/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 MyBool = MyTrue  | MyFalse ;

data Tup0 = Tup0 ;

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
elemTup0 = pt any esEsTup0;

esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
esEsTup0 Tup0 Tup0 = MyTrue;

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);

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

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
pt f g x = f (g 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 MyBool = MyTrue  | MyFalse ;

data Tup0 = Tup0 ;

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
elemTup0 = pt any esEsTup0;

esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
esEsTup0 Tup0 Tup0 = MyTrue;

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);

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

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
pt f g x = f (g 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 MyBool = MyTrue  | MyFalse ;

data Tup0 = Tup0 ;

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemTup0 :: Tup0  ->  List Tup0  ->  MyBool;
elemTup0 = pt any esEsTup0;

esEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
esEsTup0 Tup0 Tup0 = MyTrue;

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);

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

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="elemTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="elemTup0 vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="elemTup0 vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt any esEsTup0 vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="any (esEsTup0 vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="pt or (map (esEsTup0 vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="or (map (esEsTup0 vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr pePe MyFalse (map (esEsTup0 vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];23[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 10[label="",style="solid", color="burlywood", weight=3];
24[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr pePe MyFalse (map (esEsTup0 vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr pePe MyFalse (map (esEsTup0 vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr pePe MyFalse (Cons (esEsTup0 vx3 vx40) (map (esEsTup0 vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 16[label="",style="dashed", color="red", weight=0];
14[label="pePe (esEsTup0 vx3 vx40) (foldr pePe MyFalse (map (esEsTup0 vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="MyFalse",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr pePe MyFalse (map (esEsTup0 vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="pePe (esEsTup0 vx3 vx40) vx5",fontsize=16,color="burlywood",shape="triangle"];25[label="vx3/Tup0",fontsize=10,color="white",style="solid",shape="box"];16 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 19[label="",style="solid", color="burlywood", weight=3];
18[label="vx41",fontsize=16,color="green",shape="box"];19[label="pePe (esEsTup0 Tup0 vx40) vx5",fontsize=16,color="burlywood",shape="box"];26[label="vx40/Tup0",fontsize=10,color="white",style="solid",shape="box"];19 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 20[label="",style="solid", color="burlywood", weight=3];
20[label="pePe (esEsTup0 Tup0 Tup0) vx5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="pePe MyTrue vx5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="MyTrue",fontsize=16,color="green",shape="box"];}

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

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

   new_foldr(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)

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(vx3, Cons(vx40, vx41)) -> new_foldr(vx3, vx41)
The graph contains the following edges 1 >= 1, 2 > 2


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

(8)
YES