10.69/4.48	YES
12.94/5.06	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
12.94/5.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.94/5.06	
12.94/5.06	
12.94/5.06	H-Termination with start terms of the given HASKELL could be proven:
12.94/5.06	
12.94/5.06	(0) HASKELL
12.94/5.06	(1) IFR [EQUIVALENT, 0 ms]
12.94/5.06	(2) HASKELL
12.94/5.06	(3) BR [EQUIVALENT, 0 ms]
12.94/5.06	(4) HASKELL
12.94/5.06	(5) COR [EQUIVALENT, 0 ms]
12.94/5.06	(6) HASKELL
12.94/5.06	(7) Narrow [SOUND, 0 ms]
12.94/5.06	(8) AND
12.94/5.06	    (9) QDP
12.94/5.06	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
12.94/5.06	        (11) AND
12.94/5.06	            (12) QDP
12.94/5.06	                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.94/5.06	                (14) YES
12.94/5.06	            (15) QDP
12.94/5.06	                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.94/5.06	                (17) YES
12.94/5.06	    (18) QDP
12.94/5.06	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.94/5.06	        (20) YES
12.94/5.06	
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(0)
12.94/5.06	Obligation:
12.94/5.06	mainModule Main 
12.94/5.06	module Maybe where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	module List where {
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	infix 5 \\;
12.94/5.06	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.94/5.06	(\\) = foldl (flip delete);
12.94/5.06	
12.94/5.06	delete :: Eq a => a  ->  [a]  ->  [a];
12.94/5.06	delete = deleteBy (==);
12.94/5.06	
12.94/5.06	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.94/5.06	deleteBy _ _ [] = [];
12.94/5.06	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.94/5.06	
12.94/5.06	}
12.94/5.06	module Main where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(1) IFR (EQUIVALENT)
12.94/5.06	If Reductions:
12.94/5.06	The following If expression
12.94/5.06	"if eq x y then ys else y : deleteBy eq x ys"
12.94/5.06	is transformed to
12.94/5.06	"deleteBy0 ys y eq x True = ys;
12.94/5.06	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.94/5.06	"
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(2)
12.94/5.06	Obligation:
12.94/5.06	mainModule Main 
12.94/5.06	module Maybe where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	module List where {
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	infix 5 \\;
12.94/5.06	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.94/5.06	(\\) = foldl (flip delete);
12.94/5.06	
12.94/5.06	delete :: Eq a => a  ->  [a]  ->  [a];
12.94/5.06	delete = deleteBy (==);
12.94/5.06	
12.94/5.06	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.94/5.06	deleteBy _ _ [] = [];
12.94/5.06	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.94/5.06	
12.94/5.06	deleteBy0 ys y eq x True = ys;
12.94/5.06	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.94/5.06	
12.94/5.06	}
12.94/5.06	module Main where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(3) BR (EQUIVALENT)
12.94/5.06	Replaced joker patterns by fresh variables and removed binding patterns.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(4)
12.94/5.06	Obligation:
12.94/5.06	mainModule Main 
12.94/5.06	module Maybe where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	module List where {
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	infix 5 \\;
12.94/5.06	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.94/5.06	(\\) = foldl (flip delete);
12.94/5.06	
12.94/5.06	delete :: Eq a => a  ->  [a]  ->  [a];
12.94/5.06	delete = deleteBy (==);
12.94/5.06	
12.94/5.06	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.94/5.06	deleteBy vy vz [] = [];
12.94/5.06	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.94/5.06	
12.94/5.06	deleteBy0 ys y eq x True = ys;
12.94/5.06	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.94/5.06	
12.94/5.06	}
12.94/5.06	module Main where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(5) COR (EQUIVALENT)
12.94/5.06	Cond Reductions:
12.94/5.06	The following Function with conditions
12.94/5.06	"undefined |Falseundefined;
12.94/5.06	"
12.94/5.06	is transformed to
12.94/5.06	"undefined  = undefined1;
12.94/5.06	"
12.94/5.06	"undefined0 True = undefined;
12.94/5.06	"
12.94/5.06	"undefined1  = undefined0 False;
12.94/5.06	"
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(6)
12.94/5.06	Obligation:
12.94/5.06	mainModule Main 
12.94/5.06	module Maybe where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	module List where {
12.94/5.06	import qualified Main;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	infix 5 \\;
12.94/5.06	(\\) :: Eq a => [a]  ->  [a]  ->  [a];
12.94/5.06	(\\) = foldl (flip delete);
12.94/5.06	
12.94/5.06	delete :: Eq a => a  ->  [a]  ->  [a];
12.94/5.06	delete = deleteBy (==);
12.94/5.06	
12.94/5.06	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.94/5.06	deleteBy vy vz [] = [];
12.94/5.06	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.94/5.06	
12.94/5.06	deleteBy0 ys y eq x True = ys;
12.94/5.06	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.94/5.06	
12.94/5.06	}
12.94/5.06	module Main where {
12.94/5.06	import qualified List;
12.94/5.06	import qualified Maybe;
12.94/5.06	import qualified Prelude;
12.94/5.06	}
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(7) Narrow (SOUND)
12.94/5.06	Haskell To QDPs
12.94/5.06	
