8.04/3.71	YES
9.46/4.19	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.46/4.19	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.46/4.19	
9.46/4.19	
9.46/4.19	H-Termination with start terms of the given HASKELL could be proven:
9.46/4.19	
9.46/4.19	(0) HASKELL
9.46/4.19	(1) BR [EQUIVALENT, 0 ms]
9.46/4.19	(2) HASKELL
9.46/4.19	(3) COR [EQUIVALENT, 0 ms]
9.46/4.19	(4) HASKELL
9.46/4.19	(5) Narrow [SOUND, 0 ms]
9.46/4.19	(6) QDP
9.46/4.19	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
9.46/4.19	(8) AND
9.46/4.19	    (9) QDP
9.46/4.19	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.46/4.19	        (11) YES
9.46/4.19	    (12) QDP
9.46/4.19	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.46/4.19	        (14) YES
9.46/4.19	    (15) QDP
9.46/4.19	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.46/4.19	        (17) YES
9.46/4.19	
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(0)
9.46/4.19	Obligation:
9.46/4.19	mainModule Main 
9.46/4.19	module Main where {
9.46/4.19	import qualified Prelude;
9.46/4.19	data List a = Cons a (List a)  | Nil ;
9.46/4.19	
9.46/4.19	data Main.Maybe a = Nothing  | Just a ;
9.46/4.19	
9.46/4.19	data MyBool = MyTrue  | MyFalse ;
9.46/4.19	
9.46/4.19	data Ordering = LT  | EQ  | GT ;
9.46/4.19	
9.46/4.19	data Tup2 a b = Tup2 a b ;
9.46/4.19	
9.46/4.19	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.46/4.19	esEsOrdering LT LT = MyTrue;
9.46/4.19	esEsOrdering LT EQ = MyFalse;
9.46/4.19	esEsOrdering LT GT = MyFalse;
9.46/4.19	esEsOrdering EQ LT = MyFalse;
9.46/4.19	esEsOrdering EQ EQ = MyTrue;
9.46/4.19	esEsOrdering EQ GT = MyFalse;
9.46/4.19	esEsOrdering GT LT = MyFalse;
9.46/4.19	esEsOrdering GT EQ = MyFalse;
9.46/4.19	esEsOrdering GT GT = MyTrue;
9.46/4.19	
9.46/4.19	lookup k Nil = lookup3 k Nil;
9.46/4.19	lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);
9.46/4.19	
9.46/4.19	lookup0 k x y xys MyTrue = lookup k xys;
9.46/4.19	
9.46/4.19	lookup1 k x y xys MyTrue = Main.Just y;
9.46/4.19	lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;
9.46/4.19	
9.46/4.19	lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);
9.46/4.19	
9.46/4.19	lookup3 k Nil = Main.Nothing;
9.46/4.19	lookup3 vy vz = lookup2 vy vz;
9.46/4.19	
9.46/4.19	otherwise :: MyBool;
9.46/4.19	otherwise = MyTrue;
9.46/4.19	
9.46/4.19	}
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(1) BR (EQUIVALENT)
9.46/4.19	Replaced joker patterns by fresh variables and removed binding patterns.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(2)
9.46/4.19	Obligation:
9.46/4.19	mainModule Main 
9.46/4.19	module Main where {
9.46/4.19	import qualified Prelude;
9.46/4.19	data List a = Cons a (List a)  | Nil ;
9.46/4.19	
9.46/4.19	data Main.Maybe a = Nothing  | Just a ;
9.46/4.19	
9.46/4.19	data MyBool = MyTrue  | MyFalse ;
9.46/4.19	
9.46/4.19	data Ordering = LT  | EQ  | GT ;
9.46/4.19	
9.46/4.19	data Tup2 a b = Tup2 a b ;
9.46/4.19	
9.46/4.19	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.46/4.19	esEsOrdering LT LT = MyTrue;
9.46/4.19	esEsOrdering LT EQ = MyFalse;
9.46/4.19	esEsOrdering LT GT = MyFalse;
9.46/4.19	esEsOrdering EQ LT = MyFalse;
9.46/4.19	esEsOrdering EQ EQ = MyTrue;
9.46/4.19	esEsOrdering EQ GT = MyFalse;
9.46/4.19	esEsOrdering GT LT = MyFalse;
9.46/4.19	esEsOrdering GT EQ = MyFalse;
9.46/4.19	esEsOrdering GT GT = MyTrue;
9.46/4.19	
9.46/4.19	lookup k Nil = lookup3 k Nil;
9.46/4.19	lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);
9.46/4.19	
9.46/4.19	lookup0 k x y xys MyTrue = lookup k xys;
9.46/4.19	
9.46/4.19	lookup1 k x y xys MyTrue = Main.Just y;
9.46/4.19	lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;
9.46/4.19	
9.46/4.19	lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);
9.46/4.19	
9.46/4.19	lookup3 k Nil = Main.Nothing;
9.46/4.19	lookup3 vy vz = lookup2 vy vz;
9.46/4.19	
9.46/4.19	otherwise :: MyBool;
9.46/4.19	otherwise = MyTrue;
9.46/4.19	
9.46/4.19	}
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(3) COR (EQUIVALENT)
9.46/4.19	Cond Reductions:
9.46/4.19	The following Function with conditions
9.46/4.19	"undefined |Falseundefined;
9.46/4.19	"
9.46/4.19	is transformed to
9.46/4.19	"undefined  = undefined1;
9.46/4.19	"
9.46/4.19	"undefined0 True = undefined;
9.46/4.19	"
9.46/4.19	"undefined1  = undefined0 False;
9.46/4.19	"
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(4)
9.46/4.19	Obligation:
9.46/4.19	mainModule Main 
9.46/4.19	module Main where {
