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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
all p = pt and (map p);

and :: List MyBool  ->  MyBool;
and = foldr asAs MyTrue;

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

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = 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);

fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
fsEsMyBool x y = not (esEsMyBool x y);

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

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
notElemMyBool = pt all fsEsMyBool;

pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
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 ;

all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
all p = pt and (map p);

and :: List MyBool  ->  MyBool;
and = foldr asAs MyTrue;

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

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = 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);

fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
fsEsMyBool x y = not (esEsMyBool x y);

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

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
notElemMyBool = pt all fsEsMyBool;

pt :: (a  ->  b)  ->  (c  ->  a)  ->  c  ->  b;
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 ;

all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
all p = pt and (map p);

and :: List MyBool  ->  MyBool;
and = foldr asAs MyTrue;

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

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = MyTrue;

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);

fsEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
fsEsMyBool x y = not (esEsMyBool x y);

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

not :: MyBool  ->  MyBool;
not MyTrue = MyFalse;
not MyFalse = MyTrue;

notElemMyBool :: MyBool  ->  List MyBool  ->  MyBool;
notElemMyBool = pt all fsEsMyBool;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="notElemMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="notElemMyBool vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="notElemMyBool vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt all fsEsMyBool vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="all (fsEsMyBool vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="pt and (map (fsEsMyBool vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="and (map (fsEsMyBool vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr asAs MyTrue (map (fsEsMyBool vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];34[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 10[label="",style="solid", color="burlywood", weight=3];
35[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr asAs MyTrue (map (fsEsMyBool vx3) (Cons vx40 vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr asAs MyTrue (map (fsEsMyBool vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr asAs MyTrue (Cons (fsEsMyBool vx3 vx40) (map (fsEsMyBool vx3) vx41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr asAs MyTrue 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="asAs (fsEsMyBool vx3 vx40) (foldr asAs MyTrue (map (fsEsMyBool vx3) vx41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr asAs MyTrue (map (fsEsMyBool vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="asAs (fsEsMyBool vx3 vx40) vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vx41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsMyBool vx3 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];36[label="vx3/MyTrue",fontsize=10,color="white",style="solid",shape="box"];19 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 20[label="",style="solid", color="burlywood", weight=3];
37[label="vx3/MyFalse",fontsize=10,color="white",style="solid",shape="box"];19 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="asAs (not (esEsMyBool MyTrue vx40)) vx5",fontsize=16,color="burlywood",shape="box"];38[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];20 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 22[label="",style="solid", color="burlywood", weight=3];
39[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];20 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="asAs (not (esEsMyBool MyFalse vx40)) vx5",fontsize=16,color="burlywood",shape="box"];40[label="vx40/MyTrue",fontsize=10,color="white",style="solid",shape="box"];21 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 24[label="",style="solid", color="burlywood", weight=3];
41[label="vx40/MyFalse",fontsize=10,color="white",style="solid",shape="box"];21 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="asAs (not (esEsMyBool MyTrue MyTrue)) vx5",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="asAs (not (esEsMyBool MyTrue MyFalse)) vx5",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="asAs (not (esEsMyBool MyFalse MyTrue)) vx5",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="asAs (not (esEsMyBool MyFalse MyFalse)) vx5",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="asAs (not MyTrue) vx5",fontsize=16,color="black",shape="triangle"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="asAs (not MyFalse) vx5",fontsize=16,color="black",shape="triangle"];27 -> 31[label="",style="solid", color="black", weight=3];
28 -> 27[label="",style="dashed", color="red", weight=0];
28[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];29 -> 26[label="",style="dashed", color="red", weight=0];
29[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];30[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="asAs MyTrue vx5",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="MyFalse",fontsize=16,color="green",shape="box"];33[label="vx5",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