookup0 k x y xys True = lookup k xys;
10.55/4.27	"
10.55/4.27	"lookup2 k ((x,y) : xys) = lookup1 k x y xys (k == x);
10.55/4.27	"
10.55/4.27	"lookup3 k [] = Nothing;
10.55/4.27	lookup3 ww wx = lookup2 ww wx;
10.55/4.27	"
10.55/4.27	The following Function with conditions
10.55/4.27	"undefined |Falseundefined;
10.55/4.27	"
10.55/4.27	is transformed to
10.55/4.27	"undefined  = undefined1;
10.55/4.27	"
10.55/4.27	"undefined0 True = undefined;
10.55/4.27	"
10.55/4.27	"undefined1  = undefined0 False;
10.55/4.27	"
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(4)
10.55/4.27	Obligation:
10.55/4.27	mainModule Main 
10.55/4.27	module Main where {
10.55/4.27	import qualified Prelude;
10.55/4.27	}
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(5) Narrow (SOUND)
10.55/4.27	Haskell To QDPs
10.55/4.27	
10.55/4.27	digraph dp_graph {
10.55/4.27	node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.55/4.27	3[label="lookup wy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.55/4.27	4[label="lookup wy3 wy4",fontsize=16,color="burlywood",shape="triangle"];778[label="wy4/wy40 : wy41",fontsize=10,color="white",style="solid",shape="box"];4 -> 778[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	778 -> 5[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	779[label="wy4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 779[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	779 -> 6[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	5[label="lookup wy3 (wy40 : wy41)",fontsize=16,color="burlywood",shape="box"];780[label="wy40/(wy400,wy401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 780[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	780 -> 7[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	6[label="lookup wy3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.55/4.27	7[label="lookup wy3 ((wy400,wy401) : wy41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.55/4.27	8[label="lookup3 wy3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
10.55/4.27	9[label="lookup2 wy3 ((wy400,wy401) : wy41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.55/4.27	10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 wy3 wy400 wy401 wy41 (wy3 == wy400)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
10.55/4.27	12[label="lookup1 wy3 wy400 wy401 wy41 (primEqInt wy3 wy400)",fontsize=16,color="burlywood",shape="box"];781[label="wy3/Pos wy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 781[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	781 -> 13[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	782[label="wy3/Neg wy30",fontsize=10,color="white",style="solid",shape="box"];12 -> 782[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	782 -> 14[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	13[label="lookup1 (Pos wy30) wy400 wy401 wy41 (primEqInt (Pos wy30) wy400)",fontsize=16,color="burlywood",shape="box"];783[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];13 -> 783[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	783 -> 15[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	784[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 784[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	784 -> 16[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	14[label="lookup1 (Neg wy30) wy400 wy401 wy41 (primEqInt (Neg wy30) wy400)",fontsize=16,color="burlywood",shape="box"];785[label="wy30/Succ wy300",fontsize=10,color="white",style="solid",shape="box"];14 -> 785[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	785 -> 17[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	786[label="wy30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 786[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	786 -> 18[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	15[label="lookup1 (Pos (Succ wy300)) wy400 wy401 wy41 (primEqInt (Pos (Succ wy300)) wy400)",fontsize=16,color="burlywood",shape="box"];787[label="wy400/Pos wy4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 787[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	787 -> 19[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	788[label="wy400/Neg wy4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 788[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	788 -> 20[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	16[label="lookup1 (Pos Zero) wy400 wy401 wy41 (primEqInt (Pos Zero) wy400)",fontsize=16,color="burlywood",shape="box"];789[label="wy400/Pos wy4000",fontsize=10,color="white",style="solid",shape="box"];16 -> 789[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	789 -> 21[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	790[label="wy400/Neg wy4000",fontsize=10,color="white",style="solid",shape="box"];16 -> 790[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	790 -> 22[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	17[label="lookup1 (Neg (Succ wy300)) wy400 wy401 wy41 (primEqInt (Neg (Succ wy300)) wy400)",fontsize=16,color="burlywood",shape="box"];791[label="wy400/Pos wy4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 791[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	791 -> 23[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	792[label="wy400/Neg wy4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 792[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	792 -> 24[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	18[label="lookup1 (Neg Zero) wy400 wy401 wy41 (primEqInt (Neg Zero) wy400)",fontsize=16,color="burlywood",shape="box"];793[label="wy400/Pos wy4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 793[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	793 -> 25[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	794[label="wy400/Neg wy4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 794[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	794 -> 26[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	19[label="lookup1 (Pos (Succ wy300)) (Pos wy4000) wy401 wy41 (primEqInt (Pos (Succ wy300)) (Pos wy4000))",fontsize=16,color="burlywood",shape="box"];795[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];19 -> 795[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	795 -> 27[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	796[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 796[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	796 -> 28[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	20[label="lookup1 (Pos (Succ wy300)) (Neg wy4000) wy401 wy41 (primEqInt (Pos (Succ wy300)) (Neg wy4000))",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
10.55/4.27	21[label="lookup1 (Pos Zero) (Pos wy4000) wy401 wy41 (primEqInt (Pos Zero) (Pos wy4000))",fontsize=16,color="burlywood",shape="box"];797[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];21 -> 797[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	797 -> 30[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	798[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 798[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	798 -> 31[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	22[label="lookup1 (Pos Zero) (Neg wy4000) wy401 wy41 (primEqInt (Pos Zero) (Neg wy4000))",fontsize=16,color="burlywood",shape="box"];799[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];22 -> 799[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	799 -> 32[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	800[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 800[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	800 -> 33[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	23[label="lookup1 (Neg (Succ wy300)) (Pos wy4000) wy401 wy41 (primEqInt (Neg (Succ wy300)) (Pos wy4000))",fontsize=16,color="black",shape="box"];23 -> 34[label="",style="solid", color="black", weight=3];
10.55/4.27	24[label="lookup1 (Neg (Succ wy300)) (Neg wy4000) wy401 wy41 (primEqInt (Neg (Succ wy300)) (Neg wy4000))",fontsize=16,color="burlywood",shape="box"];801[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];24 -> 801[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	801 -> 35[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	802[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 802[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	802 -> 36[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	25[label="lookup1 (Neg Zero) (Pos wy4000) wy401 wy41 (primEqInt (Neg Zero) (Pos wy4000))",fontsize=16,color="burlywood",shape="box"];803[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];25 -> 803[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	803 -> 37[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	804[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 804[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	804 -> 38[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	26[label="lookup1 (Neg Zero) (Neg wy4000) wy401 wy41 (primEqInt (Neg Zero) (Neg wy4000))",fontsize=16,color="burlywood",shape="box"];805[label="wy4000/Succ wy40000",fontsize=10,color="white",style="solid",shape="box"];26 -> 805[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	805 -> 39[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	806[label="wy4000/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 806[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	806 -> 40[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	27[label="lookup1 (Pos (Succ wy300)) (Pos (Succ wy40000)) wy401 wy41 (primEqInt (Pos (Succ wy300)) (Pos (Succ wy40000)))",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3];
