/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

data Ordering = LT  | EQ  | GT ;

data Tup2 a b = Tup2 a b ;

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;

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);

lookup3 k Nil = Main.Nothing;
lookup3 vy vz = lookup2 vy vz;

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 Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

data Ordering = LT  | EQ  | GT ;

data Tup2 a b = Tup2 a b ;

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;

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);

lookup3 k Nil = Main.Nothing;
lookup3 vy vz = lookup2 vy vz;

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 Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

data Ordering = LT  | EQ  | GT ;

data Tup2 a b = Tup2 a b ;

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;

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsOrdering k x);

lookup3 k Nil = Main.Nothing;
lookup3 vy vz = lookup2 vy vz;

otherwise :: MyBool;
otherwise = MyTrue;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookup vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookup vx3 vx4",fontsize=16,color="burlywood",shape="triangle"];66[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 5[label="",style="solid", color="burlywood", weight=3];
67[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup vx3 (Cons vx40 vx41)",fontsize=16,color="burlywood",shape="box"];68[label="vx40/Tup2 vx400 vx401",fontsize=10,color="white",style="solid",shape="box"];5 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="lookup vx3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="lookup vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookup3 vx3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="lookup2 vx3 (Cons (Tup2 vx400 vx401) vx41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 vx3 vx400 vx401 vx41 (esEsOrdering vx3 vx400)",fontsize=16,color="burlywood",shape="box"];69[label="vx3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 12[label="",style="solid", color="burlywood", weight=3];
70[label="vx3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 13[label="",style="solid", color="burlywood", weight=3];
71[label="vx3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="lookup1 LT vx400 vx401 vx41 (esEsOrdering LT vx400)",fontsize=16,color="burlywood",shape="box"];72[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 15[label="",style="solid", color="burlywood", weight=3];
73[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 16[label="",style="solid", color="burlywood", weight=3];
74[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9];
74 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 EQ vx400 vx401 vx41 (esEsOrdering EQ vx400)",fontsize=16,color="burlywood",shape="box"];75[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 75[label="",style="solid", color="burlywood", weight=9];
75 -> 18[label="",style="solid", color="burlywood", weight=3];
76[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 19[label="",style="solid", color="burlywood", weight=3];
77[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 20[label="",style="solid", color="burlywood", weight=3];
14[label="lookup1 GT vx400 vx401 vx41 (esEsOrdering GT vx400)",fontsize=16,color="burlywood",shape="box"];78[label="vx400/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 21[label="",style="solid", color="burlywood", weight=3];
79[label="vx400/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 22[label="",style="solid", color="burlywood", weight=3];
80[label="vx400/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 23[label="",style="solid", color="burlywood", weight=3];
15[label="lookup1 LT LT vx401 vx41 (esEsOrdering LT LT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
16[label="lookup1 LT EQ vx401 vx41 (esEsOrdering LT EQ)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
17[label="lookup1 LT GT vx401 vx41 (esEsOrdering LT GT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="lookup1 EQ LT vx401 vx41 (esEsOrdering EQ LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="lookup1 EQ EQ vx401 vx41 (esEsOrdering EQ EQ)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="lookup1 EQ GT vx401 vx41 (esEsOrdering EQ GT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="lookup1 GT LT vx401 vx41 (esEsOrdering GT LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="lookup1 GT EQ vx401 vx41 (esEsOrdering GT EQ)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="lookup1 GT GT vx401 vx41 (esEsOrdering GT GT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="lookup1 LT LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="lookup1 LT EQ vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="lookup1 LT GT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="lookup1 EQ LT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="lookup1 EQ EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="lookup1 EQ GT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="lookup1 GT LT vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="lookup1 GT EQ vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="lookup1 GT GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="Just vx401",fontsize=16,color="green",shape="box"];34[label="lookup0 LT EQ vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
35[label="lookup0 LT GT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
36[label="lookup0 EQ LT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
37[label="Just vx401",fontsize=16,color="green",shape="box"];38[label="lookup0 EQ GT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
39[label="lookup0 GT LT vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
40[label="lookup0 GT EQ vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
41[label="Just vx401",fontsize=16,color="green",shape="box"];42[label="lookup0 LT EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3];
43[label="lookup0 LT GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3];
44[label="lookup0 EQ LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3];
45[label="lookup0 EQ GT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3];
46[label="lookup0 GT LT vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3];
47[label="lookup0 GT EQ vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3];
48 -> 4[label="",style="dashed", color="red", weight=0];
48[label="lookup LT vx41",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3];
48 -> 55[label="",style="dashed", color="magenta", weight=3];
49 -> 4[label="",style="dashed", color="red", weight=0];
49[label="lookup LT vx41",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3];
49 -> 57[label="",style="dashed", color="magenta", weight=3];
50 -> 4[label="",style="dashed", color="red", weight=0];
50[label="lookup EQ vx41",fontsize=16,color="magenta"];50 -> 58[label="",style="dashed", color="magenta", weight=3];
50 -> 59[label="",style="dashed", color="magenta", weight=3];
51 -> 4[label="",style="dashed", color="red", weight=0];
51[label="lookup EQ vx41",fontsize=16,color="magenta"];51 -> 60[label="",style="dashed", color="magenta", weight=3];
51 -> 61[label="",style="dashed", color="magenta", weight=3];
52 -> 4[label="",style="dashed", color="red", weight=0];
52[label="lookup GT vx41",fontsize=16,color="magenta"];52 -> 62[label="",style="dashed", color="magenta", weight=3];
52 -> 63[label="",style="dashed", color="magenta", weight=3];
53 -> 4[label="",style="dashed", color="red", weight=0];
53[label="lookup GT vx41",fontsize=16,color="magenta"];53 -> 64[label="",style="dashed", color="magenta", weight=3];
53 -> 65[label="",style="dashed", color="magenta", weight=3];
54[label="vx41",fontsize=16,color="green",shape="box"];55[label="LT",fontsize=16,color="green",shape="box"];56[label="vx41",fontsize=16,color="green",shape="box"];57[label="LT",fontsize=16,color="green",shape="box"];58[label="vx41",fontsize=16,color="green",shape="box"];59[label="EQ",fontsize=16,color="green",shape="box"];60[label="vx41",fontsize=16,color="green",shape="box"];61[label="EQ",fontsize=16,color="green",shape="box"];62[label="vx41",fontsize=16,color="green",shape="box"];63[label="GT",fontsize=16,color="green",shape="box"];64[label="vx41",fontsize=16,color="green",shape="box"];65[label="GT",fontsize=16,color="green",shape="box"];}

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

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

   new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
   new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)
   new_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
   new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)
   new_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
   new_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)
   new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(10) 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_lookup(LT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(LT, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(LT, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(LT, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
   new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) 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_lookup(GT, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(GT, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(GT, Cons(Tup2(EQ, vx401), vx41), h) -> new_lookup(GT, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
   new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(16) 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_lookup(EQ, Cons(Tup2(GT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(EQ, Cons(Tup2(LT, vx401), vx41), h) -> new_lookup(EQ, vx41, h)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(17)
YES