12.94/5.06	digraph dp_graph {
12.94/5.06	node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.94/5.06	3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.94/5.06	4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.94/5.06	5[label="foldl (flip List.delete) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];194[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 194[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	194 -> 6[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	195[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 195[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	195 -> 7[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.94/5.06	7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.94/5.06	8 -> 5[label="",style="dashed", color="red", weight=0];
12.94/5.06	8[label="foldl (flip List.delete) (flip List.delete wu3 wu40) wu41",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	8 -> 11[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	9[label="wu3",fontsize=16,color="green",shape="box"];10[label="wu41",fontsize=16,color="green",shape="box"];11[label="flip List.delete wu3 wu40",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.94/5.06	12[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.94/5.06	13[label="List.deleteBy (==) wu40 wu3",fontsize=16,color="burlywood",shape="box"];196[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];13 -> 196[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	196 -> 14[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	197[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 197[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	197 -> 15[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.94/5.06	15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.94/5.06	16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
12.94/5.06	17[label="[]",fontsize=16,color="green",shape="box"];18[label="List.deleteBy0 wu31 wu30 primEqChar wu40 (primEqChar wu40 wu30)",fontsize=16,color="burlywood",shape="triangle"];198[label="wu40/Char wu400",fontsize=10,color="white",style="solid",shape="box"];18 -> 198[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	198 -> 19[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	19[label="List.deleteBy0 wu31 wu30 primEqChar (Char wu400) (primEqChar (Char wu400) wu30)",fontsize=16,color="burlywood",shape="box"];199[label="wu30/Char wu300",fontsize=10,color="white",style="solid",shape="box"];19 -> 199[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	199 -> 20[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	20[label="List.deleteBy0 wu31 (Char wu300) primEqChar (Char wu400) (primEqChar (Char wu400) (Char wu300))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
12.94/5.06	21[label="List.deleteBy0 wu31 (Char wu300) primEqChar (Char wu400) (primEqNat wu400 wu300)",fontsize=16,color="burlywood",shape="box"];200[label="wu400/Succ wu4000",fontsize=10,color="white",style="solid",shape="box"];21 -> 200[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	200 -> 22[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	201[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 201[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	201 -> 23[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	22[label="List.deleteBy0 wu31 (Char wu300) primEqChar (Char (Succ wu4000)) (primEqNat (Succ wu4000) wu300)",fontsize=16,color="burlywood",shape="box"];202[label="wu300/Succ wu3000",fontsize=10,color="white",style="solid",shape="box"];22 -> 202[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	202 -> 24[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	203[label="wu300/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 203[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	203 -> 25[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	23[label="List.deleteBy0 wu31 (Char wu300) primEqChar (Char Zero) (primEqNat Zero wu300)",fontsize=16,color="burlywood",shape="box"];204[label="wu300/Succ wu3000",fontsize=10,color="white",style="solid",shape="box"];23 -> 204[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	204 -> 26[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	205[label="wu300/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 205[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	205 -> 27[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	24[label="List.deleteBy0 wu31 (Char (Succ wu3000)) primEqChar (Char (Succ wu4000)) (primEqNat (Succ wu4000) (Succ wu3000))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.94/5.06	25[label="List.deleteBy0 wu31 (Char Zero) primEqChar (Char (Succ wu4000)) (primEqNat (Succ wu4000) Zero)",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.94/5.06	26[label="List.deleteBy0 wu31 (Char (Succ wu3000)) primEqChar (Char Zero) (primEqNat Zero (Succ wu3000))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.94/5.06	27[label="List.deleteBy0 wu31 (Char Zero) primEqChar (Char Zero) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.94/5.06	28 -> 141[label="",style="dashed", color="red", weight=0];