10.55/4.27	28[label="lookup1 (Pos (Succ wy300)) (Pos Zero) wy401 wy41 (primEqInt (Pos (Succ wy300)) (Pos Zero))",fontsize=16,color="black",shape="box"];28 -> 42[label="",style="solid", color="black", weight=3];
10.55/4.27	29[label="lookup1 (Pos (Succ wy300)) (Neg wy4000) wy401 wy41 False",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
10.55/4.27	30[label="lookup1 (Pos Zero) (Pos (Succ wy40000)) wy401 wy41 (primEqInt (Pos Zero) (Pos (Succ wy40000)))",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
10.55/4.27	31[label="lookup1 (Pos Zero) (Pos Zero) wy401 wy41 (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
10.55/4.27	32[label="lookup1 (Pos Zero) (Neg (Succ wy40000)) wy401 wy41 (primEqInt (Pos Zero) (Neg (Succ wy40000)))",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
10.55/4.27	33[label="lookup1 (Pos Zero) (Neg Zero) wy401 wy41 (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
10.55/4.27	34[label="lookup1 (Neg (Succ wy300)) (Pos wy4000) wy401 wy41 False",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
10.55/4.27	35[label="lookup1 (Neg (Succ wy300)) (Neg (Succ wy40000)) wy401 wy41 (primEqInt (Neg (Succ wy300)) (Neg (Succ wy40000)))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
10.55/4.27	36[label="lookup1 (Neg (Succ wy300)) (Neg Zero) wy401 wy41 (primEqInt (Neg (Succ wy300)) (Neg Zero))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
10.55/4.27	37[label="lookup1 (Neg Zero) (Pos (Succ wy40000)) wy401 wy41 (primEqInt (Neg Zero) (Pos (Succ wy40000)))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
10.55/4.27	38[label="lookup1 (Neg Zero) (Pos Zero) wy401 wy41 (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
10.55/4.27	39[label="lookup1 (Neg Zero) (Neg (Succ wy40000)) wy401 wy41 (primEqInt (Neg Zero) (Neg (Succ wy40000)))",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
10.55/4.27	40[label="lookup1 (Neg Zero) (Neg Zero) wy401 wy41 (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
10.55/4.27	41 -> 620[label="",style="dashed", color="red", weight=0];
10.55/4.27	41[label="lookup1 (Pos (Succ wy300)) (Pos (Succ wy40000)) wy401 wy41 (primEqNat wy300 wy40000)",fontsize=16,color="magenta"];41 -> 621[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	41 -> 622[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	41 -> 623[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	41 -> 624[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	41 -> 625[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	41 -> 626[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	42[label="lookup1 (Pos (Succ wy300)) (Pos Zero) wy401 wy41 False",fontsize=16,color="black",shape="box"];42 -> 57[label="",style="solid", color="black", weight=3];
10.55/4.27	43[label="lookup0 (Pos (Succ wy300)) (Neg wy4000) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];43 -> 58[label="",style="solid", color="black", weight=3];
10.55/4.27	44[label="lookup1 (Pos Zero) (Pos (Succ wy40000)) wy401 wy41 False",fontsize=16,color="black",shape="box"];44 -> 59[label="",style="solid", color="black", weight=3];
10.55/4.27	45[label="lookup1 (Pos Zero) (Pos Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];45 -> 60[label="",style="solid", color="black", weight=3];
10.55/4.27	46[label="lookup1 (Pos Zero) (Neg (Succ wy40000)) wy401 wy41 False",fontsize=16,color="black",shape="box"];46 -> 61[label="",style="solid", color="black", weight=3];
10.55/4.27	47[label="lookup1 (Pos Zero) (Neg Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];47 -> 62[label="",style="solid", color="black", weight=3];
10.55/4.27	48[label="lookup0 (Neg (Succ wy300)) (Pos wy4000) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];48 -> 63[label="",style="solid", color="black", weight=3];
10.55/4.27	49 -> 683[label="",style="dashed", color="red", weight=0];
10.55/4.27	49[label="lookup1 (Neg (Succ wy300)) (Neg (Succ wy40000)) wy401 wy41 (primEqNat wy300 wy40000)",fontsize=16,color="magenta"];49 -> 684[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	49 -> 685[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	49 -> 686[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	49 -> 687[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	49 -> 688[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	49 -> 689[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	50[label="lookup1 (Neg (Succ wy300)) (Neg Zero) wy401 wy41 False",fontsize=16,color="black",shape="box"];50 -> 66[label="",style="solid", color="black", weight=3];
10.55/4.27	51[label="lookup1 (Neg Zero) (Pos (Succ wy40000)) wy401 wy41 False",fontsize=16,color="black",shape="box"];51 -> 67[label="",style="solid", color="black", weight=3];
