/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IPR [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) QDP
(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(10) 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;
partition :: (a  ->  Bool)  ->  [a]  ->  ([a],[a]);
partition p xs = foldr (select p) ([],[]) xs;

select p x ~((ts,fs))  | p x = (x : ts,fs)
 | otherwise = (ts,x : fs);

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

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

(1) IPR (EQUIVALENT)
IrrPat Reductions:
The variables of the following irrefutable Pattern
"~(ts,fs)"
are replaced by calls to these functions
"select0 (ts,fs) = ts;
"
"select1 (ts,fs) = fs;
"

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

(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;
partition :: (a  ->  Bool)  ->  [a]  ->  ([a],[a]);
partition p xs = foldr (select p) ([],[]) xs;

select p x vw  | p x = (x : select0 vw,select1 vw)
 | otherwise = (select0 vw,x : select1 vw);

select0 (ts,fs) = ts;

select1 (ts,fs) = fs;

}
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;
partition :: (a  ->  Bool)  ->  [a]  ->  ([a],[a]);
partition p xs = foldr (select p) ([],[]) xs;

select p x vw  | p x = (x : select0 vw,select1 vw)
 | otherwise = (select0 vw,x : select1 vw);

select0 (ts,fs) = ts;

select1 (ts,fs) = fs;

}
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
"select p x vw|p x(x : select0 vw,select1 vw)|otherwise(select0 vw,x : select1 vw);
"
is transformed to
"select p x vw = select4 p x vw;
"
"select3 p x vw True = (x : select0 vw,select1 vw);
select3 p x vw False = select2 p x vw otherwise;
"
"select2 p x vw True = (select0 vw,x : select1 vw);
"
"select4 p x vw = select3 p x vw (p x);
"

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

(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;
partition :: (a  ->  Bool)  ->  [a]  ->  ([a],[a]);
partition p xs = foldr (select p) ([],[]) xs;

select p x vw = select4 p x vw;

select0 (ts,fs) = ts;

select1 (ts,fs) = fs;

select2 p x vw True = (select0 vw,x : select1 vw);

select3 p x vw True = (x : select0 vw,select1 vw);
select3 p x vw False = select2 p x vw otherwise;

select4 p x vw = select3 p x vw (p x);

}
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.partition",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.partition vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.partition vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="foldr (List.select vz3) ([],[]) vz4",fontsize=16,color="burlywood",shape="triangle"];33[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];5 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 6[label="",style="solid", color="burlywood", weight=3];
34[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldr (List.select vz3) ([],[]) (vz40 : vz41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldr (List.select vz3) ([],[]) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="List.select vz3 vz40 (foldr (List.select vz3) ([],[]) vz41)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="([],[])",fontsize=16,color="green",shape="box"];11 -> 5[label="",style="dashed", color="red", weight=0];
11[label="foldr (List.select vz3) ([],[]) vz41",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="List.select vz3 vz40 vz5",fontsize=16,color="black",shape="triangle"];10 -> 13[label="",style="solid", color="black", weight=3];
12[label="vz41",fontsize=16,color="green",shape="box"];13[label="List.select4 vz3 vz40 vz5",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14 -> 15[label="",style="dashed", color="red", weight=0];
14[label="List.select3 vz3 vz40 vz5 (vz3 vz40)",fontsize=16,color="magenta"];14 -> 16[label="",style="dashed", color="magenta", weight=3];
16[label="vz3 vz40",fontsize=16,color="green",shape="box"];16 -> 20[label="",style="dashed", color="green", weight=3];
15[label="List.select3 vz3 vz40 vz5 vz6",fontsize=16,color="burlywood",shape="triangle"];35[label="vz6/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 18[label="",style="solid", color="burlywood", weight=3];
36[label="vz6/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 19[label="",style="solid", color="burlywood", weight=3];
20[label="vz40",fontsize=16,color="green",shape="box"];18[label="List.select3 vz3 vz40 vz5 False",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="List.select3 vz3 vz40 vz5 True",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
21[label="List.select2 vz3 vz40 vz5 otherwise",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="(vz40 : List.select0 vz5,List.select1 vz5)",fontsize=16,color="green",shape="box"];22 -> 24[label="",style="dashed", color="green", weight=3];
22 -> 25[label="",style="dashed", color="green", weight=3];
23[label="List.select2 vz3 vz40 vz5 True",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="List.select0 vz5",fontsize=16,color="burlywood",shape="triangle"];37[label="vz5/(vz50,vz51)",fontsize=10,color="white",style="solid",shape="box"];24 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="List.select1 vz5",fontsize=16,color="burlywood",shape="triangle"];38[label="vz5/(vz50,vz51)",fontsize=10,color="white",style="solid",shape="box"];25 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 28[label="",style="solid", color="burlywood", weight=3];
26[label="(List.select0 vz5,vz40 : List.select1 vz5)",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
26 -> 30[label="",style="dashed", color="green", weight=3];
27[label="List.select0 (vz50,vz51)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="List.select1 (vz50,vz51)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29 -> 24[label="",style="dashed", color="red", weight=0];
29[label="List.select0 vz5",fontsize=16,color="magenta"];30 -> 25[label="",style="dashed", color="red", weight=0];
30[label="List.select1 vz5",fontsize=16,color="magenta"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="vz51",fontsize=16,color="green",shape="box"];}

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

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

   new_foldr(vz3, :(vz40, vz41), ba) -> new_foldr(vz3, vz41, ba)

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

(9) 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_foldr(vz3, :(vz40, vz41), ba) -> new_foldr(vz3, vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(10)
YES