/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 14 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL
(13) Narrow [SOUND, 0 ms]
(14) QDP
(15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(16) 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;
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;
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;
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;
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;
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;
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;
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.findIndex",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.findIndex wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.findIndex wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="Maybe.listToMaybe . List.findIndices wx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Maybe.listToMaybe (List.findIndices wx3 wx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="Maybe.listToMaybe (concatMap (List.findIndices0 wx3) (zip wx4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="Maybe.listToMaybe (concat . map (List.findIndices0 wx3))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="Maybe.listToMaybe (concat (map (List.findIndices0 wx3) (zip wx4 (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="Maybe.listToMaybe (foldr (++) [] (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) (zipWith zip0 wx4 (enumFrom (Pos Zero)))))",fontsize=16,color="burlywood",shape="box"];119[label="wx4/wx40 : wx41",fontsize=10,color="white",style="solid",shape="box"];11 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 12[label="",style="solid", color="burlywood", weight=3];
120[label="wx4/[]",fontsize=10,color="white",style="solid",shape="box"];11 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx40 : wx41) (enumFrom (Pos Zero)))))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 [] (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 (wx40 : wx41) (numericEnumFrom (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) []))",fontsize=16,color="black",shape="triangle"];15 -> 17[label="",style="solid", color="black", weight=3];
16[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"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="Maybe.listToMaybe (foldr (++) [] [])",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[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"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="Maybe.listToMaybe []",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[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"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="Nothing",fontsize=16,color="green",shape="box"];22[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"];22 -> 23[label="",style="solid", color="black", weight=3];
23[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"];23 -> 24[label="",style="solid", color="black", weight=3];
24[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"];24 -> 25[label="",style="solid", color="black", weight=3];
25 -> 26[label="",style="dashed", color="red", weight=0];
25[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="magenta"];25 -> 27[label="",style="dashed", color="magenta", weight=3];
27[label="wx3 wx40",fontsize=16,color="green",shape="box"];27 -> 31[label="",style="dashed", color="green", weight=3];
26[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) wx5 foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="triangle"];121[label="wx5/False",fontsize=10,color="white",style="solid",shape="box"];26 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 29[label="",style="solid", color="burlywood", weight=3];
122[label="wx5/True",fontsize=10,color="white",style="solid",shape="box"];26 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 30[label="",style="solid", color="burlywood", weight=3];
31[label="wx40",fontsize=16,color="green",shape="box"];29[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3];
32[label="Maybe.listToMaybe ((++) [] foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="Maybe.listToMaybe ((++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];123[label="wx41/wx410 : wx411",fontsize=10,color="white",style="solid",shape="box"];34 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 36[label="",style="solid", color="burlywood", weight=3];
124[label="wx41/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 37[label="",style="solid", color="burlywood", weight=3];
35[label="Maybe.listToMaybe (Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx410 : wx411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3];
37[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3];
38[label="Just (Pos Zero)",fontsize=16,color="green",shape="box"];39[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx410 : wx411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40 -> 15[label="",style="dashed", color="red", weight=0];
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41[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx410 : wx411) (enforceWHNF (WHNF (Pos Zero + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))))",fontsize=16,color="magenta"];41 -> 72[label="",style="dashed", color="magenta", weight=3];
41 -> 73[label="",style="dashed", color="magenta", weight=3];
41 -> 74[label="",style="dashed", color="magenta", weight=3];
41 -> 75[label="",style="dashed", color="magenta", weight=3];
72[label="wx410",fontsize=16,color="green",shape="box"];73[label="Zero",fontsize=16,color="green",shape="box"];74[label="wx411",fontsize=16,color="green",shape="box"];75[label="Zero",fontsize=16,color="green",shape="box"];71[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (enforceWHNF (WHNF (Pos wx8 + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos wx7 + fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="triangle"];71 -> 78[label="",style="solid", color="black", weight=3];