10.55/4.27	52[label="lookup1 (Neg Zero) (Pos Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];52 -> 68[label="",style="solid", color="black", weight=3];
10.55/4.27	53[label="lookup1 (Neg Zero) (Neg (Succ wy40000)) wy401 wy41 False",fontsize=16,color="black",shape="box"];53 -> 69[label="",style="solid", color="black", weight=3];
10.55/4.27	54[label="lookup1 (Neg Zero) (Neg Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];54 -> 70[label="",style="solid", color="black", weight=3];
10.55/4.27	621[label="wy40000",fontsize=16,color="green",shape="box"];622[label="wy41",fontsize=16,color="green",shape="box"];623[label="wy300",fontsize=16,color="green",shape="box"];624[label="wy40000",fontsize=16,color="green",shape="box"];625[label="wy401",fontsize=16,color="green",shape="box"];626[label="wy300",fontsize=16,color="green",shape="box"];620[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat wy78 wy79)",fontsize=16,color="burlywood",shape="triangle"];807[label="wy78/Succ wy780",fontsize=10,color="white",style="solid",shape="box"];620 -> 807[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	807 -> 681[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	808[label="wy78/Zero",fontsize=10,color="white",style="solid",shape="box"];620 -> 808[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	808 -> 682[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	57[label="lookup0 (Pos (Succ wy300)) (Pos Zero) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];57 -> 75[label="",style="solid", color="black", weight=3];
10.55/4.27	58[label="lookup0 (Pos (Succ wy300)) (Neg wy4000) wy401 wy41 True",fontsize=16,color="black",shape="box"];58 -> 76[label="",style="solid", color="black", weight=3];
10.55/4.27	59[label="lookup0 (Pos Zero) (Pos (Succ wy40000)) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];59 -> 77[label="",style="solid", color="black", weight=3];
10.55/4.27	60[label="Just wy401",fontsize=16,color="green",shape="box"];61[label="lookup0 (Pos Zero) (Neg (Succ wy40000)) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];61 -> 78[label="",style="solid", color="black", weight=3];
10.55/4.27	62[label="Just wy401",fontsize=16,color="green",shape="box"];63[label="lookup0 (Neg (Succ wy300)) (Pos wy4000) wy401 wy41 True",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3];
10.55/4.27	684[label="wy40000",fontsize=16,color="green",shape="box"];685[label="wy300",fontsize=16,color="green",shape="box"];686[label="wy40000",fontsize=16,color="green",shape="box"];687[label="wy41",fontsize=16,color="green",shape="box"];688[label="wy300",fontsize=16,color="green",shape="box"];689[label="wy401",fontsize=16,color="green",shape="box"];683[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat wy85 wy86)",fontsize=16,color="burlywood",shape="triangle"];809[label="wy85/Succ wy850",fontsize=10,color="white",style="solid",shape="box"];683 -> 809[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	809 -> 744[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	810[label="wy85/Zero",fontsize=10,color="white",style="solid",shape="box"];683 -> 810[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	810 -> 745[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	66[label="lookup0 (Neg (Succ wy300)) (Neg Zero) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];66 -> 84[label="",style="solid", color="black", weight=3];
10.55/4.27	67[label="lookup0 (Neg Zero) (Pos (Succ wy40000)) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];67 -> 85[label="",style="solid", color="black", weight=3];
10.55/4.27	68[label="Just wy401",fontsize=16,color="green",shape="box"];69[label="lookup0 (Neg Zero) (Neg (Succ wy40000)) wy401 wy41 otherwise",fontsize=16,color="black",shape="box"];69 -> 86[label="",style="solid", color="black", weight=3];
10.55/4.27	70[label="Just wy401",fontsize=16,color="green",shape="box"];681[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat (Succ wy780) wy79)",fontsize=16,color="burlywood",shape="box"];811[label="wy79/Succ wy790",fontsize=10,color="white",style="solid",shape="box"];681 -> 811[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	811 -> 746[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	812[label="wy79/Zero",fontsize=10,color="white",style="solid",shape="box"];681 -> 812[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	812 -> 747[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	682[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat Zero wy79)",fontsize=16,color="burlywood",shape="box"];813[label="wy79/Succ wy790",fontsize=10,color="white",style="solid",shape="box"];682 -> 813[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	813 -> 748[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	814[label="wy79/Zero",fontsize=10,color="white",style="solid",shape="box"];682 -> 814[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	814 -> 749[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	75[label="lookup0 (Pos (Succ wy300)) (Pos Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];75 -> 91[label="",style="solid", color="black", weight=3];