12.94/5.06	28[label="List.deleteBy0 wu31 (Char (Succ wu3000)) primEqChar (Char (Succ wu4000)) (primEqNat wu4000 wu3000)",fontsize=16,color="magenta"];28 -> 142[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	28 -> 143[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	28 -> 144[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	28 -> 145[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	28 -> 146[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	29[label="List.deleteBy0 wu31 (Char Zero) primEqChar (Char (Succ wu4000)) False",fontsize=16,color="black",shape="box"];29 -> 34[label="",style="solid", color="black", weight=3];
12.94/5.06	30[label="List.deleteBy0 wu31 (Char (Succ wu3000)) primEqChar (Char Zero) False",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
12.94/5.06	31[label="List.deleteBy0 wu31 (Char Zero) primEqChar (Char Zero) True",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
12.94/5.06	142[label="wu3000",fontsize=16,color="green",shape="box"];143[label="wu4000",fontsize=16,color="green",shape="box"];144[label="wu4000",fontsize=16,color="green",shape="box"];145[label="wu3000",fontsize=16,color="green",shape="box"];146[label="wu31",fontsize=16,color="green",shape="box"];141[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat wu20 wu21)",fontsize=16,color="burlywood",shape="triangle"];206[label="wu20/Succ wu200",fontsize=10,color="white",style="solid",shape="box"];141 -> 206[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	206 -> 177[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	207[label="wu20/Zero",fontsize=10,color="white",style="solid",shape="box"];141 -> 207[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	207 -> 178[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	34[label="Char Zero : List.deleteBy primEqChar (Char (Succ wu4000)) wu31",fontsize=16,color="green",shape="box"];34 -> 41[label="",style="dashed", color="green", weight=3];
12.94/5.06	35[label="Char (Succ wu3000) : List.deleteBy primEqChar (Char Zero) wu31",fontsize=16,color="green",shape="box"];35 -> 42[label="",style="dashed", color="green", weight=3];
12.94/5.06	36[label="wu31",fontsize=16,color="green",shape="box"];177[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) wu21)",fontsize=16,color="burlywood",shape="box"];208[label="wu21/Succ wu210",fontsize=10,color="white",style="solid",shape="box"];177 -> 208[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	208 -> 179[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	209[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];177 -> 209[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	209 -> 180[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	178[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero wu21)",fontsize=16,color="burlywood",shape="box"];210[label="wu21/Succ wu210",fontsize=10,color="white",style="solid",shape="box"];178 -> 210[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	210 -> 181[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	211[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];178 -> 211[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	211 -> 182[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	41[label="List.deleteBy primEqChar (Char (Succ wu4000)) wu31",fontsize=16,color="burlywood",shape="triangle"];212[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];41 -> 212[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	212 -> 47[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	213[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 213[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	213 -> 48[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	42[label="List.deleteBy primEqChar (Char Zero) wu31",fontsize=16,color="burlywood",shape="box"];214[label="wu31/wu310 : wu311",fontsize=10,color="white",style="solid",shape="box"];42 -> 214[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	214 -> 49[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	215[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];42 -> 215[label="",style="solid", color="burlywood", weight=9];
12.94/5.06	215 -> 50[label="",style="solid", color="burlywood", weight=3];
12.94/5.06	179[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) (Succ wu210))",fontsize=16,color="black",shape="box"];179 -> 183[label="",style="solid", color="black", weight=3];
12.94/5.06	180[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat (Succ wu200) Zero)",fontsize=16,color="black",shape="box"];180 -> 184[label="",style="solid", color="black", weight=3];
12.94/5.06	181[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero (Succ wu210))",fontsize=16,color="black",shape="box"];181 -> 185[label="",style="solid", color="black", weight=3];
12.94/5.06	182[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];182 -> 186[label="",style="solid", color="black", weight=3];
12.94/5.06	47[label="List.deleteBy primEqChar (Char (Succ wu4000)) (wu310 : wu311)",fontsize=16,color="black",shape="box"];47 -> 56[label="",style="solid", color="black", weight=3];
12.94/5.06	48[label="List.deleteBy primEqChar (Char (Succ wu4000)) []",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
12.94/5.06	49[label="List.deleteBy primEqChar (Char Zero) (wu310 : wu311)",fontsize=16,color="black",shape="box"];49 -> 58[label="",style="solid", color="black", weight=3];
12.94/5.06	50[label="List.deleteBy primEqChar (Char Zero) []",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
12.94/5.06	183 -> 141[label="",style="dashed", color="red", weight=0];
