/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 ;

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;

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

min0 x y MyTrue = y;

min1 x y MyTrue = x;
min1 x y MyFalse = min0 x y otherwise;

min2 x y = min1 x y (ltEsOrdering x y);

minOrdering :: Ordering  ->  Ordering  ->  Ordering;
minOrdering x y = min2 x y;

minimumOrdering :: List Ordering  ->  Ordering;
minimumOrdering = foldl1 minOrdering;

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 ;

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;

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

min0 x y MyTrue = y;

min1 x y MyTrue = x;
min1 x y MyFalse = min0 x y otherwise;

min2 x y = min1 x y (ltEsOrdering x y);

minOrdering :: Ordering  ->  Ordering  ->  Ordering;
minOrdering x y = min2 x y;

minimumOrdering :: List Ordering  ->  Ordering;
minimumOrdering = foldl1 minOrdering;

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 ;

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;

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

min0 x y MyTrue = y;

min1 x y MyTrue = x;
min1 x y MyFalse = min0 x y otherwise;

min2 x y = min1 x y (ltEsOrdering x y);

minOrdering :: Ordering  ->  Ordering  ->  Ordering;
minOrdering x y = min2 x y;

minimumOrdering :: List Ordering  ->  Ordering;
minimumOrdering = foldl1 minOrdering;

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="minimumOrdering",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="minimumOrdering vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl1 minOrdering vx3",fontsize=16,color="burlywood",shape="box"];53[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 53[label="",style="solid", color="burlywood", weight=9];
53 -> 5[label="",style="solid", color="burlywood", weight=3];
54[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
54 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl1 minOrdering (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl1 minOrdering Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl minOrdering vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];55[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 55[label="",style="solid", color="burlywood", weight=9];
55 -> 9[label="",style="solid", color="burlywood", weight=3];
56[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];7 -> 56[label="",style="solid", color="burlywood", weight=9];
56 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl minOrdering vx30 (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="foldl minOrdering 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 minOrdering (minOrdering 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="minOrdering vx30 vx310",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="vx311",fontsize=16,color="green",shape="box"];15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="min1 vx30 vx310 (ltEsOrdering vx30 vx310)",fontsize=16,color="burlywood",shape="box"];57[label="vx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 57[label="",style="solid", color="burlywood", weight=9];
57 -> 17[label="",style="solid", color="burlywood", weight=3];
58[label="vx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 18[label="",style="solid", color="burlywood", weight=3];
59[label="vx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 19[label="",style="solid", color="burlywood", weight=3];
17[label="min1 LT vx310 (ltEsOrdering LT vx310)",fontsize=16,color="burlywood",shape="box"];60[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 20[label="",style="solid", color="burlywood", weight=3];
61[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 21[label="",style="solid", color="burlywood", weight=3];
62[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 22[label="",style="solid", color="burlywood", weight=3];
18[label="min1 EQ vx310 (ltEsOrdering EQ vx310)",fontsize=16,color="burlywood",shape="box"];63[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 23[label="",style="solid", color="burlywood", weight=3];
64[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 24[label="",style="solid", color="burlywood", weight=3];
65[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 25[label="",style="solid", color="burlywood", weight=3];
19[label="min1 GT vx310 (ltEsOrdering GT vx310)",fontsize=16,color="burlywood",shape="box"];66[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 26[label="",style="solid", color="burlywood", weight=3];
67[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 27[label="",style="solid", color="burlywood", weight=3];
68[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 28[label="",style="solid", color="burlywood", weight=3];
20[label="min1 LT LT (ltEsOrdering LT LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="min1 LT EQ (ltEsOrdering LT EQ)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="min1 LT GT (ltEsOrdering LT GT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="min1 EQ LT (ltEsOrdering EQ LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="min1 EQ EQ (ltEsOrdering EQ EQ)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="min1 EQ GT (ltEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="min1 GT LT (ltEsOrdering GT LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="min1 GT EQ (ltEsOrdering GT EQ)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="min1 GT GT (ltEsOrdering GT GT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="min1 LT LT MyTrue",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="min1 LT EQ MyTrue",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="min1 LT GT MyTrue",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="min1 EQ LT MyFalse",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="min1 EQ EQ MyTrue",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34[label="min1 EQ GT MyTrue",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
35[label="min1 GT LT MyFalse",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
36[label="min1 GT EQ MyFalse",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
37[label="min1 GT GT MyTrue",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
38[label="LT",fontsize=16,color="green",shape="box"];39[label="LT",fontsize=16,color="green",shape="box"];40[label="LT",fontsize=16,color="green",shape="box"];41[label="min0 EQ LT otherwise",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
42[label="EQ",fontsize=16,color="green",shape="box"];43[label="EQ",fontsize=16,color="green",shape="box"];44[label="min0 GT LT otherwise",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="min0 GT EQ otherwise",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="GT",fontsize=16,color="green",shape="box"];47[label="min0 EQ LT MyTrue",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
48[label="min0 GT LT MyTrue",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
49[label="min0 GT EQ MyTrue",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
50[label="LT",fontsize=16,color="green",shape="box"];51[label="LT",fontsize=16,color="green",shape="box"];52[label="EQ",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_min1(vx30, vx310), vx311)

The TRS R consists of the following rules:

   new_min1(LT, GT) -> LT
   new_min1(GT, LT) -> LT
   new_min1(GT, GT) -> GT
   new_min1(EQ, EQ) -> EQ
   new_min1(LT, EQ) -> LT
   new_min1(EQ, LT) -> LT
   new_min1(EQ, GT) -> EQ
   new_min1(GT, EQ) -> EQ
   new_min1(LT, LT) -> LT

The set Q consists of the following terms:

   new_min1(LT, EQ)
   new_min1(EQ, LT)
   new_min1(EQ, GT)
   new_min1(GT, EQ)
   new_min1(LT, GT)
   new_min1(GT, LT)
   new_min1(EQ, EQ)
   new_min1(GT, GT)
   new_min1(LT, LT)

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_min1(vx30, vx310), vx311)
The graph contains the following edges 2 > 2


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

(8)
YES