/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 19 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) AND
    (9) QDP
        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
        (11) AND
            (12) QDP
                (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (14) YES
            (15) QDP
                (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
                (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if eq x y then ys else y : deleteBy eq x ys"
is transformed to
"deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;
"

----------------------------------------

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
infix 5 \\;
(\\) :: Eq a => [a]  ->  [a]  ->  [a];
(\\) = foldl (flip delete);

delete :: Eq a => a  ->  [a]  ->  [a];
delete = deleteBy (==);

deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

----------------------------------------

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
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];
194 -> 6[label="",style="solid", color="burlywood", weight=3];
195[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 5[label="",style="dashed", color="red", weight=0];
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];
8 -> 11[label="",style="dashed", color="magenta", weight=3];
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[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
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];
196 -> 14[label="",style="solid", color="burlywood", weight=3];
197[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 197[label="",style="solid", color="burlywood", weight=9];
197 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
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];
198 -> 19[label="",style="solid", color="burlywood", weight=3];
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];
199 -> 20[label="",style="solid", color="burlywood", weight=3];
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];
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];
200 -> 22[label="",style="solid", color="burlywood", weight=3];
201[label="wu400/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 201[label="",style="solid", color="burlywood", weight=9];
201 -> 23[label="",style="solid", color="burlywood", weight=3];
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];
202 -> 24[label="",style="solid", color="burlywood", weight=3];
203[label="wu300/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 203[label="",style="solid", color="burlywood", weight=9];
203 -> 25[label="",style="solid", color="burlywood", weight=3];
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];
204 -> 26[label="",style="solid", color="burlywood", weight=3];
205[label="wu300/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 205[label="",style="solid", color="burlywood", weight=9];
205 -> 27[label="",style="solid", color="burlywood", weight=3];
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];
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];
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];
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];
28 -> 141[label="",style="dashed", color="red", weight=0];
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];
28 -> 143[label="",style="dashed", color="magenta", weight=3];
28 -> 144[label="",style="dashed", color="magenta", weight=3];
28 -> 145[label="",style="dashed", color="magenta", weight=3];
28 -> 146[label="",style="dashed", color="magenta", weight=3];
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];
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];
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];
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];
206 -> 177[label="",style="solid", color="burlywood", weight=3];
207[label="wu20/Zero",fontsize=10,color="white",style="solid",shape="box"];141 -> 207[label="",style="solid", color="burlywood", weight=9];
207 -> 178[label="",style="solid", color="burlywood", weight=3];
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];
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];
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];
208 -> 179[label="",style="solid", color="burlywood", weight=3];
209[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];177 -> 209[label="",style="solid", color="burlywood", weight=9];
209 -> 180[label="",style="solid", color="burlywood", weight=3];
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];
210 -> 181[label="",style="solid", color="burlywood", weight=3];
211[label="wu21/Zero",fontsize=10,color="white",style="solid",shape="box"];178 -> 211[label="",style="solid", color="burlywood", weight=9];
211 -> 182[label="",style="solid", color="burlywood", weight=3];
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];
212 -> 47[label="",style="solid", color="burlywood", weight=3];
213[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 213[label="",style="solid", color="burlywood", weight=9];
213 -> 48[label="",style="solid", color="burlywood", weight=3];
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];
214 -> 49[label="",style="solid", color="burlywood", weight=3];
215[label="wu31/[]",fontsize=10,color="white",style="solid",shape="box"];42 -> 215[label="",style="solid", color="burlywood", weight=9];
215 -> 50[label="",style="solid", color="burlywood", weight=3];
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];
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];
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];
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];
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];
48[label="List.deleteBy primEqChar (Char (Succ wu4000)) []",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
49[label="List.deleteBy primEqChar (Char Zero) (wu310 : wu311)",fontsize=16,color="black",shape="box"];49 -> 58[label="",style="solid", color="black", weight=3];
50[label="List.deleteBy primEqChar (Char Zero) []",fontsize=16,color="black",shape="box"];50 -> 59[label="",style="solid", color="black", weight=3];
183 -> 141[label="",style="dashed", color="red", weight=0];
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];
183 -> 188[label="",style="dashed", color="magenta", weight=3];
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];
185 -> 184[label="",style="dashed", color="red", weight=0];
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];
56 -> 18[label="",style="dashed", color="red", weight=0];
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];
56 -> 67[label="",style="dashed", color="magenta", weight=3];
56 -> 68[label="",style="dashed", color="magenta", weight=3];
57[label="[]",fontsize=16,color="green",shape="box"];58 -> 18[label="",style="dashed", color="red", weight=0];
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];
58 -> 70[label="",style="dashed", color="magenta", weight=3];
58 -> 71[label="",style="dashed", color="magenta", weight=3];
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];
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];
191[label="List.deleteBy primEqChar (Char (Succ wu19)) wu17",fontsize=16,color="magenta"];191 -> 192[label="",style="dashed", color="magenta", weight=3];
191 -> 193[label="",style="dashed", color="magenta", weight=3];
192[label="wu19",fontsize=16,color="green",shape="box"];193[label="wu17",fontsize=16,color="green",shape="box"];}

