/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) CR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) IFR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) BR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) COR [EQUIVALENT, 13 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL
(13) Narrow [SOUND, 0 ms]
(14) AND
    (15) QDP
        (16) TransformationProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (22) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (\vv1 ->case vv1 of { 
(x,i)->  if p x then i : [] else []; 
_-> []; 
 } ) (zip xs (enumFrom 0));

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

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\ab->(a,b)"
is transformed to
"zip0 a b = (a,b);
"
The following Lambda expression
"\vv1->case vv1 of {
(x,i)  -> if p x then i : [] else [];
_  -> []}
"
is transformed to
"findIndices0 p vv1 = case vv1 of {
(x,i)  -> if p x then i : [] else [];
_  -> []}
;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));

findIndices0 p vv1 = case vv1 of { 
(x,i)->  if p x then i : [] else []; 
_-> []; 
 } ;

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

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

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case vv1 of {
(x,i)  -> if p x then i : [] else [];
_  -> []}
"
is transformed to
"findIndices00 p (x,i) = if p x then i : [] else [];
findIndices00 p _ = [];
"

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));

findIndices0 p vv1 = findIndices00 p vv1;

findIndices00 p (x,i) =  if p x then i : [] else [];
findIndices00 p _ = [];

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

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

(5) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if p x then i : [] else []"
is transformed to
"findIndices000 i True = i : [];
findIndices000 i False = [];
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : _) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));

findIndices0 p vv1 = findIndices00 p vv1;

findIndices00 p (x,i) = findIndices000 i (p x);
findIndices00 p _ = [];

findIndices000 i True = i : [];
findIndices000 i False = [];

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

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

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

(8)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : wv) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));

findIndices0 p vv1 = findIndices00 p vv1;

findIndices00 p (x,i) = findIndices000 i (p x);
findIndices00 p ww = [];

findIndices000 i True = i : [];
findIndices000 i False = [];

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

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

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

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

(10)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : wv) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom 0));

findIndices0 p vv1 = findIndices00 p vv1;

findIndices00 p (x,i) = findIndices000 i (p x);
findIndices00 p ww = [];

findIndices000 i True = i : [];
findIndices000 i False = [];

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

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

(11) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(12)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
listToMaybe :: [a]  ->  Maybe a;
listToMaybe [] = Nothing;
listToMaybe (a : wv) = Just a;

}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
elemIndex :: Eq a => a  ->  [a]  ->  Maybe Int;
elemIndex x = findIndex (== x);

findIndex :: (a  ->  Bool)  ->  [a]  ->  Maybe Int;
findIndex p = Maybe.listToMaybe . findIndices p;

findIndices :: (a  ->  Bool)  ->  [a]  ->  [Int];
findIndices p xs = concatMap (findIndices0 p) (zip xs (enumFrom (Pos Zero)));

findIndices0 p vv1 = findIndices00 p vv1;

findIndices00 p (x,i) = findIndices000 i (p x);
findIndices00 p ww = [];

findIndices000 i True = i : [];
findIndices000 i False = [];

