/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 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;
tails :: [a]  ->  [[a]];
tails [] = [] : [];
tails xxs@(_ : xs) = xxs : tails 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.

Binding Reductions:
The bind variable of the following binding Pattern
"xxs@(vz : wu)"
is replaced by the following term
"vz : wu"

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

(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;
tails :: [a]  ->  [[a]];
tails [] = [] : [];
tails (vz : wu) = (vz : wu) : tails wu;

}
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;
tails :: [a]  ->  [[a]];
tails [] = [] : [];
tails (vz : wu) = (vz : wu) : tails wu;

}
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.tails",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.tails wv3",fontsize=16,color="burlywood",shape="triangle"];10[label="wv3/wv30 : wv31",fontsize=10,color="white",style="solid",shape="box"];3 -> 10[label="",style="solid", color="burlywood", weight=9];
10 -> 4[label="",style="solid", color="burlywood", weight=3];
11[label="wv3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 11[label="",style="solid", color="burlywood", weight=9];
11 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="List.tails (wv30 : wv31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="List.tails []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="(wv30 : wv31) : List.tails wv31",fontsize=16,color="green",shape="box"];6 -> 8[label="",style="dashed", color="green", weight=3];
7[label="[] : []",fontsize=16,color="green",shape="box"];8 -> 3[label="",style="dashed", color="red", weight=0];
8[label="List.tails wv31",fontsize=16,color="magenta"];8 -> 9[label="",style="dashed", color="magenta", weight=3];
9[label="wv31",fontsize=16,color="green",shape="box"];}

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

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

   new_tails(:(wv30, wv31), ba) -> new_tails(wv31, 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_tails(:(wv30, wv31), ba) -> new_tails(wv31, ba)
The graph contains the following edges 1 > 1, 2 >= 2


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

(8)
YES