10.55/4.27	76 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	76[label="lookup (Pos (Succ wy300)) wy41",fontsize=16,color="magenta"];76 -> 92[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	76 -> 93[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	77[label="lookup0 (Pos Zero) (Pos (Succ wy40000)) wy401 wy41 True",fontsize=16,color="black",shape="box"];77 -> 94[label="",style="solid", color="black", weight=3];
10.55/4.27	78[label="lookup0 (Pos Zero) (Neg (Succ wy40000)) wy401 wy41 True",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
10.55/4.27	79 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	79[label="lookup (Neg (Succ wy300)) wy41",fontsize=16,color="magenta"];79 -> 96[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	79 -> 97[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	744[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat (Succ wy850) wy86)",fontsize=16,color="burlywood",shape="box"];815[label="wy86/Succ wy860",fontsize=10,color="white",style="solid",shape="box"];744 -> 815[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	815 -> 750[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	816[label="wy86/Zero",fontsize=10,color="white",style="solid",shape="box"];744 -> 816[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	816 -> 751[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	745[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat Zero wy86)",fontsize=16,color="burlywood",shape="box"];817[label="wy86/Succ wy860",fontsize=10,color="white",style="solid",shape="box"];745 -> 817[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	817 -> 752[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	818[label="wy86/Zero",fontsize=10,color="white",style="solid",shape="box"];745 -> 818[label="",style="solid", color="burlywood", weight=9];
10.55/4.27	818 -> 753[label="",style="solid", color="burlywood", weight=3];
10.55/4.27	84[label="lookup0 (Neg (Succ wy300)) (Neg Zero) wy401 wy41 True",fontsize=16,color="black",shape="box"];84 -> 102[label="",style="solid", color="black", weight=3];
10.55/4.27	85[label="lookup0 (Neg Zero) (Pos (Succ wy40000)) wy401 wy41 True",fontsize=16,color="black",shape="box"];85 -> 103[label="",style="solid", color="black", weight=3];
10.55/4.27	86[label="lookup0 (Neg Zero) (Neg (Succ wy40000)) wy401 wy41 True",fontsize=16,color="black",shape="box"];86 -> 104[label="",style="solid", color="black", weight=3];
10.55/4.27	746[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat (Succ wy780) (Succ wy790))",fontsize=16,color="black",shape="box"];746 -> 754[label="",style="solid", color="black", weight=3];
10.55/4.27	747[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat (Succ wy780) Zero)",fontsize=16,color="black",shape="box"];747 -> 755[label="",style="solid", color="black", weight=3];
10.55/4.27	748[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat Zero (Succ wy790))",fontsize=16,color="black",shape="box"];748 -> 756[label="",style="solid", color="black", weight=3];
10.55/4.27	749[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];749 -> 757[label="",style="solid", color="black", weight=3];
10.55/4.27	91 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	91[label="lookup (Pos (Succ wy300)) wy41",fontsize=16,color="magenta"];91 -> 110[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	91 -> 111[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	92[label="wy41",fontsize=16,color="green",shape="box"];93[label="Pos (Succ wy300)",fontsize=16,color="green",shape="box"];94 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	94[label="lookup (Pos Zero) wy41",fontsize=16,color="magenta"];94 -> 112[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	94 -> 113[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	95 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	95[label="lookup (Pos Zero) wy41",fontsize=16,color="magenta"];95 -> 114[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	95 -> 115[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	96[label="wy41",fontsize=16,color="green",shape="box"];97[label="Neg (Succ wy300)",fontsize=16,color="green",shape="box"];750[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat (Succ wy850) (Succ wy860))",fontsize=16,color="black",shape="box"];750 -> 758[label="",style="solid", color="black", weight=3];
