/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(6) HASKELL
(7) LetRed [EQUIVALENT, 0 ms]
(8) HASKELL
(9) Narrow [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) 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;
minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
minimumBy _ [] = error [];
minimumBy cmp xs = foldl1 min xs where { 
min x y = case cmp x y of { 
GT-> y; 
_-> x; 
 } ;
 };

}
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;
_  -> x}
"
is transformed to
"min0 y x GT = y;
min0 y x _ = x;
"

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

(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;
minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
minimumBy _ [] = error [];
minimumBy cmp xs = foldl1 min xs where { 
min x y = min0 y x (cmp x y);
min0 y x GT = y;
min0 y x _ = x;
 };

}
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;
minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
minimumBy vy [] = error [];
minimumBy cmp xs = foldl1 min xs where { 
min x y = min0 y x (cmp x y);
min0 y x GT = y;
min0 y x vz = x;
 };

}
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;
minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
minimumBy vy [] = error [];
minimumBy cmp xs = foldl1 min xs where { 
min x y = min0 y x (cmp x y);
min0 y x GT = y;
min0 y x vz = x;
 };

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

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

(7) LetRed (EQUIVALENT)
Let/Where Reductions:
The bindings of the following Let/Where expression
"foldl1 min xs where {
min x y = min0 y x (cmp x y);
;
min0 y x GT = y;
min0 y x vz = x;
} 
"
are unpacked to the following functions on top level
"minimumByMin0 wu y x GT = y;
minimumByMin0 wu y x vz = x;
"
"minimumByMin wu x y = minimumByMin0 wu y x (wu x y);
"

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

(8)
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;
minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
minimumBy vy [] = error [];
minimumBy cmp xs = foldl1 (minimumByMin cmp) xs;

minimumByMin wu x y = minimumByMin0 wu y x (wu x y);

minimumByMin0 wu y x GT = y;
minimumByMin0 wu y x vz = x;

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

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

(9) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.minimumBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.minimumBy wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.minimumBy wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];30[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 5[label="",style="solid", color="burlywood", weight=3];
31[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.minimumBy wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="List.minimumBy wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl1 (List.minimumByMin wv3) (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="error []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldl (List.minimumByMin wv3) wv40 wv41",fontsize=16,color="burlywood",shape="triangle"];32[label="wv41/wv410 : wv411",fontsize=10,color="white",style="solid",shape="box"];9 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 11[label="",style="solid", color="burlywood", weight=3];
33[label="wv41/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="error []",fontsize=16,color="red",shape="box"];11[label="foldl (List.minimumByMin wv3) wv40 (wv410 : wv411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldl (List.minimumByMin wv3) wv40 []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13 -> 9[label="",style="dashed", color="red", weight=0];
13[label="foldl (List.minimumByMin wv3) (List.minimumByMin wv3 wv40 wv410) wv411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
13 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="wv40",fontsize=16,color="green",shape="box"];15[label="List.minimumByMin wv3 wv40 wv410",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="wv411",fontsize=16,color="green",shape="box"];17 -> 18[label="",style="dashed", color="red", weight=0];
17[label="List.minimumByMin0 wv3 wv410 wv40 (wv3 wv40 wv410)",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
19[label="wv3 wv40 wv410",fontsize=16,color="green",shape="box"];19 -> 25[label="",style="dashed", color="green", weight=3];
19 -> 26[label="",style="dashed", color="green", weight=3];
18[label="List.minimumByMin0 wv3 wv410 wv40 wv5",fontsize=16,color="burlywood",shape="triangle"];34[label="wv5/LT",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="wv5/EQ",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];
36[label="wv5/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 24[label="",style="solid", color="burlywood", weight=3];
25[label="wv40",fontsize=16,color="green",shape="box"];26[label="wv410",fontsize=16,color="green",shape="box"];22[label="List.minimumByMin0 wv3 wv410 wv40 LT",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
23[label="List.minimumByMin0 wv3 wv410 wv40 EQ",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="List.minimumByMin0 wv3 wv410 wv40 GT",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
27[label="wv40",fontsize=16,color="green",shape="box"];28[label="wv40",fontsize=16,color="green",shape="box"];29[label="wv410",fontsize=16,color="green",shape="box"];}

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

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

   new_foldl(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)

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

(11) 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(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(12)
YES