/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

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

map :: (b  ->  a)  ->  List b  ->  List a;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

mapM f = pt sequence (map f);

pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
pt f g x = f (g x);

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);

}

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

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

map :: (b  ->  a)  ->  List b  ->  List a;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

mapM f = pt sequence (map f);

pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
pt f g x = f (g x);

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);

}

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

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

map :: (a  ->  b)  ->  List a  ->  List b;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

mapM f = pt sequence (map f);

pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
pt f g x = f (g x);

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);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="mapM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="mapM vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt sequence (map vx3) vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];32[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];6 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 7[label="",style="solid", color="burlywood", weight=3];
33[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];6 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="sequence (map vx3 (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="sequence (map vx3 Nil)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="sequence (Cons (vx3 vx40) (map vx3 vx41))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="sequence Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 13[label="",style="dashed", color="red", weight=0];
11[label="gtGtEsMaybe (vx3 vx40) (sequence1 (map vx3 vx41))",fontsize=16,color="magenta"];11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="returnMaybe Nil",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
14[label="vx3 vx40",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
13[label="gtGtEsMaybe vx5 (sequence1 (map vx3 vx41))",fontsize=16,color="burlywood",shape="triangle"];34[label="vx5/Nothing",fontsize=10,color="white",style="solid",shape="box"];13 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 17[label="",style="solid", color="burlywood", weight=3];
35[label="vx5/Just vx50",fontsize=10,color="white",style="solid",shape="box"];13 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="Just Nil",fontsize=16,color="green",shape="box"];19[label="vx40",fontsize=16,color="green",shape="box"];17[label="gtGtEsMaybe Nothing (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3];
18[label="gtGtEsMaybe (Just vx50) (sequence1 (map vx3 vx41))",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
20[label="Nothing",fontsize=16,color="green",shape="box"];21[label="sequence1 (map vx3 vx41) vx50",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22 -> 23[label="",style="dashed", color="red", weight=0];
22[label="gtGtEsMaybe (sequence (map vx3 vx41)) (sequence0 vx50)",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3];
24 -> 6[label="",style="dashed", color="red", weight=0];
24[label="sequence (map vx3 vx41)",fontsize=16,color="magenta"];24 -> 25[label="",style="dashed", color="magenta", weight=3];
23[label="gtGtEsMaybe vx6 (sequence0 vx50)",fontsize=16,color="burlywood",shape="triangle"];36[label="vx6/Nothing",fontsize=10,color="white",style="solid",shape="box"];23 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 26[label="",style="solid", color="burlywood", weight=3];
37[label="vx6/Just vx60",fontsize=10,color="white",style="solid",shape="box"];23 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="vx41",fontsize=16,color="green",shape="box"];26[label="gtGtEsMaybe Nothing (sequence0 vx50)",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="gtGtEsMaybe (Just vx60) (sequence0 vx50)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="Nothing",fontsize=16,color="green",shape="box"];29[label="sequence0 vx50 vx60",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3];
30[label="returnMaybe (Cons vx50 vx60)",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3];
31[label="Just (Cons vx50 vx60)",fontsize=16,color="green",shape="box"];}

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

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

   new_sequence(vx3, Cons(vx40, vx41), h, ba) -> new_gtGtEsMaybe(vx3, vx41, h, ba)
   new_gtGtEsMaybe(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)

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

(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_gtGtEsMaybe(vx3, vx41, h, ba) -> new_sequence(vx3, vx41, h, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4


*new_sequence(vx3, Cons(vx40, vx41), h, ba) -> new_gtGtEsMaybe(vx3, vx41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(8)
YES