/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 19 ms]
(10) HASKELL
(11) NumRed [SOUND, 0 ms]
(12) HASKELL
(13) Narrow [SOUND, 0 ms]
(14) AND
    (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;
elemIndices :: Eq a => a  ->  [a]  ->  [Int];
elemIndices x = findIndices (== x);

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

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

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

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

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 wv = [];

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

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 wv = [];

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

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 wv = [];

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.elemIndices",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.elemIndices ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.elemIndices ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.findIndices (ww3 ==) ww4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concatMap (List.findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat . map (List.findIndices0 (ww3 ==))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="concat (map (List.findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww4 (enumFrom (Pos Zero))))",fontsize=16,color="burlywood",shape="box"];195[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];10 -> 195[label="",style="solid", color="burlywood", weight=9];
195 -> 11[label="",style="solid", color="burlywood", weight=3];
196[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 196[label="",style="solid", color="burlywood", weight=9];
196 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 [] (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) [])",fontsize=16,color="black",shape="triangle"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 (ww40 : ww41) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zip0 ww40 (Pos Zero) : zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="[]",fontsize=16,color="green",shape="box"];19[label="foldr (++) [] (List.findIndices0 (ww3 ==) (zip0 ww40 (Pos Zero)) : map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
20[label="(++) List.findIndices0 (ww3 ==) (zip0 ww40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="(++) List.findIndices00 (ww3 ==) (zip0 ww40 (Pos Zero)) foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="(++) List.findIndices00 (ww3 ==) (ww40,Pos Zero) foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="(++) List.findIndices000 (Pos Zero) (ww3 == ww40) foldr (++) [] (map (List.findIndices0 (ww3 ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];197[label="ww3/False",fontsize=10,color="white",style="solid",shape="box"];23 -> 197[label="",style="solid", color="burlywood", weight=9];
197 -> 24[label="",style="solid", color="burlywood", weight=3];
198[label="ww3/True",fontsize=10,color="white",style="solid",shape="box"];23 -> 198[label="",style="solid", color="burlywood", weight=9];
198 -> 25[label="",style="solid", color="burlywood", weight=3];
24[label="(++) List.findIndices000 (Pos Zero) (False == ww40) foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];199[label="ww40/False",fontsize=10,color="white",style="solid",shape="box"];24 -> 199[label="",style="solid", color="burlywood", weight=9];
199 -> 26[label="",style="solid", color="burlywood", weight=3];
200[label="ww40/True",fontsize=10,color="white",style="solid",shape="box"];24 -> 200[label="",style="solid", color="burlywood", weight=9];
200 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="(++) List.findIndices000 (Pos Zero) (True == ww40) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];201[label="ww40/False",fontsize=10,color="white",style="solid",shape="box"];25 -> 201[label="",style="solid", color="burlywood", weight=9];
201 -> 28[label="",style="solid", color="burlywood", weight=3];
202[label="ww40/True",fontsize=10,color="white",style="solid",shape="box"];25 -> 202[label="",style="solid", color="burlywood", weight=9];
202 -> 29[label="",style="solid", color="burlywood", weight=3];
26[label="(++) List.findIndices000 (Pos Zero) (False == False) foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="(++) List.findIndices000 (Pos Zero) (False == True) foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="(++) List.findIndices000 (Pos Zero) (True == False) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="(++) List.findIndices000 (Pos Zero) (True == True) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="(++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="(++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="(++) List.findIndices000 (Pos Zero) False foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
33[label="(++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
34[label="(++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
35[label="(++) [] foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];35 -> 39[label="",style="solid", color="black", weight=3];
36[label="(++) [] foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];36 -> 40[label="",style="solid", color="black", weight=3];
37[label="(++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3];
38[label="Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];38 -> 42[label="",style="dashed", color="green", weight=3];
39[label="foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];203[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];39 -> 203[label="",style="solid", color="burlywood", weight=9];
203 -> 43[label="",style="solid", color="burlywood", weight=3];
204[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];39 -> 204[label="",style="solid", color="burlywood", weight=9];
204 -> 44[label="",style="solid", color="burlywood", weight=3];
40[label="foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];205[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];40 -> 205[label="",style="solid", color="burlywood", weight=9];
205 -> 45[label="",style="solid", color="burlywood", weight=3];
206[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];40 -> 206[label="",style="solid", color="burlywood", weight=9];
206 -> 46[label="",style="solid", color="burlywood", weight=3];
41[label="Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];41 -> 47[label="",style="dashed", color="green", weight=3];
42 -> 35[label="",style="dashed", color="red", weight=0];
42[label="[] ++ foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];43[label="foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 (ww410 : ww411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];43 -> 48[label="",style="solid", color="black", weight=3];
44[label="foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];44 -> 49[label="",style="solid", color="black", weight=3];
45[label="foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 (ww410 : ww411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];45 -> 50[label="",style="solid", color="black", weight=3];
46[label="foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];46 -> 51[label="",style="solid", color="black", weight=3];
47 -> 36[label="",style="dashed", color="red", weight=0];
47[label="[] ++ foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];48 -> 107[label="",style="dashed", color="red", weight=0];
48[label="foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 (ww410 : ww411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];48 -> 108[label="",style="dashed", color="magenta", weight=3];
48 -> 109[label="",style="dashed", color="magenta", weight=3];
48 -> 110[label="",style="dashed", color="magenta", weight=3];
