10.57/4.45	YES
12.56/4.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.56/4.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.56/4.99	
12.56/4.99	
12.56/4.99	H-Termination with start terms of the given HASKELL could be proven:
12.56/4.99	
12.56/4.99	(0) HASKELL
12.56/4.99	(1) IFR [EQUIVALENT, 0 ms]
12.56/4.99	(2) HASKELL
12.56/4.99	(3) BR [EQUIVALENT, 0 ms]
12.56/4.99	(4) HASKELL
12.56/4.99	(5) COR [EQUIVALENT, 23 ms]
12.56/4.99	(6) HASKELL
12.56/4.99	(7) Narrow [SOUND, 0 ms]
12.56/4.99	(8) QDP
12.56/4.99	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.56/4.99	(10) AND
12.56/4.99	    (11) QDP
12.56/4.99	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.56/4.99	        (13) YES
12.56/4.99	    (14) QDP
12.56/4.99	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.56/4.99	        (16) YES
12.56/4.99	    (17) QDP
12.56/4.99	        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.56/4.99	        (19) YES
12.56/4.99	
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(0)
12.56/4.99	Obligation:
12.56/4.99	mainModule Main 
12.56/4.99	module Maybe where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	module List where {
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	delete :: Eq a => a  ->  [a]  ->  [a];
12.56/4.99	delete = deleteBy (==);
12.56/4.99	
12.56/4.99	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.56/4.99	deleteBy _ _ [] = [];
12.56/4.99	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.56/4.99	
12.56/4.99	}
12.56/4.99	module Main where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(1) IFR (EQUIVALENT)
12.56/4.99	If Reductions:
12.56/4.99	The following If expression
12.56/4.99	"if eq x y then ys else y : deleteBy eq x ys"
12.56/4.99	is transformed to
12.56/4.99	"deleteBy0 ys y eq x True = ys;
12.56/4.99	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.56/4.99	"
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(2)
12.56/4.99	Obligation:
12.56/4.99	mainModule Main 
12.56/4.99	module Maybe where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	module List where {
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	delete :: Eq a => a  ->  [a]  ->  [a];
12.56/4.99	delete = deleteBy (==);
12.56/4.99	
12.56/4.99	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.56/4.99	deleteBy _ _ [] = [];
12.56/4.99	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.56/4.99	
12.56/4.99	deleteBy0 ys y eq x True = ys;
12.56/4.99	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.56/4.99	
12.56/4.99	}
12.56/4.99	module Main where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(3) BR (EQUIVALENT)
12.56/4.99	Replaced joker patterns by fresh variables and removed binding patterns.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(4)
12.56/4.99	Obligation:
12.56/4.99	mainModule Main 
12.56/4.99	module Maybe where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	module List where {
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	delete :: Eq a => a  ->  [a]  ->  [a];
12.56/4.99	delete = deleteBy (==);
12.56/4.99	
12.56/4.99	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.56/4.99	deleteBy vy vz [] = [];
12.56/4.99	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.56/4.99	
12.56/4.99	deleteBy0 ys y eq x True = ys;
12.56/4.99	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.56/4.99	
12.56/4.99	}
12.56/4.99	module Main where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(5) COR (EQUIVALENT)
12.56/4.99	Cond Reductions:
12.56/4.99	The following Function with conditions
12.56/4.99	"undefined |Falseundefined;
12.56/4.99	"
12.56/4.99	is transformed to
12.56/4.99	"undefined  = undefined1;
12.56/4.99	"
12.56/4.99	"undefined0 True = undefined;
12.56/4.99	"
12.56/4.99	"undefined1  = undefined0 False;
12.56/4.99	"
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(6)
12.56/4.99	Obligation:
12.56/4.99	mainModule Main 
12.56/4.99	module Maybe where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	module List where {
12.56/4.99	import qualified Main;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	delete :: Eq a => a  ->  [a]  ->  [a];
12.56/4.99	delete = deleteBy (==);
12.56/4.99	
12.56/4.99	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.56/4.99	deleteBy vy vz [] = [];
12.56/4.99	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.56/4.99	
12.56/4.99	deleteBy0 ys y eq x True = ys;
12.56/4.99	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.56/4.99	
12.56/4.99	}
12.56/4.99	module Main where {
12.56/4.99	import qualified List;
12.56/4.99	import qualified Maybe;
12.56/4.99	import qualified Prelude;
12.56/4.99	}
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(7) Narrow (SOUND)
12.56/4.99	Haskell To QDPs
12.56/4.99	
12.56/4.99	digraph dp_graph {
12.56/4.99	node [outthreshold=100, inthreshold=100];1[label="List.delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.56/4.99	3[label="List.delete wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.56/4.99	4[label="List.delete wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.56/4.99	5[label="List.deleteBy (==) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];58[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 58[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	58 -> 6[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	59[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 59[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	59 -> 7[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	6[label="List.deleteBy (==) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.56/4.99	7[label="List.deleteBy (==) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.56/4.99	8[label="List.deleteBy0 wu41 wu40 (==) wu3 ((==) wu3 wu40)",fontsize=16,color="burlywood",shape="box"];60[label="wu3/LT",fontsize=10,color="white",style="solid",shape="box"];8 -> 