/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 26 ms]
(6) HASKELL
(7) Narrow [SOUND, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) AND
    (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) QDP
        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (16) 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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = case cmp x y of { 
GT-> y : insertBy cmp x ys'; 
_-> x : ys; 
 } ;

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

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

(1) CR (EQUIVALENT)
Case Reductions:
The following Case expression
"case cmp x y of {
GT  -> y : insertBy cmp x ys';
_  -> x : ys}
"
is transformed to
"insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = x : ys;
"

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

(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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy _ x [] = x : [];
insertBy cmp x ys@(y : ys') = insertBy0 y cmp x ys' ys (cmp x y);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys _ = x : ys;

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

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

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

Binding Reductions:
The bind variable of the following binding Pattern
"ys@(wu : wv)"
is replaced by the following term
"wu : wv"

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

(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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = x : ys;

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

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

(5) 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
"compare x y|x == yEQ|x <= yLT|otherwiseGT;
"
is transformed to
"compare x y = compare3 x y;
"
"compare2 x y True = EQ;
compare2 x y False = compare1 x y (x <= y);
"
"compare1 x y True = LT;
compare1 x y False = compare0 x y otherwise;
"
"compare0 x y True = GT;
"
"compare3 x y = compare2 x y (x == y);
"

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

(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;
insert :: Ord a => a  ->  [a]  ->  [a];
insert e ls = insertBy compare e ls;

insertBy :: (a  ->  a  ->  Ordering)  ->  a  ->  [a]  ->  [a];
insertBy vz x [] = x : [];
insertBy cmp x (wu : wv) = insertBy0 wu cmp x wv (wu : wv) (cmp x wu);

insertBy0 y cmp x ys' ys GT = y : insertBy cmp x ys';
insertBy0 y cmp x ys' ys vy = x : ys;

