/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 3 ms]
(4) HASKELL
(5) NumRed [SOUND, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) TransformationProof [EQUIVALENT, 0 ms]
(10) QDP
(11) DependencyGraphProof [EQUIVALENT, 0 ms]
(12) QDP
(13) UsableRulesProof [EQUIVALENT, 0 ms]
(14) QDP
(15) QReductionProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(18) YES


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

(0)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
genericIndex :: Integral b => [a]  ->  b  ->  a;
genericIndex (x : _) 0 = x;
genericIndex (_ : xs) n  | n > 0 = genericIndex xs (n - 1)
 | otherwise = error [];
genericIndex _ _ = error [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
genericIndex :: Integral b => [a]  ->  b  ->  a;
genericIndex (x : xw) 0 = x;
genericIndex (xx : xs) n  | n > 0 = genericIndex xs (n - 1)
 | otherwise = error [];
genericIndex xy xz = error [];

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"
The following Function with conditions
"genericIndex (x : xw) 0 = x;
genericIndex (xx : xs) n|n > 0genericIndex xs (n - 1)|otherwiseerror [];
genericIndex xy xz = error [];
"
is transformed to
"genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
genericIndex xy xz = genericIndex0 xy xz;
"
"genericIndex0 xy xz = error [];
"
"genericIndex1 xx xs n True = error [];
genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;
"
"genericIndex2 xx xs n True = genericIndex xs (n - 1);
genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;
"
"genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > 0);
genericIndex3 yv yw = genericIndex0 yv yw;
"
"genericIndex4 True (x : xw) yy = x;
genericIndex4 yz zu zv = genericIndex3 zu zv;
"
"genericIndex5 (x : xw) yy = genericIndex4 (yy == 0) (x : xw) yy;
genericIndex5 zw zx = genericIndex3 zw zx;
"

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

(4)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
genericIndex :: Integral a => [b]  ->  a  ->  b;
genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
genericIndex xy xz = genericIndex0 xy xz;

genericIndex0 xy xz = error [];

genericIndex1 xx xs n True = error [];
genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;

genericIndex2 xx xs n True = genericIndex xs (n - 1);
genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;

genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > 0);
genericIndex3 yv yw = genericIndex0 yv yw;

genericIndex4 True (x : xw) yy = x;
genericIndex4 yz zu zv = genericIndex3 zu zv;

genericIndex5 (x : xw) yy = genericIndex4 (yy == 0) (x : xw) yy;
genericIndex5 zw zx = genericIndex3 zw zx;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(5) NumRed (SOUND)
Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
----------------------------------------

(6)
Obligation:
mainModule Main 
module Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
genericIndex :: Integral b => [a]  ->  b  ->  a;
genericIndex (x : xw) yy = genericIndex5 (x : xw) yy;
genericIndex (xx : xs) n = genericIndex3 (xx : xs) n;
genericIndex xy xz = genericIndex0 xy xz;

genericIndex0 xy xz = error [];

genericIndex1 xx xs n True = error [];
genericIndex1 xx xs n False = genericIndex0 (xx : xs) n;

genericIndex2 xx xs n True = genericIndex xs (n - fromInt (Pos (Succ Zero)));
genericIndex2 xx xs n False = genericIndex1 xx xs n otherwise;

genericIndex3 (xx : xs) n = genericIndex2 xx xs n (n > fromInt (Pos Zero));
genericIndex3 yv yw = genericIndex0 yv yw;

genericIndex4 True (x : xw) yy = x;
genericIndex4 yz zu zv = genericIndex3 zu zv;

genericIndex5 (x : xw) yy = genericIndex4 (yy == fromInt (Pos Zero)) (x : xw) yy;
genericIndex5 zw zx = genericIndex3 zw zx;

}
module Main where {
import qualified List;
import qualified Maybe;
import qualified Prelude;
}