60[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	60 -> 10[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	61[label="wu3/EQ",fontsize=10,color="white",style="solid",shape="box"];8 -> 61[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	61 -> 11[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	62[label="wu3/GT",fontsize=10,color="white",style="solid",shape="box"];8 -> 62[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	62 -> 12[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	9[label="[]",fontsize=16,color="green",shape="box"];10[label="List.deleteBy0 wu41 wu40 (==) LT ((==) LT wu40)",fontsize=16,color="burlywood",shape="box"];63[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];10 -> 63[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	63 -> 13[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	64[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];10 -> 64[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	64 -> 14[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	65[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];10 -> 65[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	65 -> 15[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	11[label="List.deleteBy0 wu41 wu40 (==) EQ ((==) EQ wu40)",fontsize=16,color="burlywood",shape="box"];66[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 66[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	66 -> 16[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	67[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 67[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	67 -> 17[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	68[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 68[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	68 -> 18[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	12[label="List.deleteBy0 wu41 wu40 (==) GT ((==) GT wu40)",fontsize=16,color="burlywood",shape="box"];69[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 69[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	69 -> 19[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	70[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 70[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	70 -> 20[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	71[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 71[label="",style="solid", color="burlywood", weight=9];
12.56/4.99	71 -> 21[label="",style="solid", color="burlywood", weight=3];
12.56/4.99	13[label="List.deleteBy0 wu41 LT (==) LT ((==) LT LT)",fontsize=16,color="black",shape="box"];13 -> 22[label="",style="solid", color="black", weight=3];
12.56/4.99	14[label="List.deleteBy0 wu41 EQ (==) LT ((==) LT EQ)",fontsize=16,color="black",shape="box"];14 -> 23[label="",style="solid", color="black", weight=3];
12.56/4.99	15[label="List.deleteBy0 wu41 GT (==) LT ((==) LT GT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
12.56/4.99	16[label="List.deleteBy0 wu41 LT (==) EQ ((==) EQ LT)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
12.56/4.99	17[label="List.deleteBy0 wu41 EQ (==) EQ ((==) EQ EQ)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
12.56/4.99	18[label="List.deleteBy0 wu41 GT (==) EQ ((==) EQ GT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
12.56/4.99	19[label="List.deleteBy0 wu41 LT (==) GT ((==) GT LT)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
12.56/4.99	20[label="List.deleteBy0 wu41 EQ (==) GT ((==) GT EQ)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
12.56/4.99	21[label="List.deleteBy0 wu41 GT (==) GT ((==) GT GT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
12.56/4.99	22[label="List.deleteBy0 wu41 LT (==) LT True",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
12.56/4.99	23[label="List.deleteBy0 wu41 EQ (==) LT False",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
12.56/4.99	24[label="List.deleteBy0 wu41 GT (==) LT False",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
12.56/4.99	25[label="List.deleteBy0 wu41 LT (==) EQ False",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
12.56/4.99	26[label="List.deleteBy0 wu41 EQ (==) EQ True",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
12.56/4.99	27[label="List.deleteBy0 wu41 GT (==) EQ False",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
12.56/4.99	28[label="List.deleteBy0 wu41 LT (==) GT False",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
12.56/4.99	29[label="List.deleteBy0 wu41 EQ (==) GT False",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
12.56/4.99	30[label="List.deleteBy0 wu41 GT (==) GT True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
12.56/4.99	31[label="wu41",fontsize=16,color="green",shape="box"];32[label="EQ : List.deleteBy (==) LT wu41",fontsize=16,color="green",shape="box"];32 -> 40[label="",style="dashed", color="green", weight=3];
12.56/4.99	33[label="GT : List.deleteBy (==) LT wu41",fontsize=16,color="green",shape="box"];33 -> 41[label="",style="dashed", color="green", weight=3];
12.56/4.99	34[label="LT : List.deleteBy (==) EQ wu41",fontsize=16,color="green",shape="box"];34 -> 42[label="",style="dashed", color="green", weight=3];
12.56/4.99	35[label="wu41",fontsize=16,color="green",shape="box"];36[label="GT : List.deleteBy (==) EQ wu41",fontsize=16,color="green",shape="box"];36 -> 43[label="",style="dashed", color="green", weight=3];
12.56/4.99	37[label="LT : List.deleteBy (==) GT wu41",fontsize=16,color="green",shape="box"];37 -> 44[label="",style="dashed", color="green", weight=3];
12.56/4.99	38[label="EQ : List.deleteBy (==) GT wu41",fontsize=16,color="green",shape="box"];38 -> 45[label="",style="dashed", color="green", weight=3];
