/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 22 ms]
(10) HASKELL
(11) NumRed [SOUND, 11 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
    (21) QDP
        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (23) 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"];266[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];10 -> 266[label="",style="solid", color="burlywood", weight=9];
266 -> 11[label="",style="solid", color="burlywood", weight=3];
267[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 267[label="",style="solid", color="burlywood", weight=9];
267 -> 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="box"];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="triangle"];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="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="(++) List.findIndices000 (Pos Zero) (primEqChar ww3 ww40) foldr (++) [] (map (List.findIndices0 (primEqChar ww3)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];268[label="ww3/Char ww30",fontsize=10,color="white",style="solid",shape="box"];24 -> 268[label="",style="solid", color="burlywood", weight=9];
268 -> 25[label="",style="solid", color="burlywood", weight=3];
25[label="(++) List.findIndices000 (Pos Zero) (primEqChar (Char ww30) ww40) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];269[label="ww40/Char ww400",fontsize=10,color="white",style="solid",shape="box"];25 -> 269[label="",style="solid", color="burlywood", weight=9];
269 -> 26[label="",style="solid", color="burlywood", weight=3];
26[label="(++) List.findIndices000 (Pos Zero) (primEqChar (Char ww30) (Char ww400)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27 -> 59[label="",style="dashed", color="red", weight=0];
27[label="(++) List.findIndices000 (Pos Zero) (primEqNat ww30 ww400) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];27 -> 60[label="",style="dashed", color="magenta", weight=3];
27 -> 61[label="",style="dashed", color="magenta", weight=3];
27 -> 62[label="",style="dashed", color="magenta", weight=3];
60[label="ww30",fontsize=16,color="green",shape="box"];61[label="ww400",fontsize=16,color="green",shape="box"];62[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];270[label="ww41/ww410 : ww411",fontsize=10,color="white",style="solid",shape="box"];62 -> 270[label="",style="solid", color="burlywood", weight=9];
270 -> 67[label="",style="solid", color="burlywood", weight=3];
271[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];62 -> 271[label="",style="solid", color="burlywood", weight=9];
271 -> 68[label="",style="solid", color="burlywood", weight=3];
59[label="(++) List.findIndices000 (Pos Zero) (primEqNat ww3000 ww40000) ww5",fontsize=16,color="burlywood",shape="triangle"];272[label="ww3000/Succ ww30000",fontsize=10,color="white",style="solid",shape="box"];59 -> 272[label="",style="solid", color="burlywood", weight=9];
272 -> 69[label="",style="solid", color="burlywood", weight=3];
273[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 273[label="",style="solid", color="burlywood", weight=9];
273 -> 70[label="",style="solid", color="burlywood", weight=3];
67[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="(++) List.findIndices000 (Pos Zero) (primEqNat (Succ ww30000) ww40000) ww5",fontsize=16,color="burlywood",shape="box"];274[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];69 -> 274[label="",style="solid", color="burlywood", weight=9];
274 -> 73[label="",style="solid", color="burlywood", weight=3];
275[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 275[label="",style="solid", color="burlywood", weight=9];
275 -> 74[label="",style="solid", color="burlywood", weight=3];
70[label="(++) List.findIndices000 (Pos Zero) (primEqNat Zero ww40000) ww5",fontsize=16,color="burlywood",shape="box"];276[label="ww40000/Succ ww400000",fontsize=10,color="white",style="solid",shape="box"];70 -> 276[label="",style="solid", color="burlywood", weight=9];
276 -> 75[label="",style="solid", color="burlywood", weight=3];
277[label="ww40000/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 277[label="",style="solid", color="burlywood", weight=9];
277 -> 76[label="",style="solid", color="burlywood", weight=3];
71[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];71 -> 77[label="",style="solid", color="black", weight=3];
72[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) [])",fontsize=16,color="black",shape="triangle"];72 -> 78[label="",style="solid", color="black", weight=3];
73[label="(++) List.findIndices000 (Pos Zero) (primEqNat (Succ ww30000) (Succ ww400000)) ww5",fontsize=16,color="black",shape="box"];73 -> 79[label="",style="solid", color="black", weight=3];
74[label="(++) List.findIndices000 (Pos Zero) (primEqNat (Succ ww30000) Zero) ww5",fontsize=16,color="black",shape="box"];74 -> 80[label="",style="solid", color="black", weight=3];
75[label="(++) List.findIndices000 (Pos Zero) (primEqNat Zero (Succ ww400000)) ww5",fontsize=16,color="black",shape="box"];75 -> 81[label="",style="solid", color="black", weight=3];
76[label="(++) List.findIndices000 (Pos Zero) (primEqNat Zero Zero) ww5",fontsize=16,color="black",shape="box"];76 -> 82[label="",style="solid", color="black", weight=3];
77[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (Pos Zero + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];77 -> 83[label="",style="solid", color="black", weight=3];