12.94/5.06	183[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) (primEqNat wu200 wu210)",fontsize=16,color="magenta"];183 -> 187[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	183 -> 188[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	184[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) False",fontsize=16,color="black",shape="triangle"];184 -> 189[label="",style="solid", color="black", weight=3];
12.94/5.06	185 -> 184[label="",style="dashed", color="red", weight=0];
12.94/5.06	185[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) False",fontsize=16,color="magenta"];186[label="List.deleteBy0 wu17 (Char (Succ wu18)) primEqChar (Char (Succ wu19)) True",fontsize=16,color="black",shape="box"];186 -> 190[label="",style="solid", color="black", weight=3];
12.94/5.06	56 -> 18[label="",style="dashed", color="red", weight=0];
12.94/5.06	56[label="List.deleteBy0 wu311 wu310 primEqChar (Char (Succ wu4000)) (primEqChar (Char (Succ wu4000)) wu310)",fontsize=16,color="magenta"];56 -> 66[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	56 -> 67[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	56 -> 68[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	57[label="[]",fontsize=16,color="green",shape="box"];58 -> 18[label="",style="dashed", color="red", weight=0];
12.94/5.06	58[label="List.deleteBy0 wu311 wu310 primEqChar (Char Zero) (primEqChar (Char Zero) wu310)",fontsize=16,color="magenta"];58 -> 69[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	58 -> 70[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	58 -> 71[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	59[label="[]",fontsize=16,color="green",shape="box"];187[label="wu200",fontsize=16,color="green",shape="box"];188[label="wu210",fontsize=16,color="green",shape="box"];189[label="Char (Succ wu18) : List.deleteBy primEqChar (Char (Succ wu19)) wu17",fontsize=16,color="green",shape="box"];189 -> 191[label="",style="dashed", color="green", weight=3];
12.94/5.06	190[label="wu17",fontsize=16,color="green",shape="box"];66[label="wu310",fontsize=16,color="green",shape="box"];67[label="wu311",fontsize=16,color="green",shape="box"];68[label="Char (Succ wu4000)",fontsize=16,color="green",shape="box"];69[label="wu310",fontsize=16,color="green",shape="box"];70[label="wu311",fontsize=16,color="green",shape="box"];71[label="Char Zero",fontsize=16,color="green",shape="box"];191 -> 41[label="",style="dashed", color="red", weight=0];
12.94/5.06	191[label="List.deleteBy primEqChar (Char (Succ wu19)) wu17",fontsize=16,color="magenta"];191 -> 192[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	191 -> 193[label="",style="dashed", color="magenta", weight=3];
12.94/5.06	192[label="wu19",fontsize=16,color="green",shape="box"];193[label="wu17",fontsize=16,color="green",shape="box"];}
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(8)
12.94/5.06	Complex Obligation (AND)
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(9)
12.94/5.06	Obligation:
12.94/5.06	Q DP problem:
12.94/5.06	The TRS P consists of the following rules:
12.94/5.06	
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.94/5.06	   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.94/5.06	   new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.94/5.06	   new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	   new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	   new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.94/5.06	
12.94/5.06	R is empty.
12.94/5.06	Q is empty.
12.94/5.06	We have to consider all minimal (P,Q,R)-chains.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(10) DependencyGraphProof (EQUIVALENT)
12.94/5.06	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(11)
12.94/5.06	Complex Obligation (AND)
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(12)
12.94/5.06	Obligation:
12.94/5.06	Q DP problem:
12.94/5.06	The TRS P consists of the following rules:
12.94/5.06	
12.94/5.06	   new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))
12.94/5.06	
12.94/5.06	R is empty.
12.94/5.06	Q is empty.
12.94/5.06	We have to consider all minimal (P,Q,R)-chains.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(13) QDPSizeChangeProof (EQUIVALENT)
12.94/5.06	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. 
12.94/5.06	
12.94/5.06	From the DPs we obtained the following set of size-change graphs:
12.94/5.06	*new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))
12.94/5.06	The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
12.94/5.06	
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(14)
12.94/5.06	YES
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(15)
12.94/5.06	Obligation:
12.94/5.06	Q DP problem:
12.94/5.06	The TRS P consists of the following rules:
12.94/5.06	
12.94/5.06	   new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	   new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	   new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.94/5.06	   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.94/5.06	   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.94/5.06	
12.94/5.06	R is empty.
12.94/5.06	Q is empty.
12.94/5.06	We have to consider all minimal (P,Q,R)-chains.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(16) QDPSizeChangeProof (EQUIVALENT)
12.94/5.06	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. 
