10.21/4.44	YES
12.47/5.00	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.47/5.00	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.47/5.00	
12.47/5.00	
12.47/5.00	H-Termination with start terms of the given HASKELL could be proven:
12.47/5.00	
12.47/5.00	(0) HASKELL
12.47/5.00	(1) IFR [EQUIVALENT, 0 ms]
12.47/5.00	(2) HASKELL
12.47/5.00	(3) BR [EQUIVALENT, 0 ms]
12.47/5.00	(4) HASKELL
12.47/5.00	(5) COR [EQUIVALENT, 17 ms]
12.47/5.00	(6) HASKELL
12.47/5.00	(7) Narrow [SOUND, 0 ms]
12.47/5.00	(8) QDP
12.47/5.00	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
12.47/5.00	(10) AND
12.47/5.00	    (11) QDP
12.47/5.00	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.47/5.00	        (13) YES
12.47/5.00	    (14) QDP
12.47/5.00	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.47/5.00	        (16) YES
12.47/5.00	
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(0)
12.47/5.00	Obligation:
12.47/5.00	mainModule Main 
12.47/5.00	module Maybe where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	module List where {
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	delete :: Eq a => a  ->  [a]  ->  [a];
12.47/5.00	delete = deleteBy (==);
12.47/5.00	
12.47/5.00	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.47/5.00	deleteBy _ _ [] = [];
12.47/5.00	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.47/5.00	
12.47/5.00	}
12.47/5.00	module Main where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(1) IFR (EQUIVALENT)
12.47/5.00	If Reductions:
12.47/5.00	The following If expression
12.47/5.00	"if eq x y then ys else y : deleteBy eq x ys"
12.47/5.00	is transformed to
12.47/5.00	"deleteBy0 ys y eq x True = ys;
12.47/5.00	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.47/5.00	"
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(2)
12.47/5.00	Obligation:
12.47/5.00	mainModule Main 
12.47/5.00	module Maybe where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	module List where {
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	delete :: Eq a => a  ->  [a]  ->  [a];
12.47/5.00	delete = deleteBy (==);
12.47/5.00	
12.47/5.00	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.47/5.00	deleteBy _ _ [] = [];
12.47/5.00	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.47/5.00	
12.47/5.00	deleteBy0 ys y eq x True = ys;
12.47/5.00	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.47/5.00	
12.47/5.00	}
12.47/5.00	module Main where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(3) BR (EQUIVALENT)
12.47/5.00	Replaced joker patterns by fresh variables and removed binding patterns.
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(4)
12.47/5.00	Obligation:
12.47/5.00	mainModule Main 
12.47/5.00	module Maybe where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	module List where {
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	delete :: Eq a => a  ->  [a]  ->  [a];
12.47/5.00	delete = deleteBy (==);
12.47/5.00	
12.47/5.00	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.47/5.00	deleteBy vy vz [] = [];
12.47/5.00	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.47/5.00	
12.47/5.00	deleteBy0 ys y eq x True = ys;
12.47/5.00	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.47/5.00	
12.47/5.00	}
12.47/5.00	module Main where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(5) COR (EQUIVALENT)
12.47/5.00	Cond Reductions:
12.47/5.00	The following Function with conditions
12.47/5.00	"undefined |Falseundefined;
12.47/5.00	"
12.47/5.00	is transformed to
12.47/5.00	"undefined  = undefined1;
12.47/5.00	"
12.47/5.00	"undefined0 True = undefined;
12.47/5.00	"
12.47/5.00	"undefined1  = undefined0 False;
12.47/5.00	"
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(6)
12.47/5.00	Obligation:
12.47/5.00	mainModule Main 
12.47/5.00	module Maybe where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	module List where {
12.47/5.00	import qualified Main;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	delete :: Eq a => a  ->  [a]  ->  [a];
12.47/5.00	delete = deleteBy (==);
12.47/5.00	
12.47/5.00	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.47/5.00	deleteBy vy vz [] = [];
12.47/5.00	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.47/5.00	
12.47/5.00	deleteBy0 ys y eq x True = ys;
12.47/5.00	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.47/5.00	
12.47/5.00	}
12.47/5.00	module Main where {
12.47/5.00	import qualified List;
12.47/5.00	import qualified Maybe;
12.47/5.00	import qualified Prelude;
12.47/5.00	}
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(7) Narrow (SOUND)
12.47/5.00	Haskell To QDPs
12.47/5.00	
12.47/5.00	digraph dp_graph {