10.55/4.27	751[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat (Succ wy850) Zero)",fontsize=16,color="black",shape="box"];751 -> 759[label="",style="solid", color="black", weight=3];
10.55/4.27	752[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat Zero (Succ wy860))",fontsize=16,color="black",shape="box"];752 -> 760[label="",style="solid", color="black", weight=3];
10.55/4.27	753[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];753 -> 761[label="",style="solid", color="black", weight=3];
10.55/4.27	102 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	102[label="lookup (Neg (Succ wy300)) wy41",fontsize=16,color="magenta"];102 -> 121[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	102 -> 122[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	103 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	103[label="lookup (Neg Zero) wy41",fontsize=16,color="magenta"];103 -> 123[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	103 -> 124[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	104 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	104[label="lookup (Neg Zero) wy41",fontsize=16,color="magenta"];104 -> 125[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	104 -> 126[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	754 -> 620[label="",style="dashed", color="red", weight=0];
10.55/4.27	754[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 (primEqNat wy780 wy790)",fontsize=16,color="magenta"];754 -> 762[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	754 -> 763[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	755[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 False",fontsize=16,color="black",shape="triangle"];755 -> 764[label="",style="solid", color="black", weight=3];
10.55/4.27	756 -> 755[label="",style="dashed", color="red", weight=0];
10.55/4.27	756[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 False",fontsize=16,color="magenta"];757[label="lookup1 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 True",fontsize=16,color="black",shape="box"];757 -> 765[label="",style="solid", color="black", weight=3];
10.55/4.27	110[label="wy41",fontsize=16,color="green",shape="box"];111[label="Pos (Succ wy300)",fontsize=16,color="green",shape="box"];112[label="wy41",fontsize=16,color="green",shape="box"];113[label="Pos Zero",fontsize=16,color="green",shape="box"];114[label="wy41",fontsize=16,color="green",shape="box"];115[label="Pos Zero",fontsize=16,color="green",shape="box"];758 -> 683[label="",style="dashed", color="red", weight=0];
10.55/4.27	758[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 (primEqNat wy850 wy860)",fontsize=16,color="magenta"];758 -> 766[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	758 -> 767[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	759[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 False",fontsize=16,color="black",shape="triangle"];759 -> 768[label="",style="solid", color="black", weight=3];
10.55/4.27	760 -> 759[label="",style="dashed", color="red", weight=0];
10.55/4.27	760[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 False",fontsize=16,color="magenta"];761[label="lookup1 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 True",fontsize=16,color="black",shape="box"];761 -> 769[label="",style="solid", color="black", weight=3];
10.55/4.27	121[label="wy41",fontsize=16,color="green",shape="box"];122[label="Neg (Succ wy300)",fontsize=16,color="green",shape="box"];123[label="wy41",fontsize=16,color="green",shape="box"];124[label="Neg Zero",fontsize=16,color="green",shape="box"];125[label="wy41",fontsize=16,color="green",shape="box"];126[label="Neg Zero",fontsize=16,color="green",shape="box"];762[label="wy790",fontsize=16,color="green",shape="box"];763[label="wy780",fontsize=16,color="green",shape="box"];764[label="lookup0 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 otherwise",fontsize=16,color="black",shape="box"];764 -> 770[label="",style="solid", color="black", weight=3];
10.55/4.27	765[label="Just wy76",fontsize=16,color="green",shape="box"];766[label="wy860",fontsize=16,color="green",shape="box"];767[label="wy850",fontsize=16,color="green",shape="box"];768[label="lookup0 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 otherwise",fontsize=16,color="black",shape="box"];768 -> 771[label="",style="solid", color="black", weight=3];
10.55/4.27	769[label="Just wy83",fontsize=16,color="green",shape="box"];770[label="lookup0 (Pos (Succ wy74)) (Pos (Succ wy75)) wy76 wy77 True",fontsize=16,color="black",shape="box"];770 -> 772[label="",style="solid", color="black", weight=3];
10.55/4.27	771[label="lookup0 (Neg (Succ wy81)) (Neg (Succ wy82)) wy83 wy84 True",fontsize=16,color="black",shape="box"];771 -> 773[label="",style="solid", color="black", weight=3];
