/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 4 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 [a];
replicateM n x = sequence (replicate n x);

}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\xs->return (x : xs)"
is transformed to
"sequence0 x xs = return (x : xs);
"
The following Lambda expression
"\x->sequence cs >>= sequence0 x"
is transformed to
"sequence1 cs x = sequence cs >>= sequence0 x;
"

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

(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 [a];
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 [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 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 ww wx = take0 ww wx;
"
"take2 n vx True = [];
take2 n vx False = take1 n vx;
"
"take3 n vx = take2 n vx (n <= 0);
take3 wy wz = take1 wy wz;
"

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

(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 [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 xu = xu : repeatXs xu;
"

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

(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 b => Int  ->  b 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 [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 xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Monad.replicateM xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="sequence (replicate xv3 xv4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (take xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="sequence (take3 xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="sequence (take2 xv3 (repeat xv4) (xv3 <= Pos Zero))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="sequence (take2 xv3 (repeat xv4) (compare xv3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
10[label="sequence (take2 xv3 (repeat xv4) (not (compare xv3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="sequence (take2 xv3 (repeat xv4) (not (primCmpInt xv3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];118[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 12[label="",style="solid", color="burlywood", weight=3];
119[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="sequence (take2 (Pos xv30) (repeat xv4) (not (primCmpInt (Pos xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];120[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];12 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 14[label="",style="solid", color="burlywood", weight=3];
121[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 15[label="",style="solid", color="burlywood", weight=3];
13[label="sequence (take2 (Neg xv30) (repeat xv4) (not (primCmpInt (Neg xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];122[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];13 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 16[label="",style="solid", color="burlywood", weight=3];
123[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 17[label="",style="solid", color="burlywood", weight=3];
14[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpInt (Pos (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="sequence (take2 (Pos Zero) (repeat xv4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (primCmpInt (Neg (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="sequence (take2 (Neg Zero) (repeat xv4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpNat (Succ xv300) Zero == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="sequence (take2 (Pos Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="sequence (take2 (Neg Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="sequence (take2 (Pos Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="sequence (take2 (Neg Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not True))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="sequence (take2 (Pos Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29[label="sequence (take2 (Neg Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
30[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
31[label="sequence []",fontsize=16,color="black",shape="triangle"];31 -> 35[label="",style="solid", color="black", weight=3];
32 -> 31[label="",style="dashed", color="red", weight=0];
32[label="sequence []",fontsize=16,color="magenta"];33 -> 31[label="",style="dashed", color="red", weight=0];
33[label="sequence []",fontsize=16,color="magenta"];34[label="sequence (take1 (Pos (Succ xv300)) (repeat xv4))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="return []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="sequence (take1 (Pos (Succ xv300)) (repeatXs xv4))",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="Just []",fontsize=16,color="green",shape="box"];38[label="sequence (take1 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
39[label="sequence (take0 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
40[label="sequence (xv4 : take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
41[label="xv4 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="burlywood",shape="box"];124[label="xv4/Nothing",fontsize=10,color="white",style="solid",shape="box"];41 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 42[label="",style="solid", color="burlywood", weight=3];
125[label="xv4/Just xv40",fontsize=10,color="white",style="solid",shape="box"];41 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 43[label="",style="solid", color="burlywood", weight=3];
42[label="Nothing >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs Nothing))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="Just xv40 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)))",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="Nothing",fontsize=16,color="green",shape="box"];45[label="sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40))) xv40",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3];
46 -> 68[label="",style="dashed", color="red", weight=0];
46[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40))) >>= sequence0 xv40",fontsize=16,color="magenta"];46 -> 69[label="",style="dashed", color="magenta", weight=3];
69[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)))",fontsize=16,color="black",shape="box"];69 -> 89[label="",style="solid", color="black", weight=3];