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

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

(13) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.elemIndex",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.elemIndex wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.elemIndex wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.findIndex (wx3 ==) wx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Maybe.listToMaybe . List.findIndices (wx3 ==)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Maybe.listToMaybe (List.findIndices (wx3 ==) wx4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Maybe.listToMaybe (concatMap (List.findIndices0 (wx3 ==)) (zip wx4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="Maybe.listToMaybe (concat . map (List.findIndices0 (wx3 ==)))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="Maybe.listToMaybe (concat (map (List.findIndices0 (wx3 ==)) (zip wx4 (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zip wx4 (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx4 (enumFrom (Pos Zero)))))",fontsize=16,color="burlywood",shape="box"];709[label="wx4/wx40 : wx41",fontsize=10,color="white",style="solid",shape="box"];12 -> 709[label="",style="solid", color="burlywood", weight=9];
709 -> 13[label="",style="solid", color="burlywood", weight=3];
710[label="wx4/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 710[label="",style="solid", color="burlywood", weight=9];
710 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 (wx40 : wx41) (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 [] (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 (wx40 : wx41) (numericEnumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) []))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 (wx40 : wx41) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="Maybe.listToMaybe (foldr (++) [] [])",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zip0 wx40 (Pos Zero) : zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="Maybe.listToMaybe []",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="Maybe.listToMaybe (foldr (++) [] (List.findIndices0 (wx3 ==) (zip0 wx40 (Pos Zero)) : map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="Nothing",fontsize=16,color="green",shape="box"];23[label="Maybe.listToMaybe ((++) List.findIndices0 (wx3 ==) (zip0 wx40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="Maybe.listToMaybe ((++) List.findIndices00 (wx3 ==) (zip0 wx40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3];
25[label="Maybe.listToMaybe ((++) List.findIndices00 (wx3 ==) (wx40,Pos Zero) foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (wx3 == wx40) foldr (++) [] (map (List.findIndices0 (wx3 ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqChar wx3 wx40) foldr (++) [] (map (List.findIndices0 (primEqChar wx3)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];711[label="wx3/Char wx30",fontsize=10,color="white",style="solid",shape="box"];27 -> 711[label="",style="solid", color="burlywood", weight=9];
711 -> 28[label="",style="solid", color="burlywood", weight=3];
28[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqChar (Char wx30) wx40) foldr (++) [] (map (List.findIndices0 (primEqChar (Char wx30))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];712[label="wx40/Char wx400",fontsize=10,color="white",style="solid",shape="box"];28 -> 712[label="",style="solid", color="burlywood", weight=9];
712 -> 29[label="",style="solid", color="burlywood", weight=3];
29[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqChar (Char wx30) (Char wx400)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char wx30))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat wx30 wx400) foldr (++) [] (map (List.findIndices0 (primEqChar (Char wx30))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];713[label="wx30/Succ wx300",fontsize=10,color="white",style="solid",shape="box"];30 -> 713[label="",style="solid", color="burlywood", weight=9];
713 -> 31[label="",style="solid", color="burlywood", weight=3];
714[label="wx30/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 714[label="",style="solid", color="burlywood", weight=9];
714 -> 32[label="",style="solid", color="burlywood", weight=3];
31[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat (Succ wx300) wx400) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];715[label="wx400/Succ wx4000",fontsize=10,color="white",style="solid",shape="box"];31 -> 715[label="",style="solid", color="burlywood", weight=9];
715 -> 33[label="",style="solid", color="burlywood", weight=3];
716[label="wx400/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 716[label="",style="solid", color="burlywood", weight=9];
716 -> 34[label="",style="solid", color="burlywood", weight=3];
32[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat Zero wx400) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];717[label="wx400/Succ wx4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 717[label="",style="solid", color="burlywood", weight=9];
717 -> 35[label="",style="solid", color="burlywood", weight=3];
718[label="wx400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 718[label="",style="solid", color="burlywood", weight=9];
718 -> 36[label="",style="solid", color="burlywood", weight=3];