10.55/4.27	772 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	772[label="lookup (Pos (Succ wy74)) wy77",fontsize=16,color="magenta"];772 -> 774[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	772 -> 775[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	773 -> 4[label="",style="dashed", color="red", weight=0];
10.55/4.27	773[label="lookup (Neg (Succ wy81)) wy84",fontsize=16,color="magenta"];773 -> 776[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	773 -> 777[label="",style="dashed", color="magenta", weight=3];
10.55/4.27	774[label="wy77",fontsize=16,color="green",shape="box"];775[label="Pos (Succ wy74)",fontsize=16,color="green",shape="box"];776[label="wy84",fontsize=16,color="green",shape="box"];777[label="Neg (Succ wy81)",fontsize=16,color="green",shape="box"];}
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(6)
10.55/4.27	Obligation:
10.55/4.27	Q DP problem:
10.55/4.27	The TRS P consists of the following rules:
10.55/4.27	
10.55/4.27	   new_lookup(Pos(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Neg(Zero), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup12(wy81, wy82, wy83, wy84, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	   new_lookup10(wy74, wy75, wy76, wy77, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	   new_lookup(Neg(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Succ(wy860), bb) -> new_lookup11(wy81, wy82, wy83, wy84, wy850, wy860, bb)
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Pos(Zero), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Pos(wy4000), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Succ(wy790), h) -> new_lookup1(wy74, wy75, wy76, wy77, wy780, wy790, h)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Zero, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	   new_lookup(Neg(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup11(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Neg(wy4000), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Zero, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup1(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Zero, Succ(wy860), bb) -> new_lookup12(wy81, wy82, wy83, wy84, bb)
10.55/4.27	   new_lookup(Pos(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Zero, Succ(wy790), h) -> new_lookup10(wy74, wy75, wy76, wy77, h)
10.55/4.27	
10.55/4.27	R is empty.
10.55/4.27	Q is empty.
10.55/4.27	We have to consider all minimal (P,Q,R)-chains.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(7) DependencyGraphProof (EQUIVALENT)
10.55/4.27	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(8)
10.55/4.27	Complex Obligation (AND)
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(9)
10.55/4.27	Obligation:
10.55/4.27	Q DP problem:
10.55/4.27	The TRS P consists of the following rules:
10.55/4.27	
10.55/4.27	   new_lookup(Neg(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	   new_lookup(Neg(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	
10.55/4.27	R is empty.
10.55/4.27	Q is empty.
10.55/4.27	We have to consider all minimal (P,Q,R)-chains.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(10) QDPSizeChangeProof (EQUIVALENT)
10.55/4.27	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. 
10.55/4.27	
10.55/4.27	From the DPs we obtained the following set of size-change graphs:
10.55/4.27	*new_lookup(Neg(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Neg(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Neg(Zero), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(11)
10.55/4.27	YES
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(12)
10.55/4.27	Obligation:
10.55/4.27	Q DP problem:
10.55/4.27	The TRS P consists of the following rules:
10.55/4.27	
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Pos(Zero), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Neg(wy4000), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup(Pos(Succ(wy300)), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup1(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Succ(wy790), h) -> new_lookup1(wy74, wy75, wy76, wy77, wy780, wy790, h)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Zero, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	   new_lookup1(wy74, wy75, wy76, wy77, Zero, Succ(wy790), h) -> new_lookup10(wy74, wy75, wy76, wy77, h)
10.55/4.27	   new_lookup10(wy74, wy75, wy76, wy77, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	
10.55/4.27	R is empty.
10.55/4.27	Q is empty.
10.55/4.27	We have to consider all minimal (P,Q,R)-chains.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(13) QDPSizeChangeProof (EQUIVALENT)
10.55/4.27	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. 
