/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 16 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;
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;
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;
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;
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;
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;
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;
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.findIndices",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.findIndices ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.findIndices ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="concatMap (List.findIndices0 ww3) (zip ww4 (enumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concat . map (List.findIndices0 ww3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="concat (map (List.findIndices0 ww3) (zip ww4 (enumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="foldr (++) [] (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) (zipWith zip0 ww4 (enumFrom (Pos Zero))))",fontsize=16,color="burlywood",shape="box"];115[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];9 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 10[label="",style="solid", color="burlywood", weight=3];
116[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 11[label="",style="solid", color="burlywood", weight=3];
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11[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (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 (ww40 : ww41) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr (++) [] (map (List.findIndices0 ww3) [])",fontsize=16,color="black",shape="triangle"];13 -> 15[label="",style="solid", color="black", weight=3];
14[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"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
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21[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"];21 -> 22[label="",style="solid", color="black", weight=3];
22 -> 23[label="",style="dashed", color="red", weight=0];
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24[label="ww3 ww40",fontsize=16,color="green",shape="box"];24 -> 28[label="",style="dashed", color="green", weight=3];
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117 -> 26[label="",style="solid", color="burlywood", weight=3];
118[label="ww5/True",fontsize=10,color="white",style="solid",shape="box"];23 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 27[label="",style="solid", color="burlywood", weight=3];
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27[label="(++) List.findIndices000 (Pos Zero) True foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3];
29[label="(++) [] foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="(++) (Pos Zero : []) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
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119 -> 33[label="",style="solid", color="burlywood", weight=3];
120[label="ww41/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 34[label="",style="solid", color="burlywood", weight=3];
32[label="Pos Zero : [] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww41 (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];32 -> 35[label="",style="dashed", color="green", weight=3];
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34[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3];
35 -> 29[label="",style="dashed", color="red", weight=0];
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36[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww410 : ww411) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];36 -> 68[label="",style="dashed", color="magenta", weight=3];
36 -> 69[label="",style="dashed", color="magenta", weight=3];
36 -> 70[label="",style="dashed", color="magenta", weight=3];
36 -> 71[label="",style="dashed", color="magenta", weight=3];
37 -> 13[label="",style="dashed", color="red", weight=0];
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76[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww4110 : ww4111) (enforceWHNF (WHNF (primPlusInt (Pos ww7) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos ww8) (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];76 -> 77[label="",style="solid", color="black", weight=3];
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79[label="foldr (++) [] (map (List.findIndices0 ww3) (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"];79 -> 80[label="",style="solid", color="black", weight=3];
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82[label="(++) List.findIndices0 ww3 (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];82 -> 83[label="",style="solid", color="black", weight=3];
83[label="(++) List.findIndices00 ww3 (zip0 ww4110 (Pos (primPlusNat ww7 (Succ Zero)))) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];83 -> 84[label="",style="solid", color="black", weight=3];
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85 -> 86[label="",style="dashed", color="red", weight=0];
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87[label="ww3 ww4110",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
86[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) ww9 foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="triangle"];121[label="ww9/False",fontsize=10,color="white",style="solid",shape="box"];86 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 89[label="",style="solid", color="burlywood", weight=3];
122[label="ww9/True",fontsize=10,color="white",style="solid",shape="box"];86 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 90[label="",style="solid", color="burlywood", weight=3];
91[label="ww4110",fontsize=16,color="green",shape="box"];89[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) False foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];89 -> 92[label="",style="solid", color="black", weight=3];
