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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

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;

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

max0 x y MyTrue = x;

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

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

maxMyBool :: MyBool  ->  MyBool  ->  MyBool;
maxMyBool x y = max2 x y;

maximumMyBool :: List MyBool  ->  MyBool;
maximumMyBool = foldl1 maxMyBool;

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 ;

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;

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

max0 x y MyTrue = x;

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

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

maxMyBool :: MyBool  ->  MyBool  ->  MyBool;
maxMyBool x y = max2 x y;

maximumMyBool :: List MyBool  ->  MyBool;
maximumMyBool = foldl1 maxMyBool;

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 ;

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;

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

max0 x y MyTrue = x;

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

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

maxMyBool :: MyBool  ->  MyBool  ->  MyBool;
maxMyBool x y = max2 x y;

maximumMyBool :: List MyBool  ->  MyBool;
maximumMyBool = foldl1 maxMyBool;

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="maximumMyBool",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="maximumMyBool vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 maxMyBool vx3",fontsize=16,color="burlywood",shape="box"];33[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 5[label="",style="solid", color="burlywood", weight=3];
34[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 maxMyBool (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 maxMyBool Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl maxMyBool vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];35[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 9[label="",style="solid", color="burlywood", weight=3];
36[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl maxMyBool vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl maxMyBool 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 maxMyBool (maxMyBool 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="maxMyBool 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 (ltEsMyBool vx30 vx310)",fontsize=16,color="burlywood",shape="box"];37[label="vx30/MyTrue",fontsize=10,color="white",style="solid",shape="box"];16 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 17[label="",style="solid", color="burlywood", weight=3];
38[label="vx30/MyFalse",fontsize=10,color="white",style="solid",shape="box"];16 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="max1 MyTrue vx310 (ltEsMyBool MyTrue vx310)",fontsize=16,color="burlywood",shape="box"];39[label="vx310/MyTrue",fontsize=10,color="white",style="solid",shape="box"];17 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 19[label="",style="solid", color="burlywood", weight=3];
40[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="max1 MyFalse vx310 (ltEsMyBool MyFalse vx310)",fontsize=16,color="burlywood",shape="box"];41[label="vx310/MyTrue",fontsize=10,color="white",style="solid",shape="box"];18 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 21[label="",style="solid", color="burlywood", weight=3];
42[label="vx310/MyFalse",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="max1 MyTrue MyTrue (ltEsMyBool MyTrue MyTrue)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="max1 MyTrue MyFalse (ltEsMyBool MyTrue MyFalse)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="max1 MyFalse MyTrue (ltEsMyBool MyFalse MyTrue)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="max1 MyFalse MyFalse (ltEsMyBool MyFalse MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="max1 MyTrue MyTrue MyTrue",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="max1 MyTrue MyFalse MyFalse",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="max1 MyFalse MyTrue MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="max1 MyFalse MyFalse MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="MyTrue",fontsize=16,color="green",shape="box"];28[label="max0 MyTrue MyFalse otherwise",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3];
29[label="MyTrue",fontsize=16,color="green",shape="box"];30[label="MyFalse",fontsize=16,color="green",shape="box"];31[label="max0 MyTrue MyFalse MyTrue",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
32[label="MyTrue",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(MyFalse, MyFalse) -> MyFalse
   new_max1(MyTrue, MyTrue) -> MyTrue
   new_max1(MyTrue, MyFalse) -> MyTrue
   new_max1(MyFalse, MyTrue) -> MyTrue

The set Q consists of the following terms:

   new_max1(MyTrue, MyTrue)
   new_max1(MyFalse, MyFalse)
   new_max1(MyTrue, MyFalse)
   new_max1(MyFalse, MyTrue)

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