33[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat (Succ wx300) (Succ wx4000)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat (Succ wx300) Zero) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
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36[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat Zero Zero) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
37 -> 594[label="",style="dashed", color="red", weight=0];
37[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (primEqNat wx300 wx4000) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];37 -> 595[label="",style="dashed", color="magenta", weight=3];
37 -> 596[label="",style="dashed", color="magenta", weight=3];
37 -> 597[label="",style="dashed", color="magenta", weight=3];
37 -> 598[label="",style="dashed", color="magenta", weight=3];
37 -> 599[label="",style="dashed", color="magenta", weight=3];
37 -> 600[label="",style="dashed", color="magenta", weight=3];
38[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="triangle"];38 -> 43[label="",style="solid", color="black", weight=3];
39[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];39 -> 44[label="",style="solid", color="black", weight=3];
40[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];40 -> 45[label="",style="solid", color="black", weight=3];
595[label="wx300",fontsize=16,color="green",shape="box"];596[label="Zero",fontsize=16,color="green",shape="box"];597[label="Zero",fontsize=16,color="green",shape="box"];598[label="wx4000",fontsize=16,color="green",shape="box"];599[label="wx300",fontsize=16,color="green",shape="box"];600[label="wx41",fontsize=16,color="green",shape="box"];594[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos wx31) (primEqNat wx32 wx33) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx34)))) (zipWith zip0 wx35 (numericEnumFrom $! Pos wx36 + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="triangle"];719[label="wx32/Succ wx320",fontsize=10,color="white",style="solid",shape="box"];594 -> 719[label="",style="solid", color="burlywood", weight=9];
719 -> 679[label="",style="solid", color="burlywood", weight=3];
720[label="wx32/Zero",fontsize=10,color="white",style="solid",shape="box"];594 -> 720[label="",style="solid", color="burlywood", weight=9];
720 -> 680[label="",style="solid", color="burlywood", weight=3];
43[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
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731 -> 356[label="",style="solid", color="burlywood", weight=3];
355[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) (primEqChar (Char Zero) wx4110) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];732[label="wx4110/Char wx41100",fontsize=10,color="white",style="solid",shape="box"];355 -> 732[label="",style="solid", color="burlywood", weight=9];
732 -> 357[label="",style="solid", color="burlywood", weight=3];
356[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx16 (Succ Zero))) (primEqChar (Char (Succ wx300)) (Char wx41100)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx16 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];356 -> 358[label="",style="solid", color="black", weight=3];
357[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) (primEqChar (Char Zero) (Char wx41100)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];357 -> 359[label="",style="solid", color="black", weight=3];
358 -> 594[label="",style="dashed", color="red", weight=0];
358[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx16 (Succ Zero))) (primEqNat (Succ wx300) wx41100) foldr (++) [] (map (List.findIndices0 (primEqChar (Char (Succ wx300)))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx16 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];358 -> 613[label="",style="dashed", color="magenta", weight=3];
358 -> 614[label="",style="dashed", color="magenta", weight=3];
358 -> 615[label="",style="dashed", color="magenta", weight=3];
358 -> 616[label="",style="dashed", color="magenta", weight=3];
358 -> 617[label="",style="dashed", color="magenta", weight=3];
358 -> 618[label="",style="dashed", color="magenta", weight=3];
359[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) (primEqNat Zero wx41100) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];733[label="wx41100/Succ wx411000",fontsize=10,color="white",style="solid",shape="box"];359 -> 733[label="",style="solid", color="burlywood", weight=9];
733 -> 362[label="",style="solid", color="burlywood", weight=3];
734[label="wx41100/Zero",fontsize=10,color="white",style="solid",shape="box"];359 -> 734[label="",style="solid", color="burlywood", weight=9];
734 -> 363[label="",style="solid", color="burlywood", weight=3];
613[label="Succ wx300",fontsize=16,color="green",shape="box"];614 -> 398[label="",style="dashed", color="red", weight=0];
614[label="primPlusNat wx16 (Succ Zero)",fontsize=16,color="magenta"];614 -> 681[label="",style="dashed", color="magenta", weight=3];
615 -> 398[label="",style="dashed", color="red", weight=0];
615[label="primPlusNat wx16 (Succ Zero)",fontsize=16,color="magenta"];615 -> 682[label="",style="dashed", color="magenta", weight=3];
616[label="wx41100",fontsize=16,color="green",shape="box"];617[label="wx300",fontsize=16,color="green",shape="box"];618[label="wx4111",fontsize=16,color="green",shape="box"];362[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) (primEqNat Zero (Succ wx411000)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];362 -> 366[label="",style="solid", color="black", weight=3];