78[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (enforceWHNF (WHNF (primPlusInt (Pos wx8) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos wx7) (fromInt (Pos (Succ Zero)))))))))",fontsize=16,color="black",shape="box"];78 -> 79[label="",style="solid", color="black", weight=3];
79[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (enforceWHNF (WHNF (primPlusInt (Pos wx8) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos wx7) (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];79 -> 80[label="",style="solid", color="black", weight=3];
80[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (enforceWHNF (WHNF (Pos (primPlusNat wx8 (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat wx8 (Succ Zero))))))))",fontsize=16,color="black",shape="box"];80 -> 81[label="",style="solid", color="black", weight=3];
81[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (numericEnumFrom (Pos (primPlusNat wx8 (Succ Zero)))))))",fontsize=16,color="black",shape="box"];81 -> 82[label="",style="solid", color="black", weight=3];
82[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx4110 : wx4111) (Pos (primPlusNat wx8 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];82 -> 83[label="",style="solid", color="black", weight=3];
83[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zip0 wx4110 (Pos (primPlusNat wx8 (Succ Zero))) : zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];83 -> 84[label="",style="solid", color="black", weight=3];
84[label="Maybe.listToMaybe (foldr (++) [] (List.findIndices0 wx3 (zip0 wx4110 (Pos (primPlusNat wx8 (Succ Zero)))) : map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];84 -> 85[label="",style="solid", color="black", weight=3];
85[label="Maybe.listToMaybe ((++) List.findIndices0 wx3 (zip0 wx4110 (Pos (primPlusNat wx8 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];85 -> 86[label="",style="solid", color="black", weight=3];
86[label="Maybe.listToMaybe ((++) List.findIndices00 wx3 (zip0 wx4110 (Pos (primPlusNat wx8 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];86 -> 87[label="",style="solid", color="black", weight=3];
87[label="Maybe.listToMaybe ((++) List.findIndices00 wx3 (wx4110,Pos (primPlusNat wx8 (Succ Zero))) foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];87 -> 88[label="",style="solid", color="black", weight=3];
88 -> 89[label="",style="dashed", color="red", weight=0];
88[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx8 (Succ Zero))) (wx3 wx4110) foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];88 -> 90[label="",style="dashed", color="magenta", weight=3];
90[label="wx3 wx4110",fontsize=16,color="green",shape="box"];90 -> 94[label="",style="dashed", color="green", weight=3];
89[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos (primPlusNat wx8 (Succ Zero))) wx9 foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 wx4111 (numericEnumFrom $! Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="triangle"];125[label="wx9/False",fontsize=10,color="white",style="solid",shape="box"];89 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 92[label="",style="solid", color="burlywood", weight=3];
126[label="wx9/True",fontsize=10,color="white",style="solid",shape="box"];89 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 93[label="",style="solid", color="burlywood", weight=3];
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127 -> 99[label="",style="solid", color="burlywood", weight=3];
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128 -> 100[label="",style="solid", color="burlywood", weight=3];
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102[label="Maybe.listToMaybe (foldr (++) [] (map (List.findIndices0 wx3) (zipWith zip0 (wx41110 : wx41111) (Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos (primPlusNat wx8 (Succ Zero)) + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];102 -> 105[label="",style="solid", color="black", weight=3];
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129 -> 106[label="",style="solid", color="burlywood", weight=3];
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130 -> 107[label="",style="solid", color="burlywood", weight=3];
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131 -> 115[label="",style="solid", color="burlywood", weight=3];
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132 -> 116[label="",style="solid", color="burlywood", weight=3];
115[label="primPlusNat (Succ wx800) Zero",fontsize=16,color="black",shape="box"];115 -> 117[label="",style="solid", color="black", weight=3];
116[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];116 -> 118[label="",style="solid", color="black", weight=3];
117[label="Succ wx800",fontsize=16,color="green",shape="box"];118[label="Zero",fontsize=16,color="green",shape="box"];}

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

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_listToMaybe(wx8, wx3, :(wx41110, wx41111), ba) -> new_listToMaybe0(wx3, wx41110, wx41111, new_primPlusNat(wx8), new_primPlusNat(wx8), ba)
   new_listToMaybe0(wx3, wx4110, wx4111, wx8, wx7, ba) -> new_listToMaybe(wx8, wx3, wx4111, ba)

The TRS R consists of the following rules:

   new_primPlusNat(Succ(wx80)) -> Succ(Succ(new_primPlusNat0(wx80)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(wx800)) -> Succ(wx800)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

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

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

(15) 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_listToMaybe0(wx3, wx4110, wx4111, wx8, wx7, ba) -> new_listToMaybe(wx8, wx3, wx4111, ba)
The graph contains the following edges 4 >= 1, 1 >= 2, 3 >= 3, 6 >= 4


*new_listToMaybe(wx8, wx3, :(wx41110, wx41111), ba) -> new_listToMaybe0(wx3, wx41110, wx41111, new_primPlusNat(wx8), new_primPlusNat(wx8), ba)
The graph contains the following edges 2 >= 1, 3 > 2, 3 > 3, 4 >= 6


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

(16)
YES