12.47/5.00	node [outthreshold=100, inthreshold=100];1[label="List.delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.47/5.00	3[label="List.delete wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.47/5.00	4[label="List.delete wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.47/5.00	5[label="List.deleteBy (==) wu3 wu4",fontsize=16,color="burlywood",shape="box"];186[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 186[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	186 -> 6[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	187[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 187[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	187 -> 7[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	6[label="List.deleteBy (==) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.47/5.00	7[label="List.deleteBy (==) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.47/5.00	8[label="List.deleteBy0 wu41 wu40 (==) wu3 ((==) wu3 wu40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.47/5.00	9[label="[]",fontsize=16,color="green",shape="box"];10[label="List.deleteBy0 wu41 wu40 primEqChar wu3 (primEqChar wu3 wu40)",fontsize=16,color="burlywood",shape="triangle"];188[label="wu3/Char wu30",fontsize=10,color="white",style="solid",shape="box"];10 -> 188[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	188 -> 11[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	11[label="List.deleteBy0 wu41 wu40 primEqChar (Char wu30) (primEqChar (Char wu30) wu40)",fontsize=16,color="burlywood",shape="box"];189[label="wu40/Char wu400",fontsize=10,color="white",style="solid",shape="box"];11 -> 189[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	189 -> 12[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	12[label="List.deleteBy0 wu41 (Char wu400) primEqChar (Char wu30) (primEqChar (Char wu30) (Char wu400))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.47/5.00	13[label="List.deleteBy0 wu41 (Char wu400) primEqChar (Char wu30) (primEqNat wu30 wu400)",fontsize=16,color="burlywood",shape="box"];190[label="wu30/Succ wu300",fontsize=10,color="white",style="solid",shape="box"];13 -> 190[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	190 -> 14[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	191[label="wu30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 191[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	191 -> 15[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	14[label="List.deleteBy0 wu41 (Char wu400) primEqChar (Char (Succ wu300)) (primEqNat (Succ wu300) wu400)",fontsize=16,color="burlywood",shape="box"];192[label="wu400/Succ wu4000",fontsize=10,color="white",style="solid",shape="box"];14 -> 192[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	192 -> 16[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	193[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 193[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	193 -> 17[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	15[label="List.deleteBy0 wu41 (Char wu400) primEqChar (Char Zero) (primEqNat Zero wu400)",fontsize=16,color="burlywood",shape="box"];194[label="wu400/Succ wu4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 194[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	194 -> 18[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	195[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 195[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	195 -> 19[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	16[label="List.deleteBy0 wu41 (Char (Succ wu4000)) primEqChar (Char (Succ wu300)) (primEqNat (Succ wu300) (Succ wu4000))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
12.47/5.00	17[label="List.deleteBy0 wu41 (Char Zero) primEqChar (Char (Succ wu300)) (primEqNat (Succ wu300) Zero)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
12.47/5.00	18[label="List.deleteBy0 wu41 (Char (Succ wu4000)) primEqChar (Char Zero) (primEqNat Zero (Succ wu4000))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.47/5.00	19[label="List.deleteBy0 wu41 (Char Zero) primEqChar (Char Zero) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.47/5.00	20 -> 133[label="",style="dashed", color="red", weight=0];
12.47/5.00	20[label="List.deleteBy0 wu41 (Char (Succ wu4000)) primEqChar (Char (Succ wu300)) (primEqNat wu300 wu4000)",fontsize=16,color="magenta"];20 -> 134[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	20 -> 135[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	20 -> 136[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	20 -> 137[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	20 -> 138[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	21[label="List.deleteBy0 wu41 (Char Zero) primEqChar (Char (Succ wu300)) False",fontsize=16,color="black",shape="box"];21 -> 26[label="",style="solid", color="black", weight=3];
12.47/5.00	22[label="List.deleteBy0 wu41 (Char (Succ wu4000)) primEqChar (Char Zero) False",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
12.47/5.00	23[label="List.deleteBy0 wu41 (Char Zero) primEqChar (Char Zero) True",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
12.47/5.00	134[label="wu4000",fontsize=16,color="green",shape="box"];135[label="wu300",fontsize=16,color="green",shape="box"];136[label="wu4000",fontsize=16,color="green",shape="box"];137[label="wu41",fontsize=16,color="green",shape="box"];138[label="wu300",fontsize=16,color="green",shape="box"];133[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat wu20 wu21)",fontsize=16,color="burlywood",shape="triangle"];196[label="wu20/Succ wu200",fontsize=10,color="white",style="solid",shape="box"];133 -> 196[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	196 -> 169[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	197[label="wu20/Zero",fontsize=10,color="white",style="solid",shape="box"];133 -> 197[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	197 -> 170[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	26[label="Char Zero : List.deleteBy primEqChar (Char (Succ wu300)) wu41",fontsize=16,color="green",shape="box"];26 -> 33[label="",style="dashed", color="green", weight=3];
