/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 26 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


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(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;
intersperse :: a  ->  [a]  ->  [a];
intersperse _ [] = [];
intersperse _ (x : []) = x : [];
intersperse sep (x : xs) = x : sep : intersperse sep xs;

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

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

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(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;
intersperse :: a  ->  [a]  ->  [a];
intersperse vy [] = [];
intersperse vz (x : []) = x : [];
intersperse sep (x : xs) = x : sep : intersperse sep xs;

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

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

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

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

(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;
intersperse :: a  ->  [a]  ->  [a];
intersperse vy [] = [];
intersperse vz (x : []) = x : [];
intersperse sep (x : xs) = x : sep : intersperse sep xs;

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

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="List.intersperse",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.intersperse wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="List.intersperse wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];14[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];4 -> 14[label="",style="solid", color="burlywood", weight=9];
14 -> 5[label="",style="solid", color="burlywood", weight=3];
15[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 15[label="",style="solid", color="burlywood", weight=9];
15 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="List.intersperse wu3 (wu40 : wu41)",fontsize=16,color="burlywood",shape="box"];16[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];5 -> 16[label="",style="solid", color="burlywood", weight=9];
16 -> 7[label="",style="solid", color="burlywood", weight=3];
17[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
17 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="List.intersperse wu3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="List.intersperse wu3 (wu40 : wu410 : wu411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="List.intersperse wu3 (wu40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="wu40 : wu3 : List.intersperse wu3 (wu410 : wu411)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
11[label="wu40 : []",fontsize=16,color="green",shape="box"];12 -> 4[label="",style="dashed", color="red", weight=0];
12[label="List.intersperse wu3 (wu410 : wu411)",fontsize=16,color="magenta"];12 -> 13[label="",style="dashed", color="magenta", weight=3];
13[label="wu410 : wu411",fontsize=16,color="green",shape="box"];}

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

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_intersperse(wu3, :(wu40, :(wu410, wu411)), ba) -> new_intersperse(wu3, :(wu410, wu411), ba)

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

(7) 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_intersperse(wu3, :(wu40, :(wu410, wu411)), ba) -> new_intersperse(wu3, :(wu410, wu411), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(8)
YES