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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

data Tup0 = Tup0 ;

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

gtGt0 q vv = q;

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

gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);

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_Maybe (map f);

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

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

sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);

}

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

(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 ;

data Tup0 = Tup0 ;

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

gtGt0 q vv = q;

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

gtGtMaybe :: Main.Maybe b  ->  Main.Maybe a  ->  Main.Maybe a;
gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);

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_Maybe (map f);

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

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

sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);

}

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

(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 ;

data Tup0 = Tup0 ;

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

gtGt0 q vv = q;

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

gtGtMaybe :: Main.Maybe a  ->  Main.Maybe b  ->  Main.Maybe b;
gtGtMaybe p q = gtGtEsMaybe p (gtGt0 q);

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_Maybe (map f);

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

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

sequence_Maybe :: List (Main.Maybe a)  ->  Main.Maybe Tup0;
sequence_Maybe = foldr gtGtMaybe (returnMaybe Tup0);

}

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

(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_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt sequence_Maybe (map vy3) vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="sequence_Maybe (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];28[label="vy4/Cons vy40 vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 8[label="",style="solid", color="burlywood", weight=3];
29[label="vy4/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 9[label="",style="solid", color="burlywood", weight=3];
8[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 (Cons vy40 vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 Nil)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldr gtGtMaybe (returnMaybe Tup0) (Cons (vy3 vy40) (map vy3 vy41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr gtGtMaybe (returnMaybe Tup0) Nil",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="gtGtMaybe (vy3 vy40) (foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy41))",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3];
13[label="returnMaybe Tup0",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
15 -> 7[label="",style="dashed", color="red", weight=0];
15[label="foldr gtGtMaybe (returnMaybe Tup0) (map vy3 vy41)",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
14[label="gtGtMaybe (vy3 vy40) vy5",fontsize=16,color="black",shape="triangle"];14 -> 18[label="",style="solid", color="black", weight=3];
16[label="Just Tup0",fontsize=16,color="green",shape="box"];17[label="vy41",fontsize=16,color="green",shape="box"];18 -> 19[label="",style="dashed", color="red", weight=0];
18[label="gtGtEsMaybe (vy3 vy40) (gtGt0 vy5)",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
20[label="vy3 vy40",fontsize=16,color="green",shape="box"];20 -> 24[label="",style="dashed", color="green", weight=3];
19[label="gtGtEsMaybe vy6 (gtGt0 vy5)",fontsize=16,color="burlywood",shape="triangle"];30[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 22[label="",style="solid", color="burlywood", weight=3];
31[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];19 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 23[label="",style="solid", color="burlywood", weight=3];
24[label="vy40",fontsize=16,color="green",shape="box"];22[label="gtGtEsMaybe Nothing (gtGt0 vy5)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="gtGtEsMaybe (Just vy60) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
25[label="Nothing",fontsize=16,color="green",shape="box"];26[label="gtGt0 vy5 vy60",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3];
27[label="vy5",fontsize=16,color="green",shape="box"];}

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

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

   new_foldr(vy3, Cons(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, 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_foldr(vy3, Cons(vy40, vy41), h, ba) -> new_foldr(vy3, vy41, h, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4


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

(8)
YES