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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

(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 Tup2 b a = Tup2 b a ;

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = 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 (esEsMyBool 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 Tup2 a b = Tup2 a b ;

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = 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 (esEsMyBool 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 Tup2 a b = Tup2 a b ;

esEsMyBool :: MyBool  ->  MyBool  ->  MyBool;
esEsMyBool MyFalse MyFalse = MyTrue;
esEsMyBool MyFalse MyTrue = MyFalse;
esEsMyBool MyTrue MyFalse = MyFalse;
esEsMyBool MyTrue MyTrue = 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 (esEsMyBool 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"];34[label="vx4/Cons vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 5[label="",style="solid", color="burlywood", weight=3];
35[label="vx4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup vx3 (Cons vx40 vx41)",fontsize=16,color="burlywood",shape="box"];36[label="vx40/Tup2 vx400 vx401",fontsize=10,color="white",style="solid",shape="box"];5 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 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 (esEsMyBool vx3 vx400)",fontsize=16,color="burlywood",shape="box"];37[label="vx3/MyTrue",fontsize=10,color="white",style="solid",shape="box"];11 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 12[label="",style="solid", color="burlywood", weight=3];
38[label="vx3/MyFalse",fontsize=10,color="white",style="solid",shape="box"];11 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 13[label="",style="solid", color="burlywood", weight=3];
12[label="lookup1 MyTrue vx400 vx401 vx41 (esEsMyBool MyTrue vx400)",fontsize=16,color="burlywood",shape="box"];39[label="vx400/MyTrue",fontsize=10,color="white",style="solid",shape="box"];12 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 14[label="",style="solid", color="burlywood", weight=3];
40[label="vx400/MyFalse",fontsize=10,color="white",style="solid",shape="box"];12 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 15[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 MyFalse vx400 vx401 vx41 (esEsMyBool MyFalse vx400)",fontsize=16,color="burlywood",shape="box"];41[label="vx400/MyTrue",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 16[label="",style="solid", color="burlywood", weight=3];
42[label="vx400/MyFalse",fontsize=10,color="white",style="solid",shape="box"];13 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 17[label="",style="solid", color="burlywood", weight=3];
14[label="lookup1 MyTrue MyTrue vx401 vx41 (esEsMyBool MyTrue MyTrue)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="lookup1 MyTrue MyFalse vx401 vx41 (esEsMyBool MyTrue MyFalse)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="lookup1 MyFalse MyTrue vx401 vx41 (esEsMyBool MyFalse MyTrue)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="lookup1 MyFalse MyFalse vx401 vx41 (esEsMyBool MyFalse MyFalse)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="lookup1 MyTrue MyTrue vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="lookup1 MyTrue MyFalse vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="lookup1 MyFalse MyTrue vx401 vx41 MyFalse",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="lookup1 MyFalse MyFalse vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="Just vx401",fontsize=16,color="green",shape="box"];23[label="lookup0 MyTrue MyFalse vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="lookup0 MyFalse MyTrue vx401 vx41 otherwise",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="Just vx401",fontsize=16,color="green",shape="box"];26[label="lookup0 MyTrue MyFalse vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
27[label="lookup0 MyFalse MyTrue vx401 vx41 MyTrue",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28 -> 4[label="",style="dashed", color="red", weight=0];
28[label="lookup MyTrue vx41",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
28 -> 31[label="",style="dashed", color="magenta", weight=3];
29 -> 4[label="",style="dashed", color="red", weight=0];
29[label="lookup MyFalse vx41",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
29 -> 33[label="",style="dashed", color="magenta", weight=3];
30[label="vx41",fontsize=16,color="green",shape="box"];31[label="MyTrue",fontsize=16,color="green",shape="box"];32[label="vx41",fontsize=16,color="green",shape="box"];33[label="MyFalse",fontsize=16,color="green",shape="box"];}

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

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

   new_lookup(MyFalse, Cons(Tup2(MyTrue, vx401), vx41), h) -> new_lookup(MyFalse, vx41, h)
   new_lookup(MyTrue, Cons(Tup2(MyFalse, vx401), vx41), h) -> new_lookup(MyTrue, 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 2 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

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

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


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

(14)
YES