----------------------------------------

(8)
Complex Obligation (AND)

----------------------------------------

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
   new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))
   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
   new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
   new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
   new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(11)
Complex Obligation (AND)

----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_deleteBy01(:(wu310, wu311), Char(Succ(wu3000)), Char(Zero)) -> new_deleteBy01(wu311, wu310, Char(Zero))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


----------------------------------------

(14)
YES

----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
   new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
   new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
   new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
   new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
   new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(16) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_deleteBy01(:(wu310, wu311), Char(Zero), Char(Succ(wu4000))) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
The graph contains the following edges 1 > 1, 1 > 2, 3 >= 3


*new_deleteBy01(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy0(wu31, wu3000, wu4000, wu4000, wu3000)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 3 > 4, 2 > 5


*new_deleteBy(wu4000, :(wu310, wu311)) -> new_deleteBy01(wu311, wu310, Char(Succ(wu4000)))
The graph contains the following edges 2 > 1, 2 > 2


*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy(wu19, wu17)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy00(wu17, wu18, wu19) -> new_deleteBy(wu19, wu17)
The graph contains the following edges 3 >= 1, 1 >= 2


*new_deleteBy0(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy0(wu17, wu18, wu19, wu200, wu210)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5


*new_deleteBy0(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy00(wu17, wu18, wu19)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3


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(17)
YES

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(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy1(wu40, wu3), wu41)

The TRS R consists of the following rules:

   new_deleteBy04(wu17, wu18, wu19) -> :(Char(Succ(wu18)), new_deleteBy2(wu19, wu17))
   new_deleteBy3(:(wu310, wu311)) -> new_deleteBy02(wu311, wu310, Char(Zero))
   new_deleteBy2(wu4000, :(wu310, wu311)) -> new_deleteBy02(wu311, wu310, Char(Succ(wu4000)))
   new_deleteBy3([]) -> []
   new_deleteBy03(wu17, wu18, wu19, Succ(wu200), Zero) -> new_deleteBy04(wu17, wu18, wu19)
   new_deleteBy03(wu17, wu18, wu19, Zero, Succ(wu210)) -> new_deleteBy04(wu17, wu18, wu19)
   new_deleteBy02(wu31, Char(Zero), Char(Zero)) -> wu31
   new_deleteBy02(wu31, Char(Zero), Char(Succ(wu4000))) -> :(Char(Zero), new_deleteBy2(wu4000, wu31))
   new_deleteBy03(wu17, wu18, wu19, Zero, Zero) -> wu17
   new_deleteBy02(wu31, Char(Succ(wu3000)), Char(Succ(wu4000))) -> new_deleteBy03(wu31, wu3000, wu4000, wu4000, wu3000)
   new_deleteBy1(wu40, []) -> []
   new_deleteBy2(wu4000, []) -> []
   new_deleteBy1(wu40, :(wu30, wu31)) -> new_deleteBy02(wu31, wu30, wu40)
   new_deleteBy03(wu17, wu18, wu19, Succ(wu200), Succ(wu210)) -> new_deleteBy03(wu17, wu18, wu19, wu200, wu210)
   new_deleteBy02(wu31, Char(Succ(wu3000)), Char(Zero)) -> :(Char(Succ(wu3000)), new_deleteBy3(wu31))

The set Q consists of the following terms:

   new_deleteBy04(x0, x1, x2)
   new_deleteBy2(x0, :(x1, x2))
   new_deleteBy03(x0, x1, x2, Zero, Succ(x3))
   new_deleteBy1(x0, [])
   new_deleteBy2(x0, [])
   new_deleteBy03(x0, x1, x2, Zero, Zero)
   new_deleteBy3(:(x0, x1))
   new_deleteBy3([])
   new_deleteBy1(x0, :(x1, x2))
   new_deleteBy02(x0, Char(Zero), Char(Zero))
   new_deleteBy02(x0, Char(Zero), Char(Succ(x1)))
   new_deleteBy02(x0, Char(Succ(x1)), Char(Succ(x2)))
   new_deleteBy03(x0, x1, x2, Succ(x3), Succ(x4))
   new_deleteBy03(x0, x1, x2, Succ(x3), Zero)
   new_deleteBy02(x0, Char(Succ(x1)), Char(Zero))

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy1(wu40, wu3), wu41)
The graph contains the following edges 2 > 2


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(20)
YES