/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data Main.Maybe a = Nothing  | Just a ;

gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
gtGtEsMaybe (Main.Just x) k = k x;
gtGtEsMaybe Main.Nothing k = Main.Nothing;

returnMaybe :: a  ->  Main.Maybe a;
returnMaybe = Main.Just;

sequence Nil = returnMaybe Nil;
sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);

sequence0 x xs = returnMaybe (Cons x xs);

sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);

}

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data Main.Maybe a = Nothing  | Just a ;

gtGtEsMaybe :: Main.Maybe b  ->  (b  ->  Main.Maybe a)  ->  Main.Maybe a;
gtGtEsMaybe (Main.Just x) k = k x;
gtGtEsMaybe Main.Nothing k = Main.Nothing;

returnMaybe :: a  ->  Main.Maybe a;
returnMaybe = Main.Just;

sequence Nil = returnMaybe Nil;
sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);

sequence0 x xs = returnMaybe (Cons x xs);

sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);

}

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(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data Main.Maybe a = Nothing  | Just a ;

gtGtEsMaybe :: Main.Maybe a  ->  (a  ->  Main.Maybe b)  ->  Main.Maybe b;
gtGtEsMaybe (Main.Just x) k = k x;
gtGtEsMaybe Main.Nothing k = Main.Nothing;

returnMaybe :: a  ->  Main.Maybe a;
returnMaybe = Main.Just;

sequence Nil = returnMaybe Nil;
sequence (Cons c cs) = gtGtEsMaybe c (sequence1 cs);

sequence0 x xs = returnMaybe (Cons x xs);

sequence1 cs x = gtGtEsMaybe (sequence cs) (sequence0 x);

}

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(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sequence",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sequence vx3",fontsize=16,color="burlywood",shape="triangle"];23[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 4[label="",style="solid", color="burlywood", weight=3];
24[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];3 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="sequence (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="sequence Nil",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="gtGtEsMaybe vx30 (sequence1 vx31)",fontsize=16,color="burlywood",shape="box"];25[label="vx30/Nothing",fontsize=10,color="white",style="solid",shape="box"];6 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 8[label="",style="solid", color="burlywood", weight=3];
26[label="vx30/Just vx300",fontsize=10,color="white",style="solid",shape="box"];6 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="returnMaybe Nil",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="gtGtEsMaybe Nothing (sequence1 vx31)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="gtGtEsMaybe (Just vx300) (sequence1 vx31)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="Just Nil",fontsize=16,color="green",shape="box"];11[label="Nothing",fontsize=16,color="green",shape="box"];12[label="sequence1 vx31 vx300",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13 -> 14[label="",style="dashed", color="red", weight=0];
13[label="gtGtEsMaybe (sequence vx31) (sequence0 vx300)",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
15 -> 3[label="",style="dashed", color="red", weight=0];
15[label="sequence vx31",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="gtGtEsMaybe vx4 (sequence0 vx300)",fontsize=16,color="burlywood",shape="triangle"];27[label="vx4/Nothing",fontsize=10,color="white",style="solid",shape="box"];14 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 17[label="",style="solid", color="burlywood", weight=3];
28[label="vx4/Just vx40",fontsize=10,color="white",style="solid",shape="box"];14 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 18[label="",style="solid", color="burlywood", weight=3];
16[label="vx31",fontsize=16,color="green",shape="box"];17[label="gtGtEsMaybe Nothing (sequence0 vx300)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="gtGtEsMaybe (Just vx40) (sequence0 vx300)",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="sequence0 vx300 vx40",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="returnMaybe (Cons vx300 vx40)",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="Just (Cons vx300 vx40)",fontsize=16,color="green",shape="box"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_sequence(Cons(Main.Just(vx300), vx31), h) -> new_sequence(vx31, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(7) 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(Cons(Main.Just(vx300), vx31), h) -> new_sequence(vx31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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(8)
YES