/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 12 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;
}
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"];98[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];10 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 11[label="",style="solid", color="burlywood", weight=3];
99[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];10 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 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"];100[label="ww3/()",fontsize=10,color="white",style="solid",shape="box"];23 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 24[label="",style="solid", color="burlywood", weight=3];
24[label="(++) List.findIndices000 (Pos Zero) (() == ww40) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];101[label="ww40/()",fontsize=10,color="white",style="solid",shape="box"];24 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 25[label="",style="solid", color="burlywood", weight=3];
25[label="(++) List.findIndices000 (Pos Zero) (() == ()) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="(++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 (() ==)) (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[label="(++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];27 -> 28[label="",style="solid", color="black", weight=3];
28[label="Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];28 -> 29[label="",style="dashed", color="green", weight=3];
29 -> 55[label="",style="dashed", color="red", weight=0];
29[label="[] ++ foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];29 -> 56[label="",style="dashed", color="magenta", weight=3];
29 -> 57[label="",style="dashed", color="magenta", weight=3];
56[label="Zero",fontsize=16,color="green",shape="box"];57[label="ww41",fontsize=16,color="green",shape="box"];55[label="[] ++ foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww411 (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];55 -> 59[label="",style="solid", color="black", weight=3];
59[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww411 (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="triangle"];102[label="ww411/ww4110 : ww4111",fontsize=10,color="white",style="solid",shape="box"];59 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 60[label="",style="solid", color="burlywood", weight=3];
103[label="ww411/[]",fontsize=10,color="white",style="solid",shape="box"];59 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 61[label="",style="solid", color="burlywood", weight=3];
60[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3];
61[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 [] (numericEnumFrom $! Pos ww5 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3];
62[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (Pos ww5 + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos ww5 + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3];
63 -> 14[label="",style="dashed", color="red", weight=0];
63[label="foldr (++) [] (map (List.findIndices0 (() ==)) [])",fontsize=16,color="magenta"];63 -> 65[label="",style="dashed", color="magenta", weight=3];
64[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos ww5 + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos ww5 + fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];64 -> 66[label="",style="solid", color="black", weight=3];
65[label="()",fontsize=16,color="green",shape="box"];66[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww5) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos ww5) (fromInt (Pos (Succ Zero))))))))",fontsize=16,color="black",shape="box"];66 -> 67[label="",style="solid", color="black", weight=3];
67[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww5) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos ww5) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];67 -> 68[label="",style="solid", color="black", weight=3];
68[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (Pos (primPlusNat ww5 (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat ww5 (Succ Zero)))))))",fontsize=16,color="black",shape="box"];68 -> 69[label="",style="solid", color="black", weight=3];
69[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (numericEnumFrom (Pos (primPlusNat ww5 (Succ Zero))))))",fontsize=16,color="black",shape="box"];69 -> 70[label="",style="solid", color="black", weight=3];
70[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 (ww4110 : ww4111) (Pos (primPlusNat ww5 (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];70 -> 71[label="",style="solid", color="black", weight=3];
71[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero))) : zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];71 -> 72[label="",style="solid", color="black", weight=3];