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

(7) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.genericIndex",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.genericIndex zy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.genericIndex zy3 zy4",fontsize=16,color="burlywood",shape="triangle"];58[label="zy3/zy30 : zy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 58[label="",style="solid", color="burlywood", weight=9];
58 -> 5[label="",style="solid", color="burlywood", weight=3];
59[label="zy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 59[label="",style="solid", color="burlywood", weight=9];
59 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.genericIndex (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="List.genericIndex [] zy4",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.genericIndex5 (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="List.genericIndex0 [] zy4",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="List.genericIndex4 (zy4 == fromInt (Pos Zero)) (zy30 : zy31) zy4",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="error []",fontsize=16,color="black",shape="triangle"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="List.genericIndex4 (primEqInt zy4 (fromInt (Pos Zero))) (zy30 : zy31) zy4",fontsize=16,color="burlywood",shape="box"];60[label="zy4/Pos zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 13[label="",style="solid", color="burlywood", weight=3];
61[label="zy4/Neg zy40",fontsize=10,color="white",style="solid",shape="box"];11 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="error []",fontsize=16,color="red",shape="box"];13[label="List.genericIndex4 (primEqInt (Pos zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Pos zy40)",fontsize=16,color="burlywood",shape="box"];62[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];13 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 15[label="",style="solid", color="burlywood", weight=3];
63[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="List.genericIndex4 (primEqInt (Neg zy40) (fromInt (Pos Zero))) (zy30 : zy31) (Neg zy40)",fontsize=16,color="burlywood",shape="box"];64[label="zy40/Succ zy400",fontsize=10,color="white",style="solid",shape="box"];14 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 17[label="",style="solid", color="burlywood", weight=3];
65[label="zy40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="List.genericIndex4 (primEqInt (Pos (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
16[label="List.genericIndex4 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
17[label="List.genericIndex4 (primEqInt (Neg (Succ zy400)) (fromInt (Pos Zero))) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
18[label="List.genericIndex4 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
19[label="List.genericIndex4 (primEqInt (Pos (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="List.genericIndex4 (primEqInt (Pos Zero) (Pos Zero)) (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="List.genericIndex4 (primEqInt (Neg (Succ zy400)) (Pos Zero)) (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="List.genericIndex4 (primEqInt (Neg Zero) (Pos Zero)) (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="List.genericIndex4 False (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
24[label="List.genericIndex4 True (zy30 : zy31) (Pos Zero)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
25[label="List.genericIndex4 False (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
26[label="List.genericIndex4 True (zy30 : zy31) (Neg Zero)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
27[label="List.genericIndex3 (zy30 : zy31) (Pos (Succ zy400))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="zy30",fontsize=16,color="green",shape="box"];29[label="List.genericIndex3 (zy30 : zy31) (Neg (Succ zy400))",fontsize=16,color="black",shape="box"];29 -> 32[label="",style="solid", color="black", weight=3];
30[label="zy30",fontsize=16,color="green",shape="box"];31[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (Pos (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (Neg (Succ zy400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (compare (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (compare (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
35[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
36[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
37[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpInt (Pos (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
38[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (primCmpInt (Neg (Succ zy400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (primCmpNat (Succ zy400) Zero == GT)",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) (LT == GT)",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3];
41[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) (GT == GT)",fontsize=16,color="black",shape="box"];41 -> 43[label="",style="solid", color="black", weight=3];
42[label="List.genericIndex2 zy30 zy31 (Neg (Succ zy400)) False",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3];
43[label="List.genericIndex2 zy30 zy31 (Pos (Succ zy400)) True",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
44[label="List.genericIndex1 zy30 zy31 (Neg (Succ zy400)) otherwise",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
45 -> 4[label="",style="dashed", color="red", weight=0];
45[label="List.genericIndex zy31 (Pos (Succ zy400) - fromInt (Pos (Succ Zero)))",fontsize=16,color="magenta"];45 -> 47[label="",style="dashed", color="magenta", weight=3];
45 -> 48[label="",style="dashed", color="magenta", weight=3];
46[label="List.genericIndex1 zy30 zy31 (Neg (Succ zy400)) True",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3];
47[label="zy31",fontsize=16,color="green",shape="box"];48[label="Pos (Succ zy400) - fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];48 -> 50[label="",style="solid", color="black", weight=3];
49 -> 10[label="",style="dashed", color="red", weight=0];
49[label="error []",fontsize=16,color="magenta"];50[label="primMinusInt (Pos (Succ zy400)) (fromInt (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];50 -> 51[label="",style="solid", color="black", weight=3];
51[label="primMinusInt (Pos (Succ zy400)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];51 -> 52[label="",style="solid", color="black", weight=3];
52[label="primMinusNat (Succ zy400) (Succ Zero)",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
53[label="primMinusNat zy400 Zero",fontsize=16,color="burlywood",shape="box"];66[label="zy400/Succ zy4000",fontsize=10,color="white",style="solid",shape="box"];53 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 54[label="",style="solid", color="burlywood", weight=3];
67[label="zy400/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 55[label="",style="solid", color="burlywood", weight=3];
54[label="primMinusNat (Succ zy4000) Zero",fontsize=16,color="black",shape="box"];54 -> 56[label="",style="solid", color="black", weight=3];
55[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];55 -> 57[label="",style="solid", color="black", weight=3];
56[label="Pos (Succ zy4000)",fontsize=16,color="green",shape="box"];57[label="Pos Zero",fontsize=16,color="green",shape="box"];}

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

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) -> new_genericIndex(zy31, new_primMinusNat(zy400), ba)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(9) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule new_genericIndex(:(zy30, zy31), Pos(Succ(zy400)), ba) -> new_genericIndex(zy31, new_primMinusNat(zy400), ba) at position [1] we obtained the following new rules [LPAR04]:

   (new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3),new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3))
   (new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3),new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3))


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

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
   new_genericIndex(:(y0, y1), Pos(Succ(Zero)), y3) -> new_genericIndex(y1, Pos(Zero), y3)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(11) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

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

   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)

The TRS R consists of the following rules:

   new_primMinusNat(Succ(zy4000)) -> Pos(Succ(zy4000))
   new_primMinusNat(Zero) -> Pos(Zero)

The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(13) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)

R is empty.
The set Q consists of the following terms:

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)

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

(15) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   new_primMinusNat(Succ(x0))
   new_primMinusNat(Zero)


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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)

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

(17) 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_genericIndex(:(y0, y1), Pos(Succ(Succ(x0))), y3) -> new_genericIndex(y1, Pos(Succ(x0)), y3)
The graph contains the following edges 1 > 1, 3 >= 3


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

(18)
YES