12.94/5.06	
12.94/5.06	From the DPs we obtained the following set of size-change graphs:
12.94/5.06	*new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
12.94/5.06	The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	The graph contains the following edges 2 > 1, 2 > 2
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
12.94/5.06	The graph contains the following edges 3 >= 1, 1 >= 2
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
12.94/5.06	The graph contains the following edges 3 >= 1, 1 >= 2
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
12.94/5.06	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5
12.94/5.06	
12.94/5.06	
12.94/5.06	*new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
12.94/5.06	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
12.94/5.06	
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(17)
12.94/5.06	YES
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(18)
12.94/5.06	Obligation:
12.94/5.06	Q DP problem:
12.94/5.06	The TRS P consists of the following rules:
12.94/5.06	
12.94/5.06	   new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy1(wu40, wu3), wu41)
12.94/5.06	
12.94/5.06	The TRS R consists of the following rules:
12.94/5.06	
12.94/5.06	   new_deleteBy04(wu17, wu18, wu19) -> :(Char(Succ(wu18)), new_deleteBy2(wu19, wu17))
12.94/5.06	   new_deleteBy3(:(wu310, wu311)) -> new_deleteBy02(wu311, wu310, Char(Zero))
12.94/5.06	   new_deleteBy2(wu4000, :(wu310, wu311)) -> new_deleteBy02(wu311, wu310, Char(Succ(wu4000)))
12.94/5.06	   new_deleteBy3([]) -> []
12.94/5.06	   new_deleteBy03(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy04(wu17, wu18, wu19)
12.94/5.06	   new_deleteBy03(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy04(wu17, wu18, wu19)
12.94/5.06	   new_deleteBy02(wu31, Char(Zero), Char(Zero)) -> wu31
12.94/5.06	   new_deleteBy02(wu31, Char(Zero), Char(Succ(wu4000))) -> :(Char(Zero), new_deleteBy2(wu4000, wu31))
12.94/5.06	   new_deleteBy03(wu17, wu18, wu19, Zero, Zero) -> wu17
12.94/5.06	   new_deleteBy02(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy03(wu31, wu3000, wu4000, wu4000, wu3000)
12.94/5.06	   new_deleteBy1(wu40, []) -> []
12.94/5.06	   new_deleteBy2(wu4000, []) -> []
12.94/5.06	   new_deleteBy1(wu40, :(wu30, wu31)) -> new_deleteBy02(wu31, wu30, wu40)
12.94/5.06	   new_deleteBy03(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy03(wu17, wu18, wu19, wu200, wu210)
12.94/5.06	   new_deleteBy02(wu31, Char(Succ(wu3000)), Char(Zero)) -> :(Char(Succ(wu3000)), new_deleteBy3(wu31))
12.94/5.06	
12.94/5.06	The set Q consists of the following terms:
12.94/5.06	
12.94/5.06	   new_deleteBy04(x0, x1, x2)
12.94/5.06	   new_deleteBy2(x0, :(x1, x2))
12.94/5.06	   new_deleteBy03(x0, x1, x2, Zero, Succ(x3))
12.94/5.06	   new_deleteBy1(x0, [])
12.94/5.06	   new_deleteBy2(x0, [])
12.94/5.06	   new_deleteBy03(x0, x1, x2, Zero, Zero)
12.94/5.06	   new_deleteBy3(:(x0, x1))
12.94/5.06	   new_deleteBy3([])
12.94/5.06	   new_deleteBy1(x0, :(x1, x2))
12.94/5.06	   new_deleteBy02(x0, Char(Zero), Char(Zero))
12.94/5.06	   new_deleteBy02(x0, Char(Zero), Char(Succ(x1)))
12.94/5.06	   new_deleteBy02(x0, Char(Succ(x1)), Char(Succ(x2)))
12.94/5.06	   new_deleteBy03(x0, x1, x2, Succ(x3), Succ(x4))
12.94/5.06	   new_deleteBy03(x0, x1, x2, Succ(x3), Zero)
12.94/5.06	   new_deleteBy02(x0, Char(Succ(x1)), Char(Zero))
12.94/5.06	
12.94/5.06	We have to consider all minimal (P,Q,R)-chains.
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(19) QDPSizeChangeProof (EQUIVALENT)
12.94/5.06	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. 
12.94/5.06	
12.94/5.06	From the DPs we obtained the following set of size-change graphs:
12.94/5.06	*new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy1(wu40, wu3), wu41)
12.94/5.06	The graph contains the following edges 2 > 2
12.94/5.06	
12.94/5.06	
12.94/5.06	----------------------------------------
12.94/5.06	
12.94/5.06	(20)
12.94/5.06	YES
13.04/6.77	EOF