9.46/4.19	import qualified Prelude;
9.46/4.19	data List a = Cons a (List a)  | Nil ;
9.46/4.19	
9.46/4.19	data Main.Maybe a = Nothing  | Just a ;
9.46/4.19	
9.46/4.19	data MyBool = MyTrue  | MyFalse ;
9.46/4.19	
9.46/4.19	data Ordering = LT  | EQ  | GT ;
9.46/4.19	
9.46/4.19	data Tup2 a b = Tup2 a b ;
9.46/4.19	
9.46/4.19	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
9.46/4.19	esEsOrdering LT LT = MyTrue;
9.46/4.19	esEsOrdering LT EQ = MyFalse;
9.46/4.19	esEsOrdering LT GT = MyFalse;
9.46/4.19	esEsOrdering EQ LT = MyFalse;
9.46/4.19	esEsOrdering EQ EQ = MyTrue;
9.46/4.19	esEsOrdering EQ GT = MyFalse;
9.46/4.19	esEsOrdering GT LT = MyFalse;
9.46/4.19	esEsOrdering GT EQ = MyFalse;
9.46/4.19	esEsOrdering GT GT = MyTrue;
9.46/4.19	
9.46/4.19	lookup k Nil = lookup3 k Nil;
9.46/4.19	lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);
9.46/4.19	
9.46/4.19	lookup0 k x y xys MyTrue = lookup k xys;
9.46/4.19	
9.46/4.19	lookup1 k x y xys MyTrue = Main.Just y;
9.46/4.19	lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;
9.46/4.19	
9.46/4.19	lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);
9.46/4.19	
9.46/4.19	lookup3 k Nil = Main.Nothing;
9.46/4.19	lookup3 vy vz = lookup2 vy vz;
9.46/4.19	
9.46/4.19	otherwise :: MyBool;
9.46/4.19	otherwise = MyTrue;
9.46/4.19	
9.46/4.19	}
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(5) Narrow (SOUND)
9.46/4.19	Haskell To QDPs
9.46/4.19	
9.46/4.19	digraph dp_graph {
9.46/4.19	node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.46/4.19	3[label="lookup vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.46/4.19	4[label="lookup vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];66[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 66[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	66 -> 5[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	67[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 67[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	67 -> 6[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	5[label="lookup vx3 (Cons vx40 vx41)",fontsize=16,color="burlywood",shape="box"];68[label="vx40/Tup2 vx400 vx401",fontsize=10,color="white",style="solid",shape="box"];5 -> 68[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	68 -> 7[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	6[label="lookup vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.46/4.19	7[label="lookup vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.46/4.19	8[label="lookup3 vx3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.46/4.19	9[label="lookup2 vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.46/4.19	10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 vx3 vx400 vx401 vx41 (esEsOrdering vx3 vx400)",fontsize=16,color="burlywood",shape="box"];69[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	69 -> 12[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	70[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 70[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	70 -> 13[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	71[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	71 -> 14[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	12[label="lookup1 LT vx400 vx401 vx41 (esEsOrdering LT vx400)",fontsize=16,color="burlywood",shape="box"];72[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 72[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	72 -> 15[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	73[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 73[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	73 -> 16[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	74[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	74 -> 17[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	13[label="lookup1 EQ vx400 vx401 vx41 (esEsOrdering EQ vx400)",fontsize=16,color="burlywood",shape="box"];75[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 75[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	75 -> 18[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	76[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 76[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	76 -> 19[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	77[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	77 -> 20[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	14[label="lookup1 GT vx400 vx401 vx41 (esEsOrdering GT vx400)",fontsize=16,color="burlywood",shape="box"];78[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 78[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	78 -> 21[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	79[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	79 -> 22[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	80[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