363[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) (primEqNat Zero Zero) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];363 -> 367[label="",style="solid", color="black", weight=3];
681[label="wx16",fontsize=16,color="green",shape="box"];398[label="primPlusNat wx18 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];735[label="wx18/Succ wx180",fontsize=10,color="white",style="solid",shape="box"];398 -> 735[label="",style="solid", color="burlywood", weight=9];
735 -> 406[label="",style="solid", color="burlywood", weight=3];
736[label="wx18/Zero",fontsize=10,color="white",style="solid",shape="box"];398 -> 736[label="",style="solid", color="burlywood", weight=9];
736 -> 407[label="",style="solid", color="burlywood", weight=3];
682[label="wx16",fontsize=16,color="green",shape="box"];366[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) False foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];366 -> 371[label="",style="solid", color="black", weight=3];
367[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx18 (Succ Zero))) True foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];367 -> 372[label="",style="solid", color="black", weight=3];
406[label="primPlusNat (Succ wx180) (Succ Zero)",fontsize=16,color="black",shape="box"];406 -> 423[label="",style="solid", color="black", weight=3];
407[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];407 -> 424[label="",style="solid", color="black", weight=3];
371[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];371 -> 378[label="",style="solid", color="black", weight=3];
372[label="Maybe.listToMaybe ((++) (Pos (primPlusNat wx18 (Succ Zero)) : []) foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];372 -> 379[label="",style="solid", color="black", weight=3];
423[label="Succ (Succ (primPlusNat wx180 Zero))",fontsize=16,color="green",shape="box"];423 -> 438[label="",style="dashed", color="green", weight=3];
424[label="Succ Zero",fontsize=16,color="green",shape="box"];378[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];737[label="wx4111/wx41110 : wx41111",fontsize=10,color="white",style="solid",shape="box"];378 -> 737[label="",style="solid", color="burlywood", weight=9];
737 -> 386[label="",style="solid", color="burlywood", weight=3];
738[label="wx4111/[]",fontsize=10,color="white",style="solid",shape="box"];378 -> 738[label="",style="solid", color="burlywood", weight=9];
738 -> 387[label="",style="solid", color="burlywood", weight=3];
379[label="Maybe.listToMaybe (Pos (primPlusNat wx18 (Succ Zero)) : [] ++ foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];379 -> 388[label="",style="solid", color="black", weight=3];
438[label="primPlusNat wx180 Zero",fontsize=16,color="burlywood",shape="box"];739[label="wx180/Succ wx1800",fontsize=10,color="white",style="solid",shape="box"];438 -> 739[label="",style="solid", color="burlywood", weight=9];
739 -> 446[label="",style="solid", color="burlywood", weight=3];
740[label="wx180/Zero",fontsize=10,color="white",style="solid",shape="box"];438 -> 740[label="",style="solid", color="burlywood", weight=9];
740 -> 447[label="",style="solid", color="burlywood", weight=3];
386[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 (wx41110 : wx41111) (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];386 -> 396[label="",style="solid", color="black", weight=3];
387[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 [] (numericEnumFrom $! Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];387 -> 397[label="",style="solid", color="black", weight=3];
388[label="Just (Pos (primPlusNat wx18 (Succ Zero)))",fontsize=16,color="green",shape="box"];388 -> 398[label="",style="dashed", color="green", weight=3];
446[label="primPlusNat (Succ wx1800) Zero",fontsize=16,color="black",shape="box"];446 -> 453[label="",style="solid", color="black", weight=3];
447[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];447 -> 454[label="",style="solid", color="black", weight=3];
396[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 (wx41110 : wx41111) (Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];396 -> 405[label="",style="solid", color="black", weight=3];
397 -> 70[label="",style="dashed", color="red", weight=0];
397[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) []))",fontsize=16,color="magenta"];453[label="Succ wx1800",fontsize=16,color="green",shape="box"];454[label="Zero",fontsize=16,color="green",shape="box"];405 -> 326[label="",style="dashed", color="red", weight=0];
405[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 (primEqChar (Char Zero))) (zipWith zip0 (wx41110 : wx41111) (enforceWHNF (WHNF (Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat wx18 (Succ Zero)) + fromInt (Pos (Succ Zero))))))))",fontsize=16,color="magenta"];405 -> 419[label="",style="dashed", color="magenta", weight=3];
405 -> 420[label="",style="dashed", color="magenta", weight=3];
405 -> 421[label="",style="dashed", color="magenta", weight=3];
405 -> 422[label="",style="dashed", color="magenta", weight=3];
419[label="wx41111",fontsize=16,color="green",shape="box"];420 -> 398[label="",style="dashed", color="red", weight=0];
420[label="primPlusNat wx18 (Succ Zero)",fontsize=16,color="magenta"];421[label="wx41110",fontsize=16,color="green",shape="box"];422 -> 398[label="",style="dashed", color="red", weight=0];
422[label="primPlusNat wx18 (Succ Zero)",fontsize=16,color="magenta"];}

