/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 20 ms]
(6) HASKELL
(7) Narrow [EQUIVALENT, 37 ms]
(8) 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;
"

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

(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 (EQUIVALENT)
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="box"];14[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];5 -> 14[label="",style="solid", color="burlywood", weight=9];
14 -> 6[label="",style="solid", color="burlywood", weight=3];
15[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 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="burlywood",shape="box"];16[label="ww3/()",fontsize=10,color="white",style="solid",shape="box"];8 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 10[label="",style="solid", color="burlywood", weight=3];
9[label="ww3 : []",fontsize=16,color="green",shape="box"];10[label="List.insertBy0 ww40 compare () ww41 (ww40 : ww41) (compare () ww40)",fontsize=16,color="burlywood",shape="box"];17[label="ww40/()",fontsize=10,color="white",style="solid",shape="box"];10 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 11[label="",style="solid", color="burlywood", weight=3];
11[label="List.insertBy0 () compare () ww41 (() : ww41) (compare () ())",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="List.insertBy0 () compare () ww41 (() : ww41) EQ",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="() : () : ww41",fontsize=16,color="green",shape="box"];}

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

(8)
YES