72[label="foldr (++) [] (List.findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) : map (List.findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];72 -> 73[label="",style="solid", color="black", weight=3];
73 -> 74[label="",style="dashed", color="red", weight=0];
73[label="(++) List.findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];73 -> 75[label="",style="dashed", color="magenta", weight=3];
75 -> 59[label="",style="dashed", color="red", weight=0];
75[label="foldr (++) [] (map (List.findIndices0 (() ==)) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww5 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];75 -> 76[label="",style="dashed", color="magenta", weight=3];
75 -> 77[label="",style="dashed", color="magenta", weight=3];
74[label="(++) List.findIndices0 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) ww6",fontsize=16,color="black",shape="triangle"];74 -> 78[label="",style="solid", color="black", weight=3];
76[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];104[label="ww5/Succ ww50",fontsize=10,color="white",style="solid",shape="box"];76 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 79[label="",style="solid", color="burlywood", weight=3];
105[label="ww5/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 80[label="",style="solid", color="burlywood", weight=3];
77[label="ww4111",fontsize=16,color="green",shape="box"];78 -> 81[label="",style="dashed", color="red", weight=0];
78[label="(++) List.findIndices00 (() ==) (zip0 ww4110 (Pos (primPlusNat ww5 (Succ Zero)))) ww6",fontsize=16,color="magenta"];78 -> 82[label="",style="dashed", color="magenta", weight=3];
79[label="primPlusNat (Succ ww50) (Succ Zero)",fontsize=16,color="black",shape="box"];79 -> 83[label="",style="solid", color="black", weight=3];
80[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];80 -> 84[label="",style="solid", color="black", weight=3];
82 -> 76[label="",style="dashed", color="red", weight=0];
82[label="primPlusNat ww5 (Succ Zero)",fontsize=16,color="magenta"];81[label="(++) List.findIndices00 (() ==) (zip0 ww4110 (Pos ww7)) ww6",fontsize=16,color="black",shape="triangle"];81 -> 85[label="",style="solid", color="black", weight=3];
83[label="Succ (Succ (primPlusNat ww50 Zero))",fontsize=16,color="green",shape="box"];83 -> 86[label="",style="dashed", color="green", weight=3];
84[label="Succ Zero",fontsize=16,color="green",shape="box"];85[label="(++) List.findIndices00 (() ==) (ww4110,Pos ww7) ww6",fontsize=16,color="black",shape="box"];85 -> 87[label="",style="solid", color="black", weight=3];
86[label="primPlusNat ww50 Zero",fontsize=16,color="burlywood",shape="box"];106[label="ww50/Succ ww500",fontsize=10,color="white",style="solid",shape="box"];86 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 88[label="",style="solid", color="burlywood", weight=3];
107[label="ww50/Zero",fontsize=10,color="white",style="solid",shape="box"];86 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 89[label="",style="solid", color="burlywood", weight=3];
87[label="(++) List.findIndices000 (Pos ww7) (() == ww4110) ww6",fontsize=16,color="burlywood",shape="box"];108[label="ww4110/()",fontsize=10,color="white",style="solid",shape="box"];87 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 90[label="",style="solid", color="burlywood", weight=3];
88[label="primPlusNat (Succ ww500) Zero",fontsize=16,color="black",shape="box"];88 -> 91[label="",style="solid", color="black", weight=3];
89[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];89 -> 92[label="",style="solid", color="black", weight=3];
90[label="(++) List.findIndices000 (Pos ww7) (() == ()) ww6",fontsize=16,color="black",shape="box"];90 -> 93[label="",style="solid", color="black", weight=3];
91[label="Succ ww500",fontsize=16,color="green",shape="box"];92[label="Zero",fontsize=16,color="green",shape="box"];93[label="(++) List.findIndices000 (Pos ww7) True ww6",fontsize=16,color="black",shape="box"];93 -> 94[label="",style="solid", color="black", weight=3];
94[label="(++) (Pos ww7 : []) ww6",fontsize=16,color="black",shape="box"];94 -> 95[label="",style="solid", color="black", weight=3];
95[label="Pos ww7 : [] ++ ww6",fontsize=16,color="green",shape="box"];95 -> 96[label="",style="dashed", color="green", weight=3];
96[label="[] ++ ww6",fontsize=16,color="black",shape="box"];96 -> 97[label="",style="solid", color="black", weight=3];
97[label="ww6",fontsize=16,color="green",shape="box"];}

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

   new_foldr(:(ww4110, ww4111), ww5) -> new_foldr(ww4111, 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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(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_foldr(:(ww4110, ww4111), ww5) -> new_foldr(ww4111, new_primPlusNat(ww5))
The graph contains the following edges 1 > 1


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(16)
YES