/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (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;
inits :: [a]  ->  [[a]];
inits [] = [] : [];
inits (x : xs) = ([] : []) ++ map (: x) (inits 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;
inits :: [a]  ->  [[a]];
inits [] = [] : [];
inits (x : xs) = ([] : []) ++ map (: x) (inits 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;
inits :: [a]  ->  [[a]];
inits [] = [] : [];
inits (x : xs) = ([] : []) ++ map (: x) (inits 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.inits",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="List.inits vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];3 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 4[label="",style="solid", color="burlywood", weight=3];
21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="List.inits (vy30 : vy31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="List.inits []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6 -> 8[label="",style="dashed", color="red", weight=0];
6[label="([] : []) ++ map (vy30 :) (List.inits vy31)",fontsize=16,color="magenta"];6 -> 9[label="",style="dashed", color="magenta", weight=3];
7[label="[] : []",fontsize=16,color="green",shape="box"];9 -> 3[label="",style="dashed", color="red", weight=0];
9[label="List.inits vy31",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="([] : []) ++ map (vy30 :) vy4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
10[label="vy31",fontsize=16,color="green",shape="box"];11[label="[] : [] ++ map (vy30 :) vy4",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
12[label="[] ++ map (vy30 :) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="map (vy30 :) vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 14[label="",style="solid", color="burlywood", weight=3];
23[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="map (vy30 :) (vy40 : vy41)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="map (vy30 :) []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="(vy30 : vy40) : map (vy30 :) vy41",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="[]",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0];
18[label="map (vy30 :) vy41",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
19[label="vy41",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_inits(:(vy30, vy31), ba) -> new_inits(vy31, ba)

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

(8) 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_inits(:(vy30, vy31), ba) -> new_inits(vy31, ba)
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

   new_map(vy30, :(vy40, vy41), ba) -> new_map(vy30, vy41, 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_map(vy30, :(vy40, vy41), ba) -> new_map(vy30, vy41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(12)
YES