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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

data Ordering = LT  | EQ  | GT ;

data Tup0 = Tup0 ;

compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
compareTup0 Tup0 Tup0 = EQ;

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;

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

foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
foldl1 f (Cons x xs) = foldl f x xs;

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

ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;

max0 x y MyTrue = x;

max1 x y MyTrue = y;
max1 x y MyFalse = max0 x y otherwise;

max2 x y = max1 x y (ltEsTup0 x y);

maxTup0 :: Tup0  ->  Tup0  ->  Tup0;
maxTup0 x y = max2 x y;

maximumTup0 :: List Tup0  ->  Tup0;
maximumTup0 = foldl1 maxTup0;

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

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(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 ;

data Tup0 = Tup0 ;

compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
compareTup0 Tup0 Tup0 = EQ;

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;

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

foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
foldl1 f (Cons x xs) = foldl f x xs;

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

ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;

max0 x y MyTrue = x;

max1 x y MyTrue = y;
max1 x y MyFalse = max0 x y otherwise;

max2 x y = max1 x y (ltEsTup0 x y);

maxTup0 :: Tup0  ->  Tup0  ->  Tup0;
maxTup0 x y = max2 x y;

maximumTup0 :: List Tup0  ->  Tup0;
maximumTup0 = foldl1 maxTup0;

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

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(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 ;

data Tup0 = Tup0 ;

compareTup0 :: Tup0  ->  Tup0  ->  Ordering;
compareTup0 Tup0 Tup0 = EQ;

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;

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

foldl1 :: (a  ->  a  ->  a)  ->  List a  ->  a;
foldl1 f (Cons x xs) = foldl f x xs;

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

ltEsTup0 :: Tup0  ->  Tup0  ->  MyBool;
ltEsTup0 x y = fsEsOrdering (compareTup0 x y) GT;

max0 x y MyTrue = x;

max1 x y MyTrue = y;
max1 x y MyFalse = max0 x y otherwise;

max2 x y = max1 x y (ltEsTup0 x y);

maxTup0 :: Tup0  ->  Tup0  ->  Tup0;
maxTup0 x y = max2 x y;

maximumTup0 :: List Tup0  ->  Tup0;
maximumTup0 = foldl1 maxTup0;

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

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="maximumTup0",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="maximumTup0 vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 maxTup0 vx3",fontsize=16,color="burlywood",shape="box"];25[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 5[label="",style="solid", color="burlywood", weight=3];
26[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 maxTup0 (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 maxTup0 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl maxTup0 vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];27[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 9[label="",style="solid", color="burlywood", weight=3];
28[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl maxTup0 vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl maxTup0 vx30 Nil",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11 -> 7[label="",style="dashed", color="red", weight=0];
11[label="foldl maxTup0 (maxTup0 vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
11 -> 14[label="",style="dashed", color="magenta", weight=3];
12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="maxTup0 vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="max2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="max1 vx30 vx310 (ltEsTup0 vx30 vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
17[label="max1 vx30 vx310 (fsEsOrdering (compareTup0 vx30 vx310) GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
18[label="max1 vx30 vx310 (not (esEsOrdering (compareTup0 vx30 vx310) GT))",fontsize=16,color="burlywood",shape="box"];29[label="vx30/Tup0",fontsize=10,color="white",style="solid",shape="box"];18 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 19[label="",style="solid", color="burlywood", weight=3];
19[label="max1 Tup0 vx310 (not (esEsOrdering (compareTup0 Tup0 vx310) GT))",fontsize=16,color="burlywood",shape="box"];30[label="vx310/Tup0",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 20[label="",style="solid", color="burlywood", weight=3];
20[label="max1 Tup0 Tup0 (not (esEsOrdering (compareTup0 Tup0 Tup0) GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
21[label="max1 Tup0 Tup0 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
22[label="max1 Tup0 Tup0 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
23[label="max1 Tup0 Tup0 MyTrue",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
24[label="Tup0",fontsize=16,color="green",shape="box"];}

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

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

   new_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311)

The TRS R consists of the following rules:

   new_max1(Tup0, Tup0) -> Tup0

The set Q consists of the following terms:

   new_max1(Tup0, Tup0)

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_foldl(vx30, Cons(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311)
The graph contains the following edges 2 > 2


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

(8)
YES