12.47/5.00	27[label="Char (Succ wu4000) : List.deleteBy primEqChar (Char Zero) wu41",fontsize=16,color="green",shape="box"];27 -> 34[label="",style="dashed", color="green", weight=3];
12.47/5.00	28[label="wu41",fontsize=16,color="green",shape="box"];169[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) wu21)",fontsize=16,color="burlywood",shape="box"];198[label="wu21/Succ wu210",fontsize=10,color="white",style="solid",shape="box"];169 -> 198[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	198 -> 171[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	199[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];169 -> 199[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	199 -> 172[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	170[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero wu21)",fontsize=16,color="burlywood",shape="box"];200[label="wu21/Succ wu210",fontsize=10,color="white",style="solid",shape="box"];170 -> 200[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	200 -> 173[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	201[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];170 -> 201[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	201 -> 174[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	33[label="List.deleteBy primEqChar (Char (Succ wu300)) wu41",fontsize=16,color="burlywood",shape="triangle"];202[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];33 -> 202[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	202 -> 39[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	203[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];33 -> 203[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	203 -> 40[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	34[label="List.deleteBy primEqChar (Char Zero) wu41",fontsize=16,color="burlywood",shape="box"];204[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];34 -> 204[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	204 -> 41[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	205[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 205[label="",style="solid", color="burlywood", weight=9];
12.47/5.00	205 -> 42[label="",style="solid", color="burlywood", weight=3];
12.47/5.00	171[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) (Succ wu210))",fontsize=16,color="black",shape="box"];171 -> 175[label="",style="solid", color="black", weight=3];
12.47/5.00	172[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) Zero)",fontsize=16,color="black",shape="box"];172 -> 176[label="",style="solid", color="black", weight=3];
12.47/5.00	173[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero (Succ wu210))",fontsize=16,color="black",shape="box"];173 -> 177[label="",style="solid", color="black", weight=3];
12.47/5.00	174[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];174 -> 178[label="",style="solid", color="black", weight=3];
12.47/5.00	39[label="List.deleteBy primEqChar (Char (Succ wu300)) (wu410 : wu411)",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3];
12.47/5.00	40[label="List.deleteBy primEqChar (Char (Succ wu300)) []",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3];
12.47/5.00	41[label="List.deleteBy primEqChar (Char Zero) (wu410 : wu411)",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3];
12.47/5.00	42[label="List.deleteBy primEqChar (Char Zero) []",fontsize=16,color="black",shape="box"];42 -> 51[label="",style="solid", color="black", weight=3];
12.47/5.00	175 -> 133[label="",style="dashed", color="red", weight=0];
12.47/5.00	175[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat wu200 wu210)",fontsize=16,color="magenta"];175 -> 179[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	175 -> 180[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	176[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) False",fontsize=16,color="black",shape="triangle"];176 -> 181[label="",style="solid", color="black", weight=3];
12.47/5.00	177 -> 176[label="",style="dashed", color="red", weight=0];
12.47/5.00	177[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) False",fontsize=16,color="magenta"];178[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) True",fontsize=16,color="black",shape="box"];178 -> 182[label="",style="solid", color="black", weight=3];
12.47/5.00	48 -> 10[label="",style="dashed", color="red", weight=0];
12.47/5.00	48[label="List.deleteBy0 wu411 wu410 primEqChar (Char (Succ wu300)) (primEqChar (Char (Succ wu300)) wu410)",fontsize=16,color="magenta"];48 -> 58[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	48 -> 59[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	48 -> 60[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	49[label="[]",fontsize=16,color="green",shape="box"];50 -> 10[label="",style="dashed", color="red", weight=0];
12.47/5.00	50[label="List.deleteBy0 wu411 wu410 primEqChar (Char Zero) (primEqChar (Char Zero) wu410)",fontsize=16,color="magenta"];50 -> 61[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	50 -> 62[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	50 -> 63[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	51[label="[]",fontsize=16,color="green",shape="box"];179[label="wu200",fontsize=16,color="green",shape="box"];180[label="wu210",fontsize=16,color="green",shape="box"];181[label="Char (Succ wu18) : List.deleteBy primEqChar (Char (Succ wu19)) wu17",fontsize=16,color="green",shape="box"];181 -> 183[label="",style="dashed", color="green", weight=3];