78 -> 16[label="",style="dashed", color="red", weight=0];
78[label="foldr (++) [] []",fontsize=16,color="magenta"];79 -> 59[label="",style="dashed", color="red", weight=0];
79[label="(++) List.findIndices000 (Pos Zero) (primEqNat ww30000 ww400000) ww5",fontsize=16,color="magenta"];79 -> 84[label="",style="dashed", color="magenta", weight=3];
79 -> 85[label="",style="dashed", color="magenta", weight=3];
80[label="(++) List.findIndices000 (Pos Zero) False ww5",fontsize=16,color="black",shape="triangle"];80 -> 86[label="",style="solid", color="black", weight=3];
81 -> 80[label="",style="dashed", color="red", weight=0];
81[label="(++) List.findIndices000 (Pos Zero) False ww5",fontsize=16,color="magenta"];82[label="(++) List.findIndices000 (Pos Zero) True ww5",fontsize=16,color="black",shape="box"];82 -> 87[label="",style="solid", color="black", weight=3];
83[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];83 -> 88[label="",style="solid", color="black", weight=3];
84[label="ww30000",fontsize=16,color="green",shape="box"];85[label="ww400000",fontsize=16,color="green",shape="box"];86[label="(++) [] ww5",fontsize=16,color="black",shape="triangle"];86 -> 89[label="",style="solid", color="black", weight=3];
87[label="(++) (Pos Zero : []) ww5",fontsize=16,color="black",shape="box"];87 -> 90[label="",style="solid", color="black", weight=3];
88[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos Zero) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];88 -> 91[label="",style="solid", color="black", weight=3];
89[label="ww5",fontsize=16,color="green",shape="box"];90[label="Pos Zero : [] ++ ww5",fontsize=16,color="green",shape="box"];90 -> 92[label="",style="dashed", color="green", weight=3];
91[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (enforceWHNF (WHNF (Pos (primPlusNat Zero (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero)))))))",fontsize=16,color="black",shape="box"];91 -> 93[label="",style="solid", color="black", weight=3];
92 -> 86[label="",style="dashed", color="red", weight=0];
92[label="[] ++ ww5",fontsize=16,color="magenta"];93 -> 172[label="",style="dashed", color="red", weight=0];
93[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww410 : ww411) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero))))))",fontsize=16,color="magenta"];93 -> 173[label="",style="dashed", color="magenta", weight=3];
93 -> 174[label="",style="dashed", color="magenta", weight=3];
93 -> 175[label="",style="dashed", color="magenta", weight=3];
173[label="ww410",fontsize=16,color="green",shape="box"];174[label="Zero",fontsize=16,color="green",shape="box"];175[label="ww411",fontsize=16,color="green",shape="box"];172[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero))))))",fontsize=16,color="black",shape="triangle"];172 -> 177[label="",style="solid", color="black", weight=3];
177[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 (ww4110 : ww4111) (Pos (primPlusNat ww7 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];177 -> 178[label="",style="solid", color="black", weight=3];
178[label="foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero))) : zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];178 -> 179[label="",style="solid", color="black", weight=3];
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181[label="(++) List.findIndices00 (primEqChar (Char ww30)) (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];181 -> 182[label="",style="solid", color="black", weight=3];
182[label="(++) List.findIndices00 (primEqChar (Char ww30)) (ww4110,Pos (primPlusNat ww7 (Succ Zero))) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];182 -> 183[label="",style="solid", color="black", weight=3];
183[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqChar (Char ww30) ww4110) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];278[label="ww4110/Char ww41100",fontsize=10,color="white",style="solid",shape="box"];183 -> 278[label="",style="solid", color="burlywood", weight=9];
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184[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqChar (Char ww30) (Char ww41100)) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];184 -> 185[label="",style="solid", color="black", weight=3];
185 -> 218[label="",style="dashed", color="red", weight=0];
185[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat ww30 ww41100) foldr (++) [] (map (List.findIndices0 (primEqChar (Char ww30))) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];185 -> 219[label="",style="dashed", color="magenta", weight=3];
185 -> 220[label="",style="dashed", color="magenta", weight=3];
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279 -> 226[label="",style="solid", color="burlywood", weight=3];
280[label="ww4111/[]",fontsize=10,color="white",style="solid",shape="box"];221 -> 280[label="",style="solid", color="burlywood", weight=9];
280 -> 227[label="",style="solid", color="burlywood", weight=3];
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281 -> 228[label="",style="solid", color="burlywood", weight=3];
282[label="ww3000/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 282[label="",style="solid", color="burlywood", weight=9];
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228[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat (Succ ww30000) ww4110000) ww8",fontsize=16,color="burlywood",shape="box"];283[label="ww4110000/Succ ww41100000",fontsize=10,color="white",style="solid",shape="box"];228 -> 283[label="",style="solid", color="burlywood", weight=9];
