/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 23 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;
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;
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;
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;
"

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

(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;
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.insertBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.insertBy ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.insertBy ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="List.insertBy ww3 ww4 ww5",fontsize=16,color="burlywood",shape="triangle"];24[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];5 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 6[label="",style="solid", color="burlywood", weight=3];
25[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="List.insertBy ww3 ww4 (ww50 : ww51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="List.insertBy ww3 ww4 []",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.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) (ww3 ww4 ww50)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="ww4 : []",fontsize=16,color="green",shape="box"];11[label="ww3 ww4 ww50",fontsize=16,color="green",shape="box"];11 -> 17[label="",style="dashed", color="green", weight=3];
11 -> 18[label="",style="dashed", color="green", weight=3];
10[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) ww6",fontsize=16,color="burlywood",shape="triangle"];26[label="ww6/LT",fontsize=10,color="white",style="solid",shape="box"];10 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 14[label="",style="solid", color="burlywood", weight=3];
27[label="ww6/EQ",fontsize=10,color="white",style="solid",shape="box"];10 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 15[label="",style="solid", color="burlywood", weight=3];
28[label="ww6/GT",fontsize=10,color="white",style="solid",shape="box"];10 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 16[label="",style="solid", color="burlywood", weight=3];
17[label="ww4",fontsize=16,color="green",shape="box"];18[label="ww50",fontsize=16,color="green",shape="box"];14[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) LT",fontsize=16,color="black",shape="box"];14 -> 19[label="",style="solid", color="black", weight=3];
15[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) EQ",fontsize=16,color="black",shape="box"];15 -> 20[label="",style="solid", color="black", weight=3];
16[label="List.insertBy0 ww50 ww3 ww4 ww51 (ww50 : ww51) GT",fontsize=16,color="black",shape="box"];16 -> 21[label="",style="solid", color="black", weight=3];
19[label="ww4 : ww50 : ww51",fontsize=16,color="green",shape="box"];20[label="ww4 : ww50 : ww51",fontsize=16,color="green",shape="box"];21[label="ww50 : List.insertBy ww3 ww4 ww51",fontsize=16,color="green",shape="box"];21 -> 22[label="",style="dashed", color="green", weight=3];
22 -> 5[label="",style="dashed", color="red", weight=0];
22[label="List.insertBy ww3 ww4 ww51",fontsize=16,color="magenta"];22 -> 23[label="",style="dashed", color="magenta", weight=3];
23[label="ww51",fontsize=16,color="green",shape="box"];}

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

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

   new_insertBy0(ww50, ww3, ww4, ww51, ba) -> new_insertBy(ww3, ww4, ww51, ba)
   new_insertBy(ww3, ww4, :(ww50, ww51), ba) -> new_insertBy0(ww50, ww3, ww4, ww51, 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_insertBy(ww3, ww4, :(ww50, ww51), ba) -> new_insertBy0(ww50, ww3, ww4, ww51, ba)
The graph contains the following edges 3 > 1, 1 >= 2, 2 >= 3, 3 > 4, 4 >= 5


*new_insertBy0(ww50, ww3, ww4, ww51, ba) -> new_insertBy(ww3, ww4, ww51, ba)
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4


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

(10)
YES