90[label="(++) List.findIndices000 (Pos (primPlusNat ww7 (Succ Zero))) True foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];90 -> 93[label="",style="solid", color="black", weight=3];
92[label="(++) [] foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="triangle"];92 -> 94[label="",style="solid", color="black", weight=3];
93[label="(++) (Pos (primPlusNat ww7 (Succ Zero)) : []) foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];93 -> 95[label="",style="solid", color="black", weight=3];
94[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="burlywood",shape="box"];123[label="ww4111/ww41110 : ww41111",fontsize=10,color="white",style="solid",shape="box"];94 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 96[label="",style="solid", color="burlywood", weight=3];
124[label="ww4111/[]",fontsize=10,color="white",style="solid",shape="box"];94 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 97[label="",style="solid", color="burlywood", weight=3];
95[label="Pos (primPlusNat ww7 (Succ Zero)) : [] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="green",shape="box"];95 -> 98[label="",style="dashed", color="green", weight=3];
95 -> 99[label="",style="dashed", color="green", weight=3];
96[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww41110 : ww41111) (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3];
97[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 [] (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3];
98[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="burlywood",shape="triangle"];125[label="ww7/Succ ww70",fontsize=10,color="white",style="solid",shape="box"];98 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 102[label="",style="solid", color="burlywood", weight=3];
126[label="ww7/Zero",fontsize=10,color="white",style="solid",shape="box"];98 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 103[label="",style="solid", color="burlywood", weight=3];
99 -> 92[label="",style="dashed", color="red", weight=0];
99[label="[] ++ foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 ww4111 (numericEnumFrom $! Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];100 -> 67[label="",style="dashed", color="red", weight=0];
100[label="foldr (++) [] (map (List.findIndices0 ww3) (zipWith zip0 (ww41110 : ww41111) (Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos (primPlusNat ww7 (Succ Zero)) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="magenta"];100 -> 104[label="",style="dashed", color="magenta", weight=3];
100 -> 105[label="",style="dashed", color="magenta", weight=3];
100 -> 106[label="",style="dashed", color="magenta", weight=3];
100 -> 107[label="",style="dashed", color="magenta", weight=3];
101 -> 13[label="",style="dashed", color="red", weight=0];
101[label="foldr (++) [] (map (List.findIndices0 ww3) [])",fontsize=16,color="magenta"];102[label="primPlusNat (Succ ww70) (Succ Zero)",fontsize=16,color="black",shape="box"];102 -> 108[label="",style="solid", color="black", weight=3];
103[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];103 -> 109[label="",style="solid", color="black", weight=3];
104 -> 98[label="",style="dashed", color="red", weight=0];
104[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];105[label="ww41110",fontsize=16,color="green",shape="box"];106 -> 98[label="",style="dashed", color="red", weight=0];
106[label="primPlusNat ww7 (Succ Zero)",fontsize=16,color="magenta"];107[label="ww41111",fontsize=16,color="green",shape="box"];108[label="Succ (Succ (primPlusNat ww70 Zero))",fontsize=16,color="green",shape="box"];108 -> 110[label="",style="dashed", color="green", weight=3];
109[label="Succ Zero",fontsize=16,color="green",shape="box"];110[label="primPlusNat ww70 Zero",fontsize=16,color="burlywood",shape="box"];127[label="ww70/Succ ww700",fontsize=10,color="white",style="solid",shape="box"];110 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 111[label="",style="solid", color="burlywood", weight=3];
128[label="ww70/Zero",fontsize=10,color="white",style="solid",shape="box"];110 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 112[label="",style="solid", color="burlywood", weight=3];
111[label="primPlusNat (Succ ww700) Zero",fontsize=16,color="black",shape="box"];111 -> 113[label="",style="solid", color="black", weight=3];
112[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];112 -> 114[label="",style="solid", color="black", weight=3];
113[label="Succ ww700",fontsize=16,color="green",shape="box"];114[label="Zero",fontsize=16,color="green",shape="box"];}

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

   new_psPs(ww7, ww3, :(ww41110, ww41111), ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
   new_psPs0(ww3, :(ww41110, ww41111), ww7, ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
   new_foldr(ww3, ww4110, ww4111, ww7, ww8, ba) -> new_psPs(ww7, ww3, ww4111, ba)
   new_psPs(ww7, ww3, ww4111, ba) -> new_psPs0(ww3, ww4111, ww7, ba)

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.
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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(ww3, ww4110, ww4111, ww7, ww8, ba) -> new_psPs(ww7, ww3, ww4111, ba)
The graph contains the following edges 4 >= 1, 1 >= 2, 3 >= 3, 6 >= 4


*new_psPs(ww7, ww3, ww4111, ba) -> new_psPs0(ww3, ww4111, ww7, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 1 >= 3, 4 >= 4


*new_psPs(ww7, ww3, :(ww41110, ww41111), ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
The graph contains the following edges 2 >= 1, 3 > 2, 3 > 3, 4 >= 6


*new_psPs0(ww3, :(ww41110, ww41111), ww7, ba) -> new_foldr(ww3, ww41110, ww41111, new_primPlusNat(ww7), new_primPlusNat(ww7), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 4 >= 6


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(16)
YES