10.55/4.27	
10.55/4.27	From the DPs we obtained the following set of size-change graphs:
10.55/4.27	*new_lookup(Pos(Succ(wy300)), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup1(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4, 1 > 5, 2 > 6, 3 >= 7
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Zero, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	The graph contains the following edges 4 >= 2, 7 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup10(wy74, wy75, wy76, wy77, h) -> new_lookup(Pos(Succ(wy74)), wy77, h)
10.55/4.27	The graph contains the following edges 4 >= 2, 5 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup1(wy74, wy75, wy76, wy77, Succ(wy780), Succ(wy790), h) -> new_lookup1(wy74, wy75, wy76, wy77, wy780, wy790, h)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup1(wy74, wy75, wy76, wy77, Zero, Succ(wy790), h) -> new_lookup10(wy74, wy75, wy76, wy77, h)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Pos(Succ(wy300)), :(@2(Pos(Zero), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Pos(Succ(wy300)), :(@2(Neg(wy4000), wy401), wy41), ba) -> new_lookup(Pos(Succ(wy300)), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(14)
10.55/4.27	YES
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(15)
10.55/4.27	Obligation:
10.55/4.27	Q DP problem:
10.55/4.27	The TRS P consists of the following rules:
10.55/4.27	
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Pos(wy4000), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Neg(Zero), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	   new_lookup(Neg(Succ(wy300)), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup11(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Succ(wy860), bb) -> new_lookup11(wy81, wy82, wy83, wy84, wy850, wy860, bb)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Zero, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	   new_lookup11(wy81, wy82, wy83, wy84, Zero, Succ(wy860), bb) -> new_lookup12(wy81, wy82, wy83, wy84, bb)
10.55/4.27	   new_lookup12(wy81, wy82, wy83, wy84, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	
10.55/4.27	R is empty.
10.55/4.27	Q is empty.
10.55/4.27	We have to consider all minimal (P,Q,R)-chains.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(16) QDPSizeChangeProof (EQUIVALENT)
10.55/4.27	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. 
10.55/4.27	
10.55/4.27	From the DPs we obtained the following set of size-change graphs:
10.55/4.27	*new_lookup(Neg(Succ(wy300)), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup11(wy300, wy40000, wy401, wy41, wy300, wy40000, ba)
10.55/4.27	The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4, 1 > 5, 2 > 6, 3 >= 7
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Zero, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	The graph contains the following edges 4 >= 2, 7 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup12(wy81, wy82, wy83, wy84, bb) -> new_lookup(Neg(Succ(wy81)), wy84, bb)
10.55/4.27	The graph contains the following edges 4 >= 2, 5 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup11(wy81, wy82, wy83, wy84, Succ(wy850), Succ(wy860), bb) -> new_lookup11(wy81, wy82, wy83, wy84, wy850, wy860, bb)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup11(wy81, wy82, wy83, wy84, Zero, Succ(wy860), bb) -> new_lookup12(wy81, wy82, wy83, wy84, bb)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Neg(Succ(wy300)), :(@2(Pos(wy4000), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Neg(Succ(wy300)), :(@2(Neg(Zero), wy401), wy41), ba) -> new_lookup(Neg(Succ(wy300)), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(17)
10.55/4.27	YES
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(18)
10.55/4.27	Obligation:
10.55/4.27	Q DP problem:
10.55/4.27	The TRS P consists of the following rules:
10.55/4.27	
10.55/4.27	   new_lookup(Pos(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	   new_lookup(Pos(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	
10.55/4.27	R is empty.
10.55/4.27	Q is empty.
10.55/4.27	We have to consider all minimal (P,Q,R)-chains.
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(19) QDPSizeChangeProof (EQUIVALENT)
10.55/4.27	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. 
10.55/4.27	
10.55/4.27	From the DPs we obtained the following set of size-change graphs:
10.55/4.27	*new_lookup(Pos(Zero), :(@2(Neg(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	*new_lookup(Pos(Zero), :(@2(Pos(Succ(wy40000)), wy401), wy41), ba) -> new_lookup(Pos(Zero), wy41, ba)
10.55/4.27	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.55/4.27	
10.55/4.27	
10.55/4.27	----------------------------------------
10.55/4.27	
10.55/4.27	(20)
10.55/4.27	YES
10.65/4.30	EOF