9.46/4.19	80 -> 23[label="",style="solid", color="burlywood", weight=3];
9.46/4.19	15[label="lookup1 LT LT vx401 vx41 (esEsOrdering LT LT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
9.46/4.19	16[label="lookup1 LT EQ vx401 vx41 (esEsOrdering LT EQ)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
9.46/4.19	17[label="lookup1 LT GT vx401 vx41 (esEsOrdering LT GT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
9.46/4.19	18[label="lookup1 EQ LT vx401 vx41 (esEsOrdering EQ LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
9.46/4.19	19[label="lookup1 EQ EQ vx401 vx41 (esEsOrdering EQ EQ)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
9.46/4.19	20[label="lookup1 EQ GT vx401 vx41 (esEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
9.46/4.19	21[label="lookup1 GT LT vx401 vx41 (esEsOrdering GT LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
9.46/4.19	22[label="lookup1 GT EQ vx401 vx41 (esEsOrdering GT EQ)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
9.46/4.19	23[label="lookup1 GT GT vx401 vx41 (esEsOrdering GT GT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.46/4.19	24[label="lookup1 LT LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.46/4.19	25[label="lookup1 LT EQ vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.46/4.19	26[label="lookup1 LT GT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.46/4.19	27[label="lookup1 EQ LT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.46/4.19	28[label="lookup1 EQ EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.46/4.19	29[label="lookup1 EQ GT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.46/4.19	30[label="lookup1 GT LT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.46/4.19	31[label="lookup1 GT EQ vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
9.46/4.19	32[label="lookup1 GT GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
9.46/4.19	33[label="Just vx401",fontsize=16,color="green",shape="box"];34[label="lookup0 LT EQ vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
9.46/4.19	35[label="lookup0 LT GT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
9.46/4.19	36[label="lookup0 EQ LT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
9.46/4.19	37[label="Just vx401",fontsize=16,color="green",shape="box"];38[label="lookup0 EQ GT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
9.46/4.19	39[label="lookup0 GT LT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
9.46/4.19	40[label="lookup0 GT EQ vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
9.46/4.19	41[label="Just vx401",fontsize=16,color="green",shape="box"];42[label="lookup0 LT EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
9.46/4.19	43[label="lookup0 LT GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
9.46/4.19	44[label="lookup0 EQ LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
9.46/4.19	45[label="lookup0 EQ GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
9.46/4.19	46[label="lookup0 GT LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
9.46/4.19	47[label="lookup0 GT EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3];
9.46/4.19	48 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	48[label="lookup LT vx41",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	48 -> 55[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	49 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	49[label="lookup LT vx41",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	49 -> 57[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	50 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	50[label="lookup EQ vx41",fontsize=16,color="magenta"];50 -> 58[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	50 -> 59[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	51 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	51[label="lookup EQ vx41",fontsize=16,color="magenta"];51 -> 60[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	51 -> 61[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	52 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	52[label="lookup GT vx41",fontsize=16,color="magenta"];52 -> 62[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	52 -> 63[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	53 -> 4[label="",style="dashed", color="red", weight=0];
