/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) 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) QDP
(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(14) 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 b => Int  ->  b a  ->  b ();
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 a => Int  ->  a b  ->  a ();
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 vy|n <= 0[];
take vz [] = [];
take n (x : xs) = x : take (n - 1) xs;
"
is transformed to
"take n vy = take3 n vy;
take vz [] = take1 vz [];
take n (x : xs) = take0 n (x : xs);
"
"take0 n (x : xs) = x : take (n - 1) xs;
"
"take1 vz [] = [];
take1 wx wy = take0 wx wy;
"
"take2 n vy True = [];
take2 n vy False = take1 n vy;
"
"take3 n vy = take2 n vy (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 a => Int  ->  a b  ->  a ();
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 b => Int  ->  b a  ->  b ();
replicateM_ n x = sequence_ (replicate n x);

}

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

(11) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
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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];
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12[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (primCmpInt xw3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];81[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 13[label="",style="solid", color="burlywood", weight=3];
82[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 14[label="",style="solid", color="burlywood", weight=3];
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83 -> 15[label="",style="solid", color="burlywood", weight=3];
84[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 16[label="",style="solid", color="burlywood", weight=3];
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85 -> 17[label="",style="solid", color="burlywood", weight=3];
86[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 86[label="",style="solid", color="burlywood", weight=9];
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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];
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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];
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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];
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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="primretIO ()",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3];
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 -> 42[label="",style="solid", color="black", weight=3];
41[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];42[label="foldr (>>) xw5 (take1 (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 (take0 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3];
44[label="foldr (>>) xw5 (xw4 : 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 foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3];
46[label="xw4 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4)))",fontsize=16,color="black",shape="box"];46 -> 47[label="",style="solid", color="black", weight=3];
47[label="primbindIO xw4 (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))))",fontsize=16,color="burlywood",shape="box"];87[label="xw4/IO xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 48[label="",style="solid", color="burlywood", weight=3];
88[label="xw4/AProVE_IO xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 49[label="",style="solid", color="burlywood", weight=3];
89[label="xw4/AProVE_Exception xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 50[label="",style="solid", color="burlywood", weight=3];
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90 -> 51[label="",style="solid", color="burlywood", weight=3];
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58[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)) (Pos (Succ xw300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3];
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91 -> 65[label="",style="solid", color="burlywood", weight=3];
92[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 92[label="",style="solid", color="burlywood", weight=9];
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67[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (Pos (Succ xw3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];67 -> 69[label="",style="solid", color="black", weight=3];
68[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];68 -> 70[label="",style="solid", color="black", weight=3];
69[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (primCmpNat (Succ xw3000) Zero == GT)))",fontsize=16,color="black",shape="box"];69 -> 71[label="",style="solid", color="black", weight=3];
70[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];70 -> 72[label="",style="solid", color="black", weight=3];
71[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];71 -> 73[label="",style="solid", color="black", weight=3];
72[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not False))",fontsize=16,color="black",shape="box"];72 -> 74[label="",style="solid", color="black", weight=3];
73[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not True))",fontsize=16,color="black",shape="box"];73 -> 75[label="",style="solid", color="black", weight=3];
74[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) True)",fontsize=16,color="black",shape="box"];74 -> 76[label="",style="solid", color="black", weight=3];
75[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) False)",fontsize=16,color="black",shape="box"];75 -> 77[label="",style="solid", color="black", weight=3];
76[label="foldr (>>) xw5 []",fontsize=16,color="black",shape="box"];76 -> 78[label="",style="solid", color="black", weight=3];
77 -> 39[label="",style="dashed", color="red", weight=0];
77[label="foldr (>>) xw5 (take1 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)))",fontsize=16,color="magenta"];77 -> 79[label="",style="dashed", color="magenta", weight=3];
77 -> 80[label="",style="dashed", color="magenta", weight=3];
78[label="xw5",fontsize=16,color="green",shape="box"];79[label="AProVE_IO xw40",fontsize=16,color="green",shape="box"];80[label="xw3000",fontsize=16,color="green",shape="box"];}

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(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(xw5, Succ(xw3000), AProVE_IO(xw40), h) -> new_foldr(xw5, xw3000, AProVE_IO(xw40), h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(13) 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), AProVE_IO(xw40), h) -> new_foldr(xw5, xw3000, AProVE_IO(xw40), h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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(14)
YES