/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) LR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) CR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) BR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) COR [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Prelude;
catMaybes :: [Maybe a]  ->  [a];
catMaybes ls = concatMap (\vv1 ->case vv1 of { 
Just x-> x : []; 
_-> []; 
 } ) ls;

}
module Main where {
import qualified Maybe;
import qualified Prelude;
}

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

(1) LR (EQUIVALENT)
Lambda Reductions:
The following Lambda expression
"\vv1->case vv1 of {
Just x  -> x : [];
_  -> []}
"
is transformed to
"catMaybes0 vv1 = case vv1 of {
Just x  -> x : [];
_  -> []}
;
"

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

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Prelude;
catMaybes :: [Maybe a]  ->  [a];
catMaybes ls = concatMap catMaybes0 ls;

catMaybes0 vv1 = case vv1 of { 
Just x-> x : []; 
_-> []; 
 } ;

}
module Main where {
import qualified Maybe;
import qualified Prelude;
}

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

(3) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case vv1 of {
Just x  -> x : [];
_  -> []}
"
is transformed to
"catMaybes00 (Just x) = x : [];
catMaybes00 _ = [];
"

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Prelude;
catMaybes :: [Maybe a]  ->  [a];
catMaybes ls = concatMap catMaybes0 ls;

catMaybes0 vv1 = catMaybes00 vv1;

catMaybes00 (Just x) = x : [];
catMaybes00 _ = [];

}
module Main where {
import qualified Maybe;
import qualified Prelude;
}

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

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

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Prelude;
catMaybes :: [Maybe a]  ->  [a];
catMaybes ls = concatMap catMaybes0 ls;

catMaybes0 vv1 = catMaybes00 vv1;

catMaybes00 (Just x) = x : [];
catMaybes00 vy = [];

}
module Main where {
import qualified Maybe;
import qualified Prelude;
}

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

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

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

(8)
Obligation:
mainModule Main 
module Maybe where {
import qualified Main;
import qualified Prelude;
catMaybes :: [Maybe a]  ->  [a];
catMaybes ls = concatMap catMaybes0 ls;

catMaybes0 vv1 = catMaybes00 vv1;

catMaybes00 (Just x) = x : [];
catMaybes00 vy = [];

}
module Main where {
import qualified Maybe;
import qualified Prelude;
}

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Maybe.catMaybes",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Maybe.catMaybes vz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="concatMap Maybe.catMaybes0 vz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="concat . map Maybe.catMaybes0",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="concat (map Maybe.catMaybes0 vz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr (++) [] (map Maybe.catMaybes0 vz3)",fontsize=16,color="burlywood",shape="triangle"];25[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 8[label="",style="solid", color="burlywood", weight=3];
26[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr (++) [] (map Maybe.catMaybes0 (vz30 : vz31))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldr (++) [] (map Maybe.catMaybes0 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr (++) [] (Maybe.catMaybes0 vz30 : map Maybe.catMaybes0 vz31)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12 -> 14[label="",style="dashed", color="red", weight=0];
12[label="(++) Maybe.catMaybes0 vz30 foldr (++) [] (map Maybe.catMaybes0 vz31)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="[]",fontsize=16,color="green",shape="box"];15 -> 7[label="",style="dashed", color="red", weight=0];
15[label="foldr (++) [] (map Maybe.catMaybes0 vz31)",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="(++) Maybe.catMaybes0 vz30 vz4",fontsize=16,color="black",shape="triangle"];14 -> 17[label="",style="solid", color="black", weight=3];
16[label="vz31",fontsize=16,color="green",shape="box"];17[label="(++) Maybe.catMaybes00 vz30 vz4",fontsize=16,color="burlywood",shape="box"];27[label="vz30/Nothing",fontsize=10,color="white",style="solid",shape="box"];17 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 18[label="",style="solid", color="burlywood", weight=3];
28[label="vz30/Just vz300",fontsize=10,color="white",style="solid",shape="box"];17 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 19[label="",style="solid", color="burlywood", weight=3];
18[label="(++) Maybe.catMaybes00 Nothing vz4",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="(++) Maybe.catMaybes00 (Just vz300) vz4",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="(++) [] vz4",fontsize=16,color="black",shape="triangle"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="(++) (vz300 : []) vz4",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="vz4",fontsize=16,color="green",shape="box"];23[label="vz300 : [] ++ vz4",fontsize=16,color="green",shape="box"];23 -> 24[label="",style="dashed", color="green", weight=3];
24 -> 20[label="",style="dashed", color="red", weight=0];
24[label="[] ++ vz4",fontsize=16,color="magenta"];}

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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(:(vz30, vz31), h) -> new_foldr(vz31, h)

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

(11) 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(:(vz30, vz31), h) -> new_foldr(vz31, h)
The graph contains the following edges 1 > 1, 2 >= 2


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

(12)
YES