/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


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


YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) IFR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) BR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) COR [EQUIVALENT, 0 ms]
(6) HASKELL
(7) Narrow [SOUND, 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 Maybe where {
import qualified List;
import qualified Main;
import qualified Prelude;
}
module List where {
import qualified Main;
import qualified Maybe;
import qualified Prelude;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;

deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
deleteFirstsBy eq = foldl (flip (deleteBy eq));

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

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

(1) IFR (EQUIVALENT)
If Reductions:
The following If expression
"if eq x y then ys else y : deleteBy eq x ys"
is transformed to
"deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq 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;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy _ _ [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
deleteFirstsBy eq = foldl (flip (deleteBy eq));

}
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.
----------------------------------------

(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;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
deleteFirstsBy eq = foldl (flip (deleteBy eq));

}
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;
"

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

(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;
deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
deleteBy vy vz [] = [];
deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);

deleteBy0 ys y eq x True = ys;
deleteBy0 ys y eq x False = y : deleteBy eq x ys;

deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
deleteFirstsBy eq = foldl (flip (deleteBy eq));

}
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.deleteFirstsBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.deleteFirstsBy wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.deleteFirstsBy wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="List.deleteFirstsBy wu3 wu4 wu5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="foldl (flip (List.deleteBy wu3)) wu4 wu5",fontsize=16,color="burlywood",shape="triangle"];30[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];6 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 7[label="",style="solid", color="burlywood", weight=3];
31[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="foldl (flip (List.deleteBy wu3)) wu4 (wu50 : wu51)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="foldl (flip (List.deleteBy wu3)) wu4 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 6[label="",style="dashed", color="red", weight=0];
9[label="foldl (flip (List.deleteBy wu3)) (flip (List.deleteBy wu3) wu4 wu50) wu51",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3];
9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="wu4",fontsize=16,color="green",shape="box"];11[label="flip (List.deleteBy wu3) wu4 wu50",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="wu51",fontsize=16,color="green",shape="box"];13[label="List.deleteBy wu3 wu50 wu4",fontsize=16,color="burlywood",shape="triangle"];32[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];13 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 14[label="",style="solid", color="burlywood", weight=3];
33[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="List.deleteBy wu3 wu50 (wu40 : wu41)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="List.deleteBy wu3 wu50 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16 -> 18[label="",style="dashed", color="red", weight=0];
16[label="List.deleteBy0 wu41 wu40 wu3 wu50 (wu3 wu50 wu40)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];19[label="wu3 wu50 wu40",fontsize=16,color="green",shape="box"];19 -> 24[label="",style="dashed", color="green", weight=3];
19 -> 25[label="",style="dashed", color="green", weight=3];
18[label="List.deleteBy0 wu41 wu40 wu3 wu50 wu6",fontsize=16,color="burlywood",shape="triangle"];34[label="wu6/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 22[label="",style="solid", color="burlywood", weight=3];
35[label="wu6/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 23[label="",style="solid", color="burlywood", weight=3];
24[label="wu50",fontsize=16,color="green",shape="box"];25[label="wu40",fontsize=16,color="green",shape="box"];22[label="List.deleteBy0 wu41 wu40 wu3 wu50 False",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="List.deleteBy0 wu41 wu40 wu3 wu50 True",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
26[label="wu40 : List.deleteBy wu3 wu50 wu41",fontsize=16,color="green",shape="box"];26 -> 28[label="",style="dashed", color="green", weight=3];
27[label="wu41",fontsize=16,color="green",shape="box"];28 -> 13[label="",style="dashed", color="red", weight=0];
28[label="List.deleteBy wu3 wu50 wu41",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
29[label="wu41",fontsize=16,color="green",shape="box"];}

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

(8)
Complex Obligation (AND)

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

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

   new_foldl(wu3, :(wu50, wu51), ba) -> new_foldl(wu3, wu51, ba)

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_foldl(wu3, :(wu50, wu51), ba) -> new_foldl(wu3, wu51, ba)
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_deleteBy(wu3, wu50, :(wu40, wu41), ba) -> new_deleteBy0(wu41, wu40, wu3, wu50, ba)
   new_deleteBy0(wu41, wu40, wu3, wu50, ba) -> new_deleteBy(wu3, wu50, wu41, ba)

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_deleteBy0(wu41, wu40, wu3, wu50, ba) -> new_deleteBy(wu3, wu50, wu41, ba)
The graph contains the following edges 3 >= 1, 4 >= 2, 1 >= 3, 5 >= 4


*new_deleteBy(wu3, wu50, :(wu40, wu41), ba) -> new_deleteBy0(wu41, wu40, wu3, wu50, ba)
The graph contains the following edges 3 > 1, 3 > 2, 1 >= 3, 2 >= 4, 4 >= 5


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

(14)
YES