12.47/5.00	182[label="wu17",fontsize=16,color="green",shape="box"];58[label="wu411",fontsize=16,color="green",shape="box"];59[label="wu410",fontsize=16,color="green",shape="box"];60[label="Char (Succ wu300)",fontsize=16,color="green",shape="box"];61[label="wu411",fontsize=16,color="green",shape="box"];62[label="wu410",fontsize=16,color="green",shape="box"];63[label="Char Zero",fontsize=16,color="green",shape="box"];183 -> 33[label="",style="dashed", color="red", weight=0];
12.47/5.00	183[label="List.deleteBy primEqChar (Char (Succ wu19)) wu17",fontsize=16,color="magenta"];183 -> 184[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	183 -> 185[label="",style="dashed", color="magenta", weight=3];
12.47/5.00	184[label="wu19",fontsize=16,color="green",shape="box"];185[label="wu17",fontsize=16,color="green",shape="box"];}
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(8)
12.47/5.00	Obligation:
12.47/5.00	Q DP problem:
12.47/5.00	The TRS P consists of the following rules:
12.47/5.00	
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.47/5.00	   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.47/5.00	   new_deleteBy01(:(wu410, wu411), Char(Succ(wu4000)), Char(Zero)) -> new_deleteBy01(wu411, wu410, Char(Zero))
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.47/5.00	   new_deleteBy01(:(wu410, wu411), Char(Zero), Char(Succ(wu300))) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	   new_deleteBy(wu300, :(wu410, wu411)) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	   new_deleteBy01(wu41, Char(Succ(wu4000)), Char(Succ(wu300))) -> new_deleteBy0(wu41, wu4000, wu300, wu300, wu4000)
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.47/5.00	
12.47/5.00	R is empty.
12.47/5.00	Q is empty.
12.47/5.00	We have to consider all minimal (P,Q,R)-chains.
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(9) DependencyGraphProof (EQUIVALENT)
12.47/5.00	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(10)
12.47/5.00	Complex Obligation (AND)
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(11)
12.47/5.00	Obligation:
12.47/5.00	Q DP problem:
12.47/5.00	The TRS P consists of the following rules:
12.47/5.00	
12.47/5.00	   new_deleteBy01(:(wu410, wu411), Char(Succ(wu4000)), Char(Zero)) -> new_deleteBy01(wu411, wu410, Char(Zero))
12.47/5.00	
12.47/5.00	R is empty.
12.47/5.00	Q is empty.
12.47/5.00	We have to consider all minimal (P,Q,R)-chains.
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(12) QDPSizeChangeProof (EQUIVALENT)
12.47/5.00	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. 
12.47/5.00	
12.47/5.00	From the DPs we obtained the following set of size-change graphs:
12.47/5.00	*new_deleteBy01(:(wu410, wu411), Char(Succ(wu4000)), Char(Zero)) -> new_deleteBy01(wu411, wu410, Char(Zero))
12.47/5.00	The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
12.47/5.00	
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(13)
12.47/5.00	YES
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(14)
12.47/5.00	Obligation:
12.47/5.00	Q DP problem:
12.47/5.00	The TRS P consists of the following rules:
12.47/5.00	
12.47/5.00	   new_deleteBy(wu300, :(wu410, wu411)) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	   new_deleteBy01(:(wu410, wu411), Char(Zero), Char(Succ(wu300))) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	   new_deleteBy01(wu41, Char(Succ(wu4000)), Char(Succ(wu300))) -> new_deleteBy0(wu41, wu4000, wu300, wu300, wu4000)
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.47/5.00	   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.47/5.00	   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.47/5.00	
12.47/5.00	R is empty.
12.47/5.00	Q is empty.
12.47/5.00	We have to consider all minimal (P,Q,R)-chains.
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(15) QDPSizeChangeProof (EQUIVALENT)
12.47/5.00	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. 
12.47/5.00	
12.47/5.00	From the DPs we obtained the following set of size-change graphs:
12.47/5.00	*new_deleteBy01(:(wu410, wu411), Char(Zero), Char(Succ(wu300))) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy01(wu41, Char(Succ(wu4000)), Char(Succ(wu300))) -> new_deleteBy0(wu41, wu4000, wu300, wu300, wu4000)
12.47/5.00	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy(wu300, :(wu410, wu411)) -> new_deleteBy01(wu411, wu410, Char(Succ(wu300)))
12.47/5.00	The graph contains the following edges 2 > 1, 2 > 2
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.47/5.00	The graph contains the following edges 3 >= 1, 1 >= 2
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.47/5.00	The graph contains the following edges 3 >= 1, 1 >= 2
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.47/5.00	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
12.47/5.00	
12.47/5.00	
12.47/5.00	*new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.47/5.00	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
12.47/5.00	
12.47/5.00	
12.47/5.00	----------------------------------------
12.47/5.00	
12.47/5.00	(16)
12.47/5.00	YES
12.68/5.10	EOF