48 -> 111[label="",style="dashed", color="magenta", weight=3];
49 -> 14[label="",style="dashed", color="red", weight=0];
49[label="foldr (++) [] (map (List.findIndices0 (False ==)) [])",fontsize=16,color="magenta"];49 -> 53[label="",style="dashed", color="magenta", weight=3];
50 -> 115[label="",style="dashed", color="red", weight=0];
50[label="foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 (ww410 : ww411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];50 -> 116[label="",style="dashed", color="magenta", weight=3];
50 -> 117[label="",style="dashed", color="magenta", weight=3];
50 -> 118[label="",style="dashed", color="magenta", weight=3];
50 -> 119[label="",style="dashed", color="magenta", weight=3];
51 -> 14[label="",style="dashed", color="red", weight=0];
51[label="foldr (++) [] (map (List.findIndices0 (True ==)) [])",fontsize=16,color="magenta"];51 -> 55[label="",style="dashed", color="magenta", weight=3];
108[label="Zero",fontsize=16,color="green",shape="box"];109[label="Zero",fontsize=16,color="green",shape="box"];110[label="ww411",fontsize=16,color="green",shape="box"];111[label="ww410",fontsize=16,color="green",shape="box"];107[label="foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 (ww4110 : ww4111) (Pos ww5 + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos ww6 + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="triangle"];107 -> 114[label="",style="solid", color="black", weight=3];
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142[label="(++) List.findIndices00 (True ==) (ww4110,Pos (primPlusNat ww7 (Succ Zero))) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];142 -> 144[label="",style="solid", color="black", weight=3];
143[label="(++) List.findIndices000 (Pos (primPlusNat ww5 (Succ Zero))) (False == ww4110) foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];207[label="ww4110/False",fontsize=10,color="white",style="solid",shape="box"];143 -> 207[label="",style="solid", color="burlywood", weight=9];
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147[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (True == False) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];147 -> 151[label="",style="solid", color="black", weight=3];
148[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (True == True) foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];148 -> 152[label="",style="solid", color="black", weight=3];
149[label="(++) List.findIndices000 (Pos (primPlusNat ww5 (Succ Zero))) True foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];149 -> 153[label="",style="solid", color="black", weight=3];
150[label="(++) List.findIndices000 (Pos (primPlusNat ww5 (Succ Zero))) False foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];150 -> 154[label="",style="solid", color="black", weight=3];
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157[label="Pos (primPlusNat ww5 (Succ Zero)) : [] ++ foldr (++) [] (map (List.findIndices0 (False ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];157 -> 161[label="",style="dashed", color="green", weight=3];
157 -> 162[label="",style="dashed", color="green", weight=3];
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211 -> 163[label="",style="solid", color="burlywood", weight=3];
212[label="ww4111/[]",fontsize=10,color="white",style="solid",shape="box"];158 -> 212[label="",style="solid", color="burlywood", weight=9];
212 -> 164[label="",style="solid", color="burlywood", weight=3];
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213 -> 165[label="",style="solid", color="burlywood", weight=3];
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214 -> 166[label="",style="solid", color="burlywood", weight=3];
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160 -> 168[label="",style="dashed", color="green", weight=3];
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215 -> 169[label="",style="solid", color="burlywood", weight=3];
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216 -> 170[label="",style="solid", color="burlywood", weight=3];
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165[label="foldr (++) [] (map (List.findIndices0 (True ==)) (zipWith zip0 (ww41110 : ww41111) (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];165 -> 173[label="",style="solid", color="black", weight=3];
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167 -> 161[label="",style="dashed", color="red", weight=0];
167[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];167 -> 175[label="",style="dashed", color="magenta", weight=3];
168 -> 155[label="",style="dashed", color="red", weight=0];
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170[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];170 -> 177[label="",style="solid", color="black", weight=3];
171 -> 107[label="",style="dashed", color="red", weight=0];
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172 -> 14[label="",style="dashed", color="red", weight=0];
172[label="foldr (++) [] (map (List.findIndices0 (False ==)) [])",fontsize=16,color="magenta"];172 -> 182[label="",style="dashed", color="magenta", weight=3];
173 -> 115[label="",style="dashed", color="red", weight=0];
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173 -> 186[label="",style="dashed", color="magenta", weight=3];
174 -> 14[label="",style="dashed", color="red", weight=0];
174[label="foldr (++) [] (map (List.findIndices0 (True ==)) [])",fontsize=16,color="magenta"];174 -> 187[label="",style="dashed", color="magenta", weight=3];
175[label="ww7",fontsize=16,color="green",shape="box"];176[label="Succ (Succ (primPlusNat ww50 Zero))",fontsize=16,color="green",shape="box"];176 -> 188[label="",style="dashed", color="green", weight=3];
177[label="Succ Zero",fontsize=16,color="green",shape="box"];178 -> 161[label="",style="dashed", color="red", weight=0];
178[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="magenta"];179 -> 161[label="",style="dashed", color="red", weight=0];
179[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="magenta"];180[label="ww41111",fontsize=16,color="green",shape="box"];181[label="ww41110",fontsize=16,color="green",shape="box"];182[label="False",fontsize=16,color="green",shape="box"];183 -> 161[label="",style="dashed", color="red", weight=0];
183[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];183 -> 189[label="",style="dashed", color="magenta", weight=3];
184 -> 161[label="",style="dashed", color="red", weight=0];
184[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];184 -> 190[label="",style="dashed", color="magenta", weight=3];
185[label="ww41111",fontsize=16,color="green",shape="box"];186[label="ww41110",fontsize=16,color="green",shape="box"];187[label="True",fontsize=16,color="green",shape="box"];188[label="primPlusNat ww50 Zero",fontsize=16,color="burlywood",shape="box"];217[label="ww50/Succ ww500",fontsize=10,color="white",style="solid",shape="box"];188 -> 217[label="",style="solid", color="burlywood", weight=9];
217 -> 191[label="",style="solid", color="burlywood", weight=3];
218[label="ww50/Zero",fontsize=10,color="white",style="solid",shape="box"];188 -> 218[label="",style="solid", color="burlywood", weight=9];
218 -> 192[label="",style="solid", color="burlywood", weight=3];
189[label="ww7",fontsize=16,color="green",shape="box"];190[label="ww7",fontsize=16,color="green",shape="box"];191[label="primPlusNat (Succ ww500) Zero",fontsize=16,color="black",shape="box"];191 -> 193[label="",style="solid", color="black", weight=3];
192[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];192 -> 194[label="",style="solid", color="black", weight=3];
193[label="Succ ww500",fontsize=16,color="green",shape="box"];194[label="Zero",fontsize=16,color="green",shape="box"];}