}
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.insert",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.insert ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.insert ww3 ww4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="List.insertBy compare ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];78[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 6[label="",style="solid", color="burlywood", weight=3];
79[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.insertBy compare ww3 (ww40 : ww41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.insertBy compare ww3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare ww3 ww40)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare3 ww3 ww40)",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="List.insertBy0 ww40 compare ww3 ww41 (ww40 : ww41) (compare2 ww3 ww40 (ww3 == ww40))",fontsize=16,color="burlywood",shape="box"];80[label="ww3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 12[label="",style="solid", color="burlywood", weight=3];
81[label="ww3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 13[label="",style="solid", color="burlywood", weight=3];
82[label="ww3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 14[label="",style="solid", color="burlywood", weight=3];
12[label="List.insertBy0 ww40 compare LT ww41 (ww40 : ww41) (compare2 LT ww40 (LT == ww40))",fontsize=16,color="burlywood",shape="box"];83[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 15[label="",style="solid", color="burlywood", weight=3];
84[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 16[label="",style="solid", color="burlywood", weight=3];
85[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 17[label="",style="solid", color="burlywood", weight=3];
13[label="List.insertBy0 ww40 compare EQ ww41 (ww40 : ww41) (compare2 EQ ww40 (EQ == ww40))",fontsize=16,color="burlywood",shape="box"];86[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 18[label="",style="solid", color="burlywood", weight=3];
87[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 19[label="",style="solid", color="burlywood", weight=3];
88[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 20[label="",style="solid", color="burlywood", weight=3];
14[label="List.insertBy0 ww40 compare GT ww41 (ww40 : ww41) (compare2 GT ww40 (GT == ww40))",fontsize=16,color="burlywood",shape="box"];89[label="ww40/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 21[label="",style="solid", color="burlywood", weight=3];
90[label="ww40/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 22[label="",style="solid", color="burlywood", weight=3];
91[label="ww40/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 23[label="",style="solid", color="burlywood", weight=3];
15[label="List.insertBy0 LT compare LT ww41 (LT : ww41) (compare2 LT LT (LT == LT))",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3];
16[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare2 LT EQ (LT == EQ))",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
17[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare2 LT GT (LT == GT))",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
18[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare2 EQ LT (EQ == LT))",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
19[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) (compare2 EQ EQ (EQ == EQ))",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
20[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare2 EQ GT (EQ == GT))",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare2 GT LT (GT == LT))",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
22[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare2 GT EQ (GT == EQ))",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
23[label="List.insertBy0 GT compare GT ww41 (GT : ww41) (compare2 GT GT (GT == GT))",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
24[label="List.insertBy0 LT compare LT ww41 (LT : ww41) (compare2 LT LT True)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
25[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare2 LT EQ False)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
26[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare2 LT GT False)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
27[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare2 EQ LT False)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) (compare2 EQ EQ True)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
29[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare2 EQ GT False)",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
30[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare2 GT LT False)",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
31[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare2 GT EQ False)",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
32[label="List.insertBy0 GT compare GT ww41 (GT : ww41) (compare2 GT GT True)",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
33[label="List.insertBy0 LT compare LT ww41 (LT : ww41) EQ",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
34[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare1 LT EQ (LT <= EQ))",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
35[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare1 LT GT (LT <= GT))",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
36[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare1 EQ LT (EQ <= LT))",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
37[label="List.insertBy0 EQ compare EQ ww41 (EQ : ww41) EQ",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
38[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare1 EQ GT (EQ <= GT))",fontsize=16,color="black",shape="box"];38 -> 47[label="",style="solid", color="black", weight=3];
39[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare1 GT LT (GT <= LT))",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3];
40[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare1 GT EQ (GT <= EQ))",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3];
41[label="List.insertBy0 GT compare GT ww41 (GT : ww41) EQ",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3];
42[label="LT : LT : ww41",fontsize=16,color="green",shape="box"];43[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) (compare1 LT EQ True)",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="List.insertBy0 GT compare LT ww41 (GT : ww41) (compare1 LT GT True)",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare1 EQ LT False)",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46[label="EQ : EQ : ww41",fontsize=16,color="green",shape="box"];47[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) (compare1 EQ GT True)",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3];
48[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare1 GT LT False)",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
49[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare1 GT EQ False)",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3];
50[label="GT : GT : ww41",fontsize=16,color="green",shape="box"];51[label="List.insertBy0 EQ compare LT ww41 (EQ : ww41) LT",fontsize=16,color="black",shape="box"];51 -> 57[label="",style="solid", color="black", weight=3];
52[label="List.insertBy0 GT compare LT ww41 (GT : ww41) LT",fontsize=16,color="black",shape="box"];52 -> 58[label="",style="solid", color="black", weight=3];
53[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare0 EQ LT otherwise)",fontsize=16,color="black",shape="box"];53 -> 59[label="",style="solid", color="black", weight=3];
54[label="List.insertBy0 GT compare EQ ww41 (GT : ww41) LT",fontsize=16,color="black",shape="box"];54 -> 60[label="",style="solid", color="black", weight=3];
55[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare0 GT LT otherwise)",fontsize=16,color="black",shape="box"];55 -> 61[label="",style="solid", color="black", weight=3];
56[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare0 GT EQ otherwise)",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3];
57[label="LT : EQ : ww41",fontsize=16,color="green",shape="box"];58[label="LT : GT : ww41",fontsize=16,color="green",shape="box"];59[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) (compare0 EQ LT True)",fontsize=16,color="black",shape="box"];59 -> 63[label="",style="solid", color="black", weight=3];
60[label="EQ : GT : ww41",fontsize=16,color="green",shape="box"];61[label="List.insertBy0 LT compare GT ww41 (LT : ww41) (compare0 GT LT True)",fontsize=16,color="black",shape="box"];61 -> 64[label="",style="solid", color="black", weight=3];
62[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) (compare0 GT EQ True)",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3];
63[label="List.insertBy0 LT compare EQ ww41 (LT : ww41) GT",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3];
64[label="List.insertBy0 LT compare GT ww41 (LT : ww41) GT",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3];
65[label="List.insertBy0 EQ compare GT ww41 (EQ : ww41) GT",fontsize=16,color="black",shape="box"];65 -> 68[label="",style="solid", color="black", weight=3];
66[label="LT : List.insertBy compare EQ ww41",fontsize=16,color="green",shape="box"];66 -> 69[label="",style="dashed", color="green", weight=3];
67[label="LT : List.insertBy compare GT ww41",fontsize=16,color="green",shape="box"];67 -> 70[label="",style="dashed", color="green", weight=3];
68[label="EQ : List.insertBy compare GT ww41",fontsize=16,color="green",shape="box"];68 -> 71[label="",style="dashed", color="green", weight=3];
69 -> 5[label="",style="dashed", color="red", weight=0];
69[label="List.insertBy compare EQ ww41",fontsize=16,color="magenta"];69 -> 72[label="",style="dashed", color="magenta", weight=3];
69 -> 73[label="",style="dashed", color="magenta", weight=3];
70 -> 5[label="",style="dashed", color="red", weight=0];
70[label="List.insertBy compare GT ww41",fontsize=16,color="magenta"];70 -> 74[label="",style="dashed", color="magenta", weight=3];
70 -> 75[label="",style="dashed", color="magenta", weight=3];
71 -> 5[label="",style="dashed", color="red", weight=0];
71[label="List.insertBy compare GT ww41",fontsize=16,color="magenta"];71 -> 76[label="",style="dashed", color="magenta", weight=3];
71 -> 77[label="",style="dashed", color="magenta", weight=3];
72[label="ww41",fontsize=16,color="green",shape="box"];73[label="EQ",fontsize=16,color="green",shape="box"];74[label="ww41",fontsize=16,color="green",shape="box"];75[label="GT",fontsize=16,color="green",shape="box"];76[label="ww41",fontsize=16,color="green",shape="box"];77[label="GT",fontsize=16,color="green",shape="box"];}

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

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

   new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)
   new_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
   new_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)

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

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(10)
Complex Obligation (AND)

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

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)

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

(12) 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_insertBy(EQ, :(LT, ww41)) -> new_insertBy(EQ, ww41)
The graph contains the following edges 1 >= 1, 2 > 2


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

(13)
YES

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

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

   new_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
   new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)

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

(15) 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_insertBy(GT, :(EQ, ww41)) -> new_insertBy(GT, ww41)
The graph contains the following edges 1 >= 1, 2 > 2


*new_insertBy(GT, :(LT, ww41)) -> new_insertBy(GT, ww41)
The graph contains the following edges 1 >= 1, 2 > 2


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

(16)
YES