/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) NumRed [SOUND, 0 ms]
(10) HASKELL
(11) Narrow [SOUND, 0 ms]
(12) AND
    (13) QDP
        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (15) YES
    (16) QDP
        (17) TransformationProof [EQUIVALENT, 0 ms]
        (18) QDP
        (19) UsableRulesProof [EQUIVALENT, 0 ms]
        (20) QDP
        (21) QReductionProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad a => Int  ->  a b  ->  a ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\_->q"
is transformed to
"gtGt0 q _ = q;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad b => Int  ->  b a  ->  b ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(3) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad b => Int  ->  b a  ->  b ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(5) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"take n vx|n <= 0[];
take vy [] = [];
take n (x : xs) = x : take (n - 1) xs;
"
is transformed to
"take n vx = take3 n vx;
take vy [] = take1 vy [];
take n (x : xs) = take0 n (x : xs);
"
"take0 n (x : xs) = x : take (n - 1) xs;
"
"take1 vy [] = [];
take1 wx wy = take0 wx wy;
"
"take2 n vx True = [];
take2 n vx False = take1 n vx;
"
"take3 n vx = take2 n vx (n <= 0);
take3 wz xu = take1 wz xu;
"

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad b => Int  ->  b a  ->  b ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"xs where {
xs  = x : xs;
} 
"
are unpacked to the following functions on top level
"repeatXs xv = xv : repeatXs xv;
"