12.56/4.99	39[label="wu41",fontsize=16,color="green",shape="box"];40 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	40[label="List.deleteBy (==) LT wu41",fontsize=16,color="magenta"];40 -> 46[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	40 -> 47[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	41 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	41[label="List.deleteBy (==) LT wu41",fontsize=16,color="magenta"];41 -> 48[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	41 -> 49[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	42 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	42[label="List.deleteBy (==) EQ wu41",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	42 -> 51[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	43 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	43[label="List.deleteBy (==) EQ wu41",fontsize=16,color="magenta"];43 -> 52[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	43 -> 53[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	44 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	44[label="List.deleteBy (==) GT wu41",fontsize=16,color="magenta"];44 -> 54[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	44 -> 55[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	45 -> 5[label="",style="dashed", color="red", weight=0];
12.56/4.99	45[label="List.deleteBy (==) GT wu41",fontsize=16,color="magenta"];45 -> 56[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	45 -> 57[label="",style="dashed", color="magenta", weight=3];
12.56/4.99	46[label="wu41",fontsize=16,color="green",shape="box"];47[label="LT",fontsize=16,color="green",shape="box"];48[label="wu41",fontsize=16,color="green",shape="box"];49[label="LT",fontsize=16,color="green",shape="box"];50[label="wu41",fontsize=16,color="green",shape="box"];51[label="EQ",fontsize=16,color="green",shape="box"];52[label="wu41",fontsize=16,color="green",shape="box"];53[label="EQ",fontsize=16,color="green",shape="box"];54[label="wu41",fontsize=16,color="green",shape="box"];55[label="GT",fontsize=16,color="green",shape="box"];56[label="wu41",fontsize=16,color="green",shape="box"];57[label="GT",fontsize=16,color="green",shape="box"];}
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(8)
12.56/4.99	Obligation:
12.56/4.99	Q DP problem:
12.56/4.99	The TRS P consists of the following rules:
12.56/4.99	
12.56/4.99	   new_deleteBy(GT, :(EQ, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	   new_deleteBy(EQ, :(GT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	   new_deleteBy(EQ, :(LT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	   new_deleteBy(LT, :(GT, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	   new_deleteBy(LT, :(EQ, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	   new_deleteBy(GT, :(LT, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	
12.56/4.99	R is empty.
12.56/4.99	Q is empty.
12.56/4.99	We have to consider all minimal (P,Q,R)-chains.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(9) DependencyGraphProof (EQUIVALENT)
12.56/4.99	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(10)
12.56/4.99	Complex Obligation (AND)
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(11)
12.56/4.99	Obligation:
12.56/4.99	Q DP problem:
12.56/4.99	The TRS P consists of the following rules:
12.56/4.99	
12.56/4.99	   new_deleteBy(LT, :(EQ, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	   new_deleteBy(LT, :(GT, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	
12.56/4.99	R is empty.
12.56/4.99	Q is empty.
12.56/4.99	We have to consider all minimal (P,Q,R)-chains.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(12) QDPSizeChangeProof (EQUIVALENT)
12.56/4.99	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. 
12.56/4.99	
12.56/4.99	From the DPs we obtained the following set of size-change graphs:
12.56/4.99	*new_deleteBy(LT, :(EQ, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	*new_deleteBy(LT, :(GT, wu41)) -> new_deleteBy(LT, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(13)
12.56/4.99	YES
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(14)
12.56/4.99	Obligation:
12.56/4.99	Q DP problem:
12.56/4.99	The TRS P consists of the following rules:
12.56/4.99	
12.56/4.99	   new_deleteBy(EQ, :(LT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	   new_deleteBy(EQ, :(GT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	
12.56/4.99	R is empty.
12.56/4.99	Q is empty.
12.56/4.99	We have to consider all minimal (P,Q,R)-chains.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(15) QDPSizeChangeProof (EQUIVALENT)
12.56/4.99	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. 
12.56/4.99	
12.56/4.99	From the DPs we obtained the following set of size-change graphs:
12.56/4.99	*new_deleteBy(EQ, :(LT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	*new_deleteBy(EQ, :(GT, wu41)) -> new_deleteBy(EQ, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(16)
12.56/4.99	YES
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(17)
12.56/4.99	Obligation:
12.56/4.99	Q DP problem:
12.56/4.99	The TRS P consists of the following rules:
12.56/4.99	
12.56/4.99	   new_deleteBy(GT, :(LT, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	   new_deleteBy(GT, :(EQ, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	
12.56/4.99	R is empty.
12.56/4.99	Q is empty.
12.56/4.99	We have to consider all minimal (P,Q,R)-chains.
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(18) QDPSizeChangeProof (EQUIVALENT)
12.56/4.99	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. 
12.56/4.99	
12.56/4.99	From the DPs we obtained the following set of size-change graphs:
12.56/4.99	*new_deleteBy(GT, :(LT, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	*new_deleteBy(GT, :(EQ, wu41)) -> new_deleteBy(GT, wu41)
12.56/4.99	The graph contains the following edges 1 >= 1, 2 > 2
12.56/4.99	
12.56/4.99	
12.56/4.99	----------------------------------------
12.56/4.99	
12.56/4.99	(19)
12.56/4.99	YES
12.73/5.04	EOF