68[label="xv5 >>= sequence0 xv40",fontsize=16,color="burlywood",shape="triangle"];126[label="xv5/Nothing",fontsize=10,color="white",style="solid",shape="box"];68 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 90[label="",style="solid", color="burlywood", weight=3];
127[label="xv5/Just xv50",fontsize=10,color="white",style="solid",shape="box"];68 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 91[label="",style="solid", color="burlywood", weight=3];
89[label="sequence (take3 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)))",fontsize=16,color="black",shape="box"];89 -> 92[label="",style="solid", color="black", weight=3];
90[label="Nothing >>= sequence0 xv40",fontsize=16,color="black",shape="box"];90 -> 93[label="",style="solid", color="black", weight=3];
91[label="Just xv50 >>= sequence0 xv40",fontsize=16,color="black",shape="box"];91 -> 94[label="",style="solid", color="black", weight=3];
92[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)) (Pos (Succ xv300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];92 -> 95[label="",style="solid", color="black", weight=3];
93[label="Nothing",fontsize=16,color="green",shape="box"];94[label="sequence0 xv40 xv50",fontsize=16,color="black",shape="box"];94 -> 96[label="",style="solid", color="black", weight=3];
95[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)) (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];95 -> 97[label="",style="solid", color="black", weight=3];
96[label="return (xv40 : xv50)",fontsize=16,color="black",shape="box"];96 -> 98[label="",style="solid", color="black", weight=3];
97[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)) (not (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];97 -> 99[label="",style="solid", color="black", weight=3];
98[label="Just (xv40 : xv50)",fontsize=16,color="green",shape="box"];99[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (Just xv40)) (not (primCmpInt (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];99 -> 100[label="",style="solid", color="black", weight=3];
100[label="sequence (take2 (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (repeatXs (Just xv40)) (not (primCmpInt (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];100 -> 101[label="",style="solid", color="black", weight=3];
101[label="sequence (take2 (primMinusNat (Succ xv300) (Succ Zero)) (repeatXs (Just xv40)) (not (primCmpInt (primMinusNat (Succ xv300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];101 -> 102[label="",style="solid", color="black", weight=3];
102[label="sequence (take2 (primMinusNat xv300 Zero) (repeatXs (Just xv40)) (not (primCmpInt (primMinusNat xv300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];128[label="xv300/Succ xv3000",fontsize=10,color="white",style="solid",shape="box"];102 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 103[label="",style="solid", color="burlywood", weight=3];
129[label="xv300/Zero",fontsize=10,color="white",style="solid",shape="box"];102 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 104[label="",style="solid", color="burlywood", weight=3];
103[label="sequence (take2 (primMinusNat (Succ xv3000) Zero) (repeatXs (Just xv40)) (not (primCmpInt (primMinusNat (Succ xv3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];103 -> 105[label="",style="solid", color="black", weight=3];
104[label="sequence (take2 (primMinusNat Zero Zero) (repeatXs (Just xv40)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];104 -> 106[label="",style="solid", color="black", weight=3];
105[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (Just xv40)) (not (primCmpInt (Pos (Succ xv3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];105 -> 107[label="",style="solid", color="black", weight=3];
106[label="sequence (take2 (Pos Zero) (repeatXs (Just xv40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];106 -> 108[label="",style="solid", color="black", weight=3];
107[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (Just xv40)) (not (primCmpNat (Succ xv3000) Zero == GT)))",fontsize=16,color="black",shape="box"];107 -> 109[label="",style="solid", color="black", weight=3];
108[label="sequence (take2 (Pos Zero) (repeatXs (Just xv40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];108 -> 110[label="",style="solid", color="black", weight=3];
109[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (Just xv40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];109 -> 111[label="",style="solid", color="black", weight=3];
110[label="sequence (take2 (Pos Zero) (repeatXs (Just xv40)) (not False))",fontsize=16,color="black",shape="box"];110 -> 112[label="",style="solid", color="black", weight=3];
111[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (Just xv40)) (not True))",fontsize=16,color="black",shape="box"];111 -> 113[label="",style="solid", color="black", weight=3];
112[label="sequence (take2 (Pos Zero) (repeatXs (Just xv40)) True)",fontsize=16,color="black",shape="box"];112 -> 114[label="",style="solid", color="black", weight=3];
113[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (Just xv40)) False)",fontsize=16,color="black",shape="box"];113 -> 115[label="",style="solid", color="black", weight=3];
114 -> 31[label="",style="dashed", color="red", weight=0];
114[label="sequence []",fontsize=16,color="magenta"];115 -> 36[label="",style="dashed", color="red", weight=0];
115[label="sequence (take1 (Pos (Succ xv3000)) (repeatXs (Just xv40)))",fontsize=16,color="magenta"];115 -> 116[label="",style="dashed", color="magenta", weight=3];
115 -> 117[label="",style="dashed", color="magenta", weight=3];
116[label="xv3000",fontsize=16,color="green",shape="box"];117[label="Just xv40",fontsize=16,color="green",shape="box"];}

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

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

   new_sequence(Succ(xv3000), Just(xv40), h) -> new_sequence(xv3000, Just(xv40), h)

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

(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_sequence(Succ(xv3000), Just(xv40), h) -> new_sequence(xv3000, Just(xv40), h)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


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

(14)
YES