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

(14)
Complex Obligation (AND)

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

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

   new_listToMaybe0(wx31, Succ(wx320), Zero, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
   new_listToMaybe0(wx31, Zero, Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe2(wx31, wx34, wx35, wx36)
   new_listToMaybe1(wx300, Char(wx41100), wx4111, wx16, wx15) -> new_listToMaybe0(new_primPlusNat(wx16), Succ(wx300), wx41100, wx300, wx4111, new_primPlusNat(wx16))
   new_listToMaybe2(wx31, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
   new_listToMaybe0(wx31, Succ(wx320), Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe0(wx31, wx320, wx330, wx34, wx35, wx36)

The TRS R consists of the following rules:

   new_primPlusNat(Succ(wx180)) -> Succ(Succ(new_primPlusNat0(wx180)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(wx1800)) -> Succ(wx1800)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

   new_primPlusNat(Succ(x0))
   new_primPlusNat0(Zero)
   new_primPlusNat0(Succ(x0))
   new_primPlusNat(Zero)

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

(16) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule new_listToMaybe1(wx300, Char(wx41100), wx4111, wx16, wx15) -> new_listToMaybe0(new_primPlusNat(wx16), Succ(wx300), wx41100, wx300, wx4111, new_primPlusNat(wx16)) we obtained the following new rules [LPAR04]:

   (new_listToMaybe1(z2, Char(x1), z4, z5, z5) -> new_listToMaybe0(new_primPlusNat(z5), Succ(z2), x1, z2, z4, new_primPlusNat(z5)),new_listToMaybe1(z2, Char(x1), z4, z5, z5) -> new_listToMaybe0(new_primPlusNat(z5), Succ(z2), x1, z2, z4, new_primPlusNat(z5)))


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

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_listToMaybe0(wx31, Succ(wx320), Zero, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
   new_listToMaybe0(wx31, Zero, Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe2(wx31, wx34, wx35, wx36)
   new_listToMaybe2(wx31, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
   new_listToMaybe0(wx31, Succ(wx320), Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe0(wx31, wx320, wx330, wx34, wx35, wx36)
   new_listToMaybe1(z2, Char(x1), z4, z5, z5) -> new_listToMaybe0(new_primPlusNat(z5), Succ(z2), x1, z2, z4, new_primPlusNat(z5))

The TRS R consists of the following rules:

   new_primPlusNat(Succ(wx180)) -> Succ(Succ(new_primPlusNat0(wx180)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(wx1800)) -> Succ(wx1800)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

   new_primPlusNat(Succ(x0))
   new_primPlusNat0(Zero)
   new_primPlusNat0(Succ(x0))
   new_primPlusNat(Zero)

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

(18) 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_listToMaybe1(z2, Char(x1), z4, z5, z5) -> new_listToMaybe0(new_primPlusNat(z5), Succ(z2), x1, z2, z4, new_primPlusNat(z5))
The graph contains the following edges 2 > 3, 1 >= 4, 3 >= 5


*new_listToMaybe0(wx31, Succ(wx320), Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe0(wx31, wx320, wx330, wx34, wx35, wx36)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6


*new_listToMaybe2(wx31, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
The graph contains the following edges 2 >= 1, 3 > 2, 3 > 3, 4 >= 4, 4 >= 5


*new_listToMaybe0(wx31, Zero, Succ(wx330), wx34, wx35, wx36) -> new_listToMaybe2(wx31, wx34, wx35, wx36)
The graph contains the following edges 1 >= 1, 4 >= 2, 5 >= 3, 6 >= 4


*new_listToMaybe0(wx31, Succ(wx320), Zero, wx34, :(wx350, wx351), wx36) -> new_listToMaybe1(wx34, wx350, wx351, wx36, wx36)
The graph contains the following edges 4 >= 1, 5 > 2, 5 > 3, 6 >= 4, 6 >= 5


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

(19)
YES

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

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_listToMaybe(Char(Succ(wx411000)), :(wx41110, wx41111), wx18, wx17) -> new_listToMaybe(wx41110, wx41111, new_primPlusNat(wx18), new_primPlusNat(wx18))

The TRS R consists of the following rules:

   new_primPlusNat(Succ(wx180)) -> Succ(Succ(new_primPlusNat0(wx180)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(wx1800)) -> Succ(wx1800)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

   new_primPlusNat(Succ(x0))
   new_primPlusNat0(Zero)
   new_primPlusNat0(Succ(x0))
   new_primPlusNat(Zero)

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

(21) 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_listToMaybe(Char(Succ(wx411000)), :(wx41110, wx41111), wx18, wx17) -> new_listToMaybe(wx41110, wx41111, new_primPlusNat(wx18), new_primPlusNat(wx18))
The graph contains the following edges 2 > 1, 2 > 2


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

(22)
YES