283 -> 232[label="",style="solid", color="burlywood", weight=3];
284[label="ww4110000/Zero",fontsize=10,color="white",style="solid",shape="box"];228 -> 284[label="",style="solid", color="burlywood", weight=9];
284 -> 233[label="",style="solid", color="burlywood", weight=3];
229[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat Zero ww4110000) ww8",fontsize=16,color="burlywood",shape="box"];285[label="ww4110000/Succ ww41100000",fontsize=10,color="white",style="solid",shape="box"];229 -> 285[label="",style="solid", color="burlywood", weight=9];
285 -> 234[label="",style="solid", color="burlywood", weight=3];
286[label="ww4110000/Zero",fontsize=10,color="white",style="solid",shape="box"];229 -> 286[label="",style="solid", color="burlywood", weight=9];
286 -> 235[label="",style="solid", color="burlywood", weight=3];
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233[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat (Succ ww30000) Zero) ww8",fontsize=16,color="black",shape="box"];233 -> 238[label="",style="solid", color="black", weight=3];
234[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat Zero (Succ ww41100000)) ww8",fontsize=16,color="black",shape="box"];234 -> 239[label="",style="solid", color="black", weight=3];
235[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) (primEqNat Zero Zero) ww8",fontsize=16,color="black",shape="box"];235 -> 240[label="",style="solid", color="black", weight=3];
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239 -> 238[label="",style="dashed", color="red", weight=0];
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247[label="ww8",fontsize=16,color="green",shape="box"];248[label="Pos (primPlusNat ww7 (Succ Zero)) : [] ++ ww8",fontsize=16,color="green",shape="box"];248 -> 250[label="",style="dashed", color="green", weight=3];
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287 -> 253[label="",style="solid", color="burlywood", weight=3];
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252 -> 257[label="",style="dashed", color="magenta", weight=3];
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254[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];254 -> 260[label="",style="solid", color="black", weight=3];
255[label="ww8",fontsize=16,color="green",shape="box"];256[label="ww41110",fontsize=16,color="green",shape="box"];257 -> 250[label="",style="dashed", color="red", weight=0];
257[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];258[label="ww41111",fontsize=16,color="green",shape="box"];259[label="Succ (Succ (primPlusNat ww70 Zero))",fontsize=16,color="green",shape="box"];259 -> 261[label="",style="dashed", color="green", weight=3];
260[label="Succ Zero",fontsize=16,color="green",shape="box"];261[label="primPlusNat ww70 Zero",fontsize=16,color="burlywood",shape="box"];289[label="ww70/Succ ww700",fontsize=10,color="white",style="solid",shape="box"];261 -> 289[label="",style="solid", color="burlywood", weight=9];
289 -> 262[label="",style="solid", color="burlywood", weight=3];
290[label="ww70/Zero",fontsize=10,color="white",style="solid",shape="box"];261 -> 290[label="",style="solid", color="burlywood", weight=9];
290 -> 263[label="",style="solid", color="burlywood", weight=3];
262[label="primPlusNat (Succ ww700) Zero",fontsize=16,color="black",shape="box"];262 -> 264[label="",style="solid", color="black", weight=3];
263[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];263 -> 265[label="",style="solid", color="black", weight=3];
264[label="Succ ww700",fontsize=16,color="green",shape="box"];265[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_psPs0(Succ(ww30000), Succ(ww400000), ww5) -> new_psPs0(ww30000, ww400000, ww5)

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

(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_psPs0(Succ(ww30000), Succ(ww400000), ww5) -> new_psPs0(ww30000, ww400000, ww5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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

(17)
YES

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

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

   new_psPs(ww7, Succ(ww30000), Succ(ww41100000), ww8) -> new_psPs(ww7, ww30000, ww41100000, ww8)

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

(19) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*new_psPs(ww7, Succ(ww30000), Succ(ww41100000), ww8) -> new_psPs(ww7, ww30000, ww41100000, ww8)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4


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

(20)
YES

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

(21)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(ww30, Char(ww41100), :(ww41110, ww41111), ww7) -> new_foldr(ww30, ww41110, ww41111, new_primPlusNat(ww7))

The TRS R consists of the following rules:

   new_primPlusNat(Succ(ww70)) -> Succ(Succ(new_primPlusNat0(ww70)))
   new_primPlusNat(Zero) -> Succ(Zero)
   new_primPlusNat0(Succ(ww700)) -> Succ(ww700)
   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.
----------------------------------------

(22) 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(ww30, Char(ww41100), :(ww41110, ww41111), ww7) -> new_foldr(ww30, ww41110, ww41111, new_primPlusNat(ww7))
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3


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

(23)
YES