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

(14)
Complex Obligation (AND)

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

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

   new_foldr(False, :(ww41110, ww41111), ww7, ww8) -> new_foldr(ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7))
   new_psPs(:(ww41110, ww41111), ww7) -> new_foldr(ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7))
   new_foldr(True, ww4111, ww7, ww8) -> new_psPs(ww4111, ww7)

The TRS R consists of the following rules:

   new_primPlusNat(Succ(ww50)) -> Succ(Succ(new_primPlusNat0(ww50)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(ww500)) -> Succ(ww500)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

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

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_foldr(False, :(ww41110, ww41111), ww7, ww8) -> new_foldr(ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7))
The graph contains the following edges 2 > 1, 2 > 2


*new_foldr(True, ww4111, ww7, ww8) -> new_psPs(ww4111, ww7)
The graph contains the following edges 2 >= 1, 3 >= 2


*new_psPs(:(ww41110, ww41111), ww7) -> new_foldr(ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7))
The graph contains the following edges 1 > 1, 1 > 2


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(17)
YES

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

   new_foldr0(True, :(ww41110, ww41111), ww5, ww6) -> new_foldr0(ww41110, ww41111, new_primPlusNat(ww5), new_primPlusNat(ww5))
   new_foldr0(False, ww4111, ww5, ww6) -> new_psPs0(ww4111, ww5)
   new_psPs0(:(ww41110, ww41111), ww5) -> new_foldr0(ww41110, ww41111, new_primPlusNat(ww5), new_primPlusNat(ww5))

The TRS R consists of the following rules:

   new_primPlusNat(Succ(ww50)) -> Succ(Succ(new_primPlusNat0(ww50)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(ww500)) -> Succ(ww500)
   new_primPlusNat0(Zero) -> Zero

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
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(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_foldr0(True, :(ww41110, ww41111), ww5, ww6) -> new_foldr0(ww41110, ww41111, new_primPlusNat(ww5), new_primPlusNat(ww5))
The graph contains the following edges 2 > 1, 2 > 2


*new_foldr0(False, ww4111, ww5, ww6) -> new_psPs0(ww4111, ww5)
The graph contains the following edges 2 >= 1, 3 >= 2


*new_psPs0(:(ww41110, ww41111), ww5) -> new_foldr0(ww41110, ww41111, new_primPlusNat(ww5), new_primPlusNat(ww5))
The graph contains the following edges 1 > 1, 1 > 2


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(20)
YES