9.46/4.19	53[label="lookup GT vx41",fontsize=16,color="magenta"];53 -> 64[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	53 -> 65[label="",style="dashed", color="magenta", weight=3];
9.46/4.19	54[label="vx41",fontsize=16,color="green",shape="box"];55[label="LT",fontsize=16,color="green",shape="box"];56[label="vx41",fontsize=16,color="green",shape="box"];57[label="LT",fontsize=16,color="green",shape="box"];58[label="vx41",fontsize=16,color="green",shape="box"];59[label="EQ",fontsize=16,color="green",shape="box"];60[label="vx41",fontsize=16,color="green",shape="box"];61[label="EQ",fontsize=16,color="green",shape="box"];62[label="vx41",fontsize=16,color="green",shape="box"];63[label="GT",fontsize=16,color="green",shape="box"];64[label="vx41",fontsize=16,color="green",shape="box"];65[label="GT",fontsize=16,color="green",shape="box"];}
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(6)
9.46/4.19	Obligation:
9.46/4.19	Q DP problem:
9.46/4.19	The TRS P consists of the following rules:
9.46/4.19	
9.46/4.19	   new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	   new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	   new_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	   new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	   new_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	   new_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	
9.46/4.19	R is empty.
9.46/4.19	Q is empty.
9.46/4.19	We have to consider all minimal (P,Q,R)-chains.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(7) DependencyGraphProof (EQUIVALENT)
9.46/4.19	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(8)
9.46/4.19	Complex Obligation (AND)
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(9)
9.46/4.19	Obligation:
9.46/4.19	Q DP problem:
9.46/4.19	The TRS P consists of the following rules:
9.46/4.19	
9.46/4.19	   new_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	   new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	
9.46/4.19	R is empty.
9.46/4.19	Q is empty.
9.46/4.19	We have to consider all minimal (P,Q,R)-chains.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(10) QDPSizeChangeProof (EQUIVALENT)
9.46/4.19	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. 
9.46/4.19	
9.46/4.19	From the DPs we obtained the following set of size-change graphs:
9.46/4.19	*new_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	*new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(11)
9.46/4.19	YES
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(12)
9.46/4.19	Obligation:
9.46/4.19	Q DP problem:
9.46/4.19	The TRS P consists of the following rules:
9.46/4.19	
9.46/4.19	   new_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	   new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	
9.46/4.19	R is empty.
9.46/4.19	Q is empty.
9.46/4.19	We have to consider all minimal (P,Q,R)-chains.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(13) QDPSizeChangeProof (EQUIVALENT)
9.46/4.19	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. 
9.46/4.19	
9.46/4.19	From the DPs we obtained the following set of size-change graphs:
9.46/4.19	*new_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	*new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(14)
9.46/4.19	YES
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(15)
9.46/4.19	Obligation:
9.46/4.19	Q DP problem:
9.46/4.19	The TRS P consists of the following rules:
9.46/4.19	
9.46/4.19	   new_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	   new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	
9.46/4.19	R is empty.
9.46/4.19	Q is empty.
9.46/4.19	We have to consider all minimal (P,Q,R)-chains.
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(16) QDPSizeChangeProof (EQUIVALENT)
9.46/4.19	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. 
9.46/4.19	
9.46/4.19	From the DPs we obtained the following set of size-change graphs:
9.46/4.19	*new_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	*new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
9.46/4.19	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.46/4.19	
9.46/4.19	
9.46/4.19	----------------------------------------
9.46/4.19	
9.46/4.19	(17)
9.46/4.19	YES
9.74/6.41	EOF