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

(8)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad a => Int  ->  a b  ->  a ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(9) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(10)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Monad;
import qualified Prelude;
}
module Main where {
import qualified Maybe;
import qualified Monad;
import qualified Prelude;
}
module Monad where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
replicateM_ :: Monad a => Int  ->  a b  ->  a ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Monad.replicateM_ xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.replicateM_ xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence_ (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="foldr (>>) (return ()) (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (>>) (return ()) (take xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="foldr (>>) (return ()) (take3 xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (xw3 <= Pos Zero))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (compare xw3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (compare xw3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (primCmpInt xw3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];159[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 159[label="",style="solid", color="burlywood", weight=9];
159 -> 13[label="",style="solid", color="burlywood", weight=3];
160[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 160[label="",style="solid", color="burlywood", weight=9];
160 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="foldr (>>) (return ()) (take2 (Pos xw30) (repeat xw4) (not (primCmpInt (Pos xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];161[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];13 -> 161[label="",style="solid", color="burlywood", weight=9];
161 -> 15[label="",style="solid", color="burlywood", weight=3];
162[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 162[label="",style="solid", color="burlywood", weight=9];
162 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="foldr (>>) (return ()) (take2 (Neg xw30) (repeat xw4) (not (primCmpInt (Neg xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];163[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];14 -> 163[label="",style="solid", color="burlywood", weight=9];
163 -> 17[label="",style="solid", color="burlywood", weight=3];
164[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 164[label="",style="solid", color="burlywood", weight=9];
164 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpNat (Succ xw300) Zero == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not True))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
32[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];32 -> 36[label="",style="solid", color="black", weight=3];
33 -> 32[label="",style="dashed", color="red", weight=0];
33[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0];
34[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];35[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeat xw4))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="return ()",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
37 -> 39[label="",style="dashed", color="red", weight=0];
37[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="magenta"];37 -> 40[label="",style="dashed", color="magenta", weight=3];
38[label="() : []",fontsize=16,color="green",shape="box"];40 -> 36[label="",style="dashed", color="red", weight=0];
40[label="return ()",fontsize=16,color="magenta"];39[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="black",shape="triangle"];39 -> 41[label="",style="solid", color="black", weight=3];
41[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
42[label="foldr (>>) xw5 (take0 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
43[label="foldr (>>) xw5 (xw4 : take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="(>>) xw4 foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3];
45[label="xw4 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4)))",fontsize=16,color="burlywood",shape="box"];165[label="xw4/xw40 : xw41",fontsize=10,color="white",style="solid",shape="box"];45 -> 165[label="",style="solid", color="burlywood", weight=9];
165 -> 46[label="",style="solid", color="burlywood", weight=3];
166[label="xw4/[]",fontsize=10,color="white",style="solid",shape="box"];45 -> 166[label="",style="solid", color="burlywood", weight=9];
166 -> 47[label="",style="solid", color="burlywood", weight=3];
46[label="xw40 : xw41 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41))))",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
47[label="[] >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs [])))",fontsize=16,color="black",shape="box"];47 -> 49[label="",style="solid", color="black", weight=3];
48 -> 73[label="",style="dashed", color="red", weight=0];
48[label="gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))) xw40 ++ (xw41 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))))",fontsize=16,color="magenta"];48 -> 74[label="",style="dashed", color="magenta", weight=3];
48 -> 75[label="",style="dashed", color="magenta", weight=3];
49[label="[]",fontsize=16,color="green",shape="box"];74[label="foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))",fontsize=16,color="black",shape="triangle"];74 -> 118[label="",style="solid", color="black", weight=3];
75 -> 119[label="",style="dashed", color="red", weight=0];
75[label="gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))) xw40",fontsize=16,color="magenta"];75 -> 120[label="",style="dashed", color="magenta", weight=3];
73[label="xw6 ++ (xw41 >>= gtGt0 xw7)",fontsize=16,color="burlywood",shape="triangle"];167[label="xw6/xw60 : xw61",fontsize=10,color="white",style="solid",shape="box"];73 -> 167[label="",style="solid", color="burlywood", weight=9];
167 -> 121[label="",style="solid", color="burlywood", weight=3];
168[label="xw6/[]",fontsize=10,color="white",style="solid",shape="box"];73 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 122[label="",style="solid", color="burlywood", weight=3];
118[label="foldr (>>) xw5 (take3 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))",fontsize=16,color="black",shape="box"];118 -> 123[label="",style="solid", color="black", weight=3];
120 -> 74[label="",style="dashed", color="red", weight=0];
120[label="foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)))",fontsize=16,color="magenta"];119[label="gtGt0 xw8 xw40",fontsize=16,color="black",shape="triangle"];119 -> 124[label="",style="solid", color="black", weight=3];
121[label="(xw60 : xw61) ++ (xw41 >>= gtGt0 xw7)",fontsize=16,color="black",shape="box"];121 -> 125[label="",style="solid", color="black", weight=3];
122[label="[] ++ (xw41 >>= gtGt0 xw7)",fontsize=16,color="black",shape="box"];122 -> 126[label="",style="solid", color="black", weight=3];
123[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)) (Pos (Succ xw300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];123 -> 127[label="",style="solid", color="black", weight=3];
124[label="xw8",fontsize=16,color="green",shape="box"];125[label="xw60 : xw61 ++ (xw41 >>= gtGt0 xw7)",fontsize=16,color="green",shape="box"];125 -> 128[label="",style="dashed", color="green", weight=3];
126[label="xw41 >>= gtGt0 xw7",fontsize=16,color="burlywood",shape="box"];169[label="xw41/xw410 : xw411",fontsize=10,color="white",style="solid",shape="box"];126 -> 169[label="",style="solid", color="burlywood", weight=9];
169 -> 129[label="",style="solid", color="burlywood", weight=3];
170[label="xw41/[]",fontsize=10,color="white",style="solid",shape="box"];126 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 130[label="",style="solid", color="burlywood", weight=3];
127[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)) (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];127 -> 131[label="",style="solid", color="black", weight=3];
128 -> 73[label="",style="dashed", color="red", weight=0];
128[label="xw61 ++ (xw41 >>= gtGt0 xw7)",fontsize=16,color="magenta"];128 -> 132[label="",style="dashed", color="magenta", weight=3];
129[label="xw410 : xw411 >>= gtGt0 xw7",fontsize=16,color="black",shape="box"];129 -> 133[label="",style="solid", color="black", weight=3];
130[label="[] >>= gtGt0 xw7",fontsize=16,color="black",shape="box"];130 -> 134[label="",style="solid", color="black", weight=3];
131[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)) (not (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];131 -> 135[label="",style="solid", color="black", weight=3];
132[label="xw61",fontsize=16,color="green",shape="box"];133 -> 73[label="",style="dashed", color="red", weight=0];
133[label="gtGt0 xw7 xw410 ++ (xw411 >>= gtGt0 xw7)",fontsize=16,color="magenta"];133 -> 136[label="",style="dashed", color="magenta", weight=3];
133 -> 137[label="",style="dashed", color="magenta", weight=3];
134[label="[]",fontsize=16,color="green",shape="box"];135[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (xw40 : xw41)) (not (primCmpInt (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];135 -> 138[label="",style="solid", color="black", weight=3];
136 -> 119[label="",style="dashed", color="red", weight=0];
136[label="gtGt0 xw7 xw410",fontsize=16,color="magenta"];136 -> 139[label="",style="dashed", color="magenta", weight=3];
136 -> 140[label="",style="dashed", color="magenta", weight=3];
137[label="xw411",fontsize=16,color="green",shape="box"];138[label="foldr (>>) xw5 (take2 (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (repeatXs (xw40 : xw41)) (not (primCmpInt (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];138 -> 141[label="",style="solid", color="black", weight=3];
139[label="xw7",fontsize=16,color="green",shape="box"];140[label="xw410",fontsize=16,color="green",shape="box"];141[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw300) (Succ Zero)) (repeatXs (xw40 : xw41)) (not (primCmpInt (primMinusNat (Succ xw300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];141 -> 142[label="",style="solid", color="black", weight=3];
142[label="foldr (>>) xw5 (take2 (primMinusNat xw300 Zero) (repeatXs (xw40 : xw41)) (not (primCmpInt (primMinusNat xw300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];171[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];142 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 143[label="",style="solid", color="burlywood", weight=3];
172[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];142 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 144[label="",style="solid", color="burlywood", weight=3];
143[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw3000) Zero) (repeatXs (xw40 : xw41)) (not (primCmpInt (primMinusNat (Succ xw3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];143 -> 145[label="",style="solid", color="black", weight=3];
144[label="foldr (>>) xw5 (take2 (primMinusNat Zero Zero) (repeatXs (xw40 : xw41)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];144 -> 146[label="",style="solid", color="black", weight=3];
145[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)) (not (primCmpInt (Pos (Succ xw3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];145 -> 147[label="",style="solid", color="black", weight=3];
146[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (xw40 : xw41)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];146 -> 148[label="",style="solid", color="black", weight=3];
147[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)) (not (primCmpNat (Succ xw3000) Zero == GT)))",fontsize=16,color="black",shape="box"];147 -> 149[label="",style="solid", color="black", weight=3];
148[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (xw40 : xw41)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];148 -> 150[label="",style="solid", color="black", weight=3];
149[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];149 -> 151[label="",style="solid", color="black", weight=3];
150[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (xw40 : xw41)) (not False))",fontsize=16,color="black",shape="box"];150 -> 152[label="",style="solid", color="black", weight=3];
151[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)) (not True))",fontsize=16,color="black",shape="box"];151 -> 153[label="",style="solid", color="black", weight=3];
152[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (xw40 : xw41)) True)",fontsize=16,color="black",shape="box"];152 -> 154[label="",style="solid", color="black", weight=3];
153[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)) False)",fontsize=16,color="black",shape="box"];153 -> 155[label="",style="solid", color="black", weight=3];
154[label="foldr (>>) xw5 []",fontsize=16,color="black",shape="box"];154 -> 156[label="",style="solid", color="black", weight=3];
155 -> 39[label="",style="dashed", color="red", weight=0];
155[label="foldr (>>) xw5 (take1 (Pos (Succ xw3000)) (repeatXs (xw40 : xw41)))",fontsize=16,color="magenta"];155 -> 157[label="",style="dashed", color="magenta", weight=3];
155 -> 158[label="",style="dashed", color="magenta", weight=3];
156[label="xw5",fontsize=16,color="green",shape="box"];157[label="xw3000",fontsize=16,color="green",shape="box"];158[label="xw40 : xw41",fontsize=16,color="green",shape="box"];}

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

(12)
Complex Obligation (AND)

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr0(xw5, xw300, :(xw40, xw41), h) -> new_foldr(xw5, xw300, xw40, xw41, h)
   new_foldr(xw5, Succ(xw3000), xw40, xw41, h) -> new_foldr0(xw5, xw3000, :(xw40, xw41), h)
   new_foldr0(xw5, Succ(xw3000), :(xw40, xw41), h) -> new_foldr0(xw5, xw3000, :(xw40, xw41), h)

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

(14) 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(xw5, Succ(xw3000), xw40, xw41, h) -> new_foldr0(xw5, xw3000, :(xw40, xw41), h)
The graph contains the following edges 1 >= 1, 2 > 2, 5 >= 4


*new_foldr0(xw5, Succ(xw3000), :(xw40, xw41), h) -> new_foldr0(xw5, xw3000, :(xw40, xw41), h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


*new_foldr0(xw5, xw300, :(xw40, xw41), h) -> new_foldr(xw5, xw300, xw40, xw41, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5


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

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(xw60, xw61), xw41, xw7, h) -> new_psPs(xw61, xw41, xw7, h)
   new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(new_gtGt0(xw7, xw410, h), xw411, xw7, h)

The TRS R consists of the following rules:

   new_gtGt0(xw8, xw40, h) -> xw8

The set Q consists of the following terms:

   new_gtGt0(x0, x1, x2)

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

(17) TransformationProof (EQUIVALENT)
By rewriting [LPAR04] the rule new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(new_gtGt0(xw7, xw410, h), xw411, xw7, h) at position [0] we obtained the following new rules [LPAR04]:

   (new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h),new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h))


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

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

   new_psPs(:(xw60, xw61), xw41, xw7, h) -> new_psPs(xw61, xw41, xw7, h)
   new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h)

The TRS R consists of the following rules:

   new_gtGt0(xw8, xw40, h) -> xw8

The set Q consists of the following terms:

   new_gtGt0(x0, x1, x2)

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

(19) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(xw60, xw61), xw41, xw7, h) -> new_psPs(xw61, xw41, xw7, h)
   new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h)

R is empty.
The set Q consists of the following terms:

   new_gtGt0(x0, x1, x2)

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

(21) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_gtGt0(x0, x1, x2)


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

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_psPs(:(xw60, xw61), xw41, xw7, h) -> new_psPs(xw61, xw41, xw7, h)
   new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h)

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

(23) 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(:(xw60, xw61), xw41, xw7, h) -> new_psPs(xw61, xw41, xw7, h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4


*new_psPs([], :(xw410, xw411), xw7, h) -> new_psPs(xw7, xw411, xw7, h)
The graph contains the following edges 3 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(24)
YES