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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

data Ordering = LT  | EQ  | GT ;

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;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = 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);

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering 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;

notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
notElemOrdering = pt all fsEsOrdering;

pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
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 Ordering = LT  | EQ  | GT ;

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;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = 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);

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering 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;

notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
notElemOrdering = pt all fsEsOrdering;

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

data Ordering = LT  | EQ  | GT ;

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;

esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
esEsOrdering LT LT = MyTrue;
esEsOrdering LT EQ = MyFalse;
esEsOrdering LT GT = MyFalse;
esEsOrdering EQ LT = MyFalse;
esEsOrdering EQ EQ = MyTrue;
esEsOrdering EQ GT = MyFalse;
esEsOrdering GT LT = MyFalse;
esEsOrdering GT EQ = MyFalse;
esEsOrdering GT GT = 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);

fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
fsEsOrdering x y = not (esEsOrdering 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;

notElemOrdering :: Ordering  ->  List Ordering  ->  MyBool;
notElemOrdering = pt all fsEsOrdering;

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

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="notElemOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="notElemOrdering vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="notElemOrdering vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt all fsEsOrdering vx3 vx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="all (fsEsOrdering vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="pt and (map (fsEsOrdering vx3)) vx4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="and (map (fsEsOrdering vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr asAs MyTrue (map (fsEsOrdering vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];45[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 10[label="",style="solid", color="burlywood", weight=3];
46[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr asAs MyTrue (map (fsEsOrdering 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 (fsEsOrdering vx3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr asAs MyTrue (Cons (fsEsOrdering vx3 vx40) (map (fsEsOrdering 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 (fsEsOrdering vx3 vx40) (foldr asAs MyTrue (map (fsEsOrdering 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 (fsEsOrdering vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="asAs (fsEsOrdering 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 (esEsOrdering vx3 vx40)) vx5",fontsize=16,color="burlywood",shape="box"];47[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 20[label="",style="solid", color="burlywood", weight=3];
48[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 21[label="",style="solid", color="burlywood", weight=3];
49[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 22[label="",style="solid", color="burlywood", weight=3];
20[label="asAs (not (esEsOrdering LT vx40)) vx5",fontsize=16,color="burlywood",shape="box"];50[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];20 -> 50[label="",style="solid", color="burlywood", weight=9];
50 -> 23[label="",style="solid", color="burlywood", weight=3];
51[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];20 -> 51[label="",style="solid", color="burlywood", weight=9];
51 -> 24[label="",style="solid", color="burlywood", weight=3];
52[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];20 -> 52[label="",style="solid", color="burlywood", weight=9];
52 -> 25[label="",style="solid", color="burlywood", weight=3];
21[label="asAs (not (esEsOrdering EQ vx40)) vx5",fontsize=16,color="burlywood",shape="box"];53[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];21 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 26[label="",style="solid", color="burlywood", weight=3];
54[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];21 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 27[label="",style="solid", color="burlywood", weight=3];
55[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];21 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 28[label="",style="solid", color="burlywood", weight=3];
22[label="asAs (not (esEsOrdering GT vx40)) vx5",fontsize=16,color="burlywood",shape="box"];56[label="vx40/LT",fontsize=10,color="white",style="solid",shape="box"];22 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 29[label="",style="solid", color="burlywood", weight=3];
57[label="vx40/EQ",fontsize=10,color="white",style="solid",shape="box"];22 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 30[label="",style="solid", color="burlywood", weight=3];
58[label="vx40/GT",fontsize=10,color="white",style="solid",shape="box"];22 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 31[label="",style="solid", color="burlywood", weight=3];
23[label="asAs (not (esEsOrdering LT LT)) vx5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="asAs (not (esEsOrdering LT EQ)) vx5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="asAs (not (esEsOrdering LT GT)) vx5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="asAs (not (esEsOrdering EQ LT)) vx5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="asAs (not (esEsOrdering EQ EQ)) vx5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="asAs (not (esEsOrdering EQ GT)) vx5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="asAs (not (esEsOrdering GT LT)) vx5",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="asAs (not (esEsOrdering GT EQ)) vx5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="asAs (not (esEsOrdering GT GT)) vx5",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="asAs (not MyTrue) vx5",fontsize=16,color="black",shape="triangle"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="asAs (not MyFalse) vx5",fontsize=16,color="black",shape="triangle"];33 -> 42[label="",style="solid", color="black", weight=3];
34 -> 33[label="",style="dashed", color="red", weight=0];
34[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];35 -> 33[label="",style="dashed", color="red", weight=0];
35[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];36 -> 32[label="",style="dashed", color="red", weight=0];
36[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];37 -> 33[label="",style="dashed", color="red", weight=0];
37[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];38 -> 33[label="",style="dashed", color="red", weight=0];
38[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];39 -> 33[label="",style="dashed", color="red", weight=0];
39[label="asAs (not MyFalse) vx5",fontsize=16,color="magenta"];40 -> 32[label="",style="dashed", color="red", weight=0];
40[label="asAs (not MyTrue) vx5",fontsize=16,color="magenta"];41[label="asAs MyFalse vx5",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="asAs MyTrue vx5",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="MyFalse",fontsize=16,color="green",shape="box"];44[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