/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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, 10 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL
(13) Narrow [EQUIVALENT, 18 ms]
(14) 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 (EQUIVALENT)
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"];33[label="wx4/wx40 : wx41",fontsize=10,color="white",style="solid",shape="box"];12 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 13[label="",style="solid", color="burlywood", weight=3];
34[label="wx4/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 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="box"];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="burlywood",shape="box"];35[label="wx3/()",fontsize=10,color="white",style="solid",shape="box"];26 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 27[label="",style="solid", color="burlywood", weight=3];
27[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (() == wx40) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="burlywood",shape="box"];36[label="wx40/()",fontsize=10,color="white",style="solid",shape="box"];27 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 28[label="",style="solid", color="burlywood", weight=3];
28[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) (() == ()) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
29[label="Maybe.listToMaybe ((++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 (() ==)) (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 ((++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="Maybe.listToMaybe (Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 wx41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="Just (Pos Zero)",fontsize=16,